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On February 7, Dublin-based Front Line Defenders launched an online and social media campaign to focus attention on the plight of 12 human rights defenders (HRDs) from Eastern Europe and Central Asia coinciding with the launch of the Winter Olympics in Sochi, Russia. The 'Rights. Risks. Change!' campaign (www.sportshrd.org) calls on the public to take solidarity action to support these 12 defenders and to pressure local officials to respect the work of HRDs. The list of the rights defenders includes the name of Anar Mammadli, head of the Azerbaijani election watchdog group, Election Monitoring and Democracy Studies Center. Mammadli was arrested on December 16 and is charged with illegal ownership, tax evasion and abuse of office. Front Line Defenders has worked for over a decade to provide practical and rapid support for human rights defenders at risk. This year Azerbaijan will hold its next local elections, which happens every five years. By law, the election date is appointed by the Central Election Commission, which must be publicized at least 60 days in advance. Overall number of municipal bodies across Azerbaijan is 1720. Members of municipalities are elected based on relative majority system in multi-mandate constituencies. Every Azerbaijani citizen aged above 21 years is eligible to run in local elections. On February 8, the supreme council of the Party of Hope held a session to discuss preparations for the local elections and other organizational issues. The trial over Ismayilli incidents continues in Shaki Court on Grave Crimes. On February 10, the son of the former minister of Labor and Social Protection, Vugar Alakbarov and his relative Elkhan Alakbarov, son of the former executive power chief of Ismayilli, Nizami Alakbarov, testified as victims. Despite the damage caused to his property, Vugar Alakbarov said he had no claims against defendants. Elkhan Alakbarov also withdrew his claims against defendants for the burning of his cars. Then the video presented as the evidence was watched. In turn, the lawyers of deputy chief of Musavat Party, Tofig Yagublu and leader of REAL Movement Ilgar Mammadov requested the court to display the videos that they had. These videos are expected to be shown at the next hearing. Riots in the district of Ismayilli broke out on 23 January 2013, after drunk employees of Chirag Hotel caused a road accident and insulted the locals. Angered residents set the hotel on fire. The next day there were clashes between the police and the protestors, who demanded dismissal of local governor whom they linked to Chirag owner. Eighteen people, including Tofig Yagublu and Ilgar Mammadov were arrested in connection with the riot. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,623 |
Q: Wix projects not compiling when I build solution in Visual Studio I just inherited a project that uses Wix for Installation setups. I noticed that when I right-click the solution and select Build, the Wix projects don't build as well. I have to right click each Wix project and click Build. Is there something I'm missing?
A: By default, WiX projects will not be built when building the 'Any CPU' platform because Windows Installer packages are CPU-specific. As a result, you need to use the following steps to update the solution build configuration to include your WiX project and its dependencies as part of a Team Foundation Build.
*
*In the solution, open Configuration Manager (Build | Configuration Manager).
*Set the 'Debug' configuration as the active configuration.
*Select the 'x86' platform that you plan to build from the drop-down list.
*Ensure that the WiX project is checked in the 'Build' column.
*Ensure that any project references that the WiX project uses are also checked in the 'Build' column.
*Set the 'Release' configuration as the active configuration.
*Repeat steps 3-5 to ensure that the WiX project and its dependencies will build for the 'Release' configuration.
*If you plan to build the 'x64' platform, repeat steps 3-7 for the 'x64' platform.
*Close Configuration Manager and save the solution.
Source: WIX Toolset Documentation
| {
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} | 6,742 |
{"url":"http:\/\/physics.stackexchange.com\/tags\/definition\/new","text":"# Tag Info\n\n## New answers tagged definition\n\n0\n\nIf your friend's energy+ yours= F1, then you would see your own energy expenditure halved, which we know cannot be the case. If your friend helps you push the object, then you are no longer applying the same force, or (lazy answer) the force is no longer localised and motivates the part of the object most subject to friction. So, once the object is first ...\n\n3\n\nAll the definitions you've posted are correct, and they aren't in conflict with each other, although they are a bit imprecise. I'll try to explain in more detail what these concepts are. Probability I'll assume we don't need to attempt to define what probability is. :) I'll just note that it's formalized in mathematics under the aegis of measure theory. ...\n\n3\n\nTo understand the difference between probability and probability density consider the difference between mass and density. Density is the mass per unit volume, so to find the mass you multiply the density by the volume: $$mass = density \\times volume$$ In some cases the density will be a function of position and we have to write it as a function of the ...\n\n-1\n\nIt has been many years since I have used this information (so please forgive my inaccuracies, but I wanted to get you PART of the way there with an answer (since no one has responded yet). The probability density is the integral (area under the curve) of the probability. I think the reason for the funky discrepancy between the two definitions is because.. ...\n\n3\n\nI'm sure everyone has had that concern when we encountered the definition for the first time, in school. There is a valid reason why this definition is still persisted with, despite the deficiency that you hit on. The most popular (and simple) forces in physics (also the ones with which we begin learning physics) are conservative forces, implying that the ...\n\n32\n\nIf you're pushing a 10-ton truck and it's not moving, you are not doing any work on the truck because the distance $ds=0$ and the nonzero force $F$ isn't enough for the product $F\\cdot ds$ to be nonzero. Your muscles may get tired so you feel that you're \"doing something\" and \"spending energy\" but it's not the work done on truck. You're just burning the ...\n\n1\n\nI'm not familiar with ISO 5725 (a 1994 revision of a 1986 document, apparently \"reviewed and confirmed\" in 2012), and it seems that I have to buy it to read it. A 2008 vocabulary of metrology put out by the BIPM and also cited by Wikipedia has definitions much closer to my intuition, and to common usage among folks I know who specialize in precision ...\n\n7\n\nThe \"shift in the meaning\" refers to some attempts to reinterpret the terminology that were made by a metrological document, ISO 5725, in 2008. That may be described as a bureaucratic effort by a few officials \u2013 really bureaucrats of a sort \u2013 and as far as I know, the \"shift in the meaning\" hasn't penetrated to the community of professionals. The people ...\n\n3\n\nWe have a perfectly unambiguous definition of temperature for canonical ensembles, and this temperature may be negative in bounded-energy systems. This kind of negative temperature is indisputable, and some would argue it has been realized in spin-inversion experiments. The problem is that there are two decent but imperfect definitions for the entropy of a ...\n\n3\n\nWhat I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact. The main disagreement seems to be about which definition of the word \"entropy\" in the context of statistical physics is \"correct\". Definition is an ...\n\n2\n\nSimply the thermodynamical quantities used in the original paper were not suitable for that problem. They in particular calculated $T=\\frac{\\partial U}{\\partial S}$ where $U$ is the internal energy and $S$ the entropy. However a wrong definition of entropy has been used. Mathematicians has proved that the use of that specific entropy was wrong and that ...\n\n4\n\nIt's the differential relationship between internal energy and entropy: \\begin{align} dU &= T\\,dS + \\cdots \\\\ \\frac{\\partial S}{\\partial U} &= \\frac 1T \\end{align} As energy is added to a system, its internal entropy changes. Remember that the (total) entropy is $$S = k \\ln\\Omega,$$ where $\\Omega$ is the number of available microscopic states that ...\n\n0\n\nAs a metrologist, I am glad of this interest in correct notation, often not enough pondered also among metrologists but essential for understanding with each others. I would say first that any numerical value of an experimental result is always expressed as a rational, not irrational, number, because the number of digits is always bounded by the position of ...\n\n1\n\nAs mentioned above, newtonian-mechanics does not account for relativistic effects (nor quantum mechanical effects for that matter). This does not mean that the whole branch is wrong, it simply describes events under certain conditions (\"normal\" conditions, or the ones we observe in our every day life). Regarding Feynman's comment, a derivation may come in ...\n\n2\n\nHere's a way to argue it isn't the displacement of the (center of mass) of the body. Take a spring that's at rest. Apply two forces on either end so it compresses. You can do this in such a way that the center of mass of the spring doesn't move. However, the energy of the spring system has changed (the potential energy increased). In order to satisfy the ...\n\n1\n\nFirst of all, I appreciate your doubt. It is conceptual and involves critical thinking regarding the topic. In order to understand work, you need to understand the 2 physical quantities namely - Force and Displacement . Note that using only one single word Displacement in order to define work is not appropriate. (Same is for Force ) . As, Displacement ...\n\n2\n\nIt's the displacement of the point of application of the force. Elementary textbooks initially model objects as point particles. They have exactly two properties: position and mass. No others. So the displacement of the object is necessarily the same as the displacement of the point of application. Later, extended or compound objects may (or may not) be ...\n\n4\n\nIt is the displacement of the point on which the force is applied. This is it. That is the real definition, period. The books which do it otherwise are wrong. I would like you to read Resnick Halliday Krane , physics Vol. 1, it has a separate chapter devoted to this problem with most of the books available. As an example to show that this is what the ...\n\nTop 50 recent answers are included","date":"2014-07-24 03:41:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.853056013584137, \"perplexity\": 426.69196757456973}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1405997885796.93\/warc\/CC-MAIN-20140722025805-00185-ip-10-33-131-23.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/brilliant.org\/problems\/sometimes-things-are-easier-than-they-look\/","text":"# Sometimes things are easier than they look.. (=\n\nGeometry Level 4\n\nLet $$e$$ be the $$eccentricity$$ of the curve $$0.44444x\u00b2 - 5.33333x + 16 = 8y - y^2$$. Then the value of $$9.8[e]$$ $$=$$\n\n$$Note-> here~[.] ~stands ~for ~G.I.F~(Greatest~integer~function)$$","date":"2017-05-29 19:26:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9120745658874512, \"perplexity\": 406.955684409153}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463612537.91\/warc\/CC-MAIN-20170529184559-20170529204559-00262.warc.gz\"}"} | null | null |
{"url":"https:\/\/packages.tesselle.org\/alkahest\/reference\/window_sliding.html","text":"There will be $$(m - 1) \/ 2$$ points both at the beginning and at the end of the data series for which a complete $$m$$-width window cannot be obtained. To prevent data loss, progressively wider\/narrower windows are evaluated at both ends of the data series.\n\nUsage\n\nwindow_sliding(n, m, ...)\n\n# S4 method for integer,integer\nwindow_sliding(n, m, i = NULL)\n\n# S4 method for numeric,numeric\nwindow_sliding(n, m, i = NULL)\n\nArguments\n\nn\n\nAn integer giving the length of the data series (will be coerced with as.integer() and hence truncated toward zero).\n\nm\n\nAn odd integer giving the window size, i.e. the number of adjacent points to be used (will be coerced with as.integer() and hence truncated toward zero).\n\n...\n\nCurrently not used.\n\ni\n\nA vector integer specifying the indices of the data points for which windows should be returned. If NULL (the default), windows are evaluated for each data point.\n\nValue\n\nReturns a length-$$n$$\n\nlist of integer vectors (indices of the data points in each window).\n\nOther moving windows: window_tumbling()\n\nN. Frerebeau\n\nExamples\n\n## Length of the data series\nn <- 10\n\n## Progressive sliding windows\nsliding <- window_sliding(n = n, m = 5)\n\nplot(NULL, xlim = c(1, n), ylim = c(1, 10.5), xlab = \"Index\", ylab = \"Window\")\nfor (i in seq_along(sliding)) {\nw <- sliding[[i]]\ntext(x = w, y = rep(i, length(w)), labels = w, pos = 3)\nlines(w, rep(i, length(w)), type = \"l\", lwd = 2)\n}\n\n## Tumbling windows\n## (compare with drop = TRUE)\ntumbling <- window_tumbling(n = n, m = 3, drop = FALSE)\n\nplot(NULL, xlim = c(1, n), ylim = c(1, 5.5), xlab = \"Index\", ylab = \"Window\")\nfor (i in seq_along(tumbling)) {\nw <- tumbling[[i]]\ntext(x = w, y = rep(i, length(w)), labels = w, pos = 3)\nlines(w, rep(i, length(w)), type = \"l\", lwd = 2)\n}","date":"2022-10-04 14:22:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.369354248046875, \"perplexity\": 8403.638526618422}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337504.21\/warc\/CC-MAIN-20221004121345-20221004151345-00093.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:intro}
The simplest and most widely studied class of expanding interval maps, and those which we will concern ourselves, are intermediate $\beta$-transformations, namely transformations of the form $T_{\beta, \alpha} : x \mapsto \beta x + \alpha\mod 1$ acting on $[0,1]$, where $(\beta, \alpha) \in \Delta \coloneqq \{ (b, a) \in \mathbb{R}^{2} \colon b \in (1, 2) \; \text{and} \; a \in [0, 2-b]\}$. This class of transformations have motivated a wealth of results, providing practical solutions to a variety of problems. They arise as Poincar\'e maps of the geometric model of Lorenz differential equations~\cite{MR681294}, Daubechies \textsl{et al.} \cite{1011470} proposed a new approach to analog-to-digital conversion using \mbox{$\beta$-transformations}, and Jitsumatsu and Matsumura \cite{Jitsumatsu2016AT} developed a random number generator using \mbox{$\beta$-transformations}. (This random number generator passed the NIST statistical test suite.) Through their study, many new phenomena have appeared, revealing rich combinatorial and topological structures, and unexpected connections to probability theory, ergodic theory and aperiodic order; see for instance \cite{bezuglyi_kolyada_2003,Komornik:2011,ArneThesis}.
Intermediate $\beta$-transformations also have an intimate link to metric number theory in that they give rise to \mbox{non-integer} based expansions of real numbers. Given $\beta \in (1, 2)$ and $x \in [0, 1/(\beta-1)]$, an infinite word $\omega = \omega_{1}\omega_{2}\cdots$ with letters in the alphabet $\{0, 1\}$ is called a \textsl{$\beta$-expansion} of $x$ if
\begin{align*}
x = \sum_{n \in \mathbb{N}} \omega_{n} \, \beta^{-n}.
\end{align*}
Through iterating the map $T_{\beta, \alpha}$ one obtains a subset of $\{ 0, 1\}^{\mathbb{N}}$ known as the intermediate $\beta$-shift $\Omega_{\beta, \alpha}$, where each $\omega \in \Omega_{\beta, \alpha}$ is a $\beta$-expansion, and corresponds to a unique point in $[\alpha/(\beta-1), 1+\alpha/(\beta-1)]$, see \eqref{eq:commutative_diag} and the commentary following it for further details. By equipping $\Omega_{\beta, \alpha}$ with the left shift map $\sigma$, one obtains a dynamical system which is topologically conjugate to the dynamical system $\mathcal{S}_{\beta,\alpha}=(T_{\beta, \alpha}, [0, 1])$, namely one obtains a symbolic system which possess the same ergodic properties as $\mathcal{S}_{\beta,\alpha}$. Note, in this article, by topologically conjugate we mean that the conjugacy is one-to-one everywhere except on a countable set on which the conjugacy is at most finite to one.
Open dynamical systems, namely systems with holes in the state space through which mass can leak away, have received a lot of attention, see \cite{Ur86,Schme,N09,BBF2014,KKLL} and reference therein. We prove the following correspondence connecting $\mathcal{S}_{\beta,\alpha}$ and open dynamical systems driven by greedy $\beta$-transformations, namely intermediate $\beta$-transformations with no rotation factor, or equivalently when $\alpha = 0$.
\begin{theorem}\label{thm:main}
Given $(\beta, \alpha) \in \Delta$, there exist $t \in [0, 1]$ and $\beta' \in (1, 2)$ with $(T_{\beta, \alpha}, [0, 1])$ topologically conjugate to the open dynamical system $(T_{\beta', 0}\vert_{K^{+}_{\beta',0}(t)}, K^{+}_{\beta',0}(t))$, where
\begin{align*}
K^{+}_{\beta', 0}(t) \coloneqq \{ x \in[0, 1) \colon T_{\beta',0}^{n}(x)\not \in [0,t) \; \textup{for all} \; n \in \mathbb{N}_{0} \}.
\end{align*}
However, the converse does not hold, namely there exist $t \in [0, 1]$ and $\beta' \in (1, 2)$ such that there does not exist a topological conjugation between $(T_{\beta', 0}\vert_{K^{+}_{\beta',0}(t)}, K^{+}_{\beta', 0}(t))$ and $(T_{\beta, \alpha}, [0, 1])$ for any $(\beta, \alpha) \in \Delta$. Moreover, given $\beta' \in (1, 2)$ with $T_{\beta',0}^{n}(1) = 0$, for some $n \in \mathbb{N}$, there exists $\delta \in (0, \beta'^{-1})$, such that to each $t < \delta$ in the bifurcation set
\begin{align*}
E_{\beta',0}^{+} \coloneqq \{ t \in[0,1) \colon T_{\beta', 0}^{n}(t) \not\in [0, t) \; \textup{for all} \; n \in \mathbb{N}_{0} \}
\end{align*}
one may associate a unique $(\beta, \alpha) \in \Delta$ with $(T_{\beta, \alpha}, [0, 1])$ topologically conjugate to $(T_{\beta', 0}\vert_{K^{+}_{\beta',0}(t)}, K^{+}_{\beta',0}(t))$.
\end{theorem}
This result complements \cite[Proposition 3.1 and Theorem 3.5]{bundfuss_kruger_troubetzkoy_2011}. Here, it is shown that every subshift of finite type and any greedy $\beta$-shift encodes a survivor set of $x \mapsto mx \bmod 1$, for some $m \in \mathbb{N}$ with $m \geq 2$. With this and \Cref{thm:main} at hand, we have that any intermediate $\beta$-shift encodes a survivor set of the doubling map.
We employ our characterisation given in \Cref{thm:main} to (1) build a Krieger embedding theorem for intermediate \mbox{$\beta$-transformations}, and (2) obtain new metric and topological results on survivor sets of intermediate \mbox{$\beta$-transformations}.
\begin{enumerate}[label={\rm(\arabic*)},leftmargin=*]
\item {\bfseries A Krieger embedding theorem for intermediate \mbox{$\beta$-transformations.}} Subshifts, such as $\Omega_{\beta, \alpha}$, are to dynamical systems what shapes like polygons and curves are to geometry. Subshifts which can be described by a finite set of forbidden words are called \textsl{subshifts of finite type} and play an essential role in the study of dynamical systems. One reason why subshifts of finite type are so useful is that they have a simple representation using a finite directed graph. Questions concerning the subshift can then often be phrased as questions about the graph's adjacency matrix, making them more tangible, see for instance \cite{LM,brin_stuck_2002} for further details on subshifts of finite type. Moreover, in the case of greedy $\beta$-shifts (that is when $\alpha = 0$), often one first derives results for greedy $\beta$-shifts of finite type, and then one uses an approximation argument to determine the result for a general greedy $\beta$-shift, see for example \cite{DavidFarm2010,LL16}. Here we prove a Krieger embedding theorem for intermediate $\beta$-shifts. Namely, we show the following, complementing the work of \cite{LSSS} where the same result is proven except where the containment property given in Part~(iii) is reversed. Due to this reversed containment, our proof and that of \cite{LSSS}, although both of a combinatorial flavour, are substantially different.
%
\begin{corollary}\label{Cor_1}
Given $(\beta, \alpha) \in \Delta$, there exists a sequence $\{ (\beta_{n}, \alpha_{n}) \}_{n \in \mathbb{N}}$ in $\Delta$ with $\lim_{n\to \infty} (\beta_{n}, \alpha_{n}) = (\beta, \alpha)$ and
\begin{enumerate}[label={\rm(\roman*)}]
\item $\Omega_{\beta_{n}, \alpha_{n}}$ a subshift of finite type,
\item the Hausdorff distance between $\Omega_{\beta, \alpha}$ and $\Omega_{\beta_{n}, \alpha_{n}}$ converges to zero as $n$ tends to infinity, and
\item $\Omega_{\beta_{n}, \alpha_{n}} \subseteq \Omega_{\beta, \alpha}$.
\end{enumerate}
\end{corollary}
%
\noindent These results together with the results of \cite{LSSS} complements the corresponding result for the case when $\alpha = 0$ proven in \cite{P1960} and which asserts that any greedy $\beta$-shift can be approximated from \textsl{above} and \textsl{below} by a greedy $\beta$-shift of finite type.
\vspace{1em}
%
\item {\bfseries Metric and topological results on survivor sets of intermediate \mbox{$\beta$-transformations}.} Via our correspondence theorem (\Cref{thm:main}), we are able to transfer the results of \cite{KKLL} obtained for open dynamical systems driven by greedy $\beta$-transformations to general intermediate $\beta$-transformations. Specifically, we show the following, extending the results of \cite{KKLL} and complementing those of \cite{Ur86,N09}. Here we follow the notation used in \cite{KKLL}, and recall that an infinite word in the alphabet $\{0, 1\}$ is \textsl{balanced} if and only if the number of ones in any two subwords of the same length differ by at most $1$.
%
\begin{corollary}\label{Cor_2}
The bifurcation set $E_{\beta,\alpha}^{+} \coloneqq \{ t \in[0,1) \colon T_{\beta, \alpha}^{n}(t) \not\in [0, t) \; \textup{for all} \; n \in \mathbb{N}_{0} \}$ is a Lebesgue null set. Moreover, if largest lexicographic word in $\Omega_{\beta, \alpha}$ is balanced, then $E_{\beta,\alpha}^{+}$ contains no isolated points.
%
\end{corollary}
\noindent If the largest lexicographic word in $\Omega_{\beta, \alpha}$ is not balanced, then under an additional technical assumption, in \Cref{cor:isoloated_pts}, we show that there exists a $\delta > 0$, such that $E_{\beta,\alpha}^{+} \cap [0, \delta]$ contains no isolated points. Further, letting $K^{+}_{\beta, \alpha}(t)$ denote the survivor set $\{ x \in[0,1) \colon T_{\beta,\alpha}^{n}(x)\not \in [0,t) \; \text{for all} \; n \in \mathbb{N}_{0} \}$, we have:
%
\begin{corollary}\label{Cor_3}
The dimension function $\eta_{\beta, \alpha} \colon t \mapsto \dim_{\mathcal{H}}(K_{\beta,\alpha}^{+}(t))$ is a Devil staircase function, that is, $\eta_{\beta,\alpha}(0) = 1$, $\eta_{\beta,\alpha}((1-\alpha)/\beta) = 0$, $\eta_{\beta, \alpha}$ is decreasing, and $\eta_{\beta,\alpha}$ is constant Lebesgue almost everywhere.
\end{corollary}
%
\noindent With \Cref{Cor_1,Cor_3} at hand, we can also prove the following.
%
\begin{corollary}\label{Cor_E_beta_alpha}
The bifurcation set $E_{\beta,\alpha}^{+}$ has full Hausdorff dimension.
\end{corollary}
%
\end{enumerate}
The sets $K^{+}_{\beta,\alpha}(t)$ can be seen as a level sets of the set of badly approximable numbers in non-integer bases, that is,
\begin{align*}
\mathrm{BAD}_{\beta, \alpha}(0) \coloneqq \{ x \in [0,1] \colon 0 \not\in \overline{\{T_{\beta, \alpha}^n(x) \colon n\geq 0\}}\} = \bigcup_{t \in (0, 1)} K^{+}_{\beta, \alpha}(t).
\end{align*}
Moreover, for $\xi \in [0,1]$ one can study the more general set
\begin{align*}
\mathrm{BAD}_{\beta, \alpha}(\xi) \coloneqq \{ x \in [0,1] \colon \xi \not\in \overline{\{T_{\beta, \alpha}^n(x) \colon n\geq 0\}}\},
\end{align*}
which, by \Cref{Cor_3}, is a set of full Hausdorff dimension.
When $\alpha = 0$, F\"arm, Persson and Schmeling \cite{DavidFarm2010} and later Hu and Yu \cite{HY} study these sets and showed that they are winning, and hence that they have the large intersection property. To our knowledge, the present work, is the first to consider the case $\alpha \neq 0$. Before stating our results on $\mathrm{BAD}_{\beta, \alpha}(\xi)$, we recall the notion of a winning set.
In the 1960s Schmidt~\cite{S} introduced a topological game in which two players take turns in choosing balls that are a subset of the previously chosen ball. There is a target set $S$ and the objective of Player~$1$ is to make sure that the point that is present in every ball chosen during the game is in $S$. The objective of Player~$2$ is to prevent this. A set is called winning when Player~$1$ can always build a winning strategy no matter how Player~2 plays.
\begin{definition}
Let $\alpha$ and $\gamma\in (0,1)$ be fixed and suppose we have two players, Player~1 and Player~2. Let Player~2 choose a closed initial interval $B_1\subset [0,1]$ and let Player~1 and Player~2 choose nested closed intervals such that $B_1 \supset W_1 \supset B_2 \supset W_2 \supset \ldots$ and $|W_{n+1}|=\alpha |B_n|$ and $|B_{n+1}|=\gamma |W_n|$. A set $S$ is called $(\alpha,\gamma)$-winning if there is a strategy for Player~2 to ensure that $\bigcap_{i\in \mathbb{N}} W_i \subset S$. The set $S$ is called $\alpha$-winning if it is $(\alpha,\gamma)$-winning for all $\gamma\in (0,1)$ and is called winning if it is $\alpha$-winning for some $\alpha\in (0,1)$.
\end{definition}
A key attribute of winning which makes it an interesting property to study is that winning sets have full Hausdorff dimension~\cite{S}. Another, is that it persists under taking intersections, that is, for two winning sets their intersection is again winning, and hence of full Hausdorff dimension~\cite{S}; this is not true in general for sets of full Hausdorff dimension. We also note, the property of winning is preserved under bijective affine transformations.
\begin{theorem}\label{thm:main_2}
Given $(\beta, \alpha) \in \Delta$ with $\Omega_{\beta, \alpha}$ a subshift of finite type, and $\xi \in [0, 1]$, the set $\mathrm{Bad}_{\beta,\alpha}(\xi)$ is winning.
\end{theorem}
We remark that in \cite{JT2009} a similar result for $C^{2}$-expanding Markov circle maps was proven, but that intermediate $\beta$-transformations do not fall into this regime. Further, with \Cref{thm:main_2} at hand and with \cite[Theorem 1]{DavidFarm2010} in mind, we conjecture that $\mathrm{Bad}_{\beta,\alpha}(\xi)$ is winning for all $(\beta, \alpha) \in \Delta$ and $\xi \in [0, 1]$.
Our work is organised as follows. In \Cref{sec:prelim} we we present necessary definitions, preliminaries and auxiliary results. \Cref{sec:proof_thm_1_1,sec:proof_thm_1_6} are respectively devoted to proving \Cref{thm:main,thm:main_2}, and \Cref{sec:proof_cor_1_2,sec:proof_cor_1_3_4} respectively contain the proofs of \Cref{Cor_1}, and \Cref{Cor_2,Cor_3}. Additionally, in \Cref{sec:proof_cor_1_3_4}, we demonstrate how our theory may be used to numerically compute the Hausdorff dimension of $K_{\beta,\alpha}^{+}(t)$.
\section{Notation and preliminaries}\label{sec:prelim}
\subsection{Subshifts}
Let $m \geq 2$ denote a natural number and set $\Lambda = \{0, 1, \ldots, m-1\}$. We equip the space $\Lambda^\mathbb{N}$ of infinite sequences indexed by $\mathbb{N}$ with the topology induced by the \textsl{word metric} $\mathscr{D} \colon \Lambda^\mathbb{N} \times \Lambda^\mathbb{N} \to \mathbb{R}$ given by
\begin{align*}
\mathscr{D}(\omega, \nu) \coloneqq
\begin{cases}
0 & \text{if} \; \omega = \nu,\\
2^{- \lvert\omega \wedge \nu\rvert + 1} & \text{otherwise}.
\end{cases}
\end{align*}
Here, $\rvert \omega \wedge \nu \lvert \coloneqq \min \, \{ \, n \in \mathbb{N} \colon \omega_{n} \neq \nu_n \}$, for $\omega$ and $\nu \in \Lambda^{\mathbb{N}}$ with $\omega \neq \nu$, where for an element $\omega \in \Lambda^{\mathbb{N}}$ we write $\omega=\omega_1\omega_2\cdots$. Note, when equipping $\Lambda$ with the discrete topology, the topology induced by $\mathscr{D}$ on $\Lambda^{\mathbb{N}}$ coincides with the product topology on $\Lambda^{\mathbb{N}}$. We let $\sigma \colon \Lambda^{\mathbb{N}} \to \Lambda^{\mathbb{N}}$ denote the \textsl{left-shift map} defined by $\sigma(\omega_{1} \omega_{2} \cdots) \coloneqq \omega_{2} \omega_{3} \cdots$, and for $n \in \mathbb{N}$, we set $\omega\rvert_{n} = \omega_{1} \omega_{2} \cdots \omega_{n}$. A \textsl{subshift} is any closed set $\Omega \subseteq \Lambda^\mathbb{N}$ with $\sigma(\Omega) \subseteq \Omega$. Given a subshift $\Omega$, we set $\Omega\vert_{0} = \{ \varepsilon\}$, where $\varepsilon$ denotes the empty word, and for $n \in \mathbb{N}$, we set
\begin{align*}
\Omega\lvert_{n} \coloneqq \left\{ \omega_{1} \cdots \omega_{n} \in \Lambda^{n} \colon \,\text{there exists} \; \xi \in \Omega \; \text{with} \; \xi|_n = \omega_{1} \cdots \omega_{n} \right\}
\end{align*}
and write $\Omega^{*} \coloneqq \bigcup_{n \in \mathbb{N}_{0}} \Omega\lvert_{n}$ for the collection of all finite words. We denote by $\lvert \Omega\vert_{n} \rvert$ the cardinality of $\Omega\vert_{n}$, and for $\omega \in \Omega\vert_{n}$, we set $\lvert \omega \rvert = n$. We extend the domain of $\sigma$ to $\Omega^{*}$, by setting $\sigma(\varepsilon) \coloneqq \varepsilon$, and for $n \in \mathbb{N}$, letting
\begin{align*}
\sigma(\omega_{1} \omega_{2} \cdots \omega_{n}) \coloneqq
\begin{cases}
\omega_{2} \omega_{3} \cdots \omega_{n} & \text{if} \; n \neq 1,\\
\varepsilon & \text{otherwise}.
\end{cases}
\end{align*}
For $\omega = \omega_{1} \cdots \omega_{\lvert \omega \rvert} \in \Omega^{*}$ and $\xi = \xi_{1} \xi_{2} \cdots \in \Omega \cup \Omega^{*}$ we denote the concatenation $\omega_{1} \cdots \omega_{\lvert \omega \rvert} \ \xi_{1} \xi_{2} \cdots$ by $\omega \ \xi$.
\begin{definition}
A subshift $\Omega$ is said to be \textsl{of finite type} if there exists $M \in \mathbb{N}$ such that, $\omega_{n - M + 1} \cdots \omega_{n} \ \xi_{1} \cdots \xi_{m} \in \Omega^{*}$, for all $\omega_{1} \cdots \omega_{n}$ and $\xi_{1} \cdots \xi_{m} \in \Omega^{*}$ with $n, m \in \mathbb{N}$ and $n \geq M$, if and only if $\omega_{1} \cdots \omega_{n} \ \xi_{1} \cdots \xi_{m} \in \Omega^{*}$.
\end{definition}
The following result gives an equivalent condition for when a subshift is of finite type.
\begin{theorem}[{\cite[Theorem 2.1.8]{LM}}]
A subshift $\Omega \subseteq \Lambda^{\mathbb{N}}$ is of finite type if and only if there exists a finite set $F \subset \Omega^{*}$ with $\Omega = \mathcal{X}_{F}$, where $\mathcal{X}_{F} \coloneqq \{ \omega \in \Lambda^{\mathbb{N}} \colon \sigma^{m}(\omega)\vert_{\lvert \xi \rvert} \neq \xi \; \text{for all} \; \xi \in F \; \text{and} \; m \in \mathbb{N}\}$.
\end{theorem}
Two subshifts $\Omega$ and $\Psi$ are said to be \textsl{topologically conjugate} if there exists a $\phi \colon \Omega \to \Psi$ that is surjective, one-to-one everywhere except on a countable set on which it is at most finite-to-one, and $\sigma \circ \phi(\omega) = \phi \circ \sigma(\omega)$ for all $\omega \in \Omega$. We call $\phi$ the \textsl{conjugacy}. In the case that $m = 2$, a particular conjugacy which we will make use of is the \textsl{reflection map} $R$ defined by $R(\omega_{1} \omega_{2} \cdots) = (1-\omega_{1})(1-\omega_{2})\cdots$ for $\omega = \omega_{1} \omega_{2} \cdots \in \{0,1\}^{\mathbb{N}}$. This concept of two subshifts being topologically conjugate, naturally extends to general dynamical systems, see for instance \cite{LM,brin_stuck_2002}.
An infinite word $\omega = \omega_{1} \omega_{2} \cdots \in \Lambda^{\mathbb{N}}$ is called \textsl{periodic} with \textsl{period} $n \in \mathbb{N}$ if and only if, for all $m \in \mathbb{N}$, we have $\omega_{1} \cdots \omega_{n} = \omega_{(m - 1)n + 1} \cdots \omega_{m n}$, in which case we write $\omega = \omega\vert_{n}^{\infty}$, and denote the smallest period of $\omega$ by $\operatorname{per}(\omega)$. Similarly, an infinite word $\omega = \omega_{1} \omega_{2} \cdots \in \Lambda^{\mathbb{N}}$ is called \textsl{eventually periodic} with \textsl{period} $n \in \mathbb{N}$ if there exists $k \in \mathbb{N}$ such that, for all $m \in \mathbb{N}$, we have $\omega_{k+1} \cdots \omega_{k+n} = \omega_{k+(m - 1)n + 1} \cdots \omega_{k+ m n}$, in which case we write $\omega = \omega_{1} \cdots \omega_{k} (\omega_{k+1} \cdots \omega_{k+n})^\infty$.
\subsection{Intermediate \texorpdfstring{$\beta$}{beta}-shifts}\label{sec:beta-shifts}
For $(\beta, \alpha) \in \Delta$ we set $p = p_{\beta, \alpha} = (1-\alpha)/\beta$ and define the \textsl{upper $T_{\beta, \alpha}$-expansion} $\tau_{\beta, \alpha}^{+}(x)$ of $x \in [0, 1]$ to be the infinite word $\omega_{1} \omega_{2} \cdots \in \{ 0, 1\}^{\mathbb{N}}$, where, for $n \in \mathbb{N}$,
\begin{align}\label{eq:upper_kneading}
\omega_{n} \coloneqq
\begin{cases}
0 & \quad \text{if } T_{\beta,\alpha}^{n-1}(x) < p,\\
1 & \quad \text{otherwise,}
\end{cases}
\end{align}
and define the \textsl{lower $T_{\beta, \alpha}$-expansion} $x$ to be $\tau^{-}_{\beta, \alpha}(x) \coloneqq \lim_{y \nearrow x} \tau_{\beta,\alpha}^{+}(y)$. Note, one can also define $\tau^{-}_{\beta, \alpha}(x)$ analogously to $\tau^{+}_{\beta, \alpha}(x)$ by using the map $T_{\beta,\alpha}^{-} \colon x \mapsto \beta x + \alpha$ if $x \leq p$, and $x \mapsto \beta x + \alpha - 1$ otherwise, in replace of of $T_{\beta, \alpha}$, and by changing the \textsl{less than}, to \textsl{less than or equal to} in \eqref{eq:upper_kneading}, see \cite[Section 2.2]{LSSS}. With this in mind, and for ease of notation, sometimes we may write $T_{\beta,\alpha}^{+}$ for $T_{\beta, \alpha}$.
We denote the images of $[0,1)$ under $\tau_{\beta, \alpha}^{+}$ by $\Omega^{+}_{\beta, \alpha}$, the image of $(0,1]$ under $\tau_{\beta, \alpha}^{-}$ by $\Omega^{-}_{\beta, \alpha}$, and set $\Omega_{\beta, \alpha} \coloneqq \Omega_{\beta, \alpha}^{+} \cup \Omega_{\beta, \alpha}^{-}$. We refer to $\Omega_{\beta, \alpha}$ as an intermediate $\beta$-shift and define the \textsl{upper} and \textsl{lower kneading invariants} of $\Omega_{\beta,\alpha}$ to be the infinite words $\tau^{\pm}_{\beta, \alpha}(p)$, respectively. The following result shows that $\tau^{\pm}_{\beta, \alpha}(p)$ completely determine $\Omega_{\beta,\alpha}$.
\begin{theorem}[{\cite{P1960,HS:1990,AM:1996,KS:2012,BHV:2011}}]\label{thm:Structure}
For $(\beta, \alpha) \in \Delta$, the spaces $\Omega_{\beta, \alpha}^{\pm}$ are completely determined by upper and lower kneading invariants of $\Omega_{\beta, \alpha}$, namely
\begin{align*}
\Omega_{\beta, \alpha}^{+} &= \{ \omega \in \{ 0, 1\}^{\mathbb{N}} \colon \tau_{\beta, \alpha}^{+}(0) \preceq \sigma^{n}(\omega) \prec \tau_{\beta, \alpha}^{-}(p) \; \textup{or} \; \tau_{\beta, \alpha}^{+}(p) \preceq \sigma^{n}(\omega) \prec \tau_{\beta, \alpha}^{-}(1) \; \textup{for all} \; n \in \mathbb{N}_{0} \},\\
%
\Omega_{\beta, \alpha}^{-} &= \{ \omega \in \{ 0, 1\}^{\mathbb{N}} \colon \tau_{\beta, \alpha}^{+}(0) \prec \sigma^{n}(\omega) \preceq \tau_{\beta, \alpha}^{-}(p) \; \textup{or} \; \tau_{\beta, \alpha}^{+}(p) \prec \sigma^{n}(\omega) \preceq \tau_{\beta, \alpha}^{-}(1) \; \textup{for all} \; n \in \mathbb{N}_{0} \}.
\end{align*}
Here, $\prec$, $\preceq$, $\succ$ and $\succeq$ denote the lexicographic orderings on $\{ 0 ,1\}^{\mathbb{N}}$. Moreover, the cardinality of $\Omega_{\beta, \alpha}^{\pm}$ is equal to that of the continuum, and $\Omega_{\beta, \alpha}$ is closed with respect to the metric $\mathscr{D}$. Hence, $\Omega_{\beta, \alpha}$ is a subshift.
\end{theorem}
This result establishes the importance of the kneading invariants of $\Omega_{\beta, \alpha}$, for a given $(\beta, \alpha) \in \Delta$, and so it is natural to ask, for a fixed $\beta \in (1, 2)$, if they are monotonic or continuous in $\alpha$. The following proposition answers this.
\begin{proposition}[{\cite{BHV:2011,Cooperband2018ContinuityOE}}]\label{prop:mon_cont_kneading}
Let $\beta \in (1,2)$ be fixed.
\begin{enumerate}[label={\rm(\arabic*)}]
\item The maps $a \mapsto \tau_{\beta,a}^{\pm}(p_{\beta, a})$ are strictly increasing with respect to the lexicographic ordering.
\item The map $a \mapsto \tau_{\beta, a}^{+}(p_{\beta, a})$ is right continuous, and the map $a \mapsto \tau_{\beta,a}^{-}(p_{\beta, a})$ is left continuous.
\item If $\alpha \neq 0$ and $\tau_{\beta, \alpha}^{+}(p_{\beta, \alpha})$ is not periodic, then $a \mapsto \tau_{\beta,a}^{+}(p_{\beta, a})$ is continuous at $\alpha$, and if $\alpha \neq \beta-1$ and $\tau_{\beta,\alpha}^{-}(p_{\beta, \alpha})$ is not periodic, then $a \mapsto \tau_{\beta,a}^{-}(p_{\beta, a})$ is continuous at $\alpha$.
\item If $\tau_{\beta,\alpha}^{+}(p_{\beta, \alpha})$ is periodic with period $M$, for a given $\alpha \in (0, 2-\beta]$, then given $m \in \mathbb{N}$, there exists a real number $\delta > 0$ so that, $\tau_{\beta,\alpha-\delta'}^{+}(p_{\beta, \alpha-\delta'})\vert_{m} = \tau_{\beta,\alpha}^{+}(p_{\beta, \alpha})\vert_{M}\tau_{\beta,\alpha}^{-}(p_{\beta, \alpha})\vert_{m-M}$, for all $\delta' \in (0, \delta)$.
\item If $\tau_{\beta,\alpha}^{-}(p_{\beta, \alpha})$ is periodic with period $M$, for a given $\alpha \in [0, 2-\beta)$, then given $m \in \mathbb{N}$, there exists a real number $\delta > 0$ so that, $\tau_{\beta,\alpha+\delta'}^{-}(p_{\beta, \alpha+\delta'})\vert_{m} = \tau_{\beta,\alpha}^{-}(p_{\beta, \alpha})\vert_{M}\tau_{\beta,\alpha}^{+}(p_{\beta, \alpha})\vert_{m-M}$, for all $\delta' \in (0, \delta)$.
\end{enumerate}
\end{proposition}
Another natural question to ask is when do two infinite words in the alphabet $\{0, 1\}$ give rise to kneading invariants of an intermediate $\beta$-shift. This question was addressed in \cite{barnsley_steiner_vince_2014} where the following solution was derived and for which we require the following notation. Given $\omega$ and $\nu \in \{0,1\}^\mathbb{N}$ with $\sigma(\nu) \preceq \omega \preceq \nu \preceq \sigma(\omega)$ we set
\begin{align*}
\Omega^{+}(\omega,\nu) &\coloneqq \{ \xi \in \{0,1\}^{\mathbb{N}} \colon \sigma(\nu) \preceq \sigma^{n}(\xi) \prec \omega \; \text{or} \; \nu \preceq \sigma^{n}(\xi) \prec \sigma(\omega) \; \text{for all} \; n \in \mathbb{N}_{0} \},\\
\Omega^{-}(\omega,\nu) &\coloneqq \{ \xi \in \{0,1\}^{\mathbb{N}} \colon \sigma(\nu) \prec \sigma^n(\xi) \preceq \omega \; \text{or} \; \nu \prec \sigma^{n} (\xi) \preceq \sigma(\omega) \; \text{for all} \; n \in \mathbb{N}_{0} \}.
\end{align*}
\begin{theorem}[{\cite{barnsley_steiner_vince_2014}}]\label{thm:BSV14}
Two infinite words $\omega = \omega_{1}\omega_{2}\cdots$ and $\nu=\nu_{1}\nu_{2}\cdots \in \{0,1\}^\mathbb{N}$ are kneading invariants of an intermediate $\beta$-shift $\Omega_{\beta, \alpha}$, for some $(\beta, \alpha) \in \Delta$ if and only if the following four conditions hold.
\begin{enumerate}[label={\rm(\arabic*)}]
\item $\omega_1=0$ and $\nu_1=1$,
\item $\omega\in\Omega^-(\omega, \nu) $ and $\nu\in\Omega^+(\omega, \nu)$,
\item $\lim_{n\to\infty} \log(|\Omega^+|_n)>0$, and
\item if $\omega,\nu\in\{\xi,\zeta \}^\mathbb{N}$ for two finite words $\xi$ and $\zeta$ in the alphabet $\{0,1\}$ with length greater than or equal to three, such that $\xi_1\xi_2=01$ , $\zeta_1\zeta_2=10$, $\xi^\infty \in \Omega^-(\xi^\infty, \zeta^\infty)$ and $\zeta^\infty \in \Omega^{+}(\xi^\infty, \zeta^\infty)$, then $\omega=\xi^\infty$ and $\nu=\zeta^\infty$.
\end{enumerate}
\end{theorem}
This result together with \Cref{thm:Structure} can be seen as a generalisation of the following seminal result of Parry.
\begin{theorem}[{\cite[Corollary 1]{P1960}}]\label{thm:Parry_converse}
If $\omega \in \{0, 1\}^{\mathbb{N}}$ with $\sigma^{n}(\omega) \neq 0^\infty$ for all $n \in \mathbb{N}$, then there exists a $\beta \in (1,2)$ with $\omega = \tau_{\beta, 0}^{-}(1)$ if and only if $\sigma^{m}(\omega) \preceq \omega$ for all $m \in \mathbb{N}$.
\end{theorem}
Combining this result with \Cref{thm:Structure}, we obtain the following which will be utilised in our proof of \Cref{thm:main}.
\begin{corollary}\label{cor:From_greedy_to_intermediate}
Given $(\beta, \alpha) \in \Delta$, there exists $\beta' \in (1,2)$ such that $\tau_{\beta,\alpha}^{-}(1) = \tau_{\beta',0}^{-}(1)$.
\end{corollary}
In the sequel we will also make use of the projection $\pi_{\beta, \alpha} \colon \{ 0, 1 \}^{\mathbb{N}} \to [0, 1]$ defined by
\begin{align*}
\pi_{\beta, \alpha}(\omega_{1} \omega_{2} \cdots) \coloneqq \frac{\alpha}{1 - \beta} + \sum_{k \in \mathbb{N}} \frac{\omega_{k}}{\beta^k}.
\end{align*}
We note that $\pi_{\beta, \alpha}$ is linked to the iterated function systems $( [0, 1]; f_{0} \colon x \mapsto \beta^{-1}x, f_{1} \colon x \mapsto \beta^{-1}(x+1)$ via the equality
\begin{align*}
\pi_{\beta, \alpha}(\omega_{1} \omega_{2} \cdots)
= \alpha / (1-\beta) + \lim_{n \to \infty} f_{\omega_{1}} \circ \cdots \circ f_{\omega_{n}}([0, 1]),
\end{align*}
and the iterated function system $( [0, 1]; f_{\beta, \alpha, 0} \colon x \mapsto \beta^{-1}x - \alpha\beta^{-1}, f_{\beta, \alpha, 1} \colon x \mapsto \beta^{-1}x - (\alpha-1)\beta^{-1}$ via the equality
\begin{align}\label{eq:alt_IFS}
\pi_{\beta, \alpha}(\omega_{1} \omega_{2} \cdots)
=\lim_{n \to \infty} f_{\beta,\alpha,\omega_{1}} \circ \cdots \circ f_{\beta,\alpha,\omega_{n}}([0, 1]).
\end{align}
We refer the reader to \cite{F:1990} for further details on iterated function systems. An important property of $\pi_{\beta, \alpha}$ is that the following diagrams commute.
\begin{align}\label{eq:commutative_diag}
\begin{aligned}
\xymatrix@C+2pc{
\Omega^{+}_{\beta, \alpha}
\ar@/_/[d]_{\pi_{\beta, \alpha}}
\ar[r]^{\sigma} &
\Omega_{\beta, \alpha}^{+}
\ar@/^/[d]^{\pi_{\beta, \alpha}} \\
{[0, 1)}
\ar@/_/[u]_{\tau_{\beta,\alpha}^{+}}
\ar[r]_{T_{\beta, \alpha}} &
\ar@/^/[u]^{\tau_{\beta,\alpha}^{+}}
[0, 1)}
\end{aligned}
\qquad\qquad
\begin{aligned}
\xymatrix@C+2pc{
\Omega^{-}_{\beta, \alpha}
\ar@/_/[d]_{\pi_{\beta, \alpha}}
\ar[r]^{\sigma} &
\Omega_{\beta, \alpha}^{-}
\ar@/^/[d]^{\pi_{\beta, \alpha}} \\
{(0, 1]}
\ar@/_/[u]_{\tau_{\beta,\alpha}^{-}}
\ar[r]_{T^{-}_{\beta, \alpha}} &
\ar@/^/[u]^{\tau_{\beta,\alpha}^{-}}
(0, 1]}
\end{aligned}
\end{align}
This result is verifiable from the definitions of the involved maps and a sketch of a proof can be found in \cite{BHV:2011}. It also yields that each $\omega \in \Omega_{\beta, \alpha}$ is a $\beta$-expansion, and corresponds to the unique point
\begin{align*}
\sum_{k \in \mathbb{N}} \frac{\omega_{k}}{\beta^{k}} = \pi_{\beta,\alpha}(\omega) - \frac{\alpha}{1-\beta}
\end{align*}
in $[\alpha/(\beta-1), 1+\alpha/(\beta-1)]$. A particular expansion which we will make use of is $\tau_{\beta,0}^{-}(1)$ which is referred to as the \textsl{quasi-greedy $\beta$-expansion of $1$}.
The commutativity of the diagrams given in \eqref{eq:commutative_diag} also implies that the dynamical systems $(\Omega_{\beta, \alpha}, \sigma)$ and $([0, 1), T_{\beta, \alpha})$ are topologically conjugate. This in tandem with \Cref{thm:Structure}, yields that the upper and lower kneading invariants completely determine the dynamics of $T_{\beta, \alpha}$. Additionally, we have the following.
\begin{theorem}[{\cite{GH,BHV:2011}}]\label{thm:Laurent}
Let $(\beta, \alpha) \in \Delta$ be fixed. If $\omega = \omega_{1}\omega_{2} \cdots $ and $\nu = \nu_{1} \nu_{2} \cdots $, respectively, denote the upper and lower kneading invariants of $\Omega_{\beta, \alpha}$, then $\beta$ is the maximal real root of the Laurent series
\begin{align*}
\pi_{z,\alpha}(\omega) - \pi_{z,\alpha}(\nu)
= \sum_{k \in \mathbb{N}} (\omega_{k} - \nu_{k})\,z^{-k}.
\end{align*}
\end{theorem}
The above allows us to transfer results on the dynamical system $(\Omega_{\beta, \alpha}, \sigma)$ to $([0, 1], T_{\beta, \alpha})$ and vice versa. We will utilise this in the proofs of our main results, and thus will make use of the following symbolic representations of $K_{\beta, \alpha}^{+}(t)$ and $E_{\beta, \alpha}^{+}$ defined in \Cref{sec:intro}. For $(\beta, \alpha) \in \Delta$ and $t \in [0,1]$, we set
\begin{align*}
\mathcal{K}^{+}_{\beta,\alpha}(t) &\coloneqq \{ \omega \in \{0,1\}^\mathbb{N} \colon \tau_{\beta,\alpha}^{+}(t) \preceq \sigma^{n}(\omega) \prec \tau_{\beta,\alpha}^{-}(1) \; \text{for all} \; n \in \mathbb{N}_{0} \}
%
\intertext{and we let}
%
\mathcal{E}_{\beta,\alpha}^{+} &\coloneqq \{ \omega \in \{0,1\}^\mathbb{N} \colon \omega \preceq \sigma^{n}(\omega) \prec \tau_{\beta,\alpha}^{-}(1) \; \text{for all} \; n \in \mathbb{N}_{0} \}.
\end{align*}
In addition to this, we will utilise the following Ledrappier-Young formula due to Raith \cite{R}. For $t \in (0,1]$,
\begin{align}\label{eq:entopen}
\dim_{H}(K_{\beta,\alpha}(t)) = \frac{h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K_{\beta,\alpha}(t)})}{\log(\beta)}.
\end{align}
where $K_{\beta,\alpha}(t) \coloneqq \{ x \in[0,1] \colon T_{\beta,\alpha}^{n}(x)\not \in (0,t) \; \text{for all} \; n \in \mathbb{N}_{0} \}$ and where $h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)})$ denotes the topological entropy of the dynamical system $(K^{+}_{\beta,\alpha}(t), T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)})$. Here, for a given subset $L \subseteq [0, 1]$ we set
\begin{align*}
h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{L}) \coloneqq \lim_{n \to \infty} \frac{\log (\lvert \tau_{\beta,\alpha}^{+}(L)\vert_{n}\rvert)}{n}
\end{align*}
see \cite{LM,brin_stuck_2002,Walters_1982}, and reference therein, for further details on topological entropy. More specifically, in \Cref{sec:proof_cor_1_3_4}, we apply the following result, which is a consequence of Raith's Ledrappier-Young formula \eqref{eq:entopen} and \cite[Proposition 2.6]{KKLL}.
\begin{proposition}\label{prop:ent}
For $(\alpha, \beta) \in \Delta$ and $t \in [0,1]$,
\begin{align}\label{eq:entropy}
h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K_{\beta,\alpha}(t)}) = h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)})
\end{align}
and hence
\begin{align}\label{eq:ent}
\dim_{H}(K^{+}_{\beta,\alpha}(t)) = \frac{h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)})}{\log(\beta)}.
\end{align}
\end{proposition}
\begin{proof}
Since the set $K_{\beta,\alpha}(t)\backslash K^{+}_{\beta,\alpha}(t)$ is countable, we have $\dim_H(K_{\beta,\alpha}(t))=\dim_H(K^{+}_{\beta,\alpha}(t))$. The Ledrappier-Young formula given in \eqref{eq:ent} is therefore a direct consequence of \eqref{eq:entopen} and \eqref{eq:entropy}. The proof of \eqref{eq:entropy}, follows from a small adaptation of the proof of \cite[Proposition 2.6]{KKLL}, which we present below.
If $0 \not\in K_{\beta,\alpha}(t)$ or $t=0$, then $\mathcal{K}^{+}_{\beta,\alpha}(t) =\mathcal{K}_{\beta,\alpha}(t)$, and if $t\not \in E^{+}_{\beta,\alpha}$ then there exists a $t^*>t$ with $K^{+}_{\beta,\alpha}(t)=K^{+}_{\beta,\alpha}(t^*)$. The first of these two statements follows directly from the definition of $K_{\beta,\alpha}(t)$ and $K_{\beta,\alpha}^{+}(t)$, and the second can be seen to hold as follows. For every $t \not\in E^{+}_{\beta,\alpha}$ there exists a smallest natural number $N$ such that $T_{\beta,\alpha}^N(t) \in [0,t)$. Let $\delta = \min \{ p_{\beta, \alpha} - T_{\beta,\alpha}^{n}(t) \colon n \in \{ 0, 1, \dots, N-1 \} \; \text{and} \; T_{\beta,\alpha}^{n}(t) < p_{\beta, \alpha} \}$ and set $\varepsilon = \min\{t-T_{\beta,\alpha}^N(t), \delta\}/\beta^{N}$. By construction, for all $s\in (t,t+\varepsilon)$, we have that $T_{\beta,\alpha}^N(s)\in [0,t) \subset [0, s)$, and hence that $s\not\in K^{+}_{\beta,\alpha}(s)$ and $s\not\in K^{+}_{\beta,\alpha}(t) $. This implies that $K^{+}_{\beta,\alpha}(t)= K^{+}_{\beta,\alpha}(s)$ for all $s\in (t,t+\varepsilon)$. In fact letting $t^{*} = \inf \{ s \in E^{+}_{\beta,\alpha} \colon s > t \}$, a similar justification yields that $K^{+}_{\beta,\alpha}(t)= K^{+}_{\beta,\alpha}(s)$ for all $s \in (t,t^*)$.
All-in-all, this all implies that $t^* \in E^{+}_{\beta,\alpha}$.
Therefore, it suffices to show that, if $t \in E^{+}_{\beta,\alpha} \setminus \{0\}$ and $0\in K_{\beta,\alpha}(t)$, then $h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K_{\beta,\alpha}(t)})= h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)})$. To this end, suppose that $t\in E^{+}_{\beta,\alpha} \setminus \{0\}$. In which case $\sigma^{n}(\tau^{+}_{\beta,\alpha}(t)) \neq \tau^{+}_{\beta,\alpha}(0)$, and so setting
\begin{align*}
\mathcal{K}_{\beta,\alpha}^{0}(t) &\coloneqq \{ \omega \in \{0,1\}^{\mathbb{N}} \colon \text{there exists} \; m \in \mathbb{N}_{0} \; \text{with} \; \sigma^{m}(\omega) = \tau^{+}_{\beta,\alpha}(0) \; \text{and} \; \tau^{+}_{\beta,\alpha}(t) \prec \sigma^{n}(\omega) \prec \tau^{-}_{\beta,\alpha}(1) \; \text{for all} \; n \in \mathbb{N}_{0} \}
\end{align*}
and letting $\mathcal{K}_{\beta,\alpha}(t)$ denote the symbolic representation of $K_{\beta,\alpha}(t)$, namely letting
\begin{align*}
\mathcal{K}_{\beta,\alpha}(t) &\coloneqq \{ \omega \in \{0,1\}^{\mathbb{N}} \colon \sigma^{n}(\omega) = \tau^{+}_{\beta,\alpha}(0) \; \text{or} \; \tau^{+}_{\beta,\alpha}(t) \preceq \sigma^{n}(\omega) \prec \tau^{-}_{\beta,\alpha}(1) \; \text{for all} \; n \in \mathbb{N}_{0} \},
\end{align*}
we have that $\mathcal{K}_{\beta,\alpha}(t)\setminus \mathcal{K}_{\beta,\alpha}^{+}(t) = \mathcal{K}_{\beta,\alpha}^{0}(t)$. Let $k \in \mathbb{N}$ be fixed, and let $\zeta \in \mathcal{K}_{\beta,\alpha}^{0}(t)\vert_{k}$ with $\zeta \neq \tau^{+}_{\beta,\alpha}(0)\vert_{k}$. By construction, there exists $\omega \in \mathcal{K}_{\beta,\alpha}^{0}(t)$ with $\omega\vert_{k} = \zeta$. Let $j$ be the smallest natural number such that $\sigma^{j}(\omega) = \tau^{+}_{\beta,\alpha}(0)$ and set $\nu = \omega\vert_{j-1}\xi_{j}$, where $\xi_{j}$ denotes the $j$-th letter of $\tau^{+}_{\beta,\alpha}(0)$. Observe that
\begin{align}\label{eq:ent_proof}
\tau^{+}_{\beta,\alpha}(t)\vert_{j-i} \preceq \sigma^{i}(\nu) \preceq \tau^{+}_{\beta,\alpha}(1)\vert_{j-i}
\end{align}
for all $i \in \{0, 1, \dots, j-1\}$. Let $i^{*} \in \{0, 1, \dots, j-1\}$ be the smallest integer such that $\sigma^{i^{*}}(\nu) = \tau^{+}_{\beta,\alpha}(t)\vert_{j-i^{*}}$, and if strict inequality holds in the lower bound of \eqref{eq:ent_proof} for all $i \in \{0, 1, \dots, j-1\}$, then set $i^{*}=j$. By the minimality of $i^{*}$, we have $\nu \, \sigma^{j-i^{*}+1}(\tau^{+}_{\beta,\alpha}(t)) = \nu\vert_{i^{*}} \tau^{+}_{\beta,\alpha}(t) \in \mathcal{K}_{\beta,\alpha}^{+}(t)$. Noting, $\nu\vert_{j-1} = \zeta\vert_{j-1}$ if $j \leq k$, and $\nu\vert_{k} = \zeta\vert_{k}$ if $j \geq k+1$, we have that
\begin{align}\label{eq:ent_proof_2}
\lvert \mathcal{K}_{\beta,\alpha}^{0}(t)\vert_{k} \rvert
\leq 1 + 2 \sum_{j = 0}^{k} \, \lvert \mathcal{K}_{\beta,\alpha}^{+}(t)\vert_{j} \rvert
\leq 3 (k+1) \lvert \mathcal{K}_{\beta,\alpha}^{+}(t)\vert_{k} \rvert.
\end{align}
Since $\mathcal{K}_{\beta,\alpha}(t)\setminus \mathcal{K}_{\beta,\alpha}^{+}(t) = \mathcal{K}_{\beta,\alpha}^{0}(t)$, and since by definition,
\begin{align*}
h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K_{\beta,\alpha}(t)})
= \lim_{n \to \infty} \frac{\log(\mathcal{K}_{\beta,\alpha}(t)\vert_{n})}{n}
\quad \text{and} \quad
h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)})
= \lim_{n \to \infty} \frac{\log(\mathcal{K}^{+}_{\beta,\alpha}(t)\vert_{n})}{n},
\end{align*}
the inequality given in \eqref{eq:ent_proof_2} implies that $h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^{+}_{\beta,\alpha}(t)}) \geq h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K_{\beta,\alpha}(t)})$. As $K^{+}_{\beta,\alpha}(t) \subseteq K_{\beta,\alpha}(t)$ and as the entropy of a subsystem cannot exceed that of its parent system, the result follows.
\end{proof}
\subsection{\texorpdfstring{$\beta$}{beta}-shifts of finite type}
In \cite{LSS16} a study of when an intermediate $\beta$-shift is of finite type was carried out. This work was continued in \cite{LSSS} where it was shown that any intermediate $\beta$-shift can be \textsl{approximated from below} by an intermediate $\beta$-shift is of finite type. These results are summarised in the following theorem.
\begin{theorem}[{\cite{LSSS,LSS16}}]\label{thm:LSSS}
An intermediate $\beta$-shift $\Omega_{\beta,\alpha}$ is a subshift of finite type if and only if the kneading invariants $\tau_{\beta,\alpha}^{+}(p_{\beta,\alpha})$ and $\tau_{\beta,\alpha}^{-}(p_{\beta,\alpha})$ are periodic. Moreover, given $(\beta, \alpha) \in \Delta$ and $\epsilon > 0$, there exists a $(\beta', \alpha') \in \Delta$, with $0 \leq \beta' - \beta < \epsilon$ and $\lvert \alpha - \alpha' \rvert < \epsilon$ and such that
\begin{enumerate}[label={\rm(\arabic*)}]
\item $\Omega_{\beta', \alpha'}$ is a subshift of finite type,
\item the Hausdorff distance between $\Omega_{\beta, \alpha}$ and $\Omega_{\beta', \alpha'}$ is less than $\epsilon$, and
\item $\Omega_{\beta, \alpha} \subseteq \Omega_{\beta', \alpha'}$.
\end{enumerate}
\end{theorem}
\subsection{Transitivity of intermediate \texorpdfstring{$\beta$}{beta}-transformations}
An interval map $T \colon [0,1] \to [0,1]$ is said to be \textsl{transitive} if for any open subinterval $U$ of $[0,1]$ there exists an $m \in \mathbb{N}$ with $\bigcup_{k = 1}^{m} T^{k}(U) = (0,1)$. The property of transitivity will play an important part in our proof of \Cref{thm:main_2}, and thus we will utilise the following result of \cite{G1990,Palmer79} on non-transitive intermediate $\beta$-transformations. Note the contrast in the structure of the set of $(\beta, \alpha) \in \Delta$ with $T_{\beta,\alpha}$ transitive and the set of $(\beta, \alpha) \in \Delta$ with $\Omega_{\beta, \alpha}$ of finite type, namely that the former is has positive $2$-dimensional Lebesgue measure and the latter is countable.
\begin{theorem}[{\cite{G1990,Palmer79}}]\label{thm:G1990+Palmer79}
Let $\Delta_{\operatorname{trans}}$ denote the set of $(\beta, \alpha) \in \Delta$ with $T_{\beta,\alpha}$ transitive. The sets $\Delta_{\operatorname{trans}}$ and $\Delta \setminus \Delta_{\operatorname{trans}}$ have positive Lebesgue measure. Moreover, given $(\beta, \alpha) \in \Delta \setminus \Delta_{\operatorname{trans}}$, there exist
\begin{enumerate}[label={\rm(\roman*)}]
\item a natural number $n \geq 2$ and $k \in \{1,\ldots, n-1\}$ with $n$ and $k$ co-prime,
\item a sequence of points $\{ b_{0}, b_{1}, \ldots, b_{2n-1}\}$ in $(0,1)$ with $b_{i} < b_{i+1}$ for all $i \in \{0, 1, \ldots, 2n-2\}$, and
\item an $\tilde{\alpha} \in [0, 2-\beta^{n}]$,
\end{enumerate}
such that
\begin{enumerate}[label={\rm(\arabic*)}]
\item the transformation $T_{\beta^{n},\tilde{\alpha}}$ is transitive,
\item $T_{\beta,\alpha}^{n}(J_{i}) = J_{i}$ and $T_{\beta,\alpha}(J_{i}) = J_{i+k \bmod{n}}$, for all $i \in \{ 0, 1, \ldots, n-1 \}$, and
\item $T_{\beta,\alpha}^{n}\vert_{J_{i}}$ is topologically conjugate to $T_{\beta^{n},\tilde{\alpha}}$, for all $i \in \{ 0, 1, \ldots, n-1 \}$, where the conjugation is linear.
\end{enumerate}
Here, $J_{0} = [0, b_{0}] \cup [b_{2n-1}, 1]$, and $J_{i} = [b_{2i-1}, b_{2i}]$, for all $i \in \{1, 2, \ldots, n-1\}$. Further, there exists a $T_{\beta,\alpha}$-periodic point $q$ in $\mathscr{J}=\bigcup_{i = 0}^{n-2} [b_{2i}, b_{2i+1}]$, such that the orbit of $q$ under $T_{\beta,\alpha}$ is contained in $\mathscr{J}$, and for all $x$ in $\mathscr{J}$ but not in the orbit of $q$, there exists an $m \in \mathbb{N}$ such that $T_{\beta,\alpha}^{m}(x) \in \bigcup_{i = 0}^{n-1} J_{i}$.
\end{theorem}
\subsection{A sufficient condition for a dynamical set to be winning}\label{sec:HY}
To prove \Cref{thm:main_2} we not only appeal to the results of \cite{G1990,Palmer79}, but also to \cite[Theorem 2.1]{HY}, where a sufficient condition for certain dynamical sets to be winning is given. In order to state \cite[Theorem 2.1]{HY} we require the following notation. A partition of $[0,1]$ is a collection of finitely many intervals $\{ I(i) \}_{i \in \Lambda}$, where $\Lambda = \{0, 1, \ldots, m-1\}$ for some $m \in \mathbb{N}$, with pairwise disjoint interiors such that $[0,1] = \bigcup_{i \in \Lambda} I(i)$. Here, we assume that the intervals are ordered, namely, that if $i$ and $j \in \{0, 1, \ldots, m-1\}$ with $i < j$, then, for all $x \in I(i)$ and $y \in I(j)$, we have that $x \leq y$. Let $T \colon [0, 1] \to [0,1]$ and let $\{ I(i) \}_{i \in \Lambda}$ denote a partition of $[0, 1]$, such that $T$ restricted to $I(i)$ is monotonic and continuous for all $i \in \Lambda$. For $\xi = \xi_{1} \xi_{2} \cdots \xi_{n} \in \Lambda^{n}$, for some $n \in \mathbb{N}$, we set
\begin{align*}
I(\xi) = \bigcap_{i = 1}^{n} \ \{ x \in [0, 1] \colon T^{i-1}(x) \in I(\xi_{i}) \}.
\end{align*}
If $I(\xi)$ is non-empty, we call $I(\xi)$ a \textsl{level $n$ cylinder set} of $T$, and $\xi$ an \textsl{admissible word of length $n$} with respect to the partition $\{ I(i)\}_{i \in \Lambda}$. For $n \in \mathbb{N}_{0}$, we denote by $\Omega_{T}\vert_{n}$ the set of all admissible words of length $n$, where by convention $\Omega_{T}\vert_{0} = \{ \varepsilon \}$, and set $\Omega_{T}^{*} = \bigcup_{n \in \mathbb{N}_{0}} \Omega_{T}\vert_{n}$.
When $T = T_{\beta, \alpha}$ and when our partition is $\{ I(0) = [0,p_{\beta, \alpha}), I(1)=[p_{\beta, \alpha},1] \}$, for some $(\beta, \alpha) \in \Delta$, we have $\Omega_{T}\vert_{n} =\Omega_{\beta, \alpha}\vert_{n}$. Further, for $\xi \in \{0, 1\}^{*}$, we have $I(\xi)$ is non-empty if and only if there exists an $\omega \in \Omega_{\beta, \alpha}$ with $\omega\vert_{\lvert \xi \rvert} = \xi$, and $\overline{I(\xi)} = \pi_{\beta, \alpha}( \{ \omega \in \Omega_{\beta, \alpha} \colon \omega\vert_{n} = \xi \} )$, where $\overline{I(\xi)}$ denotes the closure of $I(\xi)$.
For $\xi$ and $\nu \in \Omega_{T}^{*}$ and $c > 0$ a real number, we say that $\xi$ is \textsl{$\nu$-extendable} if the concatenation $\xi\nu$ is admissible, and say that the cylinder sets $I(\xi)$ and $I(\nu)$ are \textsl{$c$-comparable} if $c \leq \lvert I(\xi) \rvert / \lvert I(\nu) \rvert \leq 1/c$. We call $T$ is \textsl{piecewise locally $C^{1+\delta}$ expanding} if $T$ restricted to $I(i)$ is differentiable for all $i \in \Lambda$, and
\begin{enumerate}[label={\rm(\arabic*)}]
\item there exists a real number $\eta > 0$, such that $\lvert T'(x) \rvert > \eta$ for all $x \in I(i)$ and $i \in \Lambda$, and there exists $k \in \mathbb{N}$ and a real number $\lambda > 1$ such that $\lvert (T^{k})'(x) \rvert \geq \lambda$ for all $\xi \in \Omega_{T}\vert_{k}$ and $x \in I(\xi)$, and
\item there exist two positive constants $\delta$ and $c$ such that for all $i \in \Lambda$ and all $x$ and $y \in I(i)$,
\begin{align*}
\left\lvert \frac{T'(x)}{T'(y)} - 1 \right\rvert \leq c \lvert x - y \rvert^{\delta}.
\end{align*}
\end{enumerate}
We call $T$ \textsl{Markov} if for all $i$ and $j \in \Lambda$, either $T(I(i)) \cap I(j) = \emptyset$, or $I(j) \subseteq T(I(i))$. Letting $(\beta, \alpha) \in \Delta$ with $\Omega_{\beta, \alpha}$ a subshift of finite type, we set $A = \{ a_{1}, a_{2}, \ldots, a_{n} \}$ to be the set of ordered points of
\begin{align*}
\{ \pi_{\beta, \alpha}(\sigma^{k}(\tau_{\beta,\alpha}^{+}(p_{\beta, \alpha}))) \colon k \in \mathbb{N}\} \cup \{ \pi_{\beta, \alpha}(\sigma^{k}(\tau_{\beta,\alpha}^{-}(p_{\beta, \alpha}))) \colon k \in \mathbb{N} \}.
\end{align*}
The transformation $T_{\beta, \alpha}$ is a piecewise locally $C^{1+\delta}$ expanding Markov map with respect to the partition
\begin{align}\label{eq:Markov_Partition}
P_{\beta,\alpha} = \{[a_{1}, a_{2}), \ldots, [a_{n-2}, a_{n-1}), [a_{n-1}, a_{n}]\}.
\end{align}
Letting $T$ be a piecewise locally $C^{1+\delta}$ expanding map with respect to the partition $\{ I(i) \}_{i \in \Lambda}$, then for $x \in [0, 1]$, there exists an infinite word $\omega = \omega_{1} \omega_{2} \cdots \in \Lambda^{\mathbb{N}}$, with $\omega \vert_{k} \in \Omega_{T}^{*}$ for all $k \in \mathbb{N}$ and such that $\{ x \} = \bigcap_{k \in \mathbb{N}} \overline{I(\omega\vert_{n})}$. We call $\omega$ a \textsl{symbolic representation} of $x$ with respect to the partition $\{ I(i)\}_{i \in \Lambda}$. In the case that $T = T_{\beta,\alpha}$, for some $(\beta,\alpha) \in \Delta$, for a point $x \in [0, 1]$, symbolic representations of $x$ with respect to the partition $\{ [0,p_{\beta, \alpha}), [p_{\beta,\alpha},1] \}$ are $\tau_{\beta, \alpha}^{\pm}(x)$. Note, all but a countable set of points have a unique symbolic representation, we denote this countable set by $E = E_{T}$.
For a fixed $x \in [0,1]$ and $\gamma \in (0, 1)$, let $\omega$ denote a symbolic representation of $x$. We denote the following geometric condition by $H_{x, \gamma}$:
\begin{align}\label{eq:condition_1}
\adjustlimits
\lim_{i \to \infty} \sup_{\; u\,:\,u\;\text{and}\;u\omega\vert_{i}\in\Omega_{T}^{*}} \frac{\lvert I(u\omega\vert_{i}) \rvert}{\lvert I(u) \rvert} = 0
\end{align}
and there exists a natural number $i^{*}$ and a real number $c>0$ such that if $i \in \mathbb{N}$ with $i \geq i^{*}$, for all $\nu$ and $\eta \in \Omega_{T}^{*}$ which are $\omega\vert_{i}$-extendable with $I(\nu)$ and $I(\eta)$ being $\gamma/4$-comparable, either
\begin{align}\label{eq:condition_2}
\operatorname{dist}(I(\nu\omega\vert_{i}), I(\eta\omega\vert_{i})) = 0
\quad \text{or} \quad
\operatorname{dist}(I(\nu\omega\vert_{i}), I(\eta\omega\vert_{i})) \geq c \operatorname{dist}(I(\nu), I(\eta)).
\end{align}
\begin{theorem}[{\cite{HY}}]\label{thm_HY_Thm_2.1}
Let $T$ be a piecewise locally $C^{1+\delta}$ expanding map with respect to the partition $\{ I(i) \}_{i \in \Lambda}$, and let $x \in [0, 1]$ with symbolic representation $\omega$.
\begin{enumerate}[label={\rm(\arabic*)}]
\item If $H_{x, \gamma}$ is satisfied for some $\gamma \in (0, 1)$ , then the set $\{ y \in [0,1] \colon T^{k}(y) \not\in I(\omega\vert_{m}) \; \text{for all} \; k \in \mathbb{N}_{0} \} \cup E$ is $\rm ( 1/2, \gamma)$-winning for some natural number $m$.
%
\item If $x \not\in E$ and if $H_{x, \gamma}$ is satisfied for any $\gamma \in (0, 1)$, then the set ${\rm BAD}_{T}(x) = \{ y \in [0, 1] \colon x \not\in \overline{\{T^k(y) \colon k \in \mathbb{N}_{0}\}}\}$ is $1/2$-winning.
\end{enumerate}
\end{theorem}
We conclude this section with the following proposition which we use in conjunction with \Cref{thm:G1990+Palmer79,thm_HY_Thm_2.1} and the fact that the property of winning is preserved under bijective affine transformations to prove \Cref{thm:main_2}.
\begin{proposition}\label{prop:alpha-winning_transport}
Let $T$ be a piecewise locally $C^{1+\delta}$ expanding interval map, let $x \in [0, 1]$ and set
\begin{align*}
\mathrm{BAD}(T,x) \coloneqq \{ y \in [0,1] \colon x \not\in \overline{\{T^{n}(y) \colon n \in \mathbb{N}\}}\}.
\end{align*}
If, for a fixed $k \in \mathbb{N}$, we have that $\mathrm{BAD}(T^k, T^m(x))$ is winning for all $m \in \{ 0, 1, \ldots, k\}$, then $\mathrm{BAD}(T,x)$ is winning.
\end{proposition}
\begin{proof}
This follows from the fact that winning is preserved under taking countable intersections and since, by construction, $\mathrm{BAD}(T^k,x)\cap \mathrm{BAD}(T^k,T(x))\cap \cdots \cap \mathrm{BAD}(T^k,T^k(x))\subset \mathrm{BAD}(T,x)$.
\end{proof}
\section{Intermediate \texorpdfstring{$\beta$}{beta}-shifts as greedy \texorpdfstring{$\beta$}{beta}-shifts: Proof of \texorpdfstring{\Cref{thm:main}}{Theorem 1.1}}\label{sec:proof_thm_1_1}
In proving \Cref{thm:main} and \Cref{Cor_2,Cor_3}, we will investigate the question, given a fixed $\beta \in (1, 2)$, for which $\omega \in \{0,1\}^\mathbb{N}$ does there exist $(\beta', \alpha') \in \Delta$ with $\Omega^+_{\beta', \alpha'} = \{ \nu \in \{0,1\}^\mathbb{N} \colon \omega \preceq \sigma^{n}(\nu) \prec \tau_{\beta,0}^{-}(1) \; \text{for all} \; n \in \mathbb{N} \}$? Not only is this question interesting in its own right, but in classifying such words, we will be able to transfer the results from \cite{KKLL} to the setting of the intermediate $\beta$-transformations. With this in mind, we let $\mathcal{A}_\beta$ denote the set of all such words and set $\rho = \inf_{n \in \mathbb{N}_{0}} \pi_{\beta,0}(\sigma^{n}(\tau_{\beta,0}^{-}(1)))$. Further, we utilise the following notation. We let $t_{\beta, 0, c} \in (0,1)$ be such that $\dim_H(K^{+}_{\beta,0}(t))>0$ for all $t < t_{\beta, 0, c}$ and $\dim_H(K^{+}_{\beta, 0}(t)) = 0$ for all $t > t_{\beta, 0, c}$, and set $\mathcal{T}_{\beta, 0, c} = \tau_{\beta,0}^{+}(t_{\beta, c})$.
\begin{proof}[Proof of \texorpdfstring{\Cref{thm:main}}{Theorem 1.1}]
For $\beta\in(1,2)$, let $\mathcal{B}_{\beta} \coloneqq \{ \omega \in \mathcal{E}_{\beta,0}^{+} \colon \pi_{\beta,0}(\omega) \leq \rho \; \text{and} \; \omega \prec \mathcal{T}_{\beta, 0, c}\}$. By \Cref{cor:From_greedy_to_intermediate} and the commutativity of the diagram given in \eqref{eq:commutative_diag}, observe that it is sufficient to show
\begin{enumerate}
\item $\mathcal{A}_\beta\subseteq \mathcal{B}_\beta$ with equality holding for Lebesgue almost every $\beta\in(1,2)$,
\item there exist $\beta\in (1,2)$ such that $\mathcal{A}_\beta \neq \mathcal{B}_\beta$, and
\item if the quasi-greedy $\beta$-expansion of $1$ is periodic, then $\mathcal{A}_\beta = \mathcal{B}_\beta$.
\end{enumerate}
To show $\mathcal{A}_\beta\subseteq \mathcal{B}_\beta$, let $\beta \in (1, 2)$ and $\eta \in \mathcal{A}_\beta$ be fixed. Setting $\omega = 1\eta$ and $\nu = 0\tau_\beta^-(1)$, we observe that they meet Conditions~(1)--(4) of \Cref{thm:BSV14} . Condition~(2) of \Cref{thm:BSV14} gives $\nu \in \Omega^{+}(\omega, \nu)$, and so, for a given $n \in \mathbb{N}_{0}$,
\begin{align*}
\eta \preceq \sigma^{n}(\eta) \prec 0 \tau_{\beta,0}^{-}(1) \quad \text{or} \quad 1\eta \preceq \sigma^{n}(\eta) \prec \tau_{\beta,0}^{-}(1),
\end{align*}
yielding that $\eta \preceq \sigma^{n}(\eta) \prec \tau_{\beta,0}^{-}(1)$ for all $n \in \mathbb{N}_{0}$, namely that $\eta \in \mathcal{E}_{\beta,0}^{+}$. Condition~(2) of \Cref{thm:BSV14} also gives $\omega\in \Omega^{-}(\omega, \nu)$, and so, for a given $n \in \mathbb{N}_{0}$,
\begin{align*}
\eta \prec \sigma^{n}(\tau_{\beta,0}^{-}(1)) \preceq 0 \tau_{\beta,0}^{-}(1) \quad \text{or} \quad 1\eta \prec \sigma^{n}(\tau_{\beta,0}^{-}(1)) \preceq \tau_{\beta,0}^{-}(1).
\end{align*}
This implies that $\eta \prec \sigma^{n}(\tau_\beta^-(1))$ for all $n \in \mathbb{N}_{0}$, and so $\pi_{\beta,0}(\eta)\leq \rho$. Since Condition~(3) of \Cref{thm:BSV14} holds, the topological entropy of $(\Omega(\omega,\nu), \sigma)$ is positive, and thus $\eta \prec \mathcal{T}_{\beta, 0, c}$. Therefore, $\eta \in \mathcal{B}_\beta$, and hence $\mathcal{A}_\beta\subseteq \mathcal{B}_\beta$.
To see that $\mathcal{A}_\beta = \mathcal{B}_\beta$ for Lebesgue almost every $\beta \in (1,2)$, from the concluding remarks of \cite{Schme} we know that, for Lebesgue almost all $\beta \in (1,2)$ there is no bound on the length of blocks of consecutive zeros in the quasi-greedy $\beta$-expansion of $1$, namely $\tau_{\beta,0}^-(1)$. This implies that $\rho = 0$, and hence that $\mathcal{B}_\beta=\{ 0^\infty \}$. Since $\tau_{\beta,0}^{\pm}(0) = 0^\infty$ it follows that $0^\infty \in \mathcal{A}_\beta$, and thus that $\mathcal{B}_\beta \subseteq \mathcal{A}_\beta$ for Lebesgue almost every $\beta \in (1,2)$. This in tandem with the fact that $\mathcal{A}_\beta\subseteq \mathcal{B}_\beta$ for all $\beta \in (1,2)$, yields that $\mathcal{A}_\beta = \mathcal{B}_\beta$ for Lebesgue almost every $\beta \in (1,2)$.
Let $\beta$ denote the algebraic number with minimal polynomial $x^5-x^4-x^3-2x^2+x+1$. An elementary calculation yields that $\tau_{\beta,0}^{-}(1)=11(100)^{\infty}$. We claim that $\xi = 00(011)^\infty \in \mathcal{B}_\beta$, but that $\xi \not\in \mathcal{A}_\beta$, namely that $\mathcal{A}_\beta \subsetneq \mathcal{B}_\beta$. It is readily verifiable that $\xi \in \mathcal{E}_\beta^+$ and also, since $\rho=\pi_{\beta,0}((001)^\infty)$, that $\pi_{\beta,0}(\xi) < \rho$. This yields that $\{001, 011 \}^\mathbb{N} \subset \mathcal{K}^{+}_{\beta,0}(\pi_{\beta,0}(\xi))$, and hence that $h_{\operatorname{top}}(\sigma\vert_{\mathcal{K}^{+}_{\beta,0}(\pi_{\beta,0}(\xi))}) > 0$. In other words, we have $\xi \prec \mathcal{T}_{\beta,c}$, and so $\xi \in \mathcal{B}_\beta $. By way of contradiction, suppose that $\xi \in \mathcal{A}_\beta$. Set $\omega=0\tau_{\beta,0}^{-}(1)=011(100)^\infty$ and $\nu = 1\xi = 100(011)^\infty$. In which case $\omega$ and $\nu \in \{ \chi, \zeta\}^\mathbb{N}$ with $\chi=011$ and $\zeta=100$. Noting that $\chi$ and $\zeta$ are words of length three in the alphabet $\{0,1\}$, that $\chi\vert_{2} = 01$, $\zeta\vert_{2} = 10$, that $\chi^\infty \in \Omega^{-}(\chi^\infty, \zeta^\infty)$ and $\zeta^\infty \in \Omega^{+}(\chi^\infty, \zeta^\infty)$, but that $\omega=\chi\zeta^\infty \neq \chi^\infty$, contradicting Condition~(4) of \Cref{thm:BSV14}.
It remains to prove that if $\tau_{\beta,0}^-(1)$ is periodic, then $\mathcal{A}_\beta = \mathcal{B}_\beta$. To this end, fix $\beta \in (1, 2)$ with $\tau_{\beta,0}^{-}(1)$ periodic. Let $\xi \in \mathcal{B}_\beta $ and set $\nu=1 \xi$ and $\omega = 0\tau_\beta^-(1)$. By assumption, $\xi \in \mathcal{E}_{\beta,0}^+$, and so $\sigma(\nu) \preceq \sigma^{n}(\nu) \prec \sigma(\omega)$ and therefore $\nu \in \Omega^{+}(\omega, \nu)$, and since $\tau_{\beta,0}^{-}(1)$ is the quasi greedy $\beta$-expansion of $1$ in base $\beta$, we have $\sigma^{n}(\omega) \preceq \sigma(\omega)$ for all $n \in \mathbb{N}_{0}$. As $\pi_{\beta,0}(\xi) < \rho$ we have $ \sigma(\nu) \prec \sigma^n(\omega)$, and so $\omega \in \Omega^{-}(\omega, \nu)$. Thus, $\omega$ and $\nu$ satisfy Conditions~(1) and~(2) of \Cref{thm:BSV14}. Condition~(3) of \Cref{thm:BSV14} follows from $\xi \prec \mathcal{T}_{\beta,c}$. To conclude the proof, it suffices to show that $\omega$ and $\nu$ satisfy Condition~(4) of \Cref{thm:BSV14}. Suppose there exist $\chi$ and $\zeta \in \{0,1\}^{*}$ of length at least three with
\begin{align*}
\chi\vert_{2} = 01, \quad
\zeta\vert_{2} = 10, \quad
\chi^{\infty} \in \Omega^{-}(\chi^{\infty},\zeta^{\infty}),
\quad \text{and} \quad
\zeta^{\infty} \in \Omega^{+}(\chi^{\infty},\zeta^{\infty}),
\end{align*}
and such that $\omega$ and $\nu \in \{ \chi, \zeta \}^{\mathbb{N}}$. By our assumption and construction, in particular, since $\omega$ is periodic and since $\chi^{\infty} \in \Omega^{-}(\chi^{\infty},\zeta^{\infty})$, we have $\omega = \chi^\infty$. By way of contradiction, suppose that $\nu \neq \zeta^\infty$. In which case, there exists $n \in \mathbb{N}_{0}$ such that $\sigma^{n}(\nu)\vert_{\lvert \chi \rvert + \lvert \zeta \rvert} = \chi \zeta$. Noting that $\chi\vert_{2} = 01$ and $\zeta\vert_{2} = 10$, this yields $\sigma(\omega) \prec \sigma^{n+1}(\nu)$ contradicting the fact that $\omega$ and $\nu$ satisfy Condition~(2) of \Cref{thm:BSV14}.
\end{proof}
\section{A Krieger embedding theorem for intermediate \texorpdfstring{$\beta$}{beta}-transformations: Proof of \texorpdfstring{\Cref{Cor_1}}{Corollary 1.2}}\label{sec:proof_cor_1_2}
To prove \Cref{Cor_1} we first show the following special case.
\begin{theorem}\label{thm:one_perioidc}
Let $(\beta,\alpha)\in\Delta$ be such that $\nu = \tau^{+}_{\beta, \alpha}(p_{\beta,\alpha})$ is not periodic and $\omega = \tau^{-}_{\beta, \alpha}(p_{\beta,\alpha})$ is periodic. There exists a sequence $((\beta_{n}, \alpha_{n}))_{n\in \mathbb{N}}$ in $\Delta$ with $\lim_{n\to \infty}\beta_{n} = \beta$ and $\lim_{n \to \infty}\alpha_{n} = \alpha$ and such that
\begin{enumerate}[label={\rm(\roman*)}]
\item[\rm (1)] $\Omega_{\beta_{n},\alpha_n}$ is a subshift of finite type,
\item[\rm (2)] the Hausdorff distance between $\Omega_{\beta, \alpha}$ and $\Omega_{\beta_{n}, \alpha_{n}}$ converges to zero as $n$ tends to infinity, and
\item[\rm (3)] $\Omega_{\beta_{n},\alpha_n}\subseteq\Omega_{\beta,\alpha}$.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove this using \Cref{thm:main} and the results of \cite{KKLL}. By \Cref{thm:main}, there exist $\beta' \in (1, 2)$ and $t \in E_{\beta',0}^+$ such that $\mathcal{K}_{\beta',0}^+(t)=\Omega^+_{\beta,\alpha}$ with $\tau_{\beta',0}^{+}(t) = \sigma(\nu)$. Our goal is to find a monotonically decreasing sequence $(t_i)_{i \in \mathbb{N}}$ converging to $t$ with $t_i\in E_{\beta',0}^+$ and $t_i$ is a $T_{\beta', 0}^+$-periodic point for all $i \in \mathbb{N}$. We will first prove that $t$ is not isolated from above. For this we use the following. A finite word $s \in \{0, 1\}^{*}$ is Lyndon if $s^\infty \prec \sigma^n(s^\infty)$ for all $n \in \mathbb{N}$ with $n \neq 0 \bmod \lvert s \rvert$, and set $L_{\beta} \coloneqq \{ s\in \{0, 1\}^* \colon s \; \text{is a Lyndon word and} \; s^\infty \in \Omega_{\beta', 0} \}$.
For $s \in L_{\beta}$, let $I_{s}$ denote the half-open interval $[\pi_{\beta',0}(s0^\infty), \pi_{\beta',0}(s^\infty))$. \Cref{thm:Structure} in combination with our hypothesis that $\tau_{\beta,\alpha}^{-}(p_{\beta,\alpha}) = 0\tau_{\beta,\alpha}^{-}(1)$ is periodic, yields there exists a shortest finite word $\zeta$ with $\tau_{\beta,\alpha}^{-}(1) = \zeta^\infty$. Letting $n$ be the length of $\zeta$, we set $\zeta^\prime$ to be the lexicographical smallest element of the set $\{ \zeta_{k} \cdots \zeta_{n} \zeta_{1} \cdots \zeta_{k-1} \colon k \in \{2, \ldots, n \}\}$, and set $y = \pi_{\beta',0}(\zeta^\prime 0^\infty)$. By construction $\zeta^{\prime}$ is a Lyndon. Since by our hypothesis $\nu = \tau^{+}_{\beta, \alpha}(p_{\beta,\alpha})$ is not periodic and since $t < \pi_{\beta',0}({\zeta^{\prime}}^{\infty})$, we observe that $t < y$.
For $s \in L_{\beta}$, by the Lyndon property of $s$, if $x \in I_s$, then $x \not\in E_{\beta',0}^{+}$, which implies $E_{\beta',0}^{+} \cap (0,y) \subseteq (0,y) \backslash \bigcup_{s\in L_\beta} I_s$. In fact we claim $(0,y) \backslash \bigcup_{s\in L_\beta} I_s = E_{\beta',0}^+ \cap (0,y)$. In order to prove this, let $x\in (0,y) \backslash \bigcup_{s\in L_\beta} I_s$ and suppose $x \notin E_{\beta',0}^+$. Under this hypothesis, there exists a minimal $n \in \mathbb{N}$ such that $\sigma^{n}(\tau_{\beta', 0}^{+}(x)) \prec \tau_{\beta', 0}^{+}(x)$. By the minimality of $n$, we have that $\xi = \tau_{\beta', 0}^{+}(x)\vert_{n}$ is a Lyndon word, and that $\tau_{\beta', 0}^{+}(x) \prec \xi^{\infty}$. If $\xi^\infty \not\in \Sigma_{\beta^\prime,0}$, then there exists $j \in \{ 1, 2, \ldots, n \}$ such that $\tau_{\beta',0}^{-}(1)\prec\sigma^j(\xi^\infty)$ where equality is excluded since $x<y$. Set $k = \tau_\beta^{-}(1) \wedge \sigma^j(\xi^\infty)$, and notice $k>n-j$; otherwise $\tau_{\beta', 0}^{+}(x)$ would not be admissible. This yields $\tau_{\beta',0}^{-}(1)=\xi_{j+1}\xi_{j+2}\cdots \xi_n (\xi_1\cdots \xi_n)^l \omega_1 \omega_2 \cdots$ with $l$ possibly $0$ but chosen so that $\omega_1 \cdots \omega_n \neq \xi_1 \cdots \xi_n$; note this is possible since $\nu$ is not periodic. Thus, $\sigma^{n-j+ln}(\tau_{\beta',0}^{-}(1)) \prec \sigma^{n-j+ln}(\sigma^{j}(\xi^\infty))=\xi^\infty$ and $\omega_1\cdots \omega_n \prec \xi$. Hence, $\sigma^{n-j+ln}(\tau_{\beta',0}^{-}(1)) \prec \xi 0^\infty \prec \tau_{\beta',0}^{-}(x)$, contradicting the fact that we choose $x\in (0,y) \backslash \bigcup_{s\in L_\beta} I_s$. It therefore follows that $E_{\beta',0}^{+} \cap (0,y) = (0,y) \backslash \bigcup_{s\in L_\beta} I_s$ as required.
Suppose that $t$ cannot be approximated from above by elements in $E_{\beta',0}^+$, that is, there exists a real number $\epsilon > 0$ with $(t,t+\epsilon)\cap E_{\beta',0}^+=\emptyset$. Since $E_{\beta',0}^+ \cap (0,y) = (0,y) \backslash \bigcup_{s\in L_\beta} I_s$, there exists a Lyndon word $s$ with $(t,t+\epsilon) \subset I_s$, but as $I_s$ is closed from the left, $t\in I_s$, contradicting our hypothesis that $t\in E_{\beta',0}^{+}$. This implies $t$ can be approximated from above by elements in $E_{\beta',0}^{+}$, namely there exists a monotonically decreasing sequence $(t_i^\prime)_{i\in \mathbb{N}}$ of real numbers converging to $t$ with $t_i^\prime \in E_{\beta',0}^+$, for all $i \in \mathbb{N}$. If $t_i^\prime$ is not $T_{\beta',0}$-periodic for some $i \in \mathbb{}N$, then by \cite[Lemmanta~3.4 and~3.5]{KKLL}, there exists a monotonically increasing sequence of $T_{\beta',0}$-periodic points $(s_{i, j}^\prime)_{j \in \mathbb{N}}$ converging to $t_i^\prime$ with $s_{i, j} \in E_{\beta',0}^+$. For $i \in \mathbb{N}$, setting $t_i=t_i^\prime$ whenever $t_i^\prime$ is $T_{\beta',0}$-periodic, and otherwise setting $t_i=s_{i,j}^\prime$ where $s_{i,j}^\prime$ is chosen so that $t < s_{i,j}^\prime < t_i^\prime$, the sequence $(t_i)_{i \in \mathbb{N}}$ converges to $t$ from above and $t_i$ is $T_{\beta',0}$-periodic.
Since $\omega$ is periodic with respect to the left shift map, \Cref{thm:main} implies, for each $i \in \mathbb{N}$, there exists $(\beta_{i}, \alpha_{i}) \in \Delta$ with $K_{\beta'0}^+(t_i)=\Omega^+_{\beta_{i},\alpha_i}$. Since both $\omega$ and $\tau^{+}_{\beta',0}(t_i)$ are periodic, \Cref{thm:LSSS} yields that $\Omega_{\beta_{i},\alpha_i}$ is of subshift of finite type. Further, since $\mathcal{K}_{\beta',0}^{+}(t_i) \subseteq \mathcal{K}^{+}_{\beta',0}(t)$, it follows that $\Omega_{\beta_{i},\alpha_{i}} \subseteq \Omega_{\beta,\alpha}$ for all $i \in \mathbb{N}$.
\end{proof}
\begin{proof}[{Proof of \Cref{Cor_1}}]
Assume the setting of \Cref{Cor_1} and for ease of notation set $p = p_{\beta, \alpha}$, $\nu = \tau_{\beta, \alpha}^{+}(p)$ and $\omega = \tau_{\beta, \alpha}^{-}(p)$. By \Cref{thm:LSSS}, we have that $\Omega_{\beta, \alpha}$ is a subshift of finite type if and only if $\omega$ and $\nu$ are periodic. Since the subshift of finite type property is preserved by topological conjugation, and observing that $\Omega^{\pm}_{\beta, \alpha}$ and $\Omega^{\mp}_{\beta, 2-\beta-\alpha}$ are topologically conjugate, with conjugation map $R$, with out loss of generality we may assume that $\nu$ is not periodic. We consider the case, when $\omega$ is periodic and when $\omega$ is not periodic separately. The former of these two cases follows from \Cref{thm:one_perioidc}, and so all that remains is to show the result for the latter case, namely when $\omega$ is not periodic. To this end, assume that $\omega$ and $\nu$ are both not periodic. Let $n \in \mathbb{N}$ be fixed, set $O_{n}^{\pm}(p) = \{ (T_{\beta, \alpha}^{\pm})^{k}(p) \colon k \in \{ 0, 1, \ldots, n-1 \} \}$, and let $\beta' \in (1, \beta)$ be such that
\begin{align}\label{eq:def_beta_prime}
(1-\alpha)/\beta' + (\beta+1)^{n}(\beta - \beta') < \min \{ x \in O_{n}^{+}(p) \cup O_{n}^{-}(p) \colon x > p \}.
\end{align}
(As defined in \Cref{sec:beta-shifts}, we let $T_{\beta,\alpha}^{-} \colon x \mapsto \beta x + \alpha$ if $x \leq p$, and $x \mapsto \beta x + \alpha - 1$ otherwise, and for ease of notation, we write $T_{\beta,\alpha}^{+}$ for $T_{\beta, \alpha}$.) Setting $p' = (1-\alpha)/\beta'$, we claim, for all $k \in \{ 1, \ldots, n-1 \}$, that either
\begin{align}\label{eq:desired_inequalities}
(T_{\beta', \alpha}^{\pm})^{k}(p') \leq (T_{\beta, \alpha}^{\pm})^{k}(p) \leq p \leq p'
\quad \text{or} \quad
(T_{\beta, \alpha}^{\pm})^{k}(p) \geq (T_{\beta', \alpha}^{\pm})^{k}(p') \geq p' \geq p.
\end{align}
Hence, by definition and since $\omega$ and $\nu$ are not periodic, $\omega\vert_{n} = \tau_{\beta', \alpha}^{-}(p)\vert_{n}$ and $\nu\vert_{n} = \tau_{\beta', \alpha}^{+}(p)\vert_{n}$.
To prove this claim, note, for all $k \in \{ 1, \ldots, n-1 \}$, either
\begin{align}\label{eq:either_or}
(T_{\beta, \alpha}^{\pm}(p))^{k} < p
\quad \text{or} \quad
(T_{\beta, \alpha}^{\pm})^{k}(p) \geq \min \{ x \in O_{n}^{+}(p) \cup O_{n}^{-}(p) \colon x > p \}.
\end{align}
If $0 \leq y \leq x \leq p$, or if $p' \leq y \leq x \leq 1$, then
\begin{align}\label{eq:orbit_bound}
0 \leq T^{\pm}_{\beta, \alpha}(x) - T^{\pm}_{\beta', \alpha}(y) = \beta x - \beta'y = \beta x - \beta y + \beta y - \beta'y \leq \beta(x-y) + (\beta-\beta').
\end{align}
Observe that $T^{\pm}_{\beta, \alpha}(p) = T^{\pm}_{\beta', \alpha}(p')$ and
\begin{align*}
0 \leq (T^{\pm}_{\beta, \alpha})^{2}(p) - (T^{\pm}_{\beta', \alpha})^{2}(p')
\leq \beta - \beta'
\leq (\beta + 1)(\beta - \beta')
\leq (\beta + 1)^{2}(\beta - \beta')
\leq (\beta+1)^{n}(\beta - \beta').
\end{align*}
Suppose, by way of induction on $m$, that
\begin{align*}
0
\leq (T^{\pm}_{\beta, \alpha})^{m}(p) - (T^{\pm}_{\beta', \alpha})^{m}(p')
\leq (\beta+1)^{m}(\beta-\beta')
\leq (\beta+1)^{n}(\beta-\beta'),
\end{align*}
for some $m \in \{2, \dots, n-2\}$. Combining \eqref{eq:def_beta_prime}, \eqref{eq:either_or} and \eqref{eq:orbit_bound} with our inductive hypothesis, we have
\begin{align*}
0 \leq (T^{\pm}_{\beta, \alpha})^{m+1}(p) - (T^{\pm}_{\beta', \alpha})^{m+1}(p')
&\leq \beta((T^{\pm}_{\beta, \alpha})^{m}{p} - (T^{\pm}_{\beta', \alpha})^{m}(p')) + (\beta-\beta')\\
&\leq \beta(\beta+1)^{m}(\beta-\beta') + (\beta-\beta')
\leq (\beta+1)^{m+1}(\beta-\beta')
\leq (\beta+1)^{n}(\beta-\beta').
\end{align*}
In other words, for all $k \in \{ 1, \ldots, n-1 \}$,
\begin{align*}
0 \leq (T^{\pm}_{\beta, \alpha})^{k}(p) - (T^{\pm}_{\beta', \alpha})^{k}(p')
\leq (\beta+1)^{k}(\beta-\beta')
\leq (\beta+1)^{n}(\beta-\beta').
\end{align*}
This in tandem with \eqref{eq:def_beta_prime} and \eqref{eq:either_or} proves the claim.
We observe that $(\omega, \nu) \neq (\tau_{\beta', \alpha}^{+}(p'), \tau_{\beta', \alpha}^{-}(p'))$, for if not, then since $\beta' < \beta$, this would contradict \Cref{thm:Laurent}. This implies that $\omega \neq \tau_{\beta', \alpha}^{-}(p')$ or $\nu \neq \tau_{\beta, \alpha}^{+}(p')$. We claim that $\omega \succ \tau_{\beta', \alpha}^{-}(p')$ and $\nu \succ \tau_{\beta, \alpha}^{+}(p')$.
Consider the case when $\omega \neq \tau_{\beta', \alpha}^{-}(p')$. This implies there exists a smallest integer $m \geq n$ such that neither
\begin{align*}
(T_{\beta', \alpha}^{-})^{m}(p') \leq (T_{\beta, \alpha}^{-})^{m}(p) \leq p
\quad \text{nor} \quad
(T_{\beta, \alpha}^{-})^{m}(p) \geq (T_{\beta', \alpha}^{-})^{m}(p') \geq p'.
\end{align*}
Using the fact that if $0 \leq y \leq x < p$ or if $p' < y \leq x \leq 1$, then $T^{-}_{\beta', \alpha}(y) \leq T^{-}_{\beta, \alpha}(x)$, in tandem with \eqref{eq:desired_inequalities}, and noting that $p<p'$, we have that
\begin{align*}
\tau_{\beta', \alpha}^{-}(p')\vert_{m-2}=\omega\vert_{m-2},
\quad
(T^{-}_{\beta',\alpha})^{m}(p')<p',
\quad
(T^{-}_{\beta,\alpha})^{m}(p) > p
\quad \text{and} \quad
(T^{-}_{\beta',\alpha})^{m}(p')\leq (T^{-}_{\beta,\alpha})^{m}(p).
\end{align*}
Thus, $\tau_{\beta', \alpha}^{-}(p')\vert_{m-1} \prec \omega\vert_{m-1}$ and hence $\tau_{\beta', \alpha}^{-}(p') \prec \omega$. An analogous argument proves the claim when $\nu \neq \tau_{\beta, \alpha}^{+}(p')$.
Hence, we have shown, given an $n \in \mathbb{N}$, that there exists a positive $\delta \in \mathbb{R}$, such that, for all $\beta' \in (\beta-\delta, \beta)$,
\begin{align}\label{smaller}
\tau_{\beta', \alpha}^{\pm}(p')\vert_{n} = \tau_{\beta, \alpha}^{\pm}(p)\vert_{n}
\quad \text{and} \quad
\tau_{\beta', \alpha}^{\pm}(p') \prec \tau_{\beta, \alpha}^{\pm}(p),
\end{align}
where $p'=(1-\alpha)/\beta'$. Further, by using the fact that $\Omega^{\pm}_{\beta, \alpha}$ and $\Omega^{\mp}_{\beta, 2-\beta-\alpha}$ are topologically conjugate, with conjugating map $R$, together with \eqref{smaller}, we have that there exists a positive $\delta' \in \mathbb{R}$, such that, for all $\beta' \in (\beta-\delta', \beta)$,
\begin{align*}
\tau_{\beta', \alpha+\beta-\beta'}^{\pm}(p_{\beta', \alpha+\beta-\beta'})\vert_{n} = \tau_{\beta, \alpha}^{\pm}(p)\vert_{n}
\quad \text{and} \quad
\tau_{\beta', \alpha+\beta-\beta'}^{\pm}(p_{\beta', \alpha+\beta-\beta'}) \succ \tau_{\beta, \alpha}^{\pm}(p).
\end{align*}
Letting $\beta' \in (\beta-\min(\delta,\delta'), \beta)$ be fixed and setting
\begin{align*}
q_{1} = \sup \{ a \in (\alpha, \alpha+\beta-\beta') \colon \tau_{\beta', a}^{\pm}(p_{\beta', a}) \preceq \tau_{\beta, \alpha}^{\pm}(p)\}
\quad \text{and} \quad
q_{2} = \inf \{ a \in (\alpha, \alpha+\beta-\beta') \colon \tau_{\beta', a}^{\pm}(p_{\beta', a}) \succeq \tau_{\beta, \alpha}^{\pm}(p)\},
\end{align*}
by \Cref{prop:mon_cont_kneading}, we have $\alpha \leq q_{1} \leq q_{2} \leq \alpha+\beta-\beta'$ and $\tau_{\beta', a}^{\pm}(p_{\beta',a})\vert_{n} = \tau_{\beta, \alpha}^{\pm}(p)\vert_{n}$, for all $a \in [q_{1}, q_{2}]$. Moreover, $\tau_{\beta', a}^{-}(p_{\beta',a}) \preceq \tau_{\beta, \alpha}^{-}(p) \prec \tau_{\beta, \alpha}^{+}(p) \preceq \tau_{\beta', a}^{+}(p_{\beta',a})$, for all $a \in [q_{1}, q_{2}]$, implying one of the following sets of orderings.
\begin{align*}
\tau_{\beta', a}^{-}(p_{\beta',a}) \prec \tau_{\beta, \alpha}^{-}(p) &\prec \tau_{\beta, \alpha}^{+}(p) \prec \tau_{\beta', a}^{+}(p_{\beta', a})\\
\tau_{\beta', a}^{-}(p_{\beta',a}) = \tau_{\beta, \alpha}^{-}(p) &\prec \tau_{\beta, \alpha}^{+}(p) \prec \tau_{\beta', a}^{+}(p_{\beta', a})\\
\tau_{\beta', a}^{-}(p_{\beta',a}) \prec \tau_{\beta, \alpha}^{-}(p) &\prec \tau_{\beta, \alpha}^{+}(p) = \tau_{\beta', a}^{+}(p_{\beta', a})
\end{align*}
If either the first case occurs, the second case occurs and $\tau_{\beta', a}^{+}(p_{\beta',a})$ is not periodic, or the third case occurs and $\tau_{\beta', a}^{-}(p_{\beta',a})$ is not periodic, then an application of \Cref{prop:mon_cont_kneading} and \Cref{thm:LSSS} yields the required result.
This leaves two remaining sub-cases, namely when $\tau_{\beta', a}^{-}(p_{\beta',a}) = \tau_{\beta, \alpha}^{-}(p) \prec \tau_{\beta, \alpha}^{+}(p) \prec \tau_{\beta', a}^{+}(p_{\beta', a})$ with $\tau_{\beta', a}^{+}(p_{\beta', a})$ periodic, and when $\tau_{\beta', a}^{-}(p_{\beta',a}) \prec \tau_{\beta, \alpha}^{-}(p) \prec \tau_{\beta, \alpha}^{+}(p) = \tau_{\beta', a}^{+}(p_{\beta', a})$ with $\tau_{\beta', a}^{-}(p_{\beta', a})$ periodic. Let us consider the first of these two sub-cases; the second follows by an analogous arguments.
For ease of notation let $\nu' = \tau_{\beta', a}^{+}(p_{\beta',a})$ and note that by assumption $\omega = \tau_{\beta', a}^{-}(p_{\beta',a})$ and that $\omega \prec \nu \prec \nu'$. If the map $s \mapsto \tau_{\beta', s}^{+}(p_{\beta',s})$ is continuous at $s = a$, then an application of \Cref{prop:mon_cont_kneading} and \Cref{thm:LSSS} yields the required result; if we do not have continuity at $s = a$, by \Cref{prop:mon_cont_kneading} we have that $\nu'$ is periodic with periodic $N$, for some $N \in \mathbb{N}$, and thus an application of \Cref{thm:one_perioidc} completes the proof, alternatively we may proceed as follows.
We claim that $\nu \prec \nu'\vert_{N}\omega$. Indeed if $\nu\vert_{N} \prec \nu'\vert_{N}$, the claim follows immediately, and so let us suppose that $\nu\vert_{N} = \nu'\vert_{N}$. If $\nu_{N+1} = 0$, the claim follows, from \Cref{thm:Structure}. On the other hand, by \Cref{thm:Structure}, if $\nu_{N+1} = 1$, then $\sigma^{N}(\nu) \succeq \nu$. If $\sigma^{N}(\nu)\vert_{N} \succ \nu\vert_{N} = \nu'\vert_{N}$, then $\nu \succ \nu'$, contradicting our assumption that $\nu \prec \nu'$, and so $\sigma^{N}(\nu)\vert_{N} = \nu\vert_{N}$. This implies there exists a minimal integer $m$ such that $\nu_{m N + 1} = 0$ and $\nu\vert_{mN} = \nu'\vert_{mN}$; otherwise $\nu$ would be periodic. However, this together with \Cref{thm:Structure}, yields that $\nu \preceq \sigma^{(m-1)N}(\nu) = \nu\vert_{N}\sigma^{mN}(\nu) \preceq \nu\vert_{N}\omega = \nu'\vert_{N}\omega$, as required.
To complete the proof of this sub-case we appeal once more to \Cref{prop:mon_cont_kneading} which together with the above implies that there exists a real number $\delta > 0$ such that for all $a' \in (a-\delta,a)$ we have $\tau_{\beta',a'}^{-}(p_{\beta',a'}) \prec \omega$ and $\nu \prec \tau_{\beta',a'}^{+}(p_{\beta',a'}) \prec \nu'\vert_{N}\omega \prec \nu'$. An application of \Cref{thm:LSSS} yields the required result.
\end{proof}
In the above proof, it is critical that $\omega$ and $\nu$ are not periodic, as this allows us to construct $\beta'$, $q_{1}$ and $q_{2}$ so that $\omega$ and $\nu$ are sufficiently close to $\tau_{\beta',a'}^{-}(p_{\beta',a'})$ and $\tau_{\beta',a'}^{+}(p_{\beta',a'})$, respectively, for all $a' \in [q_{1}, q_{2}]$. However, under the assumption that $\nu$ is periodic we may not use our construction to build such $\beta'$ and hence $q_{1}$ and $q_{2}$. Indeed, the strict inequalities in \Cref{eq:either_or} no longer hold, and thus the ordering given in \eqref{smaller} fails.
\section{Survivor sets of intermediate \texorpdfstring{$\beta$}{beta}-transformations: Proof of \texorpdfstring{\Cref{Cor_2,Cor_3}}{Corollaries 1.3 and 1.4}}\label{sec:proof_cor_1_3_4}
Here, we examine open dynamical systems on the unit interval with a hole at zero and where the dynamics is driven by an intermediate \mbox{$\beta$-transformation}. With the aid of \Cref{thm:main} we can relate such open dynamical system to open dynamical systems driven by greedy \mbox{$\beta$-transformations}. This allows us to transfer the results of \cite{KKLL} and \cite{AK} on isolated points in $E_{\beta,\alpha}^+$, the Hausdorff dimension of survivor sets, and the critical point of the dimension function from the Greedy case to the intermediate case. For readability, we omit the $0$ in notation of $\pi_{\beta,0}$, $E_{\beta}^+=E_{\beta,0}^+$, and so on, and thus write $\pi_{\beta}$ for $\pi_{\beta,0}$, $E_{\beta}^+$ for $E_{\beta,0}^+$, and so forth.
By \Cref{thm:Parry_converse,thm:Structure}, given $(\beta,\alpha) \in \Delta$, there exists a unique $\beta^\prime \in (1, 2)$ with $\tau_{\beta,\alpha}^-(1)=\tau_{\beta^\prime}^-(1)$. Thus, we define a function $u \colon \Delta \to (1, 2)$ by $u(\beta,\alpha) \coloneqq \beta^\prime$, and let $\tilde{\pi}_{\beta,\alpha} \coloneqq \pi_{u(\beta,\alpha)} \circ \tau_{\beta,\alpha}^{+}$. Correlations of the systems $(T_{\beta,\alpha},[0,1])$ and $(T_{u(\beta,\alpha)},K_{u(\beta,\alpha)}^+(\tilde{\pi}_{\beta,\alpha}(0)))$ are expressed in the following proposition.
\begin{proposition}\label{prop:char}
Let $(\beta,\alpha)\in\Delta$ and let $\beta^\prime=u(\beta,\alpha)$.
\begin{enumerate}
\item $\tilde{\pi}_{\beta,\alpha}([0,1])=K_{\beta^\prime}^+(\tilde{\pi}_{\beta,\alpha}(0))$ and $\tilde{\pi}_{\beta,\alpha}(E_{\beta,\alpha}^+)=E_{\beta^\prime}^+\cap[\tilde{\pi}_{\beta,\alpha}(0),1]$.
\item For every $x\in E_{\beta,\alpha}^+$ we have that $x$ is isolated in $E_{\beta,\alpha}^+$ if and only if $\tilde{\pi}_{\beta,\alpha}(x)$ is isolated in $E_{\beta^\prime}^+$.
\item For $t\in(0,1)$, we have $ \tilde{\pi}_{\beta,\alpha}(K_{\beta,\alpha}^+(t))=K_{\beta^\prime}^+(\tilde{\pi}_{\beta,\alpha}(t)) $.
\item For $t\in(0,1)$, we have $\dim_H(K_{\beta,\alpha}^+(t))=(\log(\beta^\prime)/\log(\beta)) \dim_H(K_{\beta^\prime}^+(\tilde{\pi}_{\beta,\alpha}(t)))$.
\end{enumerate}
\end{proposition}
\newpage
\begin{proof}
Let us begin by proving Part~(1). To this end, observe that $\tilde{\pi}_{\beta,\alpha}$ is monotonic, since it is a composition of monotonic functions, and so, $\tilde{\pi}_{\beta,\alpha}(T_{\beta,\alpha}^n(x)) \geq \tilde{\pi}_{\beta,\alpha}(0)$ for all $x \in [0,1]$ and $n \in \mathbb{N}_{0}$. By the fact that the diagrams in \eqref{eq:commutative_diag} commute, we have for all $x \in [0,1]$ and $n \in \mathbb{N}$, that
\begin{align*}
T_{\beta'}^{n}(\tilde{\pi}_{\beta,\alpha}(x))
= T_{\beta'}^{n}(\pi_{\beta'}(\tau^{+}_{\alpha,\beta}(x)))
= \pi_{\beta'}(\sigma^{n}(\tau^{+}_{\alpha,\beta}(x)))
= \pi_{\beta'}(\tau^{+}_{\alpha,\beta}(T^{n}_{\beta,\alpha}(x))
= \tilde{\pi}_{\beta,\alpha}(T^{n}_{\beta,\alpha}(x)).
\end{align*}
Combining the above, we may conclude that $\tilde{\pi}_{\beta,\alpha}([0,1]) \subseteq K_{\beta^\prime}(\tilde{\pi}_{\beta,\alpha}(0))$. To prove that equality holds, namely that $\tilde{\pi}_{\beta,\alpha}([0,1]) = K_{\beta^\prime}(\tilde{\pi}_{\beta,\alpha}(0))$, using the commutativity of the diagrams in \eqref{eq:commutative_diag}, we observe that $x \in K_{\beta^\prime}(\tilde{\pi}_{\beta,\alpha}(0))$, if and only if, $\pi_{\beta'}(\tau^{+}_{\beta,\alpha}(0)) \leq \pi_{\beta'}(\sigma^{n}(\tau^{+}_{\beta'}(x)) \leq \pi_{\beta'}(\tau_{\beta'}^{-}(1))$ for all $n \in \mathbb{N}_{0}$. Since $\pi_{\beta'}$ is injective on $\Omega_{\beta'}$ and monotonic on $\{0,1\}^{\mathbb{N}}$, and since $\tau_{\beta,\alpha}^-(1)=\tau_{\beta^\prime}^-(1)$, it follows that $\tau_{\beta'}(x) \in \Omega^{+}_{\beta, \alpha}$. In other words, there exists a $y \in [0,1]$ such that $\tilde{\pi}_{\beta,\alpha}(y) = \pi_{\beta'}(\tau^{+}_{\beta,\alpha}(y)) = x$, yielding the first statement of Part~(1). Let us now prove the second statement. If $x \in \tilde{\pi}_{\beta,\alpha}(E_{\beta,\alpha}^{+})$, then there exists a $y \in E_{\beta,\alpha}^{+} \subset [0,1]$ with $\tilde{\pi}_{\beta,\alpha}(y) = x$. This in tandem with the fact that the the diagrams in \eqref{eq:commutative_diag} are commutative, and since the maps $\tau_{\beta,\alpha}^{+}$ and $\pi_{\beta'}$ are monotonic, we have
\begin{align*}
T_{\beta'}^{n}(x)
= T_{\beta'}^{n}( \pi_{\beta'}(\tau_{\beta,\alpha}^{+}(y)))
= \pi_{\beta'}(\sigma^{n}(\tau_{\beta,\alpha}^{+}(y)))
= \pi_{\beta'}(\tau_{\beta,\alpha}^{+}(T_{\beta,\alpha}^{n}(y)))
\geq \pi_{\beta'}(\tau_{\beta,\alpha}^{+}(y))
= x.
\end{align*}
Since $y \in [0,1]$ and $\tilde{\pi}_{\beta,\alpha}(y) = x$, and since $\tilde{\pi}_{\beta,\alpha}$ is monotonic, $x \geq \tilde{\pi}_{\beta,\alpha}(0)$. This, together with the fact that $E_{\beta'}^{+} \subseteq [0, 1]$, yields $\tilde{\pi}_{\beta,\alpha}(E_{\beta,\alpha}^+) \subseteq E_{\beta^\prime}^+\cap[\tilde{\pi}_{\beta,\alpha}(0),\tilde{\pi}_{\beta,\alpha}(1)]$. To see that $\tilde{\pi}_{\beta,\alpha}(E_{\beta,\alpha}^+) \supseteq E_{\beta^\prime}^+\cap[\tilde{\pi}_{\beta,\alpha}(0),1]$ let $x \in E_{\beta'}^{+}$ with $x \geq \tilde{\pi}_{\beta,\alpha}(0)$. By definition $E_{\beta'}^{+}$ and the commutativity of the diagrams in \eqref{eq:commutative_diag},
\begin{align*}
\pi_{\beta,\alpha}(\tau_{\beta,\alpha}^{+}(0))
\leq \pi_{\beta,\alpha}(\tau_{\beta'}^{+}(x))
\leq T^{n}_{\beta,\alpha}(\pi_{\beta,\alpha}(\tau_{\beta'}^{+}(x)))
\leq \pi_{\beta,\alpha} (\tau_{\beta'}^{-}(1))
\leq \pi_{\beta,\alpha} (\tau_{\beta,\alpha}^{-}(1)).
\end{align*}
In other words $\pi_{\beta,\alpha}(\tau_{\beta'}^{+}(x)) \in E_{\beta,\alpha}^+$. Since $\tilde{\pi}_{\beta,\alpha}$ is invertible on $[0,1)$ with inverse $\pi_{\beta,\alpha} \circ \tau^{+}_{\beta'}$ the result follows.
Part~(2) follows from Part~(1) using the fact that $\tilde{\pi}_{\beta,\alpha}$ is monotonic and injective on $[0,1)$. Part~(3) follows using analogous argument to those used above to proof Part~(1), and Part~(4) follows from \Cref{prop:ent} and Part~(3) in the following way.
\[
\dim_H(K_{\beta,\alpha}^+(t))=\frac{h_{\operatorname{top}}(T_{\beta,\alpha}\vert_{K^+_{\beta,\alpha}(t)})}{\log(\beta)}=\frac{h_{\operatorname{top}}(T_{\beta^\prime}\vert_{K^+_{\beta^\prime}(\tilde{\pi}_{\beta,\alpha}(t))})}{\log(\beta)}=\frac{\log(\beta^\prime)}{\log(\beta)}\dim_H(K_{\beta^\prime}^+(\tilde{\pi}_{\beta,\alpha}(t))).
\qedhere
\]
\end{proof}
For the next proposition we will require the following analogue of the map $\tilde{\pi}_{\beta,\alpha}$, namely $\tilde{\pi}_{\beta,\alpha}^{-} \coloneqq \pi_{u(\beta,\alpha)} \circ \tau_{\beta,\alpha}^{-}$. Note in the previous proposition, we could have also used the map $\tilde{\pi}_{\beta,\alpha}^{-}$ instead of $\tilde{\pi}_{\beta,\alpha}$ since they coincide on all points considered in (1)--(4). However, in the proof of \Cref{prop:char} we would need to replace $T_{\beta,\alpha}$ by $T_{\beta,\alpha}^{-}$, $T_{\beta'}$ by $T_{\beta'}^{-}$, $\tau_{\beta,\alpha}^{\pm}$ by $\tau_{\beta,\alpha}^{\mp}$ and $\tau_{\beta'}^{\pm}$ by $\tau_{\beta'}^{\mp}$, making it notionally heavy, and thus for ease of notation we use $\tilde{\pi}_{\beta,\alpha}$.
\begin{proposition}\label{prop:char(5)}
For all $(\beta,\alpha)\in\Delta$, we have that $\tilde{\pi}^{-}_{\beta,\alpha}(t_{\beta,\alpha,c})=t_{u(\beta,\alpha),c}$.
\end{proposition}
\begin{proof}
Observe that there exists a sequence of real numbers $(t_{n})_{n\in \mathbb{N}}$ with $t_n \in E_{\beta,\alpha}^{+}$ such that $t_n < t_{\beta,\alpha,c}$ and $\lim_{n\to \infty} t_{n} = t_{\beta,\alpha,c}$; otherwise the dimension function would be constant around $t_{\beta,\alpha,c}$ contradicting its definition. Define $\hat{t}_n =\tilde{\pi}^{-}_{\beta,\alpha}(t_n)$. By Proposition \ref{prop:char} Part~(3), for all $n\in \mathbb{N}$, we have $\tilde{\pi}^{-}_{\beta,\alpha}(K_{\beta,\alpha}^+(t_n))=K_{u(\beta,\alpha)}^+(\hat{t}_n)$. An application of Proposition \ref{prop:char} Part~(4) together with our remarks directly preceding this Proposition, yields for $n \in \mathbb{N}$,
\begin{align*}
\dim_H(K_{u(\beta,\alpha)}^+(\hat{t}_n))>0, \quad
\dim_H(K_{\beta,\alpha}^+(t_{\beta,\alpha,c}))=0,
\quad \text{and} \quad
\dim_H(K_{u(\beta,\alpha)}^+(\tilde{\pi}^{-}_{\beta,\alpha}(t_{\beta,\alpha,c}))=0.
\end{align*}
As $\pi_{u(\beta,\alpha)}$ is continuous and $\tau_{\beta,\alpha}^{-}$ is left continuous, $\tilde{\pi}^{-}_{\beta,\alpha}$ is left continuous, and so $\tilde{\pi}^{-}_{\beta,\alpha}(\lim_{n\to \infty}(t_n))= \lim_{n\to \infty} \tilde{\pi}^{-}_{\beta,\alpha}(t_N)$. This implies that $\tilde{\pi}^{-}_{\beta,\alpha}(t_{\beta,\alpha,c}) = \lim_{n\to \infty} \hat{t}_N$, and hence that $\tilde{\pi}^{-}_{\beta,\alpha}(t_{\beta,\alpha,c})=t_{u(\beta,\alpha),c}$.
\end{proof}
The value of $t_{u(\beta,\alpha),c}$ is explicitly given in \cite{KKLL} when $\tau_{\beta,\alpha}^-(1)$ is balanced. For all other cases see \cite{AK}.
A word $\omega = \omega_{1}\omega_{2} \cdots \in \{0,1\}^{\mathbb{N}}$ is called \textsl{balanced} if $\lvert(\omega_{n} + \omega_{n+1} + \cdots + \omega_{n+m}) - (\omega_{k+n} + \omega_{k+n+1} + \cdots + \omega_{k+n+m}) \rvert \leq 1$ for all $k$, $n$ and $m \in \mathbb{N}_{0}$ with $n \geq 1$. Following notation of \cite{KKLL} and \cite{Schme}, we let
\begin{align*}
C_3 \coloneqq\{ \beta\in(1,2) \colon \text{the length of consecutive zeros in} \; \tau_\beta^-(1) \; \text{is bounded} \}
\;\; \text{and} \;\;
C \coloneqq \{ \beta \in (1,2) \colon \tau_\beta^-(1) \; \text{is balanced} \}.
\end{align*}
For every $(\beta,\alpha) \in \Delta$ with $\alpha>0$, we have that $u(\beta,\alpha)\in C_3$. By \cite[Theorem 3.12]{KKLL}, for $\beta \in C_3$, there exists $\delta>0$ such that $E_{\beta}^+\cap[0,\delta]$ contains no isolated points. With this in mind and, for $\beta \in (1,2)$, setting $\delta(\beta) \coloneqq \sup \{ \delta \in [0, 1] \colon E_{\beta}^+\cap[0,\delta] \; \text{contains no isolated points} \}$, we have the following corollary of Proposition \ref{prop:char}.
\begin{corollary}\label{cor:isoloated_pts}
Let $(\beta,\alpha)\in \Delta$ with $\alpha>0$. If $\tilde{\pi}_{\beta,\alpha}(0)<\delta(u(\beta,\alpha))$ then there exists a $\delta>0$ such that $E_{\beta,\alpha}^+\cap[0,\delta]$ contains no isolated points. Further, if $u(\beta,\alpha) \in C$, then $\delta(u(\beta,\alpha))=1$ and $E_{\beta,\alpha}^+$ contains no isolated points.
\end{corollary}
\begin{proof}
The first statement follows from \Cref{prop:char} Parts (1) and (2), and \cite[Theorem 3.12]{KKLL}. The second statement follows from \cite[Theorem 3]{KKLL}, which states that if $\beta\in C$ then $E_\beta$ does not contain any isolated points.
\end{proof}
\begin{proof}[Proof of \texorpdfstring{\Cref{Cor_2}}{Corollary 1.3}]
In \cite{MR0166332} an absolutely continuous invariant measure of $T_{\beta,\alpha}$ is constructed, and in \cite{Hof} it is shown that this measure is ergodic. (In fact it is shown that it is maximal, and the only measure of maximal entropy.) This yields, given an $m \in \mathbb{N}$, that for almost all $x \in [0,1]$, there exists $n_{x} = n \in \mathbb{N}_{0}$ such that $T_{\beta,\alpha}^{n}(x) \in [0, 1/m)$, and hence that $K_{\beta,\alpha}^{+}(1/m)$ is a Lebesgue null. Since $E_{\beta,\alpha}^{+} \setminus \{0\} \subseteq \cup_{m=1}^{\infty} K_{\beta,\alpha}^{+}(1/m)$, by subadditivity of the Lebesgue measure, it follows that $E_{\beta,\alpha}^{+}$ is a Lebesgue null set. The statement on the isolated points of $E_{\beta,\alpha}^{+}$ follows from \Cref{cor:isoloated_pts}.
\end{proof}
\begin{proof}[Proof of \texorpdfstring{\Cref{Cor_3}}{Corollary 1.4}]
This is a direct consequence of \Cref{prop:char} Part~(4) and \cite[Theorem~A (ii)]{KKLL}.
\end{proof}
\begin{proof}[Proof of \texorpdfstring{\Cref{Cor_E_beta_alpha}}{Corollary 1.5}]
Let $(\beta, \alpha) \in \Delta$ with $\Omega_{\beta,\alpha}$ a subshift of finite type and $T_{\beta,\alpha}$ transitive, let $P_{\beta,\alpha}$ denote the Markov partition of $T_{\beta,\alpha}$ defined in \eqref{eq:Markov_Partition}, and for $n\in \mathbb{N}$, let $\Omega_{T_{\beta,\alpha}}\vert_{n}$ denote the set of all length $n$ admissible words of $T_{\beta,\alpha}$ with respect to the partition $P_{\beta,\alpha}$. Fix $n \in \mathbb{N}$ sufficient large, set $\omega$ to be lexicographically the smallest word in $\Omega_{T_{\beta,\alpha}}\vert_{n}$, let $a_{n} = a_{\beta, \alpha, n} = \sup I(\omega)$ and let $\nu \in \Omega_{T_{\beta,\alpha}}\vert_{n}$ with $\nu \succ \omega$. By transitivity and the Markov property, there exist $k \in \mathbb{N}$ and $\xi \in \Omega_{\beta,\alpha}\vert_{k}$ with $k > n$, $I(\xi) \subset I(\omega)$, $T_{\beta,\alpha}^{k-n}(I(\xi)) = I(\nu)$, and $T_{\beta,\alpha}^j(I(\xi))$ an interval and $T_{\beta,\alpha}^{j}(x) \geq a_{n}$ for all $j \in \{1,2, \dots, k - n\}$ and $x \in I(\xi)$. In other words, there exists a linear scaled copy of $K_{\beta,\alpha}^{+}(a_{n}) \cap I(\nu)$ in $I(\xi) \cap E_{\beta,\alpha}^{+}$. Namely, we have
\begin{align}\label{eq:scaling_func}
f_{\beta,\alpha,\nu,n}(K_{\beta,\alpha}^{+}(a_{n}) \cap I(\nu)) \subseteq I(\xi) \cap E_{\beta,\alpha}^{+},
\end{align}
where $f_{\beta, \alpha,\nu, n} = f_{\beta,\alpha,\chi(\xi_{1})} \circ f_{\beta,\alpha,\chi(\xi_{2})} \circ \cdots \circ f_{\beta,\alpha,\chi(\xi_{k-n})}$ and $\chi \colon \Omega_{T_{\beta,\alpha}}\vert_{1} \to \{ 0, 1\}$ is defined by
\begin{align*}
\chi(a) = \begin{cases}
0 & \text{if} \; I(a) \subseteq [0, p_{\beta,\alpha}],\\
1 & \text{otherwise.}
\end{cases}
\end{align*}
Here, we recall that $f_{\beta,\alpha,0}(x) = \beta^{-1}x-\alpha\beta^{-1}$ and $f_{\beta,\alpha, 1}(x) = \beta^{-1}x-(\alpha-1)\beta^{-1}$ for $x \in [0,1]$. With the above at hand, we may conclude that
\begin{align*}
\dim_H (E_{\beta,\alpha}^{+})
\geq \max \{ \dim_H(K_{\beta,\alpha}^{+}(a_{n}) \cap I(\nu)) \colon \nu \in \Omega_{T_{\beta,\alpha}}\vert_{n} \}
= \dim_H(K_{\beta,\alpha}^{+}(a_{n})).
\end{align*}
Since $n$ was chosen sufficiently large but arbitrarily, this in tandem with \Cref{Cor_3} implies $\dim_H (E_{\beta,\alpha}^{+}) = 1$, since $a_n$ converges to zero as $n$ tends to infinity.
Since Hausdorff dimension is preserved under taking linear transformations, an application of \Cref{thm:G1990+Palmer79,thm:LSSS}, yields for $(\beta, \alpha) \in \Delta$ with $\Omega_{\beta,\alpha}$ a subshift of finite type, that $\dim_H (E_{\beta,\alpha}^{+}) = 1$.
To conclude, let $(\beta,\alpha) \in \Delta$ be chosen arbitrarily, and let $((\beta_{n},\alpha_{n}))_{n \in \mathbb{N}}$ denote the sequence of tuples given in \Cref{Cor_1} converging to $(\beta, \alpha)$. Set $\tilde{\pi}_{\beta, \alpha}^{(n)} = \pi_{\beta,\alpha} \circ \tau^{+}_{\beta_{n},\alpha_{n}}$, and for $t$ and $s \in [0,1]$ with $t < s$, let
\begin{align*}
K_{\beta,\alpha}^{+}(t, s) \coloneqq \{ x \in[0, 1) \colon T_{\beta,\alpha}^{n}(x) \not \in [0,t) \cup (s, 1] \; \textup{for all} \; n \in \mathbb{N}_{0} \}
\end{align*}
By \Cref{Cor_1}, \Cref{thm:Structure}, and the commutativity of the diagram in \eqref{eq:commutative_diag}, we may choose $((\beta_{n},\alpha_{n}))_{n \in \mathbb{N}}$ so that $(\pi_{\beta,\alpha}^{(n)}(0))_{n \in \mathbb{N}}$ is a monotonically decreasing sequence converging to zero, and $(\pi_{\beta,\alpha}^{(n)}(1))_{n \in \mathbb{N}}$ is a monotonically increasing sequence converging to one. Thus, by construction, for $n$ and $l \in \mathbb{N}$ with $l \geq n$,
\begin{align*}
K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(0), \pi_{\beta,\alpha}^{(n)}(1)) \subseteq K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(0), \pi_{\beta,\alpha}^{(l)}(1)),
\quad \text{and} \quad
K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(0)) = \bigcup_{m \in \mathbb{N}} K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(0), \pi_{\beta,\alpha}^{(m)}(1)).
\end{align*}
Hence, by countable stability of the Hausdorff dimension and \Cref{Cor_3},
\begin{align}\label{eq:limit_hausdorff_proj_0,1}
\lim_{n \to \infty} \dim_{H}(K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(0),\pi_{\beta,\alpha}^{(n)}(1))) = 1.
\end{align}
Via analogous arguments to those given in the proof of \Cref{prop:char}, we have the following.
\begin{enumerate}
\item[($1^{*}$)] $\tilde{\pi}_{\beta,\alpha}^{(n)}([0,1])=K_{\beta,\alpha}^+(\tilde{\pi}_{\beta,\alpha}^{(n)}(0), \tilde{\pi}_{\beta,\alpha}^{(n)}(1))$ and $\tilde{\pi}_{\beta,\alpha}^{(n)}(E_{\beta_{n},\alpha_{n}}^+)=E_{\beta, \alpha}^+\cap[\tilde{\pi}_{\beta,\alpha}^{(n)}(0),\tilde{\pi}_{\beta,\alpha}^{(n)}(1)]$.
\item[($3^{*}$)] For $t\in(0,1)$, we have $ \tilde{\pi}_{\beta,\alpha}^{(n)}(K_{\beta_{n},\alpha_{n}}^+(t))=K_{\beta,\alpha}^+(\tilde{\pi}_{\beta,\alpha}^{(n)}(t), \tilde{\pi}_{\beta,\alpha}^{(n)}(1)) $.
\end{enumerate}
For $k \in \mathbb{N}$ sufficiently large and $\nu \in \Omega_{T_{\beta_{n},\alpha_{n}}}\vert_{k}$, setting $a_{n,k} = a_{\beta_{n},\alpha_{n},k}$, from the equalities given in \eqref{eq:alt_IFS} and \eqref{eq:scaling_func}, the commutativity of the diagram in \eqref{eq:commutative_diag} and ($1^{*}$), we have that
\begin{align*}
f_{\beta,\alpha, \nu, k}(
\pi_{\beta,\alpha}^{(n)}( K_{\beta_{n},\alpha_{n}}^{+}(a_{n,k}) \cap I(\nu)))
=\pi_{\beta,\alpha}^{(n)}( f_{\beta_{n},\alpha_{n}, \nu, k}( K_{\beta_{n},\alpha_{n}}^{+}(a_{n,k}) \cap I(\nu)))
\subseteq \pi_{\beta,\alpha}^{(n)}(E_{\beta_{n},\alpha_{n}}^{+})
\subseteq E_{\beta,\alpha}.
\end{align*}
This in tandem with ($3^{*}$), the fact that there exists $\nu \in \Omega_{T_{\beta_{n},\alpha_{n}}}\vert_{k}$ with
\begin{align*}
\dim_{H}(\pi_{\beta,\alpha}^{(n)}( K_{\beta_{n},\alpha_{n}}^{+}(a_{n,k}) \cap I(\nu))) = \dim_{H}(\pi_{\beta,\alpha}^{(n)}( K_{\beta_{n},\alpha_{n}}^{+}(a_{n, k}))),
\end{align*}
and that Hausdorff dimension is invariant under linear scaling, implies that
\begin{align*}
\dim_{H}(K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(a_{n,k}),
\pi_{\beta,\alpha}^{(n)}(1)))
=\dim_{H}(\pi_{\beta,\alpha}^{(n)}( K_{\beta_{n},\alpha_{n}}^{+}(a_{n,k}))) \leq \dim_{H}(E_{\beta,\alpha}).
\end{align*}
This in tandem with \Cref{Cor_3}, the equality given in \eqref{eq:limit_hausdorff_proj_0,1}, the observations that, for $n \in \mathbb{N}$, the sequence $(\pi_{\beta,\alpha}^{(n)}(a_{n,k}))_{k \in \mathbb{N}}$ is monotonically decreasing with $\lim_{k \to \infty} \pi_{\beta,\alpha}^{(n)}(a_{n,k}) =\pi_{\beta,\alpha}^{(n)}(0)$, and for $l$ and $m \in \mathbb{N}$ with $l \geq m$,
\begin{align*}
K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(a_{n,m}),
\pi_{\beta,\alpha}^{(n)}(1)) \subseteq K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(a_{n,l}), \pi_{\beta,\alpha}^{(n)}(1)),
%
\quad \text{and} \quad
%
K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(0),
\pi_{\beta,\alpha}^{(n)}(1)) = \bigcup_{k \in \mathbb{N}} K_{\beta,\alpha}^{+}(\pi_{\beta,\alpha}^{(n)}(a_{n,k}),
\pi_{\beta,\alpha}^{(n)}(1)),
\end{align*}
and the countable stability of the Hausdorff dimension, yields the required result.
\end{proof}
\subsection*{Examples and applications}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{dimhbeta110alpha0.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{dimhsymbetagoldenmean.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{dimhbeta10alpha0001.png}
\end{subfigure}
\caption{Graphs of $\eta_{\beta,\alpha}$: on the left, $\eta_{\beta}(t)$ with $\beta$ such that $\tau_\beta^-(1)=(110)^\infty$; in the middle, $\eta_{\beta,\alpha}$ for $(\beta, \alpha) \in \Delta$ with $\tau_{\beta,\alpha}^-(1)=(110)^\infty$ and $\tau_{\beta,\alpha}^+(0)=(001)^\infty$; on the right, $\eta_{\beta}$ for $(\beta, \alpha) \in \Delta$ such that $\tau_{\beta,\alpha}^-(1)=(10)^\infty$ and $\tau_{\beta,\alpha}^+(0)=(0001)^\infty $.}\label{fig:dim}
\end{figure}
Let $(\beta,\alpha) \in \Delta$ be such that $\tau^-_{\beta,\alpha}(1)=(10)^\infty$. In which case, $u(\beta,\alpha)$ is equal to the golden mean, which we denote by $G$, and belongs to the set $C$. Thus, $E^{+}_{\beta,\alpha}$ contains no isolated points. From \cite[Proposition 5.2]{KKLL} and by an elementary calculation, we have that $t_{u(\beta,\alpha),c} = G^{-2}$ and $\tau_{G}^{-}(G^{-2}) = 00(10)^\infty$. This, in tandem with \Cref{prop:char(5)}, yields $t_{\beta,\alpha,c} = \pi_{\beta, \alpha} \tau_{G}^{-}(G^{-2}) = \pi_{\beta, \alpha}(00(10)^\infty) = \alpha/(1-\beta)+1/(\beta^{3}-1)$, which one can show is equal to $(1-\alpha-\beta\alpha)\beta^{-2}$ using the fact that $\tau^-_{\beta,\alpha}(1)=(10)^\infty$. Moreover, by \Cref{prop:char} Part~(4),
\begin{align*}
\dim_H(K_{\beta,\alpha}(t))=(\log(G)/\log(\beta)) \dim_H(K_{G}(\tilde{\pi}_{\beta,\alpha}(t)),
\end{align*}
for all $t \in (0, 1)$. We now show that for a given $\beta\in(1,G)$ there exists a unique $\alpha\in(0,1/2)$ with $G = u(\beta,\alpha)$, or equivalently, that for a given $\beta\in(1,G)$ there exists a unique $\alpha\in(0,1/2)$ with $T_{\beta,\alpha}(1) = p_{\beta,\alpha}$. Using the definitions of the involved terms, $T_{\beta,\alpha}(1) = p_{\beta,\alpha}$ if and only if $\alpha=1-\beta^2/(\beta+1)$. Noting, when $\beta = G$, that $\alpha=0$, and as $\beta$ approaches $1$ from above, that $\alpha =1-\beta^2/(\beta+1)$ converges to $1/2$, yields the required result.
By \Cref{thm:LSSS}, under the assumption that $\tau^-_{\beta,\alpha}(1)=(10)^\infty$, if $\tau_{\beta,\alpha}^+(0)$ is periodic, then $\Omega_{\beta,\alpha}$ is a subshift of finite type. We now find $(\beta, \alpha) \in \Delta$ such that $\tau_{\beta,\alpha}^+(0)=(0001)^\infty$ and $\tau^-_{\beta,\alpha}(1)=(10)^\infty$. For this we, observe that
\begin{align*}
T_{\beta,\alpha}(0) = \alpha, \quad
T_{\beta,\alpha}^2(0) = \beta\alpha+\alpha, \quad
T_{\beta,\alpha}^3(0) = \beta(\beta\alpha+\alpha)+\alpha, \quad \text{and} \quad
T_{\beta,\alpha}^4(0) = \beta(\beta(\beta\alpha+\alpha)+\alpha)+\alpha-1=0.
\end{align*}
Substituting $\alpha=1-\beta^2/(\beta+1)$ into the last equality, gives
\begin{align*}
\beta(\beta(\beta(1-\beta^2/(\beta+1))+(1-\beta^2/(\beta+1)))+(1-\beta^2/(\beta+1)))+(1-\beta^2/(\beta+1))-1 = 0.
\end{align*}
This reduces to $\beta(\beta^4 -\beta^2- \beta -1) =0$. Thus, if $\beta$ is the positive real root of $\beta^4 -\beta^2- \beta -1 =0$ and $\alpha=1-\beta^2/(\beta+1)$, then $\tau_{\beta,\alpha}^+(0)=(0001)^\infty$ and $\tau^-_{\beta,\alpha}(1)=(10)^\infty$. Numerically approximating $\beta$ and $\alpha$ yields $\beta\approx 1.4656$ and $\alpha\approx 0.1288$.
We utilise the above, in particular \Cref{prop:char}, in studying the dimension function $\eta_{\beta,\alpha}$. Recall, if $t\not \in E^{+}_{\beta,\alpha}$, then there exists $t^*>t$ with $K^{+}_{\beta,\alpha}(t)=K^{+}_{\beta,\alpha}(t^*)$. Thus, it suffices to study $K^{+}_{\beta,\alpha}(t)$ for $t\in E^{+}_{\beta,\alpha}$. For a fixed $t\in E^{+}_{\beta,\alpha}$, with the aid of \Cref{thm:BSV14}, we find $(\beta^\prime,\alpha^\prime) \in \Delta$ with $\tau_{\beta,\alpha}(t)=\tau_{\beta^\prime,\alpha^\prime}(0)$ and $\tau^{-}_{\beta,\alpha}(1) = \tau^{-}_{\beta^\prime,\alpha^\prime}(1)$.
By \Cref{prop:char} Part~(4),
\begin{align*}
\eta_{\beta,\alpha}(t)=\frac{\log(u(\beta,\alpha))}{\log(\beta)} \dim_H(K_{u(\beta,\alpha)}^+(\tilde{\pi}_{\beta,\alpha}(t)))
\quad \text{and} \quad
\eta_{\beta^\prime,\alpha^\prime}(0)=\frac{\log(u(\beta^\prime,\alpha^\prime))}{\log(\beta^\prime)} \dim_H(K_{u(\beta^\prime,\alpha^\prime)}^+(\tilde{\pi}_{\beta^\prime,\alpha^\prime}(0))).
\end{align*}
Since $u(\beta^\prime,\alpha^\prime)=u(\beta,\alpha)$, $\tilde{\pi}_{\beta,\alpha}(t)=\tilde{\pi}_{\beta^\prime,\alpha^\prime}(0)$ and $\eta_{\beta^\prime,\alpha^\prime}(0)=1$, we have $\eta_{\beta,\alpha}(t)=\log(\beta^\prime)/\log(\beta)$. In summary, determining the value of $\eta_{\beta, \alpha}(t)$ reduces down to finding such $\alpha^\prime$ and $\beta^\prime$. This can performed numerically with the aid of the monotonicity and continuity of the projection maps, see Figure \ref{fig:dim} for sample numerical outputs.
\section{Winning sets of intermediate \texorpdfstring{$\beta$}{beta}-transformations: Proof of Proof of \texorpdfstring{\Cref{thm:main_2}}{Theorem 1.6}}\label{sec:proof_thm_1_6}
To show the conditions of \Cref{thm_HY_Thm_2.1} are satisfied when $T=T_{\beta, \alpha}$ for all $(\beta, \alpha) \in \Delta$ with $T_{\beta, \alpha}$ transitive and $\Omega_{\beta, \alpha}$ of finite type we use the following lemma on the geometric length of cylinder sets and the following proposition.
\begin{lemma}\label{lem:geometric_lengths_of_cylinders}
Let $(\beta, \alpha) \in \Delta$ be such that $T_{\beta, \alpha}$ is transitive and $\Omega_{\beta, \alpha}$ is a subshift of finite type. If $\nu = \nu_{1} \cdots \nu_{\lvert \nu \rvert}$ is an admissible word with respect to the partition $P_{\beta,\alpha}$, then $\rho \beta^{-\lvert \nu \rvert} \leq \lvert I(\nu) \rvert \leq \beta^{-\lvert \nu \rvert}$, where $\rho = \min \{ \lvert I(i) \rvert \colon i \in \Lambda \}$.
\end{lemma}
\begin{proof}
If $\lvert \nu \rvert = 1$, the result is a consequence of the fact that $\max\{ p_{\beta,\alpha}, 1 - p_{\beta,\alpha}\} \leq \beta^{-1}$, and that $I(\nu) \subseteq [0, p_{\beta,\alpha}]$ or $I(\nu) \subseteq [p_{\beta,\alpha},1]$. Therefore, we may assume that $\lvert \nu \rvert \geq 2$. Since $T_{\beta, \alpha}$ is Markov with respect to the partition $P_{\beta,\alpha}$, for $j \in \{ 0, 1, \ldots, \lvert \nu \rvert \}$, we have $T_{\beta,\alpha}^{j}(I(\nu))$ is an interval and $T_{\beta,\alpha}^{\lvert \nu \rvert - 1}(J(\nu)) = J(\nu_{\lvert \nu \rvert})$, where for a given admissible finite word $\omega$, we denote by $J(\omega)$ the interior of $I(\omega)$.
This implies that $\lvert I(\nu) \rvert = \beta^{-\lvert \nu \rvert + 1}\lvert I(\nu_{\lvert \nu \rvert}) \rvert$ and hence that $\rho \beta^{-\lvert \nu \rvert} \leq \lvert I(\nu_{\lvert \nu \rvert}) \rvert \beta^{-\lvert \nu \rvert + 1} = \lvert I(\nu) \rvert = \lvert I(\nu_{\lvert \nu \rvert}) \rvert \beta^{-\lvert \nu \rvert + 1} \leq \beta^{-\lvert \nu \rvert}$.
\end{proof}
\begin{proposition}\label{prop:thm_SFT+Transitive_implies_winning}
Under the hypotheses of \Cref{lem:geometric_lengths_of_cylinders}, for all $x \in [0, 1]$ and $\gamma \in (0, 1)$, we have that the geometric condition $H_{x, \gamma}$, with $T = T_{\beta, \alpha}$ and the partition $P_{\beta,\alpha}$, is satisfied.
\end{proposition}
\begin{proof}
\Cref{lem:geometric_lengths_of_cylinders} yields \eqref{eq:condition_1} of $H_{\xi, \gamma}$, thus is suffices to show that \eqref{eq:condition_2} of $H_{x, \gamma}$ is satisfied. To this end, let $n-1$ denote the cardinality of $P_{\beta,\alpha}$, and observe that, since by assumption $T_{\beta, \alpha}$ is transitive, there exists an $m = m_{\beta, \alpha} \in \mathbb{N}$, so that $J(l) \subseteq T_{\beta,\alpha}^{m}(J(k))$, for all $l$ and $k \in \{0, 1, \ldots, n-1\}$, where $J(l)$ and $J(k)$ are as defined in the proof of \Cref{lem:geometric_lengths_of_cylinders}. Further, if for two admissible words $\nu$ and $\eta$, we have that $I(\nu)$ and $I(\eta)$ are $\gamma/4$-comparable, then by \Cref{lem:geometric_lengths_of_cylinders} there exists $k_{0} \in \mathbb{N}$ with $\lvert \lvert \nu \rvert - \lvert \eta \rvert \rvert \leq k_{0}$. Letting $\omega$ denote the symbolic representation of $x$ generated by $T_{\beta, \alpha}$ with respect to the partition $P_{\beta,\alpha}$, set
\begin{align*}
M = M_{\beta, \alpha} = \begin{cases}
\max \{ \sigma^{k}(\omega) \wedge \omega \colon k \in \{1, 2, \ldots, k_{0} \}\} & \text{if} \; \omega \; \text{is not periodic},\\
\max \{ \sigma^{k}(\omega) \wedge \omega \colon k \in \{1, 2, \ldots, \operatorname{per}(\omega) -1 \}\} & \text{if} \; \omega \; \text{is periodic.}
\end{cases}
\end{align*}
Our aim is to show \eqref{eq:condition_2} of $H_{x, \gamma}$ is satisfied for all admissible words $\nu$ and $\eta$ with $I(\nu)$ and $I(\eta)$ are $\gamma/4$-comparable and all integers $i > i^{*} = k_{0} + m + M$. For this, suppose $\nu$ and $\eta$ are $\omega\vert_{i}$-extendable with $0 \leq \lvert \lvert \nu \rvert - \lvert \eta \rvert \rvert \leq k_{0}$ and $\operatorname{dist}(I(\nu\omega\vert_{i}), I(\eta\omega\vert_{i})) > 0$. We consider case when $\nu$ is not a prefix of $\eta$, and when $\nu$ is a prefix of $\eta$ separately.
For both of these cases we use the following facts. For $l \in \{ 0, 1, \ldots, n-1\}$ there exists a minimal $j \in \{ 1, 2, \ldots, m \}$ such that $T_{\beta, \alpha}^{j}(I(l))$ contains the interiors of at least two elements of $P_{\beta,\alpha}$. For $k \in \{1, 2, \ldots, \lvert \nu \rvert + i -1\}$ and $l \in \{1, 2, \ldots, \lvert \eta \rvert + i -1\}$, we have $T_{\beta, \alpha}^{k}(I(\nu \omega\vert_{i}))$ and $T_{\beta, \alpha}^{l}(I(\eta \omega\vert_{i}))$ are intervals, and $T_{\beta, \alpha}^{k}(J(\nu \omega\vert_{i})) = J(\sigma^{k}(\nu \omega\vert_{i}))$ and $T_{\beta, \alpha}^{l}(J(\eta \omega\vert_{i})) = J(\sigma^{l}(\eta \omega\vert_{i}))$.
Let us consider the first of our two cases, namely when $\nu$ is not a prefix of $\eta$. Our above two facts imply that there exist $l \in \{1, 2, \ldots, m-1\}$ and $F \subseteq \{0, 1, \ldots, n-1\}$ with $\lvert F \rvert \geq 2$ such that
\begin{align}\label{eq:HY-splitting_of_cylinders}
I(\nu \omega\vert_{l}) = \bigcup_{k \in F} I(\nu \omega\vert_{l} k)
\quad \text{and} \quad
I(\eta \omega\vert_{l}) = \bigcup_{k \in F} I(\eta \omega\vert_{l} k).
\end{align}
Since $\nu$ is not a prefix of $\eta$, there exists $j \in \{ 1, 2, \ldots, \min\{\lvert \nu \rvert, \lvert \eta \rvert \} - 1 \}$ such that $\nu\vert_{j} = \eta\vert_{j}$ and $\nu\vert_{j+1} \prec \eta\vert_{j+1}$, or $\nu\vert_{j} = \eta\vert_{j}$ and $\nu\vert_{j+1} \succ \eta\vert_{j+1}$. Suppose that $\nu\vert_{j} = \eta\vert_{j}$ and $\nu\vert_{j+1} \prec \eta\vert_{j+1}$, and that $\omega_{1} = \max F$. Letting $k \in F \setminus \{\omega_{1}\}$, for all $x \in I(\nu \omega\vert_{i})$, $y \in I(\eta \omega\vert_{l} k)$ and $z \in I(\eta \omega\vert_{i})$, that $x \leq y \leq z$. In other words, $\operatorname{dist}(I(\nu\omega\vert_{i}), I(\eta\omega\vert_{i})) \geq \lvert I(\eta \omega\vert_{l} k)\rvert$, and hence by \Cref{lem:geometric_lengths_of_cylinders},
\begin{align*}
\operatorname{dist}(I(\nu\omega\vert_{i}), I(\eta\omega\vert_{i}))
\geq \lvert I(\eta \omega\vert_{l} k)\rvert
\geq \rho \beta^{-(\lvert \eta \rvert + l + 1 )}
\geq \rho \beta^{-(\lvert \eta \rvert + m + 1)}
\geq \rho \beta^{-(m + 1)} \lvert I(\eta) \rvert.
\end{align*}
Similarly, if $\omega_{1} \neq \max\{F\}$, setting $k = \max F$, we obtain that
\begin{align*}
\operatorname{dist}(I(\nu\omega\vert_{i}), I(\eta\omega\vert_{i}))
\geq \lvert I(\nu \omega\vert_{l} k)\rvert
\geq \rho \beta^{-(\lvert \nu \rvert + l + 1 )}
\geq \rho \beta^{-(\lvert \nu \rvert + m + 1)}
\geq \rho \beta^{-(m + 1)} \lvert I(\nu) \rvert.
\end{align*}
An analogous argument yields the result when $\nu\vert_{j} = \eta\vert_{j}$ and $\nu\vert_{j+1} \succ \eta\vert_{j+1}$.
When $\nu$ is a prefix of $\eta$, the result follows using a similar reasoning as in the case when $\nu$ is not a prefix of $\eta$, but where we replace the first line of the argument, namely \eqref{eq:HY-splitting_of_cylinders}, by the following observation. By construction, there exists a $j \in \{ 1, 2, \ldots, \lvert \eta \rvert - \lvert \nu \rvert + M -1 \}$ such that $\nu \omega\vert_{j} = (\eta \omega\vert_{i})\vert_{j + \lvert \nu \rvert}$ but $\nu \omega\vert_{j+1} \neq (\eta \omega\vert_{i})\vert_{j + \lvert \nu \rvert + 1}$. In which case, by our two facts, there exists an $l \in \{0, 1, 2, \ldots, m-1\}$ and a subset of $F$ of $\{1, 2, \ldots, n-1\}$ with $\lvert F \rvert \geq 2$ such that
\[
I(\nu \omega\vert_{j+l}) = \bigcup_{k \in F}
I(\nu \omega\vert_{j+l} k)
\quad \text{and} \quad
I((\eta \omega\vert_{i})\vert_{j + \lvert \nu \rvert + l}) = \bigcup_{k \in F} I((\eta \omega\vert_{i})\vert_{j + \lvert \nu \rvert +l} k).
\qedhere
\]
\end{proof}
\begin{proof}[{Proof of \Cref{thm:main_2}}]
This is a direct consequence of \Cref{thm:G1990+Palmer79,thm_HY_Thm_2.1}, and \Cref{prop:alpha-winning_transport,prop:thm_SFT+Transitive_implies_winning}.
\end{proof}
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,315 |
{"url":"http:\/\/calculus7.org\/page\/2\/","text":"# Words that contain UIO, and best-fitting\u00a0lines\n\nIn Calculus I we spend a fair amount of time talking about how nicely the tangent line fits a smooth curve.\n\nBut truth be told, it fits only near the point of tangency. How can we find the best approximating line for a function ${f}$ on a given interval?\n\nA natural measure of quality of approximation is the maximum deviation of the curve from the line, ${E(f;\\alpha,\\beta) = \\max_{[a, b]} |f(x) - \\alpha x-\\beta|}$ where ${\\alpha,\\beta}$ are the coefficients in the line equation, to be determined. We need ${\\alpha,\\beta}$ that minimize ${E(f;\\alpha,\\beta)}$.\n\nThe Chebyshev equioscillation theorem is quite useful here. For one thing, its name contains the letter combination uio, which Scrabble players may appreciate. (Can you think of other words with this combination?) Also, its statement does not involve concepts outside of Calculus I. Specialized to the case of linear fit, it says that ${\\alpha,\\beta}$ are optimal if and only if there exist three numbers ${ x_1 in ${[a, b]}$ such that the deviations ${\\delta_i = f(x_i) - \\alpha x_i-\\beta}$\n\n\u2022 are equal to ${E(f;\\alpha,\\beta)}$ in absolute value: ${|\\delta_i| = E(f;\\alpha,\\beta)}$ for ${i=1,2,3}$\n\u2022 have alternating signs: ${\\delta_1 = -\\delta_2 = \\delta_3}$\n\nLet\u2019s consider what this means. First, ${f'(x_i) =\\alpha}$ unless ${x_i}$ is an endpoint of ${[a,b]}$. Since ${x_2}$ cannot be an endpoint, we have ${f'(x_2)=\\alpha}$.\n\nFurthermore, ${f(x) - \\alpha x }$ takes the same value at ${x_1}$ and ${x_3}$. This gives an equation for ${x_2}$\n\n$\\displaystyle f(x_1)-f'(x_2)x_1 = f(x_3)-f'(x_2) x_3 \\qquad \\qquad (1)$\n\nwhich can be rewritten in the form resembling the Mean Value Theorem:\n\n$\\displaystyle f'(x_2) = \\frac{f(x_1)-f(x_3)}{x_1-x_3} \\qquad \\qquad (2)$\n\nIf ${f'}$ is strictly monotone, there can be only one ${x_i}$ with ${f'(x_i)=\\alpha}$. Hence ${x_1=a}$ and ${x_3=b}$ in this case, and we find ${x_2}$ by solving (2). This gives ${\\alpha = f'(x_2)}$, and then ${\\beta}$ is not hard to find.\n\nHere is how I did this in Sage:\n\nvar('x a b')\nf = sin(x) # or another function\ndf = f.diff(x)\na = # left endpoint\nb = # right endpoint\n\nThat was the setup. Now the actual computation:\n\nvar('x1 x2 x3')\nx1 = a\nx3 = b\nx2 = find_root(f(x=x1)-df(x=x2)*x1 == f(x=x3)-df(x=x2)*x3, a, b)\nalpha = df(x=x2)\nbeta = 1\/2*(f(x=x1)-alpha*x1 + f(x=x2)-alpha*x2)\nshow(plot(f,a,b)+plot(alpha*x+beta,a,b,color='red'))\n\nHowever, the algorithm fails to properly fit a line to the sine function on ${[0,3\\pi\/2]}$:\n\nThe problem is, ${f'(x)=\\cos x}$ is no longer monotone, making it possible for two of ${x_i}$ to be interior points. Recalling the identities for cosine, we see that these points must be symmetric about ${x=\\pi}$. One of ${x_i}$ must still be an endpoint, so either ${x_1=a}$ (and ${x_3=2\\pi-x_2}$) or ${x_3=b}$ (and ${x_1=2\\pi-x_2}$). The first option works:\n\nThis same line is also the best fit on the full period ${[0,2\\pi]}$. It passes through ${(\\pi,0)}$ and has the slope of ${-0.2172336...}$ which is not a number I can recognize.\n\nOn the interval ${[0,4\\pi]}$, all three of the above approaches fail:\n\nLuckily we don\u2019t need a computer in this case. Whenever ${|f|}$ has at least three points of maximum with alternating signs of ${f}$, the Chebyshev equioscillation theorem implies that the best linear fit is the zero function.\n\n# Unreasonable effectiveness of the left endpoint\u00a0rule\n\nThe left endpoint rule, and its twin right endpoint rule, are ugly ducklings of integration methods. The left endpoint rule is just the average of the values of the integrand over left endpoints of equal subintervals:\n\n$\\displaystyle \\int_a^b f(x)\\,dx \\approx \\frac{b-a}{n} \\sum_{i=0}^{n-1} f(a+i\/n)$\n\nHere is its graphical representation with ${n=10}$ on the interval ${[-1,1]}$: the sample points are marked with vertical lines, the length of each line representing the weight given to that point. Every point has the same weight, actually.\n\nPrimitive, ineffective, with error ${O(1\/n)}$ for ${n}$ points used.\n\nSimpson\u2019s rule is more sophisticated, with error ${O(1\/n^4)}$. It uses weights of three sizes:\n\nGaussian quadrature uses specially designed (and difficult to compute) sample points and weights: more points toward the edges, larger weights in the middle.\n\nLet\u2019s compare these quadrature rules on the integral ${\\int_{-1}^1 e^x \\cos \\pi x\\,dx}$, using ${10}$ points as above. Here is the function:\n\n\u2022 The exact value of the integral is ${\\dfrac{e^{-1}-e}{1+\\pi^2}}$, about -0.216.\n\u2022 Simpson\u2019s rule gets within 0.0007 of the exact value. Well done!\n\u2022 Gaussian quadrature gets within 0.000000000000003 of the exact value. Amazing!\n\u2022 And the lame left endpoint rule outputs\u2026 a positive number, getting even the sign wrong! This is ridiculous. The error is more than 0.22.\n\nLet\u2019s try another integral: ${\\int_{-1}^1 \\sqrt{4+\\cos \\pi x +\\sin \\pi x} \\,dx}$, again using ${10}$ points. The function looks like this:\n\nThe integral can be evaluated exactly\u2026 sort of. In terms of elliptic integrals. And preferably not by hand:\n\n\u2022 Simpson\u2019s rule is within 0.00001 of the exact value, even better than the first time.\n\u2022 Gaussian quadrature is within 0.00000003, not as spectacular as in the first example.\n\u2022 And the stupid left endpoint rule is \u2026 accurate within 0.00000000000000005. What?\n\nThe integral of a smooth periodic function over its period amounts to integration over a circle. When translated to the circle, the elaborate placement of Gaussian sample points is\u2026 simply illogical. There is no reason to bunch them up at any particular point: there is nothing special about (-1,0) or any other point of the circle.\n\nThe only natural approach here is the simplest one: equally spaced points, equal weights. Left endpoint rule uses it and wins.\n\nIt is easy to find the minimum of ${f(x,y) = x^2+16y^2}$ if you are human. For a computer this takes more work:\n\nThe animation shows a simplified form of the Nelder-Mead algorithm: a simplex-based minimization algorithm that does not use any derivatives of ${f}$. Such algorithms are easy to come up with for functions of one variable, e.g., the bisection method. But how to minimize a function of two variables?\n\nA natural way to look for minimum is to slide along the graph in the direction opposite to ${\\nabla f}$; this is the method of steepest descent. But for computational purposes we need a discrete process, not a continuous one. Instead of thinking of a point sliding down, think of a small tetrahedron tumbling down the graph of ${f}$; this is a discrete process of flips and flops. The process amounts to the triangle of contact being replaced by another triangle with an adjacent side. The triangle is flipped in the direction away from the highest vertex.\n\nThis is already a reasonable minimization algorithm: begin with a triangle ${T}$; find the values of ${f}$ at the vertices of ${T}$; reflect the triangle away from the highest value; if the reflected point ${R}$ has a smaller value, move there; otherwise stop.\n\nBut there\u2019s a problem: the size of triangle never changes in this process. If ${T}$ is large, we won\u2019t know where the minimum is even if ${T}$ eventually covers it. If ${T}$ is small, it will be moving in tiny steps.\n\nPerhaps, instead of stopping when reflection does not work anymore, we should reduce the size of ${T}$. It is natural to contract it toward the \u201cbest\u201d vertex (the one with the smallest value of ${f}$), replacing two other vertices with the midpoints of corresponding sides. Then repeat. The stopping condition can be the values of ${f}$ at all vertices becoming very close to one another.\n\nThis looks clever, but the results are unspectacular. The algorithm is prone to converge to a non-stationary point where just by an accident the triangle attains a nearly horizontal position. The problem is that the triangle, while changing its size, does not change its shape to fit the geometry of the graph of ${f}$.\n\nThe Nelder-Mead algorithm adapts the shape of the triangle by including the possibility of stretching while flipping. Thus, the triangle can grow smaller and larger, moving faster when the path is clear, or becoming very thin to fit into a narrow passage. Here is a simplified description:\n\n\u2022 Begin with some triangle ${T}$.\n\u2022 Evaluate the function ${f}$ at each vertex. Call the vertices ${W,G,B}$ where ${W}$ is the worst one (the largest value of ${f}$) and ${B}$ is the best.\n\u2022 Reflect ${W}$ about the midpoint of the good side ${GB}$. Let ${R}$ be the reflected point.\n\u2022 If ${f(R), then we consider moving even further in the same direction, extending the line ${WR}$ beyond ${R}$ by half the length of ${WR}$. Choose between ${R}$ and ${E}$ based on where ${f}$ is smaller, and make the chosen point a new vertex of our triangle, replacing ${W}$.\n\u2022 Else, do not reflect and instead shrink the triangle toward ${B}$.\n\u2022 Repeat, stopping when we either exceed the number of iterations or all values of ${f}$ at the vertices of triangle become nearly equal.\n\n(The full version of the Nelder-Mead algorithm also includes the comparison of ${R}$ with ${G}$, and also involves trying a point inside the triangle.)\n\nThis is Rosenbrock\u2019s function ${f(x,y)=100(x^2-y)^2 + (x-1)^2}$, one of standard torture tests for minimization algorithms. Its graph has a narrow valley along the parabola ${y=x^2}$. At the bottom of the valley, the incline toward the minimum ${(1,1)}$ is relatively small, compared to steep walls surrounding the valley. The steepest descent trajectory quickly reaches the valley but dramatically slows down there, moving in tiny zig-zagging steps.\n\nThe algorithm described above gets within ${0.001}$ of the minimum in 65 steps.\n\nIn conclusion, Scilab code with this algorithm.\n\nx = -0.4:0.1:1.6; y = -2:0.1:1.4 \/\/ viewing window\n[X,Y] = meshgrid(x,y); contour(x,y,f(X,Y)',30) \/\/ contour plot\nplot([1],[1],'r+') \/\/ minimum point\ntol = 10^(-6)\nn = 0\nT = [0, -1.5 ; 1.4, -1.5; 1.5, 0.5] \/\/ initial triangle\nfor i=1:3\nvalues(i) = f(T(i,1), T(i,2))\nend\nwhile (%T)\nxpoly(T(:,1),T(:,2),'lines',1) \/\/ draw the triangle\n[values, index] = gsort(values) \/\/ sort the values\nT = T(index,:)\nif values(1)-values(3) < tol \/\/ close enough?\nmfprintf(6, \"Minimum at (%.3f, %.3f)\",T(3,1),T(3,2))\nbreak\nend\nR = T(2,:) + T(3,:) - T(1,:) \/\/ reflected\nfR = f(R(1),R(2))\nif fR < values(3)\nE = 1.5*T(2,:) + 1.5*T(3,:) - 2*T(1,:) \/\/ extended\nfE = f(E(1),E(2))\nif fE < fR\nT(1,:)=E; values(1)=fE \/\/ pick extended\nelse\nT(1,:)=R; values(1)=fR \/\/ pick reflected\nend\nelse\nfor i=1:2\nT(i,:) = (T(i,:)+T(3,:))\/2 \/\/ shrink\nvalues(i) = f(T(i,1), T(i,2))\nend\nend\nn = n+1\nif n >= 200\ndisp('Failed to converge'); break \/\/ too bad\nend\nend\n\n# Hertzsprung\u2013Russell diagram of Stack Exchange\u00a0sites\n\nThe Hertzsprung\u2013Russell diagram is a scatter plot of stars by temperature and luminosity.\n\nTraditionally, it is shown with temperature axis pointing from right to left, which I don\u2019t really like.\n\nStack Exchange family of sites is not (yet) as numerous as stars in the Universe; there are only 120 or so of them. But they can also be organized on a two-parameter log-log scatterplot. The two parameters are: total number of questions (intrinsic characteristic, like surface temperature) and the number of daily visits (luminosity in Internet terms).\n\nThe linear scale chart was not going to work, due to the supergiant size of Stack Overflow:\n\nEven on the log-log scale Stack Overflow is an outlier, but within reason:\n\nThe colors follow Stack Exchange classification: Technology, Science, Culture & Recreation, Life & Arts, Business, Professional. The largest of science sites, and the second largest overall, is Mathematics, although it trails several non-science sites in luminosity.\n\nAnnotated version of the above diagram:\n\nBoth Mathematics and MathOverflow have low traffic compared to their size: perhaps the Internet audience is just not that into math? On the other hand, the young Mathematics Educators site is way up in the traffic category.\n\nIronically, Astronomy is a low-luminosity site, trailing in traffic the much smaller Earth Science. Other annotated science sites are well established by now: Physics, Statistics, and (slow-moving) Theoretical Computer Science.\n\nThe only two non-technology sites that beat Mathematics in traffic are Gaming and English. Gaming isn\u2019t really an exception here, since all large \u201ctechnology\u201d sites revolve around computers.\n\nThree of the sites have extremely low traffic currently: Beer, Italian, and Expatriates. If this does not improve, they may be shut down\u2026 or perhaps merged into one: Beer for Italian Expatriates.\n\n(*) \u201cHertzsprung-Russel StarData\u201d by ESOhttp:\/\/www.eso.org\/public\/images\/. Licensed under CC BY 3.0 via Wikimedia Commons.\n\n# Fractal-ish monotone functions\n\nThere are several ways to construct a strictly increasing continuous function which has zero derivative almost everywhere. I like the explicit construction given by R. Salem, On some singular monotonic functions which are strictly increasing (1943).\n\nHere is a streamlined version of the construction.\n\nFix ${\\lambda\\in (0,1\/2)}$ (on the above picture ${\\lambda=1\/4}$). Let ${f_0(x)=x}$, and inductively define ${f_{n+1}}$ so that\n\n1. ${f_{n+1} (x) = f_n(x)}$ when ${x\\in 2^{-n}\\mathbb Z}$.\n2. If ${x\\in 2^{-n}\\mathbb Z}$, let ${f_{n+1}(x+2^{-n-1}) =\\lambda f_n(x) + (1-\\lambda) f_n(x+2^{-n})}$.\n3. Now that ${f_{n+1}}$ has been defined on ${2^{-n-1}\\mathbb Z}$, extend it to ${\\mathbb R}$ by linear interpolation.\n4. Let ${f=\\lim f_n}$.\n\nSince ${f(x+1)=f(x)+1}$ by construction, it suffices to understand the behavior of ${f}$ on ${[0,1]}$.\n\nEach ${f_n}$ is piecewise linear and increasing. At each step of the construction, every line segment of ${f_n}$ (say, with slope ${s}$) is replaced by two segments, with slopes ${2(1-\\lambda)s}$ and ${2\\lambda s}$. Since ${\\lambda<1\/2}$, it follows that ${f_{n+1}(x+2^{-n-1}) > f_n(x+2^{-n-1})}$. Hence, ${f_{n+1}\\ge f_n}$.\n\nSince ${f(x)=f_n(x)}$ when ${x\\in 2^{-n}\\mathbb Z}$, it is easy to understand ${f}$ by considering its values at dyadic rationals and using monotonicity. This is how one can see that:\n\n\u2022 The difference of values of ${f}$ at consecutive points of ${2^{-n}\\mathbb Z}$ is at most ${(1-r)^{n}}$. Therefore, ${f}$ is H\u00f6lder continuous with exponent ${\\alpha = - \\frac{\\log (1-r)}{\\log 2}}$.\n\u2022 The difference of values of ${f}$ at consecutive points of ${2^{-n}\\mathbb Z}$ is at least ${r^{n}}$. Therefore, ${f}$ is strictly increasing, and its inverse is H\u00f6lder continuous with exponent ${\\alpha = - \\frac{\\log r}{\\log 2}}$.\n\nIt remains to check that ${f'=0}$ almost everywhere. Since ${f}$ is monotone, it is differentiable almost everywhere. Let ${x}$ be a point of differentiability (and not a dyadic rational, though this is automatic). For each ${n}$ there is ${x_n\\in 2^{-n}\\mathbb Z}$ such that ${x_n < x< x_n+2^{-n}}$. Let ${s_n = 2^{n} (f_n(x_n+2^{-n})-f_n(x_n))}$; this is the slope of ${f_n}$ on the ${2^{-n}}$-dyadic interval containing ${x}$. Since ${f'(x)}$ exists, we must have ${f'(x) = \\lim_{n\\rightarrow\\infty} s_n}$. On the other hand, the ratio of consecutive terms of this sequence, ${s_{n+1}\/s_n}$, is always either ${2 (1-\\lambda )}$ or ${2\\lambda}$. Such a sequence cannot have a finite nonzero limit. Thus ${f'(x)=0}$.\n\nHere is another ${f}$, with ${\\lambda=1\/8}$.\n\nBy making ${\\lambda}$ very small, and being more careful with the analysis of ${f'}$, one can make the Hausdorff dimension of the complement of ${\\{x \\colon f'(x)=0\\}}$ arbitrarily small.\n\nAn interesting modification of Salem\u2019s function was introduced by Tukia in Hausdorff dimension and quasisymmetric mappings (1989). For the functions considered above, the one-sided derivatives at every dyadic rational are zero and infinity, which is a rather non-symmetric state of affair. In particular, these functions are not quasisymmetric. But Tukia showed that if one alternates between ${\\lambda}$ and ${1-\\lambda}$ at every step, the resulting homeomorphism of ${\\mathbb R}$ becomes quasisymmetric. Here is the picture of alternating construction with ${\\lambda=1\/4}$; preliminary stages of construction are in green.\n\nThis is quite similar to how the introduction of alternating signs turns Takagi curve (blancmange curve) into a quasiarc (i.e., a curve without cusps); see Sweetened and flavored dessert made from gelatinous or starchy ingredients and milk. But the fractal curves in this post are relatively mild-mannered: they are rectifiable (and thus, not really fractal).\n\nHere is the simple Scilab code I used for the above plots.\n\nr = 1\/4\nt = [0 1]\nf = t\nfor i = 1:10\nt = [t, t(1:$-1)+2^(-i)] f = [f, r*f(1:$-1)+(1-r)*f(2:\\$)]\n[t, ind] = gsort(t,'g','i')\nf = f(ind)\nend\nplot(t,f)\n\n\nTo have preliminary stages shown as well, move plot(t,f) into the loop. For Tukia\u2019s alternating version, insert the line r = 1-r into the loop.\n\nLink to preliminary version of this post.\n\n# Unreasonable effectiveness of Runge-Kutta\u00a02\n\nI used to give this example of an initial-value problem whenever numerical solutions of ODE came up in class:\n\n$\\displaystyle y' = -y^2, \\quad y(0)=1$\n\nThe exact solution is ${y(t)=1\/(1+t)}$. Using Euler\u2019s method with step ${h=1}$ (admittedly large) we get, starting with ${y_0=1}$:\n\n$\\displaystyle y_1 = y_0 + h (-y_0^2) = 0,\\quad y_2 = y_1 + h (-y_1^2) = 0, \\dots$\n\nThe numerical solution drops down to zero at once and stays there. Not good at all.\n\nThe trapezoidal version of RK2 (also known as improved Euler, or Heun\u2019s method) treats the above value of ${y_1}$ merely as prediction ${\\tilde y_1}$, and computes the actual ${y_1}$ using the average of the slopes at ${y_0}$ and ${\\tilde y_1}$:\n\n$\\displaystyle y_1 = y_0 + \\frac{h}{2} (-y_0^2 - \\tilde y_1^2) = 1 + \\frac12 (-1 - 0 ) = \\frac12$\n\n\u2026 which is right on target. Let\u2019s do it again: the prediction ${\\tilde y_2 = y_1 + h (-y_1^2) = 1\/4}$ and correction\n\n$\\displaystyle y_2 = y_1 + \\frac{h}{2} (-y_1^2 - \\tilde y_2^2) = 1 + \\frac12 \\left(-\\frac{1}{4}-\\frac{1}{16} \\right) = \\frac{11}{32}$\n\n\u2026 which is ${\\approx 0.34}$, compared to the exact solution ${1\/3}$. Nice!\n\nI like the example, because all numbers are manageable with mental arithmetics for two steps. And it\u2019s clear that ${11\/32}$ is pretty close to ${1\/3 = 11\/33}$.\n\nBut the fact that the relative error stays below ${3.7\\%}$ throughout the computation despite the large step size ${h=1}$ seems\u2026 too good to be true. Consider that for the exact solution, the expression ${y^2}$ can differ by the factor of ${4}$ between two ${t}$-values at distance ${1}$.\n\nThe midpoint version of RK2, which generally performs about as well as the trapezoidal version, is nowhere as close in this example. Indeed, using the same data as above, we get\n\n$\\displaystyle y_1 = y_0 - h \\left(\\frac{y_0+\\tilde y_1}{2}\\right)^2 = \\frac{3}{4}$\n\nand so forth: the relative error reaches ${59\\%}$ at the second step. That\u2019s 16 times worse than the trapezoidal version.\n\nWhat happens here has less to do with numerical analysis than with algebra of rational functions. Using ${h=1}$ in trapezoidal RK2, we are in effect iterating the function\n\n$\\displaystyle \\varphi(y) = y + \\frac12 (-y^2-(y-y^2)^2) = y-y^2+y^3-\\frac12 y^4$\n\nThe exact solution would be obtained by iterating\n\n$\\displaystyle \\Phi (y) = \\frac{y}{1+y} = y-y^2+ y^3 - y^4 + \\dots$\n\nTwo functions just happen to coincide at ${y=1}$, which is our starting point here.\n\nFrom there we get to ${0.5}$, and on ${[0,0.5]}$ they are really close anyway, due to another coincidence: the truncation error ${|\\varphi(y)-\\Phi(y)|}$ is ${O(y^4)}$ instead of ${O(y^3)}$ as it is normally for second-order methods.\n\nThe midpoint method with ${h=1}$ amounts to iterating\n\n$\\displaystyle \\psi(y) = y - \\left(\\frac{y+y-y^2}{2} \\right)^2 = y-y^2+y^3-\\frac14 y^4$\n\nwhich is not substantially further away from ${\\Phi}$, but does not enjoy the lucky coincidence at ${y=1}$.\n\nThe tables are turned with the initial value ${y_0=3}$. The exact solution is ${y(t) = 3\/(1+3t)}$, which drops sharply from ${t=0}$ to ${t=1}$; its slope decreases by the factor of ${16}$ during one step. Still, midpoint-RK2 does a decent job with ${h=1}$:\n\nwhile trapezoidal RK2 outputs ${y_1=-19.5}$, ${y_2=-80110}$, ${y_3 = -2\\cdot 10^{19}}$ and promptly overflows.\n\nWith a reasonable step size, like ${h=0.1}$, normality is restored: both methods perform about equally well, with ${0.13\\%}$ error for trapezoidal and ${0.21\\%}$ for midpoint.\n\n# Calculating 2*2 with high\u00a0precision\n\nThe definition of derivative,\n\n$\\displaystyle f'(x) = \\lim_{h\\rightarrow 0}\\frac{f(x+h) - f(x)}{h} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (1)$\n\nis not such a great way to actually find the derivative numerically. Its symmetric version,\n\n$\\displaystyle f'(x) = \\lim_{h\\rightarrow 0}\\frac{f(x+h) - f(x-h)}{2h} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (2)$\n\nperforms much better in computations. For example, consider the derivative ${f(x)=e^x }$ at the point ${x=1}$. We know that ${f'(1)=2.718281828\\dots}$. Numerically, with ${h=0.001}$, we get\n\n$\\displaystyle \\frac{f(1+h) - f(1)}{h} \\approx \\mathbf{2.71}9 \\dots$\n\n(error ${>10^{-3}}$) versus\n\n$\\displaystyle \\frac{f(1+h) - f(1-h)}{2h} \\approx \\mathbf{2.71828}2 \\dots$\n\n(error ${<10^{-6}}$).\n\nConsidering this, why don\u2019t we ditch (1) altogether and adopt (2) as the definition of derivative? Just say that by definition,\n\n$\\displaystyle f'(x) = \\lim_{h\\rightarrow 0}\\frac{f(x+h)-f(x-h)}{2h}$\n\nwhenever the limit exists.\n\nThis expands the class of differentiable functions: for example, ${f(x)=|x|}$ becomes differentiable with ${f'(0)=0}$. Which looks more like a feature than a bug: after all, ${f}$ has a minimum at ${0}$, and the horizontal line through the minimum is the closest thing to the tangent line that it has.\n\nAnother example: the function\n\n$\\displaystyle f(x) = \\begin{cases} x ,\\quad & x\\le 0 \\\\ 3x,\\quad & x>0 \\end{cases}$\n\nhas ${f'(0)=2}$ under this definition, because\n\n$\\displaystyle \\lim_{h\\rightarrow 0^+}\\frac{f(x+h)-f(x-h)}{2h} = \\lim_{h\\rightarrow 0^+}\\frac{3h-(-h)}{2h} = 2$\n\nand\n\n$\\displaystyle \\lim_{h\\rightarrow 0^-}\\frac{f(x+h)-f(x-h)}{2h} = \\lim_{h\\rightarrow 0^-}\\frac{h-3(-h)}{2h} = 2$\n\nThis example also makes sense: since ${f(x)=|x|+2x}$, getting ${f'(0)=0+2}$ is expected. In fact, with the new definition we still have basic derivative rules: if ${f,g}$ are differentiable, then ${f+g}$, ${f-g}$, ${fg}$, ${f\/g}$ are also differentiable (with the usual caveat about ${g\\ne 0}$) and the familiar formulas hold.\n\nLet\u2019s test the chain rule on the function ${g = f\\circ f}$. The rule says\n\n$\\displaystyle g'(0) = f'(f(0)) f'(0) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (3)$\n\nSince ${f(0)=0}$, the product on the right is ${2\\cdot 2}$. On the other hand,\n\n$\\displaystyle g(x) = \\begin{cases} x ,\\quad & x\\le 0 \\\\ 9x,\\quad & x>0 \\end{cases}$\n\nwhich implies, by a computation similar to the above, that ${g'(0)=5}$. So, if we want to have the chain rule (3), we must accept that\n\n$\\displaystyle \\mathbf{2\\cdot 2 = 5}$\n\nThis is where the desire for high numerical precision leads.\n\nPlenty of other things go wrong with the symmetric definition:\n\n\u2022 Maximum or minimum of ${f}$ may be attained where ${f'}$ exists and is nonzero.\n\u2022 A differentiable function may be discontinuous.\n\u2022 Having ${f'>0}$ everywhere does not imply that ${f}$ is increasing.\n\u2022 Mean Value Theorem fails.\n\n# Polygonal inequalities: beyond the\u00a0triangle\n\n(Related to previous post but can be read independently). The triangle inequality, one of the axioms of a metric space, can be visualized by coloring the vertices of a triangle by red and blue.\n\nThe inequality says that the sum of monochromatic distances must not exceed the sum of dichromatic distances. That is, for every assignment of the vertices to points in the space, the sum of all red-red and blue-blue distances does not exceed the sum of red-blue distances. An assignment is just a map from the set of vertices ${V}$ to the metric space ${X}$; it need not be injective.\n\nBut why stop at the triangle? We can take any number of points (vertices), color them in some way, and require the same polygonal inequality:\n\n$\\displaystyle \\text{monochromatic } \\le \\text{ dichromatic}$\n\nAlready for the square we have two distinct plausible colorings to explore: even red-blue split\n\nand predominantly red square\n\nBut it turns out that the second coloring is useless: the inequality it defines fails in every metric space with more than one point. More generally, suppose we have ${R}$ red points and ${B}$ blue ones, and ${R- B\\ge 2}$. Pick two distinct points ${a,b\\in X}$. Assign one red point to ${a}$ and all others to ${b}$. The sum of monochromatic distances is ${(R-1)\\,d(a,b)}$ while the sum of dichromatic distances is ${B\\,d(a,b)}$, which is strictly less.\n\nSo, we are limited to nearly-even colorings: those with ${|R-B|\\le 1}$. For even numbers of vertices this means even split, while odd numbers should be divided as evenly as possible: like 3+2 for the pentagon.\n\nThe inequalities turn out to be related. For every ${n}$, the ${n}$-gonal inequality implies the ${(n-2)}$-gonal inequality, because if we assign two opposite-colored vertices to the same point, their contributions cancel out. More interestingly: when ${n}$ is odd, the ${n}$-gonal inequality implies the ${(n-1)}$-gonal inequality. Indeed, suppose we have ${(n-1)}$ points, evenly colored. Add an arbitrary ${n}$th point. Whether the added point is blue or red, the ${n}$-gonal inequality holds. Averaging these two inequalities, we see the effect of the added points canceling out.\n\nSo, if the ${n}$-gonal inequality holds for all odd ${n}$, it holds for all ${n}$. This property is exactly the hypermetric property from the previous post. Except it was stated there in a different form:\n\n$\\displaystyle \\sum_{i,j}b_i b_j d(x_i , x_j ) \\le 0$\n\nfor every finite sequence of points ${x_i\\in X}$ and every choice of integers ${b_i}$ such that ${\\sum_i b_i=1}$. But if the point ${x_i}$ is repeated ${|b_i|}$ times, we can replace ${b_i}$ by ${\\mathrm{sign}\\,b_i}$. Then represent +1 as blue and -1 as red.\n\nThe hypermetric inequalities were introduced by John B. Kelly (a student of William Ted Martin) in late 1960s. He showed they are necessary for the space to be embeddable into the space ${\\ell^1}$. It would be great if they were also sufficient (and for some classes of spaces they are), but this is not so: a counterexample was given by Patrice Assouad in 1977.\n\nIt is also interesting to consider the ${n}$-gonal inequalities for even ${n}$ only. By repetition of vertices, they are equivalent to requiring\n\n$\\displaystyle \\sum_{i,j}b_i b_j d(x_i , x_j ) \\le 0 \\quad\\quad \\quad (1)$\n\nfor every finite sequence of points ${x_i\\in X}$ and every choice of integers ${b_i}$ such that ${\\sum_i b_i=0}$. But of course then we have (1) for rational ${b_i}$ (just clear the denominators), hence for all real ${b_i}$ (by approximation), as long as they sum to ${0}$. So, the requirement amounts to the matrix ${(d(x_i,d_j))}$ being negative semidefinite on the subspace ${\\sum x_i=0}$. Such metrics are called metrics of negative type.\n\nTheir relation to embeddability of the space is well-known: ${(X,d)}$ is of negative type if and only if the \u201csnowflake\u201d ${(X,\\sqrt{d})}$ isometrically embeds into a Hilbert space. In other words, we can \u201cdraw\u201d any finite metric space of negative type in a Euclidean space, with the understanding that Euclidean distances represent the square roots of the actual distances. This embedding result is a 1935 theorem of Isaac Schoenberg who is also known for connecting dots naturally (introducing splines).\n\n# Pentagrams and hypermetrics\n\nThe Wikipedia article Metric (mathematics) offers a plenty of flavors of metrics, from common to obscure: ultrametric, pseudometric, quasimetric, semimetric, premetric, hemimetric and pseudoquasimetric (I kid you not).\n\nOne flavor it does not mention is a hypermetric. This is a metric ${d}$ on a set ${X}$ such that the inequality\n\n$\\displaystyle \\sum_{i,j}b_i b_j d(x_i , x_j ) \\le 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ (1)$\n\nholds for every finite sequence of points ${x_i\\in X}$ and every choice of integers ${b_i}$ such that ${\\sum_i b_i=1}$. The requirement that ${b_i}$ be integers gives some combinatorial meaning to (1); this is not just some quadratic form being negative semidefinite.\n\nAs a warm-up, observe that (1) contains in it the triangle inequality: with ${b_1=b_2=1}$ and ${b_3=-1}$ we get ${d(x_1,x_2)-d(x_1,x_3)-d(x_2,x_3)\\le 0}$. But it appears that (1) says something about \u201cpolygons\u201d with more vertices too.\n\nTo make (1) worth thinking about, it should be satisfied by some important metric space. Such as the real line ${\\mathbb R}$, for example. It is not quite obvious that the inequality\n\n$\\displaystyle \\sum_{i,j}b_i b_j |a_i - a_j| \\le 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ (2)$\n\nholds for all reals ${a_i}$ and all integers ${b_i}$ adding up to ${1}$. It helps to order the numbers: ${a_1\\le \\dots\\le a_m}$ and focus on the contribution of a particular gap ${[a_k,a_{k+1}]}$ to the sum (2). The amount it contributes is ${|a_k-a_{k+1}|}$ multiplied by\n\n$\\displaystyle \\sum_{i\\le k k} b_j \\right) = \\left(\\sum_{i\\le k} b_i \\right) \\left(1-\\sum_{i\\le k} b_i \\right) \\le 0$\n\nbecause ${n(1-n)\\le 0}$ for every integer ${n}$. This proves (2).\n\nNow that we have one hypermetric space, ${\\mathbb R}$, other such spaces can be created easily. If ${X}$ is any set and ${f \\colon X\\rightarrow\\mathbb R}$ any function, consider ${d(x,y) = |f(x)-f(y)|}$, the pullback pseudometric on ${X}$. By applying (2) to the numbers ${f(x_i)}$, we see that ${d}$ satisfies the hypermetric inequality. Since (1) is additive in ${d}$, we can take any family of functions ${f_\\alpha \\colon X\\rightarrow\\mathbb R}$ and add together the corresponding pseudometrics. Or even integrate them against a positive measure: ${d(x,y)=\\int |f_\\alpha(x)-f_\\alpha(y)|\\,d\\mu(\\alpha)}$.\n\nFor example, the plane ${\\mathbb R^2}$ is a hypermetric space, because the distance between two points ${(x_1,y_1)}$ and ${(x_2,y_2)}$, besides the familiar form\n\n$\\displaystyle \\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$\n\ncan also be represented as an integral of the aforementioned pullbacks:\n\n$\\displaystyle \\frac12 \\int_0^\\pi \\big| (x_1-x_2)\\cos \\alpha + (y_1-y_2) \\sin\\alpha \\big| \\,d\\alpha$\n\nA similar integral representation holds in all dimensions; thus, all Euclidean spaces are hypermetric.\n\nOkay, what is not a hypermetric then? For example, the cube distance induced by the norm ${\\|x\\|_\\infty=\\max |x_i|}$ is not, in dimensions 3 and higher. Specifically, (1) fails as the five-point inequality with ${(b_1,\\dots,b_5) =(1,1,1,-1,-1)}$. I\u2019ll call it the pentagram inequality:\n\nIt says that for any five points in the space the sum of monochromatic distances does not exceed the sum of all bi-chromatic (red-blue) distances.\n\nThe pentagram inequality fails when ${x_1,\\dots,x_5}$ are the columns of the matrix\n\n$\\displaystyle \\begin{pmatrix} 1& 1& 1& 2& 0\\\\ 0& 2& 2& 1& 1\\\\ 0& 1& 0& 1& 0\\\\ \\end{pmatrix}$\n\n(first three columns blue, the last two red). Indeed, the sum of monochromatic distances is ${2+1+2+2=7}$ while the sum of bichromatic distances is ${1+1+1+1+1+1=6}$.\n\nIf the above example does not look conceptual enough, it\u2019s because I found it via computer search. I don\u2019t have much intuition for the pentagram inequality.\n\nAnyway, the example delivers another proof that taking the maximum of three numbers is hard. More precisely, there is no isometric embedding of ${\\mathbb R^3}$ with the maximum metric into ${\\ell_1}$. Unlike the earlier proof, this one does not assume the embedding is linear.\n\nA good reference for hypermetric inequalities is the book Geometry of cuts and metrics by Deza and Laurent.\n\n# Maximum of three numbers: it\u2019s harder than it\u00a0sounds\n\nThis simple identity hold for any two real numbers ${x,y}$:\n\n$\\displaystyle \\max(|x|,|y|) = \\frac12\\,(|x+y|+|x-y|) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (1)$\n\nIndeed, if ${|x|}$ realizes the maximum, then both ${x+y}$ and ${x-y}$ have the same sign as ${x}$. After opening the absolute value signs, we get ${y}$ to cancel out.\n\nSo, (1) represents ${\\max(|x|,|y|)}$, also known as the ${\\ell_\\infty}$ norm, as the sum of absolute values of linear functions. Let\u2019s try the same with ${\\max(|x|,|y|,|z|)}$. Since the right hand side of (1) is just the average of ${|\\pm x \\pm y|}$ over all possible choices of ${\\pm }$ signs, the natural thing to do is to average ${|\\pm x \\pm y \\pm z|}$ over all eight choices. The sign in front of ${x}$ can be taken to be ${+}$, which simplifies the average to\n\n$\\displaystyle \\frac14\\,(|x+y+z|+|x+y-z|+|x-y+z|+|x-y-z|) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (2)$\n\nDoes this formula give ${\\max(|x|,|y|,|z|)}$? Instead of trying random numbers, let\u2019s just plot the unit ball for the norm given by (2). If the identity works, it will be a cube. I used Maple:\n\nwith(plots): f:=(abs(x+y+z)+abs(x+y-z)+abs(x-y-z)+abs(x-y+z))\/4:\nimplicitplot3d(f=1,x=-1\/4..1\/4,y=-1\/4..1\/4,z=-1\/4..1\/4,grid=[25,25,25]);\n\nClose, but no cube. Luckily, this is my favorite Archimedean solid, the cuboctahedron.\n\nAlthough all terms of (2) look exactly the same, the resulting shape has both triangular and square faces. Where does the difference of shapes come from?\n\nMore importantly, is (2) really the best we can do? Is there some other sum of moduli of linear functions that will produce ${\\max(|x|,|y|,|z|)}$?\n\n\u2014 No.\n\nEven if negative coefficients are allowed?\n\n\u2014 Even then. (But you can come arbitrarily close.)\n\nWhat if we allow integrals with respect to an arbitrary (signed) measure, as in\n\n$\\displaystyle \\iiint |\\alpha x+\\beta y+\\gamma z|\\,d \\mu(\\alpha, \\beta, \\gamma) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (3)$\n\n\u2014 Still no. But if ${\\mu}$ is allowed to be a distribution of higher order (an object more singular than a measure), then a representation exists (W. Weil, 1976). Yes, one needs the theory of distributions to write the maximum of three numbers as a combination of linear functions.\n\nI\u2019ll only prove that there is no identity of the form\n\n$\\displaystyle \\max(|x|,|y|,|z|) = \\sum_{i=1}^N |\\alpha_i x+\\beta_i y+ \\gamma_i z|$\n\nIndeed, such an identity amounts to having an isometric embedding ${T\\colon \\ell_\\infty^3 \\rightarrow \\ell_1^N}$. The adjoint operator ${T^* \\colon \\ell_\\infty^N \\rightarrow \\ell_1^3}$ is a submetry meaning that it maps the unit ball of ${\\ell_\\infty^N }$ onto the unit ball ${\\ell_1^3}$. The unit ball of ${\\ell_\\infty^N }$ is just a cube; all of its faces are centrally symmetric, and this symmetry is preserved by linear maps. But ${\\ell_1^3}$ is an octahedron, with triangular faces. A contradiction. ${\\ \\Box}$\n\nAn aside: what if instead of averaging ${|\\pm x \\pm y|}$ over all ${\\pm }$ choices (i.e., unimodular real coefficients) we take the average over all unimodular complex coefficients? This amounts to ${\\|(x,y)\\| = \\frac{1}{2\\pi} \\int_0^{2\\pi} |x+e^{it}y|\\,dt}$. I expected something nice from this norm, but\n\nit\u2019s a strange shape whose equation involves the complete elliptic integral of second kind. 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The snow has caused disruptions for us all. The weather service tells us it should get better soon, but there is still uncertainty.
It is best to be cautious rather than sorry and even if it stops snowing, the slush in most areas will be a treacherous mess.
This Wednesday's General Meeting 2/13 is now cancelled due to the weather.
In addition, the office will close on Tuesday.
Think sun, spring and great boating weather coming soon. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,020 |
Q: Linux memory manager infringes on PCI memory My board has a Cavium Octeon NPU, running Linux kernel 2.6.34.10 that acts as a PCIe Root Complex. It is connected to PCIe switch, as are some other peripheral devices (Endpoints), among which there is Marvell's 9143 PCI-to_SATA controller based SSD.
When PCIe is initially enumerated, PCI driver on Octeon adds up the sizes of all the prefetchable memory resources and programs the PLIMIT and PBASE registers on the upstream switch port accordingly. In my case that address range is 0x80000000 - 0xEFFFFFFF.
After that, I would expect that address range to be inaccessible to kernel memory manager allocating for DMA buffers etc. And yet, I see the kernel, at some point starts sending SCSI requests to the SSD device, where scatter-gather list elements fall within this address range. I confirmed this, by looking at PCI analyzer trace. Naturally, when SSD controller receives such an address, it tries to access it (DMA read or write), and fails, because upstream switch port refuses to forward this request upstream to Root Complex, because it is programmed to think that this address would be downstream from it. (Interestingly enough, it mostly happens when I manipulate large files, I see that kernel allocated buffer addresses grow downward, until they dip below 0xEFFFFFFF)
Hence, the question: shouldn't PCI enumeration/rescan code, tell the kernel - these are PCI devices register addresses and therefore are off-limit for DMA buffer allocation? Or is it responsibility of each individual device driver to reserve its prefetchable memory? Marvell driver I use reserves regular memory BAR, but not the prefetcheable one. Is that a problem?
Thanks in advance and apologies for lengthy description.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 655 |
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"redpajama_set_name": "RedPajamaC4"
} | 4,420 |
var _ = require('underscore');
var Promise = require('bluebird');
var SearchSupport = require('./../../../core/search/search-support.js');
var NoWaitOperationPromise = require('./../../../base-services/queue/domain/no-wait-operation-promise.js');
var LoadBalancerNodeCriteria = require('./domain/node-criteria.js');
var Criteria = require('./../../../core/search/criteria.js');
module.exports = SharedLoadBalancerNodes;
/**
* @typedef SharedLoadBalancerNodeMetadata
* @type {object}
* @property {String} status - Status of the node: enabled, disabled or deleted.
* @property {String} ipAddress - The internal (private) IP address of the node server
* @property {int} privatePort - The internal (private) port of the node server
*
* @example
* {
* "status" : "enabled",
* "ipAddress" : "10.11.12.13",
* "privatePort" : 80
* }
*/
/**
* @typedef CreateNodeConfig
* @type {object}
*
* @property {string} status - Status of the node: enabled, disabled or deleted.
* @property {string} ipAddress - The internal (private) IP address of the node server
* @property {number} privatePort - The internal (private) port of the node server.
* Must be a value between 1 and 65535.
*/
/**
* Service that allow to manage load balancer nodes in CenturyLink Cloud
*
* @param loadBalancerPools
* @param loadBalancerClient
* @param queueClient
* @constructor
*/
function SharedLoadBalancerNodes(loadBalancerPools, loadBalancerClient, queueClient) {
var self = this;
function init () {
SearchSupport.call(self);
self.Status = {
ENABLED: "enabled",
DISABLED: "disabled",
DELETED: "deleted"
};
}
function loadAllNodesWithPool(pool) {
return Promise.props({
pool: Promise.resolve(pool),
nodes: loadBalancerClient.findLoadBalancerNodes(pool.id, pool.balancer.id, pool.balancer.dataCenter.id)
.then(function(nodes) {
_.each(nodes, function(node) {
node.pool = pool;
});
return nodes;
})
});
}
/**
* Method allow to create load balancer nodes
*
* @param {object} command
* @param {LoadBalancerPoolCriteria} command.pool - the search criteria
* that specifies one single target load balancer pool
* @param {Array<CreateNodeConfig>} command.nodes - the list with nodes config
*
* @returns {Promise<Array<SharedLoadBalancerNodeMetadata>>} the array of created nodes
*
* @instance
* @function create
* @memberof SharedLoadBalancerNodes
*/
self.create = function(command) {
var result = loadBalancerPools.findSingle(command.pool)
.then(loadAllNodesWithPool)
.then(function(enhancedPool) {
return modifyNodesForPool(
enhancedPool.pool,
_.asArray(enhancedPool.nodes, command.nodes),
command.nodes
);
});
return new NoWaitOperationPromise(queueClient, "Create Load Balancer Node").from(result);
};
function findPools(criteria) {
var poolCriteria = new Criteria(criteria).extractSubCriteria(function (criteria) {
return criteria.pool;
});
return loadBalancerPools.find(poolCriteria)
.then(function(pools) {
return Promise.all(_.map(pools, loadAllNodesWithPool));
});
}
function deletePools(criteria, enhancedPool) {
var pool = enhancedPool.pool;
var nodes = enhancedPool.nodes;
var nodesToDelete = filterNodes(nodes, criteria);
if (nodesToDelete.length === 0) {
return Promise.resolve(nodes);
}
var nodesLeft = _.filter(nodes, function(node) {
return nodesToDelete.indexOf(node) === -1;
});
return modifyNodesForPool(pool, _.asArray(nodesLeft), nodesToDelete);
}
/**
* Method allow to delete load balancer nodes
* @param {LoadBalancerNodeCriteria} arguments - criteria that specify set of pool nodes that will be removed
*
* @returns {Promise<Array<SharedLoadBalancerNodeMetadata>>} the array of deleted nodes
*
* @instance
* @function delete
* @memberof SharedLoadBalancerNodes
*/
self.delete = function () {
var criteria = initCriteria(arguments);
var result = findPools(criteria)
.then(function(enhancedPools) {
return Promise.settle(_.map(enhancedPools, _.partial(deletePools, criteria)));
});
return new NoWaitOperationPromise(queueClient, _.flatten, "Delete Load Balancer Node").fromInspections(result);
};
function modifyNodesForPool(pool, nodesToProcess, nodesToReturn) {
nodesToProcess = _.map(nodesToProcess, function(node) {
return _.omit(node, 'pool');
});
return loadBalancerClient
.modifyLoadBalancerNodes(pool.id, pool.balancer.id, pool.balancer.dataCenter.id, nodesToProcess)
.then(function() {
_.each(nodesToReturn, function(node) {
node.pool = pool;
});
return nodesToReturn;
});
}
function modifyNodes(enhancedPool, criteria, modificationConfig) {
var pool = enhancedPool.pool;
var nodes = enhancedPool.nodes;
var nodesToUpdate = filterNodes(nodes, criteria);
if (nodesToUpdate.length === 0) {
return Promise.resolve(nodes);
}
var nodesWithoutUpdate = _.filter(nodes, function(node) {
return nodesToUpdate.indexOf(node) === -1;
});
_.each(nodesToUpdate, function(node) {
if (modificationConfig.ipAddress) {
node.ipAddress = modificationConfig.ipAddress;
}
if (modificationConfig.privatePort) {
node.privatePort = modificationConfig.privatePort;
}
if (modificationConfig.status) {
node.status = modificationConfig.status;
}
});
return modifyNodesForPool(pool, _.asArray(nodesWithoutUpdate, nodesToUpdate), nodesToUpdate);
}
/**
* Method allow to modify load balancer nodes
*
* @param {LoadBalancerNodeCriteria} nodeCriteria - criteria that specify set of nodes that will be modified
*
* @param {CreateNodeConfig} modificationConfig - update config
*
* @return {Promise<Array<SharedLoadBalancerNodeMetadata>>} - promise that resolved by list of nodes.
* @instance
* @function modify
* @memberof SharedLoadBalancerNodes
*/
self.modify = function (nodeCriteria, modificationConfig) {
var criteria = initCriteria(nodeCriteria);
var result = findPools(criteria)
.then(function(enhancedPools) {
return Promise.settle(
_.map(
enhancedPools,
function(enhancedPool) {
if (modificationConfig instanceof Array) {
return modifyNodesForPool(enhancedPool.pool, modificationConfig, modificationConfig);
}
return modifyNodes(enhancedPool, criteria, modificationConfig);
}
)
);
});
return new NoWaitOperationPromise(queueClient, "Update Load Balancer Node").fromInspections(result);
};
/**
* Method allows to search load balancer nodes.
*
* @param {LoadBalancerNodeCriteria} arguments - criteria that specify set of balancer pools that will be searched
*
* @return {Promise<Array<SharedLoadBalancerNodeMetadata>>} - promise that resolved by list of nodes.
*
* @instance
* @function find
* @memberof SharedLoadBalancerNodes
*/
self.find = function() {
var criteria = initCriteria(arguments);
return findPools(criteria)
.then(_.partial(_.pluck, _, 'nodes'))
.then(_.flatten)
.then(_.partial(filterNodes, _, criteria));
};
function filterNodes(nodes, criteria) {
if (!nodes || nodes.length === 0) {
return [];
}
return _.filter(nodes, new LoadBalancerNodeCriteria(criteria).predicate().fn);
}
function initCriteria() {
return new LoadBalancerNodeCriteria(self._searchCriteriaFrom(arguments)).parseCriteria();
}
init();
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,534 |
\section{A promenade from subfactors to CFT meeting Thompson's group on the way}
From the very beginning the work of Jones\ has been motivated by and connected to mathematical physics.
His theory of subfactors is linked to quantum field theory and in particular to chiral conformal field theory (CFT), which has been formalized in various ways, such as vertex operator algebras or conformal nets.
Those latter mathematical objects give subfactors, and some subfactors provide conformal nets.
It is by trying to find a systematic reconstruction that Jones\ unexpectedly met Richard Thompson's groups $F$ and $T$.
We will tell this story and its repercussions by first presenting subfactors, Thompson's groups, CFT and explaining how they all became linked together.
We will then introduce Jones' technology for constructing actions of Thompson's groups and will mainly focus on unitary representations.
Finally, we will present how this latter framework led to a connection between Thompson's groups and knot theory.
\subsection{Subfactors}
Subfactors are inclusions of von Neumann algebras with trivial centre, which are called factors.
They carry a rich algebraic structure (the \textit{standard invariant}) that can be axiomatized - thanks to a reconstruction theorem due to Popa - and is described, for instance, by Jones' planar algebra \cite{Jones_index_for_subfactors,popa_system_construction_subfactor,Jones_planar_algebra}.
Structures like groups, subgroups and quantum groups can be encoded via subfactors, but also more exotic structures naturally appear in that context.
Note that the Jones polynomial (the celebrated knot/link invariant) was defined using standard invariants of subfactors, creating a long bridge from operator algebras to low dimensional topology \cite{Jones_polynome_vna}.
Planar algebras are algebraic structures for which elements are composed in the plane rather than on a line.
Compositions are encoded by planar diagrams that look like string diagrams used for monoidal categories.
It is a collection of sets $(P_n,n\geq 0)$ (usually some finite dimensional C*-algebras) that is a representation of the planar operad.
Hence, any planar tangle like the following:
$$\begin{tikzpicture}
\draw (-.25,0) circle (.2);
\draw (0,0) circle (.75);
\draw (.25,0) circle (.2);
\draw (-.75,0)--(-.45,0);
\draw (0,.75)--(.25,.2);
\draw (-.05,0)--(.05,0);
\draw (0,-.75)--(.25,-.2);
\end{tikzpicture}$$
defines a map where, informally, one can place inside the inner discs some elements of the $P_n$ (where $n$ must be the number of boundary points) giving a new element of $P_m$ with $m$ the number of boundary points of the outer disc.
Hence, the last diagram gives a map from $P_2\times P_3$ to $P_3.$
Gluing tangles by placing one into an inner disc of another provides an associative composition of maps and one can modify tangles by isotopy without changing the associated map.
\subsection{Thompson's groups}
Richard Thompson defined three groups $F\subset T\subset V$, sometimes called chameleon groups for good reason, where $F$ is the group of piecewise linear homeomorphisms of the unit interval with slopes powers of two and finitely many breakpoints at dyadic rationals \cite{Cannon-Floyd-Parry96}.
Elements of $F$ map one (standard dyadic) partition into another in an order preserving way, being affine on each subinterval.
Larger groups $T,V$ are defined similarly but their elements are allowed to permute subintervals of the associated partitions in a cyclic way, or in any possible way, respectively.
In particular, $T$ still acts by homeomorphisms but on the circle rather than on the interval.
Those groups have been extensively studied as they naturally appeared in various fields of mathematics such as infinite group theory, homotopy and dynamical systems, and follow very unusual behaviour \cite{Brin-Squier85,Brin96-chameleon}.
A famous open problem is to decide whether $F$ is amenable or not, but even more elementary questions are still open, such as whether $F$ is exact or weakly amenable in the sense of Cowling and Haagerup \cite{Cowling-Haagerup-89}.
It is surprising to meet those discrete groups while considering very continuous structures like CFT and subfactors, but we will see that $T$ appears as a discretisation of the conformal group.
Moreover, elements of $T$ can be described by diagrams of trees, suggesting a connection with Jones' planar algebras and thus with subfactors.
\subsection{From CFT to subfactors and back}
For us, a conformal net or a CFT is the collection of field algebras localized on intervals of the circle (spacetime regions), on which the diffeomorphism group acts, and that is subject to various axioms coming from physics \cite{Evans_Kawahigashi_92_sf_cft}.
Representation theory of a conformal net looks like very much the algebraic data of a subfactor and one wants to know how similar they are.
From a conformal net one can reconstruct a subfactor.
However, the converse is fairly mysterious and only specific examples have been worked out, missing the most fascinating ones: the exotic subfactors (subfactors not coming from quantum groups).
It is a fundamental question whether such a reconstruction always exists ("{Does every subfactor have something to do with a CFT?}") and Jones\ has been trying very hard to answer it \cite{Jones-Morrison-Synder14,Bischoff17,Xu18-CFT}.
One of his attempts started as follows \cite{Jones17-Thompson}:
given a subfactor we consider its planar algebra $P=(P_n ,\ n\geq 0).$
The idea is then to interpret the outer boundary of a planar tangle as the spacetime circle of a CFT.
Given any finite subset $X$ of the dyadic rationals of the unit disc we consider $P_X$, a copy of $P_{|X|}$, where all boundary points on the outer disc of planar tangles are in $X$.
This $X$ provides a partition of the unit disc.
We want to be able to refine this partition $X$ into a thinner one $Y$ by adding middle points and to embed $P_X$ inside $P_Y$ (giving us a directed system).
This is done using a fixed element $R\in P_4$ that we think of as a trident-like diagram
$$\begin{tikzpicture}[baseline=-.25cm]
\draw (0,0) circle (.25);
\draw (40 : .25) -- (40 : .75);
\draw (140 : .25) -- (140 : .75);
\draw (0,.25) -- (0,.5);
\draw (0,-.25) -- (0,-.5);
\node at (0,0) {$R$};
\end{tikzpicture}=\begin{tikzpicture}[baseline=.25cm]
\draw (0,0)--(0,1);
\draw (0,1/2)--(-1/3,1);
\draw (0,1/2)--(1/3,1);
\end{tikzpicture} .$$
{Here is one example that explains how we build a map $P_X\to P_Y.$
Consider a finite subset of points $X:=\{0,1/8,1/4,1/2,3/4\}$ of the circle identified with the torus $\mathbf{R}/\mathbf{Z}.$
Placing those points on the disc we obtain a partition with intervals $(0,1/8), (1/8,1/4), \cdots, (3/4,1)$.
Let us refine this partition by splitting the two consecutive intervals $(0,1/8)$ and $(1/8,1/4)$ in two equal halfs.
This refined partition is characterized by the larger subset of points $Y:=X\cup \{1/16,3/16\}$ in which we added the middle points of $(0,1/8)$ and $(1/8,1/4).$
Consider a planar tangle with one inner disc.
Place the points of $X$ on the inner disc and the points of $Y$ on the outer disc.
For common elements of $X$ and $Y$ we draw a straight line from the inner to the outer disc.
In order to connect the two new points of $Y$ we use our trident-like diagram.
We obtain the following tangle:
$$\begin{tikzpicture}[baseline = 0cm]
\draw (0,0) circle (.25);
\draw (0,0) circle (1.2);
\draw (45:.25)--(45:1.2);
\draw (45:.7)--(67.5:1.2);
\draw (45:.7)--(22.5:1.2);
\draw (0:.25)--(0:1.2);
\draw (90:.25)--(90:1.2);
\draw (180:.25)--(180:1.2);
\draw (-90:.25)--(-90:1.2);
\end{tikzpicture}\ .$$
By definition of the planar operad this tangle encodes a map from $P_X$ to $P_Y$ and under a certain condition on $R$ this latter map is injective.}
Continuing this process of refinement of finite partitions we obtain at the limit the dense subset of dyadic rationals of the circle and obtain an obvious notion of support defining localized field algebras exactly like in (physics) lattice theory.
Moreover, we can rotate and perform some local scale transformations but only using those behaving well with dyadic rationals.
\emph{This group of transformation is none other than Thompson's group $T$.}
Moreover, the tree-diagram description of elements of $T$ can be explicitly used to understand this action simply by sending a branching of a tree to a trident $R$ in the planar algebra.
We obtain some kind of discrete CFT with $T$ replacing the diffeomorphism group and field algebras localized on intervals of the circle.
At this point, the hope was to perform a continuum limit and obtain an honest CFT but unfortunately strong discontinuities arise and the CFT goal was out of reach \cite{Jones16-Thompson}; see also \cite{Kliesch-Koenig18}.
The story could have stopped here but in fact this failed attempt opened whole new fields of research in both mathematics and physics.
Indeed, accepting that the continuum limit cannot be done provides physical models relevant at a quantum phase transition with Thompson's group for symmetry \cite{Jones18-Hamiltonian,Osborne-Stiegemann19}.
Moreover, Jones' construction paired with models in quantum loop gravity leads to lattice-gauge theories, again with Thompson's group symmetry \cite{Brot-Stottmeister-M19,Brot-Stottmeister-Phys}.
{The physics described by Jones mathematical model is rather discontinuous and predicts different phenomena than CFT.
Jones suggested the following laboratory experiment which would confront the two theories:
set up a quantum spin chain and observe the correlation number associated to small translations.
Approach a quantum phase transition. According to CFT the correlation number stays close to one but Jones' model with Thompson group for symmetry predicts that this number becomes small.}
On the mathematical side, Jones discovered a beautiful connection between knot theory and Thompson's groups by using the planar algebra of Conway tangles \cite{Jones19-thomp-knot}.
Moreover, he provided a whole new formalism for constructing unitary representations and evaluating matrix coefficients for Thompson's groups that generalizes the planar algebraic construction \cite{Jones16-Thompson}.
\section{Actions and coefficients}
After presenting how Thompson's groups were found in between subfactors and CFT we now present the general theory for constructing groups and actions from categories and functors that we illustrate with Thompson's groups.
Note that this formalism was not developed for the sake of generality but rather to understand better Thompson's group and other related structures.
Jones's research is driven by the study of concrete and fundamental objects in mathematics such as Temperley-Lieb-Jones algebras, Haagerup's subfactor, Thompson's groups, braid groups, etc.
His approach is to use or create whatever formalism is pertinent for better understanding those objects, leading to brand new theories like subfactor theory, planar algebras and today Jones actions for groups of fractions.
We follow Jones' attitude by presenting a general formalism but always accompanied by key examples and applications.
\subsection{Groups of fractions}
The general idea is that a category gives a group and a functor an action.
Our leading example is the category $\mathcal F$ of finite ordered rooted binary forests where the objects are the natural numbers and morphisms $\mathcal F(n,m)$ the set of forests with $n$ roots and $m$ leaves that we consider as diagrams with roots on the bottom and leaves on top.
Composition is obtained by vertical concatenation.
\newcommand{\treeT}{
\begin{tikzpicture}[baseline = .4cm]
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (11/6,2/3)--(2,1);
\end{tikzpicture}
}
\newcommand{\compo}{
\begin{tikzpicture}[baseline = -.2cm, scale = .6]
\draw (1,-2)--(1,-2.5);
\draw (1,-2)--(0,0);
\draw (.5,-1)--(1,0);
\draw (1,-2)--(2,0);
\draw (0,0)--(0,1);
\draw (1,0)--(1,2/3);
\draw (1,2/3)--(2/3,1);
\draw (1,2/3)--(4/3,1);
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\end{tikzpicture}
}
For example, $$\text{ if } f= \ \begin{tikzpicture}[baseline = .4cm]
\draw (0,0)--(0,1);
\draw (1,0)--(1,2/3);
\draw (1,2/3)--(2/3,1);
\draw (1,2/3)--(4/3,1);
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\end{tikzpicture} \text{ and } t= \ \treeT \text{ , then } f\circ t = \ \begin{small}\compo\ .\end{small}$$
It has been observed that an element of Thompson's group $F$ is described by an equivalence class $\dfrac{t}{s}$ of pairs of \textit{trees} $(t,s)$ having the same number of leaves where the class $(t,s)$ is unchanged if we add a common forest on top of each tree \cite{Brown87,Cannon-Floyd-Parry96}.
This comes from the identification between finite binary rooted trees and standard dyadic partitions of the unit interval.
We often described this pair with two trees: $s$ on the bottom and $t$ reversed on top.
For example, if
\newcommand{\treeS}{
\begin{tikzpicture}[baseline = .4cm]
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\end{tikzpicture}
}
\newcommand{\treeST}{
\begin{tikzpicture}[baseline = .4cm]
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\draw (2,2)--(2,2-1/3);
\draw (2,2-1/3)--(5/3,1);
\draw (2,2-1/3)--(7/3,1);
\draw (11/6,2-2/3)--(2,1);
\end{tikzpicture}
}
\begin{equation}\label{eq:fraction}
t=\treeT \text{ and } s = \treeS \ , \text{ then } \frac{t}{s} = \treeST \ .
\end{equation}
The group structure is given by the formula $\frac{t}{s}\cdot \frac{s}{r} = \frac{t}{r}$ and thus $\left(\frac{t}{s}\right)^{-1} = \frac{s}{t}.$
This corresponds to formally inverting trees and considering morphisms from 1 to 1 inside the universal groupoid of the category of forests $\mathcal F$, where the group $F$ is identified with the automorphism group of the object $1$ inside this latter groupoid.
Groups arising in this way are called \textit{groups of fractions}.
Considering trees together with cyclic permutations (affine trees) or all permutations (symmetric trees) we obtain the larger Thompson's groups $T$ and $V$ and if we consider forests with $r$ roots instead of trees we get Higman-Thompson's groups.
Taking braids, we obtain the braid groups, and taking a topological space as a collection of objects with paths (up to homotopy) for morphisms we get the Poincar\'e group.
All of this was observed long ago in a categorical language in \cite{GabrielZisman67} and for the particular example of Thompson's groups \cite{Brown87} that was rediscovered in different terms by Jones.
\subsection{Jones actions}
Jones\ found a machine to produce in a very explicit manner \textit{actions} of groups of fractions.
Given a functor $\Phi:\mathcal F\to\mathcal D$ he constructed an action $\pi:F\curvearrowright X$ that we call a \textit{Jones action}.
Formally, for a covariant functor and a target category with sets for objects, the space $X$ is the set of \textit{fractions} $\frac{t}{x}$ that are classes of pairs $(t,x)$ with $t$ a tree, $x\in\Phi(n)$ with $n$ being the number of leaves of $t$ and where the equivalence relation is generated by $(t,x)\sim (ft,\Phi(f)x)$ for any forest $f$.
The Jones action is then defined as $\pi(\frac{s}{t}) \frac{t}{x} = \frac{s}{x}.$
We sometimes want to complete this space w.r.t.~a given metric and this is what we do if $\mathcal D$ is the category of Hilbert spaces.
Observe that $X$ is defined in the same way as the group of fractions except that now the denominator is in the target category and the equivalence relation is defined using the functor $\Phi.$
Making $\mathcal F$ monoidal by declaring that the tensor product of forests is the horizontal concatenation we obtain that $\mathcal F$ is generated by the single morphism $Y$, i.e.~the tree with two leaves.
Hence, (monoidal) functors $\Phi:\mathcal F\to\mathcal D$ correspond to morphisms $R:=\Phi(Y)\in\Hom_\mathcal D(a, a\otimes a)$ in the target category $\mathcal D$.
In particular, a Hilbert space $\mathfrak H$ and an \textit{isometry} $R:\mathfrak H\to\mathfrak H\otimes\mathfrak H$ provide a unitary representation of Thompson's group that we call a \textit{Jones representation}.
Using string diagrams to represent morphisms in a monoidal category we can interpret a functor $\Phi:\mathcal F\to\mathcal D$ as taking the diagram of a forest and associating the exact same diagram but in the different environment of the target category $\mathcal D$.
This procedure is nothing other than replacing each branching in a forest by an instance of the morphism $R$:
$$\begin{tikzpicture}
\draw (0,0) circle (.25);
\draw (40 : .25) -- (40 : .75);
\draw (140 : .25) -- (140 : .75);
\draw (0,-.25) -- (0,-.5);
\node at (0,0) {$R$};
\end{tikzpicture} \ .$$
Note that for technical reasons we might use morphisms with four boundary points rather than three (if, for instance, one wants to work with subfactor planar algebras or Conway tangles that only have even numbers of boundary points) and thus considering a map of the form:
$$\begin{tikzpicture}
\draw (0,0) circle (.25);
\draw (40 : .25) -- (40 : .75);
\draw (140 : .25) -- (140 : .75);
\draw (0,.25) -- (0,.5);
\draw (0,-.25) -- (0,-.5);
\node at (0,0) {$R$};
\end{tikzpicture}\ .$$
{We give credit to Jones for those actions even if some of the ideas were already around but certainly not the construction with a direct limit that was completely new.}
We are grateful to Matt Brin for a very nice explanation of the state of the art before Jones' work.
``{What was known was that certain automorphism groups contained Thompson's groups.
How they acted was never under investigation and the fact that the actions could be manipulated to get desired properties never even occurred to anyone.}''
\subsubsection{Planar algebraic examples}
Let us compute some coefficients with this technique.
Start with a planar algebra $P$ with a one dimensional 0-box space (diagrams without boundary points corresponds to numbers) and choose an object $R
\begin{tikzpicture}[baseline=.4cm]
\draw (0,0)--(0,1/2);
\draw (0,1/2)--(-1/3,1);
\draw (0,1/2)--(1/3,1);
\end{tikzpicture}$
satisfying
\begin{equation}\label{eq:normal}\begin{small}\begin{tikzpicture}
\draw (2,0)--(2,2);
\node at (1,1) {$=$};
\draw (0,0)--(0,1/2);
\draw (0,1/2)--(-1/3,1);
\draw (0,1/2)--(1/3,1);
\draw (0,2)--(0,3/2);
\draw (-1/3,1)--(0,3/2);
\draw (1/3,1)--(0,3/2);
\end{tikzpicture}\end{small} \ .\end{equation}
Assume that $P$ is equipped with an inner product which consists of connecting two elements of $P_n$ via $n$ strings like:
$$\begin{tikzpicture}[baseline = 0cm]
\draw (0,0) circle (.3);
\draw (1,0) circle (.3);
\node at (0,0) {$A$};
\node at (1,0) {$B^*$};
\draw (.22,-.2)--(.78,-.2);
\draw (.22,.2)--(.78,.2);
\node at (.5,0) {$\cdot$};
\end{tikzpicture} \ .$$
Then $R$ defines a unitary representation and moreover has a favourite vector called the vacuum vector corresponding to a straight line in the planar algebra.
The positive definite function associated to the vacuum vector $\Omega$ is then a closed diagram inside $P$ which is equal to the following if we consider the group element $\frac{t}{s}$ of \eqref{eq:fraction}:
\begin{equation}\label{eq:closed}\begin{tikzpicture}[baseline = .4cm]
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\draw (2,2)--(2,2-1/3);
\draw (2,2-1/3)--(5/3,1);
\draw (2,2-1/3)--(7/3,1);
\draw (11/6,2-2/3)--(2,1);
\draw (2,0) arc (-90:90:1);
\end{tikzpicture}\end{equation}
but viewed inside $P$ where it corresponds to a number (via the \textit{partition function}).
This number can be explicitly computed using \textit{skein relations} (diagrammatic rules for reducing diagrams, such as \eqref{eq:normal}, analogous to relations for a presented group) of the planar algebra chosen.
Jones\ called them \textit{wysiwyg representations} (``{what you see is what you get}'') \cite{Jones19Irred}.
Planar algebras have been extensively studied in the past two decades and we know today many interesting examples with fully understood skein relations providing candidates for wysiwyg representations.
Using a certain class of planar algebras, (trivalent categories studied by Peters, Morrison and Snyder \cite{MPS17-trivalent}), Jones\ constructed an uncountable family of \textit{mutually inequivalent} wysiwyg unitary representations that are all \textit{irreducible}.
Those latter examples emanate from the planar algebraic approximation of CFT and keep some geometric flavour.
Next we present examples that somehow forget the geometric structure of planar algebras but can be defined in a very elementary way.
\subsubsection{Analytic examples}
Let us consider the whole category of Hilbert spaces $\Hilb$ with \textit{isometries} for morphisms.
There are various monoidal structures $\odot$ we can equip $\Hilb$ with such as the classical tensor product or the direct sum.
{Free products can also be done but one has to consider a slightly different category where objects are pointed Hilbert spaces $(H,\xi)$ where $\xi$ is a unit vector and morphisms are isometries sending the chosen unit vector to the other one, see \cite{Voiculescu_dykema_nica_Free_random_variables}.}
Each case provides Jones representations by taking an isometry $R:\mathfrak H\to\mathfrak H\odot\mathfrak H$ for the chosen monoidal structure $\odot.$
The second case can be written as $R=A\oplus B$ with $A,B:\mathfrak H\to\mathfrak H$ satisfying the \textit{Pythagorean} identity:
\begin{equation}\label{eq:Pyth}A^*A+B^*B=1.\end{equation}
We call the \textit{Pythagorean algebra} the universal C*-algebra generated by this relation and observe that a representation of this latter algebra provides a Jones representation of Thompson's groups \cite{Brot-Jones18-2}.
Moreover, it has interesting quotient algebras such as the Cuntz algebra, noncommutative tori, and Connes-Landi spheres for instance \cite{Cuntz77,Connes80-Calg,Rieffel81-NCT,Connes-Landi01}.
Taking any unit vector $\xi\in \mathfrak H$ we obtain a positive definite function as matrix coefficient.
It can be computed as follows.
Consider an element of $F$ written $\frac{t}{s}$.
Place $\xi$ at the root of $s$ and make it go to the top by applying $A\oplus B$ at each branching. We obtain on top of each leaf a word in $A,B$ applied to $\xi$.
We do the same thing for $t$ and then take the sum of the inner product at each leaf.
For example, if $\frac{t}{s}$ is the example of \eqref{eq:fraction} we obtain the inner product:
$$\langle A\xi \oplus AB\xi \oplus BB\xi , AA\xi\oplus BA \xi \oplus B\xi\rangle$$
and the procedure in making $\xi$ going from the bottom to the top of the tree $s$ is described by the following diagram:
$$\begin{tikzpicture}[baseline = .4cm]
\draw (0,0)--(0,2/3);
\draw (0,2/3)--(-2/3,2);
\draw (0,2/3)--(2/3,2);
\draw (2/6,4/3)--(0,2);
\node at (0,-.3) {$\xi$};
\node at (1,1) {$\leadsto$};
\draw (2,0)--(2,2/3);
\draw (2,2/3)--(4/3,2);
\draw (2,2/3)--(8/3,2);
\draw (7/3,4/3)--(2,2);
\node at (4/3,2+.2) {$A\xi$};
\node at (7/3+.3,4/3) {$B\xi$};
\node at (3.5,1) {$\leadsto$};
\draw (5,0)--(5,2/3);
\draw (5,2/3)--(13/3,2);
\draw (5,2/3)--(17/3,2);
\draw (16/3,4/3)--(5,2);
\node at (13/3-.3,2+.2) {$A\xi$};
\node at (15/3,2+.3) {$AB\xi$};
\node at (18/3,2+.2) {$BB\xi$};
\end{tikzpicture}$$
{The formula of this coefficient for elements of the larger group $T$ is similar up to permuting cyclically the order of the vectors in the direct sum and can be extended to $V$ by considering \textit{any} permutations.}
Many interesting representations and coefficients of Thompson's groups can be created in that way.
If $A=B$ are real numbers equal to $1/\sqrt 2$, then we recover the Koopman representation $T\curvearrowright L^2(\mathbb S^1)$ induced by the usual action of $T$ on the circle.
{In more details: fix a tree $s$ and a complex number $\xi$. Following the procedure explained by the diagram above we obtain that each leaf $\ell$ of $s$ is decorated by $2^{-d^\ell_s/2} \xi$ where $d_s^\ell$ is the distance from the leaf to the root. This latter number corresponds to $\xi$ times the square root of the length of the interval $I_s^\ell$ associated to the leaf.
Taking a second tree $t$ such that $g=\frac{t}{s}\in F$ we obtain that the contribution of the inner product associated to $\xi=1$ and $g$ at a leaf $\ell$ is $2^{(d^\ell_s-d^\ell_t)/2}$ that is the square root of the slope of $g$ when restricted to $I_s^\ell$.
From this observation it is not hard to conclude.}
Then, thanks to the flexibility of Jones' formalism, we can easily deform this representation by replacing $1/\sqrt 2$ by two different real or complex numbers $v$ and $w$ with $|v|^2+|w|^2=1$ obtaining various paths between the Koopman and the trivial representations {where the former appears when $v$ or $w$ is equal to zero.}
Using the free group we obtain the map $g\in F\mapsto Measure(x\in (0,1) :\ gx=x)$ as a diagonal matrix coefficient and it is then positive definite.
Other examples arise by taking representations of quotients of the Pythagorean algebras, providing interesting family of representations.
One can also uses this approach for constructing representations of such quotient algebras:
with the help of Anna-Marie Bohman and Ruy Exel, Jones\ and I could relate precisely representations of the Cuntz and the Pythagorean algebras, obtaining new methods for practical constructions of representations of the former.
If we choose the monoidal structure to be the classical tensor product of Hilbert spaces, then any isometry $R:\mathfrak H\to\mathfrak H\otimes\mathfrak H$ provides a unitary representation of $V$.
Matrix coefficients associated to $\xi,\eta\in\mathfrak H$, $\langle \pi(\frac{t}{s}) \xi,\eta\rangle$, can be computed as above but where we need to perform an inner product of two vectors in a tensor power of $\mathfrak H$ instead of a direct sum where each tensor power factor corresponds to a leaf of the tree $t$.
Interesting and manageable examples arise when $R\xi$ is a finite sum of elementary tensors and thus matrix coefficients are then computed in an algorithmic way.
Here is one story concerning those representations and how they can be manipulated and used.
During February 2018 Jones\ and I met one week in the beautiful coastal town of Raglan in New Zealand to finish up the paper on Pythagorean representations and to enjoy the kitesurf spot a bit.
{During this stay Jones told me that the absence of Kazhdan Property (T) for $F,T,V$ could be trivially proved via his recent formalism.
Indeed, this can be done using maps like $R\xi=u\xi\otimes\zeta$ where $\zeta$ is a fixed unit vector and $u$ an isometry.
For example, this map and the pair of trees $\frac{t}{s}$ of \eqref{eq:fraction} gives the following matrix coefficient:
$$\langle \pi(\frac{t}{s}) \zeta , \zeta \rangle = \langle u\zeta\otimes u\zeta\otimes \zeta , u^2\zeta\otimes \zeta\otimes \zeta\rangle = |\langle \zeta,u\zeta\rangle|^2.$$
By considering a family of those pairs $(u,\zeta)$ and making $\langle u\zeta,\zeta\rangle$ tend to one we obtain an almost invariant vector but no invariant one in the associated Jones representation.}
{Moreover, he showed me how to create the left regular representation of $F$ via a tensor product construction where $\mathfrak H=\ell^2(\mathbb N)$ and $R\delta_n=\delta_{n+1}\otimes\delta_{n+1}=\delta_{n+1,n+1}$.
Indeed, if $t$ is a tree, then using the functor $\Phi$ we get $\Phi(t)\delta_0 = \delta_{w_t}$ where $w_t$ is the list of distances between each leaf of $t$ to its root. Since this characterizes the tree $t$ we obtain that the cyclic component of the Jones representation associated to the vector $\delta_0$ is the left regular representation of $F$.}
{Those two facts made me very excited.
Showing that Thompson's groups are not Kazhdan groups is a difficult result that stayed open for quite some time.
Jones' proof being so effortless gave hope to obtain stronger results with more elaborated techniques.
The regular representation has coefficients vanishing at infinity and thus one might be able to construct other of those kind.
In this purpose we started to think about deforming the isometry $R\delta_n=\delta_{n+1,n+1}$ obtaining paths between the trivial and the left regular representations and new coefficients.
}
Going back home during the very long journey from Raglan to Rome I only thought about those deformations.
When I landed I had more or less a full proof showing that $T$ has the Haagerup property improving the absence of Kazhdan property but only for the intermediate Thompson's group $T$.
I wrote to Jones\ about it and we decided to write a short paper giving the two proofs: $F,T,V$ are not Kazhdan groups and $T$ has the Haagerup property \cite{Brot-Jones18-1}.
Even though those results are not optimal and already known (Farley showed that $V$ has the Haagerup property \cite{Farley03}) they display the power of Jones' new techniques.
{One intriguing fact is that the maps constructed by Farley (the one associated to his cocycle) coincide with ours on Thompson's group $T$ but differ on the larger group $V$ and it is still unclear how to build them using Jones representations. Another interesting problem would be to construct Farley actions on CAT(0) cubical complexes via Jones actions using the appropriate target category \cite{Farley03bis}.}
A year later, new results were proved regarding analytical properties of groups.
Choose a group $\Gamma$ and a single group morphism $g\in\Gamma\mapsto (a_g,b_g)\in\Gamma\oplus\Gamma.$
This provides a (monoidal) functor from the category of forests to the category of groups and thus a Jones action of $V$ on a limit group. One can then consider the semidirect product.
Choosing the trivial embedding $g\in\Gamma\mapsto (g,e)$ we obtain the (permutational and restricted) wreath product $\oplus_{\mathbf{Q}_2} \Gamma\rtimes V$ where $V$ shifts the indices via the usual action $V\curvearrowright \mathbf{Q}_2$ where $\mathbf{Q}_2$ is the set of dyadic rationals on the unit circle.
Now comes a trivial but \textit{key observation}: this new group can be written as a group of fractions where the new category is basically made of forests but with leaves labelled by elements of $\Gamma.$
Compositions of forests with group elements in this latter category, constructed with the map $g\mapsto (a_g,b_g)$, is expressed by the equality:
\newcommand{\treelaw}{\begin{tikzpicture}[baseline = .4cm]
\draw (0,0)--(0,1/2);
\draw (0,1/2)--(-1/3,1);
\draw (0,1/2)--(1/3,1);
\node at (0,-.2) {$g$};
\node at (3/4,1/2) {$=$};
\draw (3/2,0)--(3/2,1/2);
\draw (3/2,1/2)--(3/2-1/3,1);
\draw (3/2,1/2)--(3/2+1/3,1);
\node at (3/2-1/3,1+.2) {$a_g$};
\node at (3/2+1/3,1+.2) {$b_g$};
\end{tikzpicture}}
$$\treelaw$$
Note that a similar construction was observed by Brin using Zappa-Sz\'ep products that he used to define the braided Thompson's group \cite{Brin07-BraidedThompson}; see also \cite{Dehornoy06}.
Since it is a group of fractions we can then apply Jones' technology for constructing representations and coefficients of this larger group.
Using this strategy I was able to show that those wreath products have the Haagerup property when $\Gamma$ has it, which was out of reach by other known approaches \cite{Brothier19WP}.
\subsection{Connection with knot theory}
Knot theory and Thompson's groups are connected using the technology presented above.
This has been very well explained in a recent expository article of Jones so I will be brief \cite{Jones19-thomp-knot}.
The connection comes from the idea to consider functors from forests to the category of Conway tangles that are roughly speaking
strings inside a box possibly attached to top and or bottom that can cross like:
$$\begin{tikzpicture}[baseline = .4cm]
\draw (0,0)--(0,1.45);
\draw (0,1.55)--(0,2);
\draw (-1,0)--(-.05,.95);
\draw (.05,1.05)--(1,2);
\draw (-.5,2) arc (180:360:.5);
\draw (-1.2,0)--(1.2,0)--(1.2,2)--(-1.2,2)--(-1.2,0);
\end{tikzpicture} \ .$$
We ask that those diagrams be invariant under isotopy and the three Reidemeister moves.
We define a functor from (binary) forests to Conway tangles by replacing each branching by one crossing as follows:
$$\begin{tikzpicture}[baseline = .4cm]
\draw (0,0)--(0,1/2);
\draw (0,1/2)--(-1/3,1);
\draw (0,1/2)--(1/3,1);
\node at (.75,.5) {$\mapsto$};
\draw (1.5,0)--(1.5,.6);
\draw (1.5,.7)--(1.5,1);
\draw (1.15,1) arc (180:360:.35);
\end{tikzpicture} \ . $$
Given an element of $g=\frac{t}{s}\in F$ with $t,s$ trees that we put $s$ on the bottom, $t$ upside down on top and connect their roots.
We then apply our transformation that replace each branching by a crossing obtaining a link.
The procedure is the following for the example of \eqref{eq:fraction}:
\newcommand{\trees}{
\begin{tikzpicture}[baseline = .8cm]
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\draw (2,2)--(2,2-1/3);
\draw (2,2-1/3)--(5/3,1);
\draw (2,2-1/3)--(7/3,1);
\draw (11/6,2-2/3)--(2,1);
\end{tikzpicture}}
\newcommand{\closetrees}{
\begin{tikzpicture}[baseline = .8cm]
\draw (2,0)--(2,1/3);
\draw (2,1/3)--(5/3,1);
\draw (2,1/3)--(7/3,1);
\draw (13/6,2/3)--(2,1);
\draw (2,2)--(2,2-1/3);
\draw (2,2-1/3)--(5/3,1);
\draw (2,2-1/3)--(7/3,1);
\draw (11/6,2-2/3)--(2,1);
\draw (2,0) arc (-90:90:1);
\end{tikzpicture}}
\newcommand{\link}{
\begin{tikzpicture}[baseline = .8cm]
\draw (2,0)--(2,1/3-.05);
\draw (2,1/3)--(5/3,1);
\draw (13/6,2/3)--(7/3,1);
\draw (2,1/3)--(13/6-.05,2/3-.05);
\draw (13/6,2/3)--(2,1);
\draw (2,2)--(2,2-1/3);
\draw (2,2-1/3)--(11/6+.05,2-2/3+.05);
\draw (11/6,2-2/3)--(5/3,1);
\draw (2,2-1/3)--(7/3,1);
\draw (11/6,2-2/3)--(2,1);
\draw (2,1/3+.1)--(11/6,2-2/3-.1 );
\draw (2+1/6,2/3+.1)--(2,2-1/3-.1);
\draw (2,0) arc (-90:90:1);
\end{tikzpicture}}
$$g = \left( \ \treeT \ , \treeS \ \right) \ \leadsto \trees\ \leadsto \ \closetrees \ \leadsto \ \link \ .$$
Smoothing out this diagram we obtain the following knot:
\begin{figure}[H]
\includegraphics[width=18mm]{smoothknot.jpg} \ .
\end{figure}
Jones\ proved the theorem that every link can be obtained in that way, concluding that ``{Thompson's group is as good as the braid groups for producing links}''.
This opened a completely new land of study in which Jones\ already suggested nine explicit research problems which explore the group structure analogy between Thompson's and braid groups \cite[Section 7]{Jones19-thomp-knot}.
Examples of problems are: finding a Markov theorem for Thompson (what are the relations between two Thompson's group elements that give the same link) or deciding the \textit{Jones-Thompson's index} of a link: what is the minimal number of leaves necessary for a couple of trees to form this link.
This latter index for links can be defined for $k$-ary trees $k\geq 2$ instead of binary trees giving a discrete parameter family of invariants for links.
The connection with links provided a new point of view on Thompson's group elements.
One can then ask whether the link associated to an element of $F$ is orientable or not.
It turns out that the set of all $g\in F$ giving an orientable link forms a subgroup $\Vec F\subset F$ known today as the \textit{Jones subgroup}.
This subgroup can be defined in a number of ways using diagram groups, skein theory, stabilizers and is even equal to a group of fractions \cite{Jones17-Thompson,Golan-Sapir17,Ren18-Thomp,Aiello-Conti-Jones18}.
Golan and Sapir were able to prove the striking result that $\Vec F$ is isomorphic to Thompson's group $F_3$ associated to $3$-adic numbers and moreover is equal to its commensurator implying that the associated quasi-regular representation is irreducible \cite{Golan-Sapir17}.
The different definitions of $\Vec F$ suggested various natural generalizations of it:
a circular Jones subgroup $\Vec T\subset T$; from the diagram group approach Golan and Sapir obtained an increasing chain of subgroups $\Vec F_n\subset F,n\geq 2$; and the stabilizer definition provided an uncountable family $F_{(I)}\subset F$ parametrized by the famous Jones' range of indices for subfactors $\{4\cos(\pi/n)^2:\ n\geq 4\}\cup [4,\infty)$ \cite{Jones_index_for_subfactors}.
Using skein theory of planar algebras Ren interpreted differently $\Vec F$ reproving that it is isomorphic to $F_3$ and interestingly, Nikkel and him showed that $\Vec T$ is \textit{not} isomorphic $T_3$ but is similar from a diagram group perspective \cite{Ren18-Thomp,Nikkel-Ren18}.
Golan and Sapir proved similar properties for the subgroups $\Vec F_n\subset F$ than for $\Vec F\subset F$ obtaining an infinite family of irreducible representations and isomorphisms $\Vec F_n \simeq F_{n+1}$ (Thompson's group associated to $(n+1)$-adic numbers) \cite{Golan-Sapir17}.
The uncountable family of $F_{(I)}$ is less exciting as it generically provides trivial subgroups, except of course for $\Vec F$ which corresponds to the first nontrivial Jones' index $2$ \cite{ABC19}.
Jones subgroup $\Vec F$ and the study around it summarize well the interplay of various fields in this brand new framework of Jones and how ideas, say from knot theory or skein theory, can be then applied in group theory and vice versa.
\section{Conclusion}
The recent technology of Jones regarding Thompson's groups has provided new perspectives and connections for and between groups of fractions, knot theory, subfactor theory and quantum field theory.
This complements previous beautiful connections that Jones made more than 35 years ago with his celebrated polynomial.
This is only the very beginning of this development and various exciting research directions remain untouched.
There have been already beautiful applications and promising techniques developed which augur a bright future.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,604 |
\section{Introduction}
Photoassociation (PA) is a process in which two colliding atoms form an excited molecule by absorbing a photon. PA spectroscopy provides a versatile tool for probing the physics of rovibrational molecular states~\cite{Jones_RMP_2006} and precisely determining the collisional properties of atoms, such as scattering length and interatomic potential coefficients~\cite{Kitagawa_Twocolor_2008}. Furthermore, the PA process can be actively used to control the strength of atomic interaction via coupling to an excited molecular state, a phenomenon called optical Feshbach resonance (OFR)~\cite{Fatemi_OFR_2000, Theis_OFR_2004, Ciurylo_OFR_2005, Enomoto_OFR_2008,Yamazaki2010,Yan2013}, and to measure pair correlations in strongly correlated atomic gas systems~\cite{Partridge_BCS_2005,Sugawa_dualMott_2011}.
Recently, there has been broad interest in studies on the PA physics of two-valence-electron atoms such as Yb~\cite{Takasu_PA1P1_2004,Tojo_PA_2006,Enomoto_OFR_2008,Enomoto_C6_2007, Kitagawa_Twocolor_2008,Borkowski_Lineshape_2009,Enomoto_PLR_2008,Borkowski_Hetero_2011, Roy_YbLi_2016}, Sr~\cite{Nagel_PASr_2005, Zelevinsky_Narrow_2006, Borkowski_Mass_2014, Stellmer_Sr2_2012, Reinaudi_Sr2_2012,Nicholson2015,Yan2013}, and Ca~\cite{Kahmann_Zeeman_2014,Tiemann_Zeeman_2015}. These atoms have narrow ${^1}S$--${^3}P$ intercombination transition, which is beneficial for the precise determination of PA resonances and enables the implementation of OFR without significant atom loss~\cite{Ciurylo_OFR_2005,Enomoto_OFR_2008,Yamazaki2010,Yan2013}. In particular, Yb atoms have rich, stable isotopes, including five spinless bosons ($^{168}$Yb, $^{170}$Yb, $^{172}$Yb, $^{174}$Yb, and $^{176}$Yb) and two fermions ($^{171}$Yb, with a nuclear spin of $i=1/2$ and $^{173}$Yb, with $i=5/2$), providing an interesting opportunity to study the mass scaling of PA physics~\cite{Kitagawa_Twocolor_2008,Borkowski_Lineshape_2009}. To date, many PA spectra of Yb atoms have been reported for bosonic isotopes~\cite{Tojo_PA_2006, Enomoto_C6_2007, Kitagawa_Twocolor_2008,Borkowski_Lineshape_2009}, fermionic $^{171}$Yb~\cite{Enomoto_PLR_2008}, and isotopic mixtures~\cite{Borkowski_Hetero_2011, Roy_YbLi_2016}. However, the complete PA spectrum of fermionic $^{173}$Yb, with its high nuclear spin, is still unknown, although a couple of PA resonances have been reported~\cite{Sugawa_dualMott_2011,Taie_Pomeranchuk_2012,Kitagawa_Twocolor_2008}. In addition, the $^{173}$Yb Fermi gas system has been discussed as a candidate platform for studies of exotic $SU(\mathcal{N}>2)$ quantum magnetism~\cite{Gorshkov_SUN_2010}. Information on the PA spectrum of $^{173}$Yb near the intercombination line is highly desirable for such a quantum simulation application~\cite{Reichenbach_OFR_2009}.
In this paper, we report the PA spectrum of a degenerate Fermi gas of $^{173}$Yb atoms near the dissociation limit of the $\ket{{^1}S_0,f=5/2}\rightarrow \ket{{^3}P_1,f'=7/2}$ intercombination transition. We measured an atom-loss spectrum as a function of the frequency of the PA light and we observed eighty PA resonances in the spectrum on the red-detuned side of the atomic resonance down to $-1$~GHz. The high density of the spectral lines can be attributed to the high nuclear spin number of $^{173}$Yb, which we confirmed by performing a multi-channel calculation of the molecular energy levels based on known spectroscopic results. To collect further spectroscopic information on the excited molecular states, we investigated the Zeeman effect in the spectrum near the frequency detuning of $-0.8$~GHz. By employing various two-component spin mixture samples, we determined the quantum numbers of the Zeeman sublevels and estimated the $g$ factor of the molecular state corresponding to the PA line at $-796$~MHz detuning. Finally, we measured the two-body loss rates under PA light for several pronounced PA resonances. Our measurement results provide a starting point for studies of the PA physics of fermionic $^{173}$Yb atoms, although further theoretical efforts will be required to interpret the measured spectra.
\section{Experiment}
\begin{center}
\begin{figure*}
\includegraphics[width=17.5cm]{Fig1_spectrum}
\caption{ Photoassociation (PA) spectrum of unpolarized $^{173}$Yb atoms at low temperature. The various colors represent different PA laser beam conditions with intentisy $I_\mathrm{PA}$ and pulse duration $\tau$: blue ($I_\mathrm{PA}=$0.74~W/cm$^2$, $\tau=100$~ms), black (0.37~W/cm$^2$, $100$~ms), red (74~mW/cm$^2$, $100$~ms), green (74~mW/cm$^2$, $50$~ms), orange (74~mW/cm$^2$, $30$~ms), and purple (37~mW/cm$^2$, $30$~ms). The barcode lines at the bottom of the plot show the numerical calculation results obtained for the adiabatic molecular potentials with $T = 1$(bottom) to $5$(top) (Fig.~2).\label{fig:spectrum}}
\end{figure*}
\end{center}
We prepared an ultracold gas sample of $^{173}$Yb atoms as described in Ref.~\cite{Lee_SOC_2017}. The atoms were first collected in a magneto-optical trap using the 556~nm $^1S_0$--${^3}P_1$ intercombination transition and transferred into a 1070 nm optical dipole trap (ODT). Then, the atoms were transported to an auxiliary vacuum chamber by moving the focus of the trapping laser beam and were loaded into a crossed ODT formed by superposing an additional 532 nm trapping laser beam. The atomic sample was evaporatively cooled by lowering the trap depth, and after cooling, it was held for an additional 0.3~s to ensure equilibrium. The final sample was an equal mixture of all six spin components of the ${^1}S_0$ ground state, containing approximately $3.1\times 10^5$ atoms. The sample temperature was measured to be $T \approx 130$~nK. The \textit{in situ} density distribution of the trapped sample was found to be well fit by a Gaussian profile with a $1/e^2$ radius of $\{ \sigma_x, \sigma_y, \sigma_z\}\approx \{12.0, 7.5, 3.8\}~\mu$m, and the central density was estimated to be $n_0\approx 1.6\times 10^{14}$~cm$^{-3}$, corresponding to a Fermi energy of $E_F = \hbar^2 (\pi^2 n_0)^{2/3} /(2m) \approx k_B\times 190$~nK, where $\hbar$ is the Planck constant $h$ divided by $2\pi$, $m$ is the atomic mass, and $k_B$ is the Boltzmann constant. To measure the sample condition, we take absorption image using the ${^1}S_0$--${^1}P_1$ transition.
PA resonances were detected via atom loss by illuminating the trapped sample with a pulsed PA laser beam. The linewidth of our PA laser was $<70$~kHz, which is sufficiently narrow to probe excited molecular states with a natural linewidth of $\Gamma_\mathrm{nat}/2\pi \approx (2\Gamma_\mathrm{a})/2\pi = 364$~kHz, where $\Gamma_a$ is the atomic linewidth of the $^1S_0$--$^3P_1$ transition~\cite{Ciurylo_OFR_2005}. The PA laser beam was $\sigma^-$--polarized and focused onto the sample with a Gaussian beam waist of $\approx 114~\mu$m, which was large to uniformly irradiate the entire sample. We obtained a PA spectrum by measuring the remaining atom number fraction $\eta_a$ as a function of the frequency $\nu$ of the PA laser beam. For each $\nu$, we determined $\eta_a$ by measuring the numbers of atoms with and without application of the PA laser beam, respectively.
\section{Results}
\subsection{PA spectrum}
Figure~\ref{fig:spectrum} shows the PA spectrum measured for $\delta \nu=\nu-\nu_0= -1 \sim 0$~GHz, where $\nu_0$ is the resonance frequency for the $\ket{^1S_0,f=5/2}\rightarrow \ket{{^3}P_1,f^\prime=7/2}$ atomic transition. In the measurement, we reduced the PA beam intensity $I_\mathrm{PA}$ and the pulse duration $\tau$ in a piecewise manner as we approach the atomic resonance to avoid power broadening and photon scattering loss effects, where $I_\mathrm{PA}=0.037-0.74$~W/cm$^2$ and $\tau=30-100$~ms. The saturation intensity for the atomic transition is $I_\mathrm{sat}=0.14$~mW/cm$^2$. The spectrum shows a high density of spectral lines, and we identified eighty PA resonances in the range of $-1~\mathrm{GHz}<\delta \nu <-38$~MHz. For $\delta \nu >-38$~MHz, it was difficult to unambiguously identify PA resonances because of high photon scattering loss near the atomic resonance. The positions $\nu_b$ and linewidths $\Gamma_b$ of the spectral lines were determined from Lorentzian line fits to the measured data and are listed in Table~\ref{tab:spectrum}. In our experiment, the ac Stark shift due to the dipole trapping beams was $< 100$~kHz and insignificant, and the thermal broadening was negligible for $k_\mathrm{B} T/h \approx$ 4~kHz.
\newcolumntype{M}[1]{>{\centering\arraybackslash}m{#1}}
\begin{centering}
\begin{table}[h]
\centering
\caption{Measured PA resonances $\nu_\mathrm{b}$ and the corresponding linewidths $\Gamma_\mathrm{b}$ from Lorentzian fits to the individual resonances depicted in Fig.~\ref{fig:spectrum}~\cite{Ciurylo_PACa_2004}. The errors represent the 95\% confidence intervals from the fits. For some resonances, no linewidth is given due to insufficient data points. Note that the PA laser beam intensity varies over the frequency detuning (see the caption of Fig.~1).}
\label{tab:spectrum}
\begin{tabular}{M{1.95cm}M{1.95cm}}
\addlinespace
\hline\hline
$\nu_\mathrm{b}$ & $\Gamma_\mathrm{b}/2\pi$ \\
(MHz) & (MHz) \\
\hline
$-997.8\pm0.1$ & $1.3\pm0.2$ \\
$-993.1\pm0.3$ & $0.6\pm0.4$ \\
$-987.4\pm0.2$ & $1.6\pm0.6$ \\
$-955.0\pm0.2$ & $1.5\pm0.5$ \\
$-945.9\pm0.1$ & $2.1\pm0.4$ \\
$-940.0\pm0.1$ & $1.2\pm0.3$ \\
$-925.4\pm0.1$ & $0.5\pm0.4$ \\
$-910.4\pm0.2$ & $2.8\pm0.5$ \\
$-895.3\pm0.1$ & $1.3\pm0.3$ \\
$-887.6\pm0.1$ & $0.5\pm0.2$ \\
$-870.2\pm0.2$ & $1.6\pm0.5$ \\
$-865.3\pm0.2$ & $0.9\pm0.4$ \\
$-859.3\pm0.2$ & $0.4\pm0.3$ \\
$-832.9\pm0.1$ & $0.8\pm0.3$ \\
$-824.5\pm0.1$ & $2.1\pm0.3$ \\
$-812.5\pm0.1$ & $1.5\pm0.4$ \\
$-796.2\pm0.1$ & $1.4\pm0.2$ \\
$-791.3\pm0.1$ & $1.3\pm0.2$ \\
$-776.1\pm0.3$ & $0.6\pm0.4$ \\
$-771.0\pm1.2$ & -- \\
$-763.3\pm0.2$ & $0.6\pm0.3$ \\
$-750.1\pm0.1$ & -- \\
$-748.3\pm0.1$ & -- \\
$-744.9\pm0.1$ & $0.5\pm0.3$ \\
$-738.9\pm0.1$ & $1.1\pm0.2$ \\
$-734.4\pm0.1$ & $0.7\pm0.3$ \\
$-732.8\pm0.1$ & $0.9\pm0.2$ \\
$-730.7\pm0.1$ & $0.5\pm0.2$ \\
$-724.1\pm0.7$ & -- \\
$-705.4\pm0.1$ & $0.8\pm0.2$ \\
$-686.5\pm0.1$ & $1.0\pm0.3$ \\
$-668.1\pm0.1$ & $1.0\pm0.2$ \\
$-659.9\pm0.1$ & $1.1\pm0.3$ \\
$-647.6\pm0.1$ & $0.5\pm0.3$ \\
$-640.5\pm0.1$ & $0.5\pm0.3$ \\
$-633.1\pm0.1$ & $1.0\pm0.3$ \\
$-622.9\pm0.1$ & $0.6\pm0.2$ \\
$-599.1\pm0.1$ & $1.4\pm0.3$ \\
$-588.5\pm0.1$ & $0.7\pm0.3$ \\
$-557.7\pm0.3$ & -- \\
\hline\hline
\end{tabular}
\hfill
\begin{tabular}{M{1.95cm}M{1.95cm}}
\addlinespace
\hline\hline
$\nu_\mathrm{b}$ & $\Gamma_\mathrm{b}/2\pi$ \\
(MHz) & (MHz) \\
\hline
$-529.8\pm0.1$ & $0.6\pm0.5$ \\
$-523.9\pm0.1$ & -- \\
$-520.3\pm0.1$ & $1.5\pm0.4$ \\
$-513.3\pm0.1$ & $1.1\pm0.3$ \\
$-496.3\pm0.2$ & -- \\
$-492.8\pm0.1$ & $1.3\pm0.4$ \\
$-483.3\pm0.1$ & $1.1\pm0.3$ \\
$-478.5\pm0.1$ & $0.9\pm0.3$ \\
$-452.0\pm0.2$ & $0.6\pm0.3$ \\
$-438.6\pm0.1$ & $1.2\pm0.5$ \\
$-432.1\pm0.1$ & $0.8\pm0.5$ \\
$-423.6\pm0.1$ & -- \\
$-405.7\pm0.1$ & $1.1\pm0.6$ \\
$-374.9\pm0.4$ & -- \\
$-363.1\pm0.1$ & -- \\
$-338.5\pm0.2$ & $1.3\pm0.5$ \\
$-316.6\pm0.4$ & $1.1\pm0.3$ \\
$-299.4\pm0.2$ & -- \\
$-295.1\pm0.1$ & $1.3\pm0.4$ \\
$-278.2\pm0.2$ & -- \\
$-270.2\pm0.4$ & $2.3\pm1.3$ \\
$-259.5\pm0.5$ & -- \\
$-251.7\pm0.5$ & -- \\
$-238.9\pm0.1$ & $0.9\pm0.7$ \\
$-226.4\pm0.5$ & -- \\
$-220.6\pm0.2$ & -- \\
$-216.1\pm0.1$ & $0.7\pm0.3$ \\
$-207.8\pm0.3$ & $1.9\pm0.9$ \\
$-197.4\pm0.7$ & $2.6\pm2.2$ \\
$-193.7\pm1.1$ & -- \\
$-190.2\pm0.3$ & $1.1\pm0.9$ \\
$-184.5\pm0.3$ & $1.9\pm0.9$ \\
$-180.0\pm0.1$ & $1.5\pm0.4$ \\
$-168.0\pm0.2$ & $1.4\pm0.9$ \\
$-137.3\pm0.2$ & $1.5\pm0.8$ \\
$-119.1\pm0.5$ & -- \\
$-80.7\pm0.2$ & $1.0\pm0.7$ \\
$-57.1\pm0.5$ & -- \\
$-49.5\pm0.2$ & $1.2\pm0.9$ \\
$-38.1\pm0.2$ & $0.9\pm0.7$ \\
\hline\hline
\end{tabular}
\end{table}
\end{centering}
\subsection{High spectral density}
To understand the observed high density of the spectral lines, we calculate the bound state energy levels for two $^{173}$Yb atoms in the $^3P_1$+$^1S_0$ channel, following the methods presented in Refs.~\cite{Zelevinsky_Narrow_2006,Reichenbach_OFR_2009}. The Hamiltonian for the two atoms is
\begin{equation} \label{eq:adiabeticpot}
\hat{H}= \frac{p_r^2}{2\mu}+\frac{\hbar^2}{2\mu r^2} R(R+1) + V_\textrm{BO}(r) + \hat{H}_\mathrm{hf},
\end{equation}
where the first and second terms represent the radial and angular kinetic energies, respectively of the nuclei of the two atoms, $V_\textrm{BO}(r)$ is the electronic Born-Oppenheimer (BO) potential, and $\hat{H}_\mathrm{hf}$ is the hyperfine interaction term. Here, $\mu$ and $r$ denote the reduced mass and radial separation of the two nuclei, respectively, and $R$ is the quantum number for the overall rotation of the atom dimer. The BO potential is given by
\begin{equation} \label{eq:VBO}
V_\textrm{BO}(r)= -\frac{C_6}{r^6}\left(1-\frac{\sigma^6}{r^6}\right)-s\frac{C^{\Omega}_3}{r^3}.
\end{equation}
The first term is the Lennard-Jones potential and the second term represents the dipole-dipole interaction, where $s=+1 (-1)$ for the \textit{gerade} (\textit{ungerade}) potential and $\Omega$ is the internuclear projection of the angular momentum $\bm{J} = \bm{j}_1 + \bm{j}_2$. Here, $\bm{j}_{k=1,2}$ is the total electronic angular momentum of atom $k$. From Ref.~\cite{Borkowski_Lineshape_2009}, we have $C_6 = 2.41(0.22)\times 10^3 E_h a_0^6$, $\sigma =8.5(1.0)a_0$, and $C_3^0 = -2C_3^1 = -0.1949(11)\times E_h a_0^3$, where $E_h$ is the Hartree energy and $a_0$ is the Bohr radius. The hyperfine interaction is described by $\hat{H}_\mathrm{hf}=A(\bm{i}_1\cdot\bm{j}_1)+B\frac{3(\bm{i}_1\cdot\bm{j}_1)^2+\frac{3}{2}(\bm{i}_1\cdot\bm{j}_1)-i_1(i_1+1)j_1(j_1+1)}{2i_1j_1(2i_1-1)(2j_1-1)}$~\cite{Reichenbach_OFR_2009}, where we assume that the $k=2$ atom belongs to the ${^1}S_0$ state, i.e., $\bm{i}_2\cdot\bm{j}_2=0$. We adopt the values of $A/h =-1094.328$~MHz and $B/h = -826.635$~MHz from Ref.~\cite{Pandey_Hyperfine_2009}.
At the low temperature of our experiment, we expect only $s$-wave ($R=0$) collisions for two fermionic $^{173}$Yb atoms in the ${^1}S_0$ ground state, and the initial ${^1}S_0$+${^1}S_0$ dimer state should have total angular momentum of $T=F=I=0,2,$ or $4$ and even spatial parity ($p=1$). Here, $\bm{F} = \bm{f}_1 + \bm{f}_2$ and $\bm{I}=\bm{i}_1+\bm{i}_2$. According to the selection rules for optical excitation, excited molecular states should have $T=1,2,3,4,$ or $5$ and odd parity ($p=-1$). In the modified Hund's case (e) that is relevant to our condition, with large spin-orbit coupling and hyperfine interaction, we count 205 different configurations of $(T,F,R)$ for the final states of the PA transition. Note that the transition from the initial $^1\Sigma_g$ molecular state to a \textit{gerade}-symmetry state is possible because the \textit{u-g} symmetry is broken in Hund's case (e)~\cite{Pique_ugsymmetry_1984}.
The adiabatic potentials for molecular states can be obtained by diagonalizing the Hamiltonian in Eq.~(1) via basis transformation between different Hund's cases~\cite{Tiesinga_PA_2005,Reichenbach_OFR_2009}. For a short distance $r$, the BO potential, which is diagonal under Hund's case (c), is dominant. In this case, the basis set is given by $\ket{\gamma} =\ket{J,\Omega,I,\iota,\Phi,(T,M_T,p)}$, where $\iota$ and $\Phi$ are the projections of $\bm{I}$ and $\bm{F}$ onto the internuclear axis, respectively, and $M_T$ is the projection of the total angular momentum onto a space-fixed quantization axis. At large $r$, i.e., when the two atoms are far apart, Hund's case (p) becomes relevant and results in a basis set consisting of the products of internal atomic states and molecular rotation as follows: $\ket{\pi}=\ket{f_1,m_1,f_2,m_2,R,(T,M_T,p)}$. In the intermediate range of $r$, we consider Hund's case (e), in which rotational and hyperfine interactions are diagonal and use a basis of $\ket{\epsilon}=\ket{f_1,f_2,F,R,(T,M_T,p)}$.
\begin{figure}
\includegraphics[width=8.0cm]{Fig2_adiabaticpot}
\caption{Adiabatic molecular potentials for a $^{173}$Yb$_2$ dimer in the $^1S_0+$$^3P_1$ channel as functions of the interatomic separation $r$. The molecular potentials for 205 different $(T,F,R)$ configurations are displayed, which are accessible via PA from the initial $s$-wave colliding atoms in the $^1S_0+$$^1S_0$ channel. At large $r$, the potentials converge to three asymptotic branches, which correspond to excited atomic states with hyperfine numbers of $f_1^\prime=3/2, 5/2$, and $7/2$. The energy offset is adjusted to the $f_1^\prime=7/2$ asymptote. The shaded region indicates the spectral range of our measurements.\label{fig:adiabaticpot}
}
\end{figure}
The calculated adiabatic potentials for the 205 channels for excited molecular states are displayed in Fig.~\ref{fig:adiabaticpot}. At small $r$, the potentials are grouped into four branches, representing the four different dipole-dipole interaction configurations, and at large $r$, they converge to three asymptotes near the dissociation limit, which correspond to $f_1^\prime=3/2, 5/2,$ and $7/2$, respectively. We note that the potential related to the $f_1^\prime=3/2$ asymptote has a local minimum near $\sim 75~a_0$, predicting purely-long-range molecular states. This is due to the large hyperfine structure of heavy Yb atoms, and purely-long-range states have been observed with $^{171}$Yb~\cite{Enomoto_PLR_2008}.
From the calculated molecular potentials, we compute the bound state energy levels using a multi-channel discrete variable representation (DVR) method~\cite{Tiesinga_PA_2005,Tiesinga_DVR_1998,Reichenbach_OFR_2009}. This calculation predicts more than 200 bound states in the range of $-1~\mathrm{GHz} < \delta \nu <-38$~MHz, whose positions are indicated in Fig.~\ref{fig:spectrum} alongside the measured PA spectrum~\cite{LBE_comment}. Considering the limited experimental sensitivity, the observed high density of the spectral lines is reasonably explained by the calculated results. With regard to the resonance positions, a better comparison might be enabled by using an iterative fitting method to tune the potential coefficients values~\cite{Borkowski_Lineshape_2009}, but because of the heavy calculation load involved, we will leave such an effort as a topic for future studies.
\begin{figure}
\includegraphics[width=8.0cm]{Fig3_Bchange}
\caption{PA spectra near $\delta \nu =-800$~MHz for various magnetic fields of $B=0$~G, $8.3$~G, $16.6$~G, $33.2$~G, and $49.8$~G. The PA laser beam was $\sigma^{-}$-polarized and the magnetic field was applied along the beam axis. All data points except those at $B=8.3$~G were obtained by averaging five independent measurements and the error bars denote their standard deviations. The data are offset for clarity.
\label{fig:Bchange}}
\end{figure}
\begin{figure}
\includegraphics[width=8.2cm]{Fig4_Bchangecomp}
\caption{PA spectra of various spin mixtures for $B=33.2$~G: (a) an unpolarized spin mixture, as shown in Fig.~\ref{fig:Bchange}; (b) two-component spin mixtures; and (c) a spin-polarized Fermi gas. The red lines show the sums of the Lorentzian fit curves to guide the eye, and the dotted vertical lines indicate the resonance positions as fitted from the spectrum of the balanced mixture. All data points were obtained by averaging five independent measurements of the same experiment and the error bars denote their standard deviations. (d) Zeeman splitting of the $-796$~MHz resonance (dotted lines), with markers representing the measured PA resonance positions.
\label{fig:Bchangecomp}}
\end{figure}
\subsection{Zeeman effect}
To facilitate the spectroscopic identification of the observed PA lines, we investigated the Zeeman effect by applying an external magnetic field $B$~\cite{Hamley_RARb_2009}. In the presence of $B$, the total angular momentum $T$ and its projection $M_T$ onto the field direction are still good quantum numbers of the system, and for low $B$ the Zeeman shift is described as $\Delta E_Z=\mu_B g B M_T$, where $\mu_B$ and $g$ are the Bohr magneton and the Lande $g$-factor of the molecular state, respectively. The number of Zeeman sublevels and the magnitude of their spectral splitting for $B$ directly reveal the quantum numbers $(T,M_T)$ of the molecular state as well as the $g$ factor value, which is sensitively determined by the interatomic potential~\cite{Kahmann_Zeeman_2014, Tiemann_Zeeman_2015}.
We applied the magnetic field along the axis of the PA laser beam and measured the PA spectra for various $B$ (Fig.~\ref{fig:Bchange}). The scan range of $\delta \nu$ was set to be from $-820~\mathrm{MHz}$ to $-780~\mathrm{MHz}$, where for $B=0$~G, three pronounced PA lines are located at $\delta \nu \approx-813$~MHz, $-796$~MHz, and $-791$~MHz, with relatively large linewidths of $>1$~MHz. The latter two PA lines have been reported in previous experiments~\cite{Kitagawa_Twocolor_2008,Taie_Pomeranchuk_2012,Sugawa_dualMott_2011}. With increasing $B$, each spectral peak broadens and splits into multiple weak peaks. The Zeeman splitting response appears relatively rapidly for the line at $-791$~MHz and slowly for the line at $-813$~MHz, reflecting the different magnitudes of $gT$ for these PA lines. Asymmetric shifting toward negative detuning is observed, which we attribute mainly to the $\sigma^{-}$ polarization of the PA light, which allows only $\Delta M_T = -1$ transitions.
For $B>30$~G, the spectrum shows a group of fully resolved Zeeman peaks with linewidths of $\sim 1$~MHz. To determine the $M_T$ numbers of the Zeeman peaks, we measured the PA spectra of two-component spin mixture samples. When such a sample is prepared with two spin components with magnetic Zeeman numbers of $m_{f_1}$ and $m_{f_2}$, the initial dimer state for $s$-wave collision in $^1S_0$+$^1S_0$ has a specific quantum number of $M_{T}=m_{f_1}+m_{f_2}$, and with a $\sigma^{-}$--polarized PA laser beam, this state can be coupled only to excited molecular states with $M_T=m_{f_1}+m_{f_2}-1$. Thus, the corresponding $M_T$ number can be assigned to Zeeman peaks that appear in the PA spectrum of such a two-component sample. In our experiment, we employed five binary spin mixtures of $m_{f_1}=-5/2$ and $m_{f_2}=\{-3/2,-1/2,1/2,3/2,5/2\}$~\cite{Lee_SOC_2017} and the PA spectra of the samples were measured at $B=33.2$~G [Fig.~\ref{fig:Bchangecomp}(b)]. As expected, each spectrum shows a subset of the Zeeman peaks observed in the PA spectrum of a fully balanced spin mixture [Fig.~\ref{fig:Bchangecomp}(a)]. To suppress unwanted optical pumping by the PA light into different spin states, we set $I_\mathrm{PA}=0.16$~W/cm$^2$ and $\tau = 100$~ms to obtain $\Gamma_\mathrm{sc}\tau \approx 0.8$, where $\Gamma_\mathrm{sc}$ is the Rayleigh scattering rate of the PA light at $\delta \nu = -800$~MHz.
The main finding from the $M_T$ analysis is that the three Zeeman peaks that are almost equally spaced in the detuning range of $-805~\mathrm{MHz}<\delta \nu <-798~\mathrm{MHz}$ have $M_T=-3,-2,$ and $-1$, respectively. We find that Zeeman peaks are also located at the positions linearly extrapolated for $M_T=0,1,2,$ and $3$ from these three Zeeman peaks and, in particular, that the peak position corresponding to $M_T=0$ coincides with the zero-field PA line at $-796$~MHz. From these observations and the fact that there is no $M_T=-4$ Zeeman peak at the corresponding expected spacing from the $M_T=-3$ peak, we infer that the total angular momentum number of the PA line at $\delta \nu=-796.2$~MHz is $T=3$. From a linear fit to the seven Zeeman peak positions, a $g$-factor of $g=0.056(3)$ can be determined, which is approximately ten times smaller than the atomic value of $g_F=0.426$ for the $^3P_1$ state. In Fig.~\ref{fig:Bchangecomp}(d), we display the PA resonance positions measured from the data in Fig.~\ref{fig:Bchange} as a function of $B$, and the Zeeman splitting lines predicted from the measured $g$-factor are found to reasonably fit the experimental data. For a high $B$ of approximately $\approx 50$~G, the PA resonance positions slightly deviate from the predictions toward a negative detuning except for $M_T=0$, indicating higher-order Zeeman effects.
Although the $M_T$ information is helpful for deciphering the linear Zeeman splitting of the PA line at $-796$~MHz, an analysis of the Zeeman effects of the other two PA lines is not straightforward. First, we observe no $M_T=-1$ Zeeman peaks for these two PA lines, although such peaks should exist because $T\geq 1$. Second, each PA spectrum for $M_T=-3$ and $-4$ shows four resonances [Fig.~4(b)], which means that our PA spectrum for a high $B$ of $>30$~G must involve Zeeman contributions from other PA lines outside the measurement range. Theoretical support will be critical for a complete understanding of the observed Zeeman effects.
\begin{figure}
\includegraphics[width=7.6cm]{Fig5_decay}
\caption{Atom loss curves as functions of the pulse duration $\tau$ of the PA light for $\delta \nu =-791$~MHz (on resonance, solid green circles) and $\delta \nu =-784$~MHz (off resonance, open blue circles) with $I_\mathrm{PA}=$0.74~W/cm$^2$. The solid lines are exponential fits to the initial $10$~ms of decay data. All data points were obtained by averaging the results of five independent runs of the same experiment.
\label{fig:decay}}
\end{figure}
\subsection{Two-body loss rate}
Finally, we characterized some of the pronounced PA resonances by measuring the two-body loss rate $K_2$ under PA light. $K_2$ contains important information such as the Franck-Condon factor for the optical transition~\cite{Jones_RMP_2006,Bohn_semianalytic_1999} and the so-called optical length $l_\mathrm{opt}$ that describes the magnitude of the change of the scattering length due to the OFR~\cite{Ciurylo_OFR_2005,Borkowski_Lineshape_2009,Nicholson2015}.
In the presence of PA light, the atom density $n$ evolves as $\dot{n}(t)=-2 K_2 n^2 - \gamma n$, where the first term represents the two-body PA process and the second term accounts for one-body decay processes such as Rayleigh photon scattering loss and background trap loss. For a case of a trapped sample, considering its inhomogeneous density distribution, the rate equation for the total number of atoms $N$ is given by $\dot{N}(t) = - 2 K_2 \frac{N^2}{V_e} - \gamma N$, where $V_e=(2\pi)^{3/2} \sigma_x \sigma_y \sigma_z$ is the effective volume of the sample for a Gaussian density distribution. To avoid nonlinear effects caused by sample heating on $K_2$ and $V_e$, we measured the decay rate $\gamma'$ of $N$ from an exponential fit to the initial $10$~ms of $N(t)$ data and calculated $K_2$ as $K_2=\frac{V_e}{2\bar{N}}(\gamma'-\gamma)$. Here $\bar{N}$ denotes the average number of atoms over the initial 10~ms and $\gamma$ was independently measured at off-resonance detuning which is more than $4\Gamma_b$ away from the PA resonance (Fig.~\ref{fig:decay}).
We measured $K_2$ for the three PA resonances at $\delta \nu=-38.1$~MHz, $-791.3$ ~MHz, and $-796.2$~MHz, and obtained $K_2=1.0(3)\times 10^{-12}$~cm$^{3}$/s with $I_\mathrm{PA}=74$~mW/cm$^{2}$, $K_2=0.5(1)\times 10^{-12}$~cm$^{3}$/s with $I_\mathrm{PA}=0.74$~W/cm$^{2}$, and $K_2=0.8(5)\times 10^{-12}$~cm$^{3}$/s with $I_\mathrm{PA}=0.74$~W/cm$^{2}$, respectively. In the cold collision limit, the optical length is given as $l_\mathrm{opt}= \eta \mu K_2 /(8\pi \hbar)$~\cite{Ciurylo_OFR_2005,Yan2013}, where $\eta$ is the enhancement factor of the molecular linewidth with respect to the natural linewidth. Assuming that $\eta$ is order of unity, our measurement results suggest that $l_\mathrm{opt} \sim 10 a_0$ at $I_\mathrm{PA}=1$~W/cm$^{2}$. It is surprising that the estimated value of $l_\mathrm{opt}$ is more than two orders of magnitude smaller than the values reported for other Yb isotopes~\cite{Reichenbach_OFR_2009,Borkowski_Lineshape_2009,Kim_OFR_2016}. It would be worthwhile to investigate the tempearture and $\delta \nu$ dependence of $K_2$ in a further systematic manner~\cite{Borkowski_Lineshape_2009}.
\section{Summary}
We have measured the PA spectrum of a degenerate Fermi gas of $^{173}$Yb atoms near the dissociation limit of the ${^1}S_0$--${^3}P_1$ intercombination transition and have characterized some of the prominent PA lines by investigating their Zeeman splitting and measuring their two-body loss rates under PA light. The high density of the spectral lines was accounted for by the calculation of the molecular energy levels based on an extended version of Hund's case (e), but further theoretical investigation will be necessary for spectral identification of the observed molecular states. This will improve our understanding of the collisional properties of Yb atoms in $^1S$+$^3P$, which are important for many applications using Yb atoms, such as optical clocks~\cite{Bloom_clock_2014,Nemitz2016} and the simulation of novel quantum magnetism~\cite{Sugawa_dualMott_2011,Gorshkov_SUN_2010}.
Finally, we note that when the Zeeman splitting of a PA line is fully understood in terms of its molecular quantum number, the spin-dependent PA transitions may find immediate use in probing interspin correlations, particularly, in optical lattice experiments~\cite{Sugawa_dualMott_2011,Hofrichter2016}. For example, the correlations between the $m_f=m$ and $-m$ spin states may be distinctively detected by using the PA resonance at $-798.4$~MHz for $B=33.2$~G. Thus, it might be worthwhile to search for an isolated $T=5$ PA line whose Zeeman lines are spectroscopically well resolved and have reasonable transition strengths for a moderate magnetic field.
\section{Acknowledgments}
We thank Paul S. Julienne and Jee Woo Park for helpful discussions. This work was supported by the Institute for Basic Science (IBS-R009-D1) and the National Research Foundation of Korea (Grant No. 2014-H1A8A1021987).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,510 |
using Argentum.Core;
using SilverScreen.Domain;
using SilverScreen.Infrastructure;
namespace SilverScreen.Application.Customer
{
public class CreateCustomer : ICommand
{
public string Name { get; set; }
public string Adress { get; set; }
public CreateCustomer(string name, string adress)
{
Name = name;
Adress = adress;
}
}
public class CreateCustomerHandler : IHandleCommand<CreateCustomer>
{
private readonly IDomainRepository _repository;
public CreateCustomerHandler(IDomainRepository repository)
{
_repository = repository;
}
public void HandleCommand(CreateCustomer command)
{
var customer = Domain.Customer.Create(command.Name, command.Adress);
_repository.Save(customer);
}
}
public class CustomerCreatedHandler : IHandleEvent<CustomerCreated>
{
public void HandleEvent(CustomerCreated evt)
{
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,623 |
Q: Object property access doesn't work in ES6 function expression I can't access my object property with this ES6 expression :
const obj =
{
name: 'john',
test: () => console.log('hello'+this.name)
}
obj.test()
I only get "hello", and with this expression it works :
const obj =
{
name: 'john',
test: function() {
console.log('hello'+this.name)
}
}
What's happening here ?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,944 |
namespace LeetABit.Binary.Blocks
{
using System;
/// <summary>
/// Represents a decorator that processes one of the two blocks depending on the specified condition.
/// </summary>
public class ConditionalOrElseBlock : ConditionalBlock
{
/// <summary>
/// Initializes a new instance of the <see cref="ConditionalOrElseBlock"/> class.
/// </summary>
/// <param name="inner">
/// Definition block that shall be decorated.
/// </param>
/// <param name="elseBlock">
/// Definition block that should be processed when the condition is not meet.
/// </param>
/// <param name="condition">
/// Function that obtains a condition for a decorator.
/// </param>
protected ConditionalOrElseBlock(IDefinitionBlock inner, IDefinitionBlock elseBlock, Func<IEvaluationContext, bool> condition)
: base(inner, condition)
{
this.ElseBlock = Requires.ArgumentNotNull(elseBlock, nameof(elseBlock));
}
/// <summary>
/// Gets a definition block that should be processed when the condition is not meet.
/// </summary>
protected IDefinitionBlock ElseBlock
{
get;
}
/// <summary>
/// Implements processing of the current definition block using specified coding context.
/// </summary>
/// <param name="context">
/// Coding context that contans current coding data.
/// </param>
/// <returns>
/// Object that represents processing result.
/// </returns>
public override Result Process(ICodingContext context)
{
_ = Requires.ArgumentNotNull(context, nameof(context));
return this.Condition(context)
? this.Inner.Process(context)
: this.ElseBlock.Process(context);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,146 |
\section{Introduction}
Modern adaptive moving mesh schemes present significant advantages over traditional fixed mesh schemes in many geophysical applications. Adaptive meshes can focus resolution in places of interest in order to make better use of available computational power, \citet{huang2010adaptive}, or can be designed to optimise computational cost and accuracy based on external factors, an example being ship and acoustic receiver locations in the prediction of underwater noise pollution from oceanic shipping activity, see \citet{Trigg2018}. In some applications one may require the mesh to change as the system evolves to better represent the underlying physics, \citet{weller2010bams}. Adaptive meshes are typically governed by a set of rules suitable to the specific problem being solved.
There are many reasons why solving a geophysical problem in a Lagrangian frame may be appropriate, see {\it e.g.} \citet{asch2016data,jablonowski2004adaptive}. The associated numerical solver will inevitably be based on a moving mesh, and some of the advantages described above of a moving mesh are delivered a fortiori by the use of such a scheme that advects the nodes with the flow. For instance, this may have the effect of naturally concentrating nodes in locations of increased activity, or better resolving coherent structures. More specifically, if the nodes are advected with the flow, node clusters can provide information where gradients are large and sinks or eddies exist, likewise node deserts can indicate where gradients are small. When the nodes are advected by the flow in this way, it is almost inevitable that they will have to change in both number and location in order to maintain solution accuracy for the numerical solver. When using an adaptive mesh that is governed by the model physics like this, the node locations and physical quantities are inexorably coupled. As a consequence the node locations can be considered part of the model state and of the model's solution history.
They key point to note is that for computational models based on such Lagrangian solvers, the node locations encode underlying physics and therefore provide information about the overall model state. As such, they are all updated together under the model evolution. But the observational data reflect underlying physics also and we would therefore expect that the optimal incorporation of such data should update the node locations as well as the values of the physical state variables. We develop here a data assimilation scheme for achieving exactly this impact of data on the mesh itself.
The process of incorporating data into physical models is called Data Assimilation (DA). A survey of DA methods can be found in \citet{budhiraja2018}. DA has become an integral tool in the geosciences and meteorology improving numerical weather prediction and as a method for parameter estimation. A review of DA in the geosciences can be found in \citet{carrassi2018}. We will focus on ensemble methods \citet{evensen2009,houtekamer2016enkf} which make use of estimated statistics from an ensemble of model runs at an analysis time step. These methods are attractive when attempting to leverage the information that node locations carry through covariances estimated from the ensemble members. That information is specifically brought in through the cross covariances between the physical values and the node locations driven by those values. In the case where the nodes are advected with the flow, these cross covariances very closely match the spatial gradient of the fluid velocities across the model domain. This encodes extra and important physical information into the DA update step. We will use observations of the physical state to update both the physical values and node locations in our approach which, in this work, will come from a twin model experiment using the models outlined in Section~\ref{modelandmesh}.
Adapting existing ensemble methods to adaptive moving meshes involves tackling some significant challenges. Ensemble DA methods rely on estimated statistics from the ensemble members and for success they must be statistically consistent. The main challenge is the fact that each ensemble member may have nodes in different locations, in different numbers, or both. Previous work along these lines has been carried out in \citet{bonan2017} for an adaptive mesh 1-d ice sheet model. In that work the adaptive mesh was conservative, in that each ensemble member has the same number of points. Observations of the ice sheet edge were also directly assimilated. In this work we consider updating node locations for a non-conservative adaptive mesh model using eulerian observations of the physical quantities of the "truth run", similar to how a satellite may take observations. A non-conservative mesh means that each ensemble member will have a {\it different number} of nodes in {\it different locations} requiring us to develop methodologies to obtain consistent and meaningful error covariance estimates. Any methodology we develop will necessarily have some disruptive effect on the individual ensemble members themselves in order to achieve a measure of statistical consistency between them. This may come through the addition or removal of nodes or the interpolation of values to specific locations. With this in mind, we define a successful method as one which improves the estimate of the truth over forecast with no DA and take special care to study the effects the method has on the ensemble members themselves. We discuss these effects in Section~\ref{results} and recommend considerations to minimise any negative effects depending on the application at hand and the desired prediction goal.
Other ensemble approaches aimed at adaptive meshes have been developed. \citet{jain2018amr} study a tsunami model which uses an adaptively refined mesh taking the union of all meshes as reference mesh to which each ensemble member is interpolated to before the update step. In \citet{du2016ensemble} a model which uses 3D unstructured adaptive mesh model for geophysical flows \citet{maddison2011fluidity,davies2011fluidity} was considered and an EnKF developed which uses the idea of a reference mesh to carry out the analysis step. The reference mesh is chosen using the idea of super-meshing \citet{farrell2009conservative} and each ensemble member is interpolated to that fixed reference mesh before the analysis step. These previous studies all concern conservative adaptive meshes.
However, in \citet{Aydodu2019npg} a fixed reference mesh is used in two 1-d models for which the mesh evolves with the flow and undergoes a ``remeshing'' step which injects new nodes should two be to far apart or removes nodes should two be too close together. This remeshing means that each ensemble member will likely have different numbers of points in different locations. In that work two reference meshes are used and are chosen based on the rules of the remeshing scheme. Ensemble members are mapped to the reference mesh before the update and mapped back to their previous meshes after. Our work goes further extending the update to the node locations themselves. We use the reference mesh only as a guide to match components of the state vector and augment our state vector with the node locations. A reasonable supposition is that avoiding the mapping scheme will help to lessen disruption of individual ensemble members providing for better estimates of the error covariances needed for the update step.
This paper is structured as follows, in Section~\ref{modelandmesh} we describe the model and adaptive mesh scheme we use in our twin model experiments. In Section~\ref{AMMENKFS} we outline the necessary ingredients for an EnKF on non conservative adaptive mesh models and describe the two implementations of such that we will compare. The results are presented Section~\ref{results} along with the optimised inflation parameters needed for the methods. We follow the results with a discussion in Section~\ref{results} on the cross covariances of the physical variables and node locations as well as the effect the adapted EnKF schemes have on the ensemble members themselves. We also present considerations on choosing the inflation parameters depending on the application and finally in Section \ref{conclusions} we present some concluding remarks and summary.
\section{Model and Mesh}\label{modelandmesh}
\subsection{Adaptive Mesh} \label{amm}
In this work we are interested in adaptive meshes that evolve with the flow of a physical system and which are non-conservative. We will make use of the same 1-d adaptive mesh scheme developed in \citet{Aydodu2019npg} as a prototype of 2-d, or 3-d, non-conservative adaptive mesh used in some modern numerical models, including the Lagrangian sea ice model neXtSIM \citet{rampal2016nextsim,rabatel2018impact}.
The mesh itself is a 1-d mesh defined on the domain $D=[0,L)$ with nodes $\{z_1, z_2, \dots, z_N\}\in D$. It is assumed that $0\leq z_i<z_{i+1}<L$ and that the positions of the nodes satisfy criteria which define a valid mesh through two tolerance parameters $\delta_1, \delta_2$. A valid mesh is one for which,
\begin{align}
\delta_1 \leq |z_{i+1}-z_i| \leq \delta_2 \quad \forall i \in \mathbb{N}: 1\leq i < N-1 \label{valid1}\\
\text{and} \quad \delta_1\leq |z_1+L-z_N| \leq \delta_2 \label{valid2}.
\end{align}
This criteria ensures that the mesh is periodic and that no two nodes are closer than $\delta_1$ or further apart than $\delta_2$. Moreover, $\delta_1$ and $\delta_2$ are chosen so that $\delta_1/\delta_2\geq2$ and are both divisors of $L$ (see \citet{Aydodu2019npg} for an extensive explanation and details on the assumptions).
The mesh points themselves evolve directly with the velocity ${\bf u}$ as,
\begin{align}
\frac{{\rm d}z_i}{{\rm d}t}={\bf u}(t,z_i).
\label{zevolove}
\end{align}
Equation~\eqref{zevolove} together with the physical model updating the velocity (along with any other model state variables) represents a coupled system of equations which can be solved alternately or simultaneously \citet{huang2010adaptive}.
Given that the node locations are a function of time $z_i=z_i(t)$, it is clear that there will be instances when the criteria for a valid mesh given in Eqs.~\eqref{valid1} and \eqref{valid2} are violated. In such cases we need a suitable remeshing scheme to enforce our criteria which is given as follows. For each $i$ if $|z_{i+1}-z_{i}|<\delta_1$, $z_{i+1}$ is deleted. Alternately, if $|z_{i+1}-z_{i}|>\delta_2$ a new point $z^*$ is inserted at the mid point between $z_{i+1}$ and $z_i$ and the points are re-indexed according to their order from left to right. The most relevant consequence of this is that the number of nodes in the mesh is not constant.
\subsection{Models and Observations} \label{models}
In this work we consider two models for use in our numerical experiments. The first is a diffusive form of Burgers' equation (BGM), \citet{burgers1948}:
\begin{equation}
{\rm {\bf BGM}}:\quad \frac{{\rm \partial} u}{{\rm \partial} t}+u\frac{{\rm \partial} u}{{\rm \partial} z}=\nu \frac{{\rm \partial}^2u}{{\rm \partial} z^2}, \quad z\in [0,1), \label{BGM}
\end{equation}
with viscosity, $\nu=0.08$ and periodic boundary and initial conditions:
\begin{align}
u(0,t)=u(1,t), \label{BGMBC} \\
u(z,0)=\sin(2\pi z)+\frac{1}{2} \sin(\pi z). \label{BGMIC}
\end{align}
The Burgers equation has been used in several DA studies \citet{cohn1993dynamics, verlaan2001, pannekoucke2018parametric}. This model is of particular interest because of the steep gradients near the shock, a motivating reason to use an adaptive mesh.
The second model is a version of the Kuramoto-Sivashinsky (KSM) equation \citet{papageorgiou1991route} given by
\begin{equation}
{\rm {\bf KSM}}: \quad \frac{{\rm \partial} u}{{\rm \partial} t}+\nu\frac{{\rm \partial} u^4}{{\rm \partial} z^4}+\frac{{\rm \partial} u^2}{{\rm \partial} z^2}+u\frac{{\rm \partial} u}{{\rm \partial} z}=0 \quad z\in[0,2\pi).
\label{KSM}
\end{equation}
The periodic boundary and initial condition are defined as:
\begin{align}
BC: u(0,t)=u(2\pi,t),\label{KSMBC}\\
IC: u(z,0)=-\sin(2\pi z).\label{KSMIC}
\end{align}
Here the viscosity $\nu=0.027$ is chosen so that we see chaotic behaviour in the model. Both models are solved using central differences and an Eulerian time stepping scheme with time steps of $10^{-3}$ for BGM and $10^{-5}$ for KSM. The tolerances used in the remeshing scheme outlined in section~\ref{amm} are $\delta_1=0.01$, $\delta=0.02$ for BGM and $\delta_1=0.02\pi$, $\delta_2=0.04 \pi$ for KSM.
Observations of the physical values are generated from high resolution ``nature'' runs for both models. For the KSM model, there is an initial spin up to $T=20$ before observations are taken and the model state at that time is used to initialise the ensemble members in the DA experiments described in section~\ref{results}. Mean zero, Gaussian distributed, white noise is added to the observations for both models and experiments carried out with differing observation error standard deviation, $\sigma_{\rm o}$. The observations are Eulerian, {\it i.e.} they are taken on a fixed-in-time regularly spaced grid on the interval $[0,L)$ and at regular time intervals. The choice of regular spatial and temporal distributions for the data is done for the sake of simplicity and it can be relaxed without impact on the algorithm setup.
\section{EnKF for an Adaptive Moving Mesh model - AMMEnKF }
\label{AMMENKFS}
The ensemble Kalman filter (EnKF) relies on estimates of error statistics using an ensemble of model runs assumed to be Gaussian distributed. The error estimates themselves are calculated using the state vector formed from each ensemble member. In the case of an Eulerian solver with a fixed mesh, this calculation is easily carried out as the number of nodes and their locations are the same for each ensemble member and thus, the dimension of the state vector is also the same for each ensemble member. In contrast for an adaptive moving mesh (AMM), the mesh node locations for each ensemble member will almost certainly be in different locations at an assimilation time. Further, due to the re-meshing outlined in Section~\ref{amm}, will have different numbers of nodes as well. This makes estimation of the error statistics less direct and lends to a need for the development of modified versions of the EnKF suited to models with solvers like those we consider here.
For a non-conservative AMM solver we see two additional steps to be necessary each with their own important considerations. The first key step needed before applying the EnKF we refer to as {\it dimension matching}, this needed to provide consistent estimations of ensemble statistics. One would need to decide to add or remove points from ensemble members to achieve the same number of components among state vectors. In addition a sub-step is that of {\it component paring}, that is, how to assign which points, which may be in different locations, are to be compared in the state vectors. The second key step comes after applying the update and we refer to it as {\it dimension return}. This would involve deciding whether or not to remove points which were added, if they were, or if points were removed, whether or not to add points back into the ensemble members. Both of these steps have the potential to disrupt the ensemble statistics and should be tailored to the model and meshing schemes.
One avenue toward an AMMEnKF involves the use of a reference mesh to which each ensemble member can be mapped and on which error statistics can be estimated. This has been explored originally in \citet{du2016ensemble} and \citet{Aydodu2019npg} in case of conservative and non-conservative meshes respectively. In \citet{Aydodu2019npg} the use of a reference mesh was explored in 1-d where the reference mesh itself is chosen based on the properties of the mesh adaptation scheme. In particular, two meshes were explored. The first is a high resolution (HR) mesh defined by the node proximity tolerance, $\delta_1$, which ensures {\it at most} one point from each ensemble member can be in any given interval of the partitioned domain. The second is a low resolution mesh (LR) defined by the node separation tolerance, $\delta_2$, which ensures each ensemble member has {\it at least} one point in any given interval of the partitioned domain. In both cases each ensemble member is mapped to the reference mesh before error statistics are calculated and then mapped back to their original meshes after the physical velocity values are updated in the analysis step. The mesh locations were not updated during analysis.
However, the node positions themselves are driven by the physical flow and as such can be considered time dependent state variables. In this work we consider updating the node locations making use of the HR partitioning of the interval domain for the same models considered in \citet{Aydodu2019npg}. The key difference between the previous and the current works is that we now augment our state vector with the node locations and update them in the analysis step. We are interested in exploring the use of the augmented state vector to leverage extra statistical information implied by the different meshes among the ensemble members. This is because, in this case, the mesh is connected to the physics and cross covariances between the physical values and the node locations says something about the system. Previous work for a conservative moving mesh was carried out in \citet{bonan2017}, there they also augment their state vector but avoid the issue of dimension matching.
We will, when needed, describe the methods in \citet{Aydodu2019npg} so that the reader may understand the relevant differences. In particular we focus on the HR method and refer to the augmented state vector as the HRA method. In both cases, HR and HRA, the analysis update is preceded and followed by two additional steps: (1) {\it dimension matching}, when the individual ensemble members (each on its own mesh) is projected onto the uniform, fixed-in-time, reference mesh, and, (2) {\it dimension return}, when the ensemble members are given each a mesh after their physical values (for HR) and their physical values and node locations (for HRA) have been updated. The full AMMEnKF procedure is detailed in the following subsection for both HR and HRA.
\subsection{Dimension matching}
\paragraph{{\bf HR Scheme}}\label{dimmatchHR}
In order to avoid the statistical consistency issues presented by having ensemble members with differing numbers of nodes at different locations, one can map each ensemble member to a reference mesh. The reference mesh can be defined on the physical domain $[0,L)$ into $M$ intervals of equal length $\Delta \gamma$,
\begin{align}
[0,L)=L_1\bigcup L_2 \bigcup \dots \bigcup L_m
\end{align}
where $L_i=[\gamma_i,\gamma_{i+1})$. In this case $\gamma_1=0$, $\gamma_i=(i-1)\Delta \gamma$ for each i. Further $\gamma_M=L-\Delta\gamma$ as $0$ and $L$ are identified on the periodic domain. The points $\gamma_i$ form the nodes of the reference grid.
The reference grid is chosen in one of two ways, to ensure that each ensemble member has {\it at most} one point in each interval, $\Delta \gamma=\delta_1$, or that each ensemble member has {\it at least}, $\Delta \gamma=\delta_2$, one point in each interval. The former is referred to as the high resolution mesh (HR) and the latter the low resolution mesh (LR).
Here we focus on the HR mesh since we partition our physical domain in the same way. The mapping from an ensemble member to the HR mesh will take the $j^{th}$ ensemble member's state vector ${ \bf x}_j=(u_1 \dots u_N, z_1 \dots z_N)_j \in \mathbb{R}^{2N}$ to the vector,
\begin{align}
{\bf x}_j=\left(\widetilde{{\bf }u},{\bm\gamma}\right)^{\rm T}_j=(\widetilde{u}_1 \dots \widetilde{u}_M, \gamma_1 \dots \gamma_M)^{\rm T}_j \in \mathbb{R}^{2M} \quad \text{with} \quad M\geq N.
\label{svHR}
\end{align}
Here $\widetilde{u}_i$ will be the physical value assigned to $\gamma_i$ through the introduction of a shifted mesh where $L_i\rightarrow \widetilde{L}_i=[\gamma_i-\delta_1/2,\gamma_i+\delta_1/2 )$ for $i=2,\dots M$. The first interval is taken to be $\widetilde{L}_1=[L-\delta_1/2,L]\bigcup[0,\delta_1 /2 )$ since we identify 0 and $L$. If there is a $z_k\in \widetilde{L_i}$, then set $\widetilde{u}_i=u_k$. If there is no such $z_k$ but $z_k\leq\gamma_i$ find $k$ such that $z_k\leq\gamma_i\leq z_{k+1}$ and set
\begin{align}
\widetilde{u}_i=\frac{u_k+u_{k+1}}{2}
\label{setutildehr}
\end{align}
if there is no such $z_k$, then set
\begin{align}
\widetilde{u}_i=\frac{u_1+u_{N}}{2}.
\label{setutildehrend}
\end{align}
This mapping is illustrated in the right branch of Fig.~\ref{embeddings}. Once each ensemble member has been mapped to the fixed reference grid, the standard EnKF can be applied.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/embed2.pdf}
\caption{Illustration of the two dimension matching schemes. In the HR scheme, points are shifted to a fixed reference mesh with empty nodes interpolated to. In the HRA scheme, points are added to empty intervals and interpolated to.}\label{embeddings}
\end{figure*}
\paragraph{{\bf HRA Scheme}}
\label{dimmatchHRA}
In the HRA setting the reference mesh is also used to choose which nodes will be compared, but with out changing their locations. We partition the domain $D=\left[0,L\right)$ into $M$ subintervals ($L_i$) each of length $\delta_1$ so that $D=\bigcup_i L_i$. Since $\delta_1$ is the node proximity tolerance we are guaranteed that each subinterval will have {\it at most} one point in it. With this we can component match nodes which fall in the same subintervals. If an ensemble member does not have a point in a given subinterval we will insert one, a {\it ghost point}, based on the nearest neighbors.
In this approach we take the state vector of the $j^{th}$ ensemble member on the reference mesh to be of the form
\begin{align}
{\bf x}_j=\left({\bf u},{\bf z}\right)^{\rm T}_j=\left(
u_1,u_2, \dots ,\widetilde{u_i},\dots, u_M, z_1, z_2 ,\dots,\widetilde{z_i},\dots,z_M\right)^{\rm T}_j\in \mathbb{R}^{2M} \quad \text{with} \quad M\geq N.
\label{svHRA}
\end{align}
where $u_i$ or $\widetilde{u}_i$ would be the value of the velocity in the $i^{th}$ sub interval of the reference mesh. A value with no tilde would mean the ensemble member had a point in that interval while a tilde implies the member did not have a point in the $i^{th}$ interval and one was inserted and a physical value interpolated to that location. The location of an interpolated point is drawn from the Gaussian distribution, $\mathcal{N}\left(\frac{\gamma_i+\gamma_{i+1}}{2},\delta_1/2\right)$, with a check that the point drawn actually resides in the interval $L_i$, if not, we draw again until it does. This is illustrated in the left branch of Fig.~\ref{embeddings}. The choice of randomly sampling the node location is done to avoid biasing node locations in intervals that are empty amongst a large proportion of the ensemble members which can happen in areas of lower velocities and larger node spacing. However, it is possible that we end up having in invalid mesh in this process. Nevertheless, we do not enforce validity at this step as there will be many cases where no location in an empty interval can be chosen for which there is not a point with in $\delta_1$ near it. This is because the intervals themselves are of size $\delta_1$.
The physical value assigned to a ghost point $\widetilde{z}_i$ is calculated by linear interpolation as:
\begin{align}
\widetilde{u}_i=\frac{b}{a+b}u_l+\frac{a}{a+b}u_r && a=\widetilde{z}_i-z_l, b=z_r-\widetilde{z}_i
\label{physicalinterp}
\end{align}
where $(z_l,u_l)$ and $(z_r,u_r)$ are the closest nodes to $\widetilde{z}_i$ to the left and right respectively and $u_l,u_r$ the corresponding physical values at those nodes. This is done from left to right which does allow for the possibility that a nearest left neighbour may have been a ghost point. However, in the case that $\delta_2=2\delta_1$ we are guaranteed each empty interval will have a non-empty interval to its left and right.
\subsection{Observation Operator}
\paragraph{{\bf HR scheme}}
For the HR method the observation operator applied to the $j^{th}$ ensemble member takes the form
\begin{align}
h\left({{\bf x}_j}\right)=
\widetilde{u}_i+\frac{z_k^o-\gamma_i}{\gamma_{i+1}-\gamma_i}(\widetilde{u}_{i+1}-\widetilde{u}_i).
\label{obsopref}
\end{align}
Where $z_k^o$ is the observation location with $\gamma_i \leq z_k^o \leq \gamma_{i+1}$.
\paragraph{{\bf HRA scheme}}
In a similar way we may define the observation operator for the HRA method as,
\begin{align}
h({\bf x}_j)=u_i+\frac{z_k^0-z_i}{z_{i+1}-z_i}(u_{i+1}-u_i).
\label{obsop}
\end{align}
Where either $z_i$, $z_{i+1}$, $u_i$ or $u_{i+1}$ could have a tilde if they were inserted due to the ensemble member having no value in the $i^{th}$ interval (see section~\ref{dimmatchHRA}).
This form of the observation operator means that we are not considering the location of the observation in the update, just the physical value. This is done since most geophysical measurements will not directly relate to a node position, since the nodes are not physical objects. Yet the physics does fundamentally drive node motion and the covariances between physical values and the node locations are non-zero in the error covariance matrix, described below and seen in Fig.~\ref{covuz}.
\subsection{Analysis using the EnKF}
Once the dimensions of the state vectors of each ensemble member have been matched the EnKF can be applied in the usual way. As in \citet{Aydodu2019npg} we will work using the stochastic version of the EnKF \citet{evensen2009}, and here as well the choice is not influential on the modification we propose for AMM models. Our method will apply equally if using deterministic EnKFs.
Let us define the forecast ensemble matrix ${\bf E}^{\rm f}$ as
\begin{align}
{\bf E}^{\rm f} = \left[ {\bf x}_1^{\rm f}, \dots , {\bf x}^{\rm f}_{ N^{\rm e}} \right] \in \mathbb{R}^{2M \times N^{\rm e}} . \label{errorcov}
\end{align}
Where the forecast state vectors ${\bf x}_j^{\rm f}$ takes the form as in Eq.~\eqref{svHR} for the fixed reference mesh case and Eq. ~\eqref{svHRA} for the augmented case where we also update node locations.
In Eq.~\eqref{errorcov}, $M$ is the number of subintervals, $L_i$, which partition the domain $D$ into subintervals of size $\delta_1$ and $N^e$ is the number of ensemble members. The vectors ${\bf x}_j^{\rm f}$ are the dimension matched state vectors taken to be the columns of ${\bf E}^{\rm f}$.
The forecast anomaly matrix ${\bf X}^{\rm f}$ takes the form
\begin{align}
{\bf X}^{\rm f} = \frac{1}{\sqrt{N^{\rm e}-1}}\left[ {\bf x}_1^f-\bar{{\bf x}}^{\rm f}, \dots , {\bf x}^{\rm f}_{N^{\rm e}}-\bar{{\bf x}}^{\rm f} \right] ,
\end{align}
where $\bar{{\bf x}}^{\rm f}$ is the forecast ensemble mean defined as,
\begin{align}
\bar{{\bf x}}^{\rm f}=\frac{1}{N^{\rm e}}\sum_{j=1}^{N^{\rm e}} {\bf x}_j^{\rm f}.
\end{align}
In the stochastic EnKF the observations are treated as random variables so that each ensemble member is compared to a slightly different perturbation of the observation vector \citet{burgers1998analysis} . That is, given an observation vector ${\bf y}$ we generate $N^{\rm e}$ observations according to,
\begin{align}
{\bf y}_j={ \bf y}+\epsilon_j \quad 1\leq j\leq N^{\rm e} \quad \epsilon_j \sim \mathcal{N}({\bf 0},{\bf R}),
\end{align}
where ${\bf R}$ is the covariance of the assumed zero mean, white-in-time observation noise $\epsilon$. We can then calculate the normalized anomaly ensemble of observations,
\begin{align}
{\bf Y}_{\rm o} &= \frac{1}{\sqrt{N^{\rm e}-1}} \left[ {\bf y}_1-{\bf y}, \dots , {\bf y}_{N^{\rm e}}-{\bf y}\right] \\
&=\frac{1}{\sqrt{N^{\rm e}-1}}\left[ \epsilon_1, \dots \epsilon_2\right],
\end{align}
which in turn defines the ensemble observation error covariance matrix,
\begin{align}
{\bf R}^e={\bf Y}_{\rm o}\left({\bf Y}_{\rm o}\right)^{\rm T}.
\end{align}
We then define the observed ensemble-anomaly matrix using our observation operator $h$ as,
\begin{align}
{\bf Y}=h({\bf E}^{\rm f})-h(\bar{{\bf E}}^{\rm f}),
\end{align}
where the operator $h$ is applied at each column of the matrix ${\bf E}^{\rm f}$.
This leads the Kalman Gain matrix, ${\bf K}$ to be,
\begin{align}
{\bf K}={\bf X}^{\rm f}{\bf Y}^{\rm T}\left[\frac{1}{N^{\rm e}-1}{\bf Y}{\bf Y}^{\rm T}+{\bf R}^{\rm e}\right]^{-1}
\label{kgain}
\end{align}
which is used, in the stochastic EnKF formulation, to individually update each ensemble member according to,
\begin{align}
{\bf x}^{\rm a}_i={\bf x}^{\rm f}_i+{\bf K}\left[{\bf y}_i-h({\bf x}^{\rm f}_i)\right] \quad 1\leq i \leq N^e.
\end{align}
With the HRA method, however, there is the possibility that an ensemble member will have an invalid mesh after the update step. For this reason the re-meshing algorithm is applied to each ensemble member after updating. The remeshing is also tasked with handling points that have moved out of the domain; although not common, it can happen.
In this work we also make use of covariance multiplicative scalar inflation \citet{anderson1999monte} in which the ensemble forecast anomaly matrix is inflated as,
\begin{align}
{\bf X}^{\rm f}\rightarrow \alpha {\bf X}^{\rm f},
\end{align}
with $\alpha \geq1$, before ${\bf X}^{\rm f}$ is used in the analysis update. This parameter is one that can be tuned through numerical experimentation, although approaches exist to make this task automatic and adaptive along the experiments (see {\it e.g.} \citet{raanes2019} and references therein).
After updating each ensemble member the mean of each analysis can be used to obtain a best estimate of the physical state of the system.
\subsection{Dimension Return}\label{dimreturn}
After the update is complete each ensemble analysis vector has its dimension returned to its pre-analysis value. For the AMMEnKF-HR scheme this is a needed step as the structure of the adaptive mesh is removed during the update step and some kind of map back to the previous mesh state before the next forecast is necessary. In the AMMEnKF-HRA scheme the mesh itself is updated and and the remeshing scheme is applied to ensure a valid mesh.
\paragraph{{\bf HR scheme}}
Following \citet{Aydodu2019npg}, in the HR case a backward map is used to return the updated ensemble members to their original meshes before forecasting again. In the forward mapping step, the mapping indices associating the nodes in the adaptive moving mesh with nodes in the reference mesh are stored in an array. These are the indices resulting from the projections on to the HR reference mesh. This allows us to map the updated physical values $\widetilde{{ \bf u}}^{\rm a}$ back to the mesh that the ensemble member came into the update step with, that is, the values updated at $\gamma_i$ are shifted back to their previous node locations. From there, the forecast is run until the next assimilation time step. It is notable that this can have the effect of introducing some amount of noise in each ensemble member as physical values determined at one location are moved to another. This is illustrated in the right branch of Fig.~\ref{backmap}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/backwardmap.pdf}
\caption{The dimension return steps. In the HRA scheme points that occupy previously empty intervals are simply deleted. In the HR scheme points which came in from the forecast step are shifted back to their original mesh locations and points inserted at the dimension matching step are deleted.}
\label{backmap}
\end{figure*}
\paragraph{{\bf HRA scheme}}
In the HRA case, after the update and remeshing, nodes that are in intervals which were previously unoccupied by a point before the update step are deleted for each ensemble member using the stored indices as in the HR case. This last deletion is not specifically necessary to the scheme and performance with and without this step is essentially equivalent. However, we include this step in our analysis as there may be some applications where keeping the dimension of the ensemble members low is desirable during the forecast step. This process is diagrammed in Fig.~\ref{backmap}.
A beneficial by-product of the mapping to and from the reference mesh in the HR scheme is that it induces additional variability among the physical values. This occurs when a value at one location is moved to another in the shift to and from the reference mesh.
The net effect is that the ensemble spread stays reasonably large, leading to the healthy functioning of the EnKF.
This is not the case in the HRA method given that physical values and their locations are updated together. As a result, the spread of the ensemble when using the HRA scheme tends to be smaller than the HR case and in fact spread collapses quickly with the augmented HRA scheme. This behaviour is shown in Fig.~\ref{INHERENT}.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/Inherent_HR_20_0_1.pdf}
\label{INHERENT_0HR}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/Inherent_HRA_20_0_1.pdf}
\label{INHERENT_0_11HR}
\end{subfigure}
\caption{Examples of the spread in the forecast ensembles and an example of an forecast ensemble member for the HR (left) and HRA (right) schemes. The forward and backward mapping of the HR scheme induces some inherent jitter increasing the spread. For the HRA case this does not happen and the ensemble members can collapse quickly. Also notable is the reduced smoothness in the ensemble member shown for the HR case caused by the mapping procedure. For these experiments $\alpha_{\rm j}=0,\alpha=1,\sigma_{\rm o}=0.01,N^{\rm e}=30$ and $I_{\rm m}=70$.} \label{INHERENT}
\end{figure*}
While little spread could also be reflecting the desired analysis convergence to the truth, in practice it is a dangerous situation as it often induces the filter to underestimate the actual error, leading to filter divergence.
We counteract this effect by adding white noise to the physical values, but leave the node locations unaltered. We shall refer to this process as jitter and it can be applied to each ensemble member after the update step. For a given ensemble member analysis vector ${\bf x}_j^{\rm a}$ the jitter is applied to its first $M$ components ({\it i.e.} to the physical values), according to,
\begin{align}
{\bf u}_j^{\rm a}={\bf u}_j^{\rm a}+\left( \mathcal{N}(0,\bm{\sigma}_J), {\bf 0} \right)^{\rm T} \quad \text{with} \quad \mathcal{N}(0,\bm{\sigma}_J), \ {\bf 0} \in \mathbb{R}^{M}\quad \text{and} \quad {\bm \sigma}_J=\alpha_J \max_{u_i,u_k \in {\bf u^{{\rm a}}}}|u_i-u_k|.
\label{jitter}
\end{align}
The scalar parameter $\alpha_J$ regulates the amount of jitter. We take $0\leq\alpha_J\leq 1$ so that we add a percentage of the maximum difference between the physical values of the ensemble members. By having $\alpha_J$ dependent on the analysis field, the jitter is adaptive and is similar to an adaptive form of additive inflation \citet{anderson1999monte}.
For comparison consistency, we also experimented by applying jitter in the HR method and found improvements in time averaged RMSE values for both schemes.
The HR and HRA algorithms are diagrammed in Fig.~\ref{Flows}.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Figures/HRFLOW.pdf}
\label{HRFLOW}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Figures/HRAFLOW.pdf}
\label{HRAFLOW}
\end{subfigure}
\caption{Illustration of the AMMEnKF-HR (Left) and AMMEnKF-HRA (Right) schemes. The details of the dimension matching, update and dimension return as well as the addition of $\alpha$ and $\alpha_J$ can be found in section~\ref{AMMENKFS}}. \label{Flows}
\end{figure*}
\section{Results and Discussion}\label{results}
\begin{table}[h]
\centering
\begin{tabular}{l| l| l| l| l| l| l| l}
\hline
& $\nu$ & $\delta_1$, $\delta_2$ & $N$ & $T$ & $\Delta t$ & $N_{obs}$ \\
\hline
BGM & 0.008 & 0.01, 0.02 & 100 & 2 & 0.05 & 10 \\
KSM & 0.027 & $0.02 \pi$, $0.04 \pi$ & 100 & 5 & 0.05 & 20 \\
\end{tabular}
\caption{The model parameter settings used in each of the DA experiments for the models described in section~\ref{models}. Here $\nu$ is the viscosity, $\delta_1$ and $\delta_2$ the node proximity and distance tolerances, $N$ the size of the reference mesh, $T$ the duration of the experiment, $\Delta t$ the integration time step, and $N_{obs}$ number of observations.}
\label{expsetup}
\end{table}
In this section we present the results of numerical experiments designed to measure the performance of the two schemes with different parameter settings in terms of a time averaged RMSE. We make use of the BGM and KSM models described in section~\ref{models} for our experiments. For the BGM model, we run for a short time from $t=0$ to $t=2$ because of the rapid dissipation in fluid velocity with our chosen viscosity parameter. The time averaged RMSE in the DA experiments is taken after $t=1$. The ensemble members are initialised by perturbing the initial condition of the nature run. For the KSM model an initial spin up until $T=20$ is done which is used as the initial condition for the DA experiments that follow. With the initial condition provided by spin up, the model is then run for 5 more time steps and observations for the nature run are taken in those last 5 time steps. The ensemble members are initialised with a perturbed version of the initial condition provided after the 20 time step spin up and the time averaged RMSE is taken in the last 4 time steps of the 5 time step integration. The dimension of the reference mesh for both schemes is $N=100$ however the dimension of the state-vector in the AMMEnKF-HRA scheme is twice that of the AMMEnKF-HR scheme since it has been augmented with the node locations. The parameters used for the BGM and KSM models in these experiments are summarised in Table~\ref{expsetup}. It is also important to note that the observations are taken at fixed time and on fixed even intervals equally dividing the spatial domain for both model cases. In addition, random white noise with standard deviation $\sigma_{\rm o}$ is added to each observation and we vary this values in our experiments described below.
\subsection{Comparison between AMMEnKF-HR and AMMEnKF-HRA}
We compare the performance of the AMMEnKF-HR introduced by \citet{Aydodu2019npg} (and recalled in section~\ref{AMMENKFS}), with that of the novel augmented formulation AMMEnKF-HRA also presented in section~\ref{AMMENKFS}.
We will use two metrics to evaluate the performance. Together with the more standard RMSE of the analysis mean, we also consider the time average RMSE for the first spatial derivative of the analysis mean, ${\rm \partial}_x \widetilde{{ \bf u}}^{\rm a}$. The gradient of the analysis field allows us to assess how well each of the methodologies preserve derivative information. This is relevant for two reasons, the first is to evaluate if and how much applying jitter to the analysed ensemble members distorts their curve smoothness. The second is that the mapping scheme in the HR case can create artificially sharp changes in function values. This will happen when mapping to the reference mesh and when the analysis vector is mapped back to the original ensemble member mesh if the original node location is sufficiently far away from a reference mesh location. These sharp changes over the domain, due to the jitter, HR mapping, or both, can disrupt local rates of change with the risk of violating conservation rules, such as incompressibility ($\nabla \cdot { \bf u}=0$), for example.
While we make no direct study of conservation laws in this work, we do evaluate the fidelity of the first derivative after the update step for each of these methods as a proxy for the potential violation of conservation laws in more realistic scenarios. In these results the time averaged RMSE's for the derivatives are obtained using the inflation parameters ($\alpha,\alpha_J$) that optimise the time averaged RMSE of the solution analysis mean. Depending on the situation, one may run similar experiments and choose a jitter and inflation that best preserve the first derivative if high fidelity of it is needed. In this work for these models though, there is not much difference in time averaged RMSE when using parameters that optimise the RMSE for the first derivative instead of solution itself.
The comparison is carried out over ranges of the three key experimental parameters: the ensemble size, $N^{\rm e}$, the initial mesh size, $I_m$, and the observation error, $\sigma_{\rm o}$.
We study the performance of the methods by running experiments with two of them kept fixed while varying the other.
For each parameter setting, the optimal jitter and inflation for each scheme are determined by running tuning experiments that identify the pair of values giving the lowest time averaged RMSE. This way we will compare the best possible configuration of each scheme. The values used in the experiments are given in Table~\ref{tab:sensitivitysettings}.
Results are shown in Fig.~\ref{optimalBGM} and \ref{optimalKSM} for the BGM and KSM model respectively.
\begin{table}[ht]
\centering
\begin{tabular}{l| l l l| l l l}
\hline
& \multicolumn{3}{c|}{BGM} & \multicolumn{3}{c}{KSM}\\
\hline
Experiment Type& $N^{\rm e}$ & $I_{\rm m}$ & $\sigma_{\rm o}$ & $N^{\rm e}$ & $I_{\rm m}$ & $\sigma_{\rm o}$ \\ \hline
Varying $N^{\rm e}$ & [20-90] & 70 & 0.01 & [20-90] & 70 & 0.798 \\
Varying $I_{\rm m}$ & 30 & [50-90] & 0.01 & 40 & [50-90] & 0.798 \\
Varying $\sigma_{\rm0}$ & 30 & 70 & [0.01-0.07] & 40 & 70 & [0.60-2.0]\\ \hline
\end{tabular}
\caption{The settings used for the three sensitivity experiments for the BGM and KSM models. For all three experiment types the ranges of $\alpha$ and $\alpha_{\rm J}$ optimised over remained the same. The range of $\alpha_{\rm J}$ differed between the model types with a range of $[0-0.1]$ for the BGM model and $[0-0.5]$ for the KSM model. The range of $\alpha$ was $[0-1.6]$ for both models.}
\label{tab:sensitivitysettings}
\end{table}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HRA_VS_HR_FUNCTION.pdf}
\label{ENSoptimalfunctionBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HRA_VS_HR_FUNCTION.pdf}
\label{INIoptimalfunctionBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HRA_VS_HR_FUNCTION.pdf}
\label{OBSoptimalfunctionBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HRA_VS_HR_DERIVATIVE.pdf}
\label{ENSoptimalderivativeBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HRA_VS_HR_DERIVATIVE.pdf}
\label{INIoptimalderivativeBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HRA_VS_HR_DERIVATIVE.pdf}
\label{OBSoptimalderivativeBGM}
\end{subfigure}
\caption{Results of the tuning experiments for the HR and HRA schemes with the BGM model. For each parameter shown along the x-axis, the optimal jitter and inflation are used to obtain a time averaged RMSE for the analysis (top panels). The time averaged RMSE values for the spatial derivatives (bottom panels) correspond to the parameter which optimise analysis RMSE (see text for details).}\label{optimalBGM}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HRA_VS_HR_FUNCTION.pdf}
\label{ENSoptimalfunctionKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HRA_VS_HR_FUNCTION.pdf}
\label{INIoptimalfunctionKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HRA_VS_HR_FUNCTION.pdf}
\label{OBSoptimalfunctionKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HRA_VS_HR_DERIVATIVE.pdf}
\label{ENSoptimalderivativeKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HRA_VS_HR_DERIVATIVE.pdf}
\label{INIoptimalderivativeKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HRA_VS_HR_DERIVATIVE.pdf}
\label{OBSoptimalderivativeKSM}
\end{subfigure}
\caption{Same as Fig.~\ref{optimalBGM} for the KSM model.}\label{optimalKSM}
\end{figure*}
First note that Fig.~\ref{optimalBGM} and \ref{optimalKSM} immediately reveal that the ensemble spread in the HR scheme is typically much larger than that of the HRA scheme even when performance is comparable, such as in the initial mesh size experiments. This is due to the inherent stochasticity of the HR scheme discussed in section~\ref{dimreturn}.
By looking at the RMSE of the analysis mean, it is evident that the HRA scheme tends to out perform the HR scheme in general and particularly for smaller ensemble sizes. This is due to the extra information carried in the cross covariances between the physical values and the node locations.
The RMSE of the spatial derivatives is also generally lower in the HRA scheme, except at ensemble size 50 for the BGM case. It is worth reiterating that we use parameters optimised for the solution itself and not the first derivatives here. We will discuss this behaviour more extensively later in this section together with other metrics used to understand this particular issue. Figures~\ref{optimalBGM} and \ref{optimalKSM} also highlight that only marginal improvements in time averaged RMSE are obtained after an ensemble size of 30 for the BGM model and 50 for the KSM model. This kind of behaviour can be observed when the ensemble size is larger than or equal to the dimension of the unstable, neutral subspace of the dynamics \citet{bocquet2017four}.
For both the BGM and KSM models we do not see much dependence on the initial mesh size but with HRA performing comparably to or better than the HR scheme in the BGM case and out performing HR in the KSM case.
We also make a comparison to the performance of each scheme with respect to increasing observation error. For the BGM case both schemes perform comparably but we see better results from the HRA scheme in the KSM case particularly with regard to the first spatial derivative of the solution. We would also like to remark that the clearer trends in the KSM experiments is likely a result of a longer time average of the RMSE available as the BGM model damps quickly limiting the experimental time window. When the observation error is large enough both models perform about the same suggesting that one might choose to accept extra computational cost of the HRA scheme when the observations are good enough to warrant doing so.
In Fig.~\ref{BGMsurf} and \ref{KSMsurf}, respectively for KSM and BGM, we show examples of the time averaged RMSE surfaces from the experiments described above as a function of inflation and jitter with the optimal pair of values marked by a red star.
The difference in the smoothness of the contour plots arises from the aforementioned fact that we run the KSM model for a longer time than the BGM model which dissipates quickly due to the chosen viscosity term. The longer run provides a larger sample of RMSE values to average over producing a smoother surface.
The need for some jitter in the HRA method (bottom panels) is highlighted by the fact that the time averaged RMSE error is higher near the x-axis ($\alpha_J=0$) for both models, but particularly with BGM). While this is also the case for the HR scheme with the BGM model the effect is less pronounced. For the HR method with the KSM model there is a region with $\alpha_J=0$ that the time averaged RMSE remains close to the one obtained using optimal jitter and inflation, this is likely achievable due to the chaos in the KSM model naturally increasing the spread.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HR_50.pdf}
\label{ENSBGMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HR_50.pdf}
\label{INIBGMSHRURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HR_0.01.pdf}
\label{OBSBGMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HRA_50.pdf}
\label{ENSBGMHRASURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HRA_50.pdf}
\label{INIBGMSHRAURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HRA_0.01.pdf}
\label{OBSBGMHRASURF}
\end{subfigure}
\caption{Contour plots for the jitter and inflation calibrations for the BGM model for the HR (top row) and HRA (bottom row) schemes for the experimental parameters. The points in the plot represent the sample points used while the red star represents the jitter and inflation with the lowest time averaged RMSE. Of particular note is that the HRA scheme has its valley of low RMSE well away from the x-axis implying that some jitter is beneficial while this is not as strong a feature with the HR method due to the inherent stochasticity added during the dimension matching step.}\label{BGMsurf}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HR_50.pdf}
\label{ENSKSMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HR_50.pdf}
\label{INIKSMSHRURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HR_0.798.pdf}
\label{OBSKSMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HRA_50.pdf}
\label{ENSKSMHRASURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HRA_50.pdf}
\label{INIKSMSHRAURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HRA_0.798.pdf}
\label{OBSKSMHRASURF}
\end{subfigure}
\caption{Same as Fig.~\ref{BGMsurf} but for the KSM model.}\label{KSMsurf}
\end{figure*}
\subsection{Error Covariance Structures}
We study here the ensemble-based forecast error covariance matrices, ${\bf X}^{\rm f} {\bf X}^{{\rm f}^{\rm T}}$, with ${\bf X}^{\rm f}$ defined in Eq.~\eqref{errorcov}. The structure of the matrices is shown in Fig.~\ref{covuz} and~\ref{covuuzz}. The size of ${\bf X}^{\rm f}{\bf X}^{{\rm f}^{\rm T}}$ for the HRA method in this case is $200\times200$ while that for the HR case is $100\times100$. With the HR method we only have covariances between physical values themselves in contrast to HRA where we have covariances between physical values, physical values and node locations and the node locations themselves. In Fig.~\ref{covuz} we show the forecast covariances between the physical values and node locations for the HRA method with both the BGM and KSM models just before the 10th and 20th assimilation steps respectively.
These error covariance matrices correspond to no jitter or inflation in an effort to understand the intrinsic differences between the methods. Also shown is the gradient of the corresponding forecast mean and the associated covariance between the physical values $u_i$ and their node location $z_i$, {\it i.e.} the diagonal of the matrix. This is done to highlight that the largest covariances occur at sharp gradients and have the same sign. This is natural since a negative gradient would imply a negative correlation between a physical value and its independent variable, likewise for a positive gradient. In fact, the shape of the diagonal closely matches that of the gradient demonstrating that including the node locations in the state vector encodes a deeper level of information into the Kalmain gain matrix.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/COVEX/GRADVSCOVBGMKSM.pdf}
\caption{ Top row: Examples showing the forecast covariances between the physical values and the node locations in the HRA method for the BGM and KSM models right before the 10th and 20th assimilation steps respectively. Bottom row: The spatial gradient of the forecast mean and covariance between $u_i$ and $z_i$ corresponding to the covariance matrices above, this highlights the extra information encoded into the Kalman gain when using the HRA scheme.}\label{covuz}
\end{figure*}
In Fig.~\ref{covuuzz} we show the covariances for the physical values for HR and HRA ({\it i.e.} the top left $100\times100$ block) as well as the HRA covariances of the node locations ({\it i.e.} the bottom right $100\times100$ block) for both the BGM and KSM models. Typically the HR covariances are higher in magnitude than that of the HRA scheme, this is because the ensemble members are compared on the same mesh in conjunction with the effect of the intrinsic stochasticity caused by the mapping to and from the reference mesh. The shock is immediately identifiable in the physical value covariances for the BGM model as a bright spot near the sharp gradient ({\it cf} Fig.~\ref{covuz}). The sharp gradients of the KSM model are also apparent in the physical error covariances. For the KSM model, there is a strong, albeit regular structure in the matrix for the HR method resulting form the fixed mesh with some long distance cross-correlations. Those long distance correlations are greatly reduced in the HRA. The correlations between the node locations themselves in the HRA scheme (rightmost panels) are very small due to the fact that they are not very far from each other since the intervals themselves are very small. This means that the extra contribution to the innovation in the HRA scheme is mainly coming from the correlations between the physical values and the node locations as opposed to the node locations themselves. This is indeed desirable since we need to inject new nodes in the embedding process and would prefer to avoid incidental biases.
\begin{figure*}
\centering
{\bf BGM }\par \medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/COVEX/BGMUUHR.pdf}
\label{BGMUUHR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/COVEX/BGMUU.pdf}
\label{BGMUU}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/COVEX/BGMZZ.pdf}
\label{BGMZZ}
\end{subfigure}
{\bf KSM }\par \medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/COVEX/KSMUUHR.pdf}
\label{KSMUUHR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/COVEX/KSMUU.pdf}
\label{KSMUU}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/COVEX/KSMZZ.pdf}
\label{KSMZZ}
\end{subfigure}
\caption{Examples of the error covariance matrix before the update step for the HR and HRA methods. The top row corresponds to the BGM model while the bottom row corresponds to the KSM model. Each row shows $\sigma_{u_i u_j}$ (left and middle columns) for the HR and HRA schemes respectively while the right column shows $\sigma_{z_i z_j}$ which only exists for the HRA scheme. }\label{covuuzz}
\end{figure*}
\subsection{Ensemble Member Fidelity}
As discussed earlier the addition of jitter can disrupt the shape of the ensemble members while still improving the analysis mean. This may be problematic if the ensemble members are used to feed information to another model component.
As an example where ensemble member fidelity may be important, we consider the Heterogeneous Multiscale Method (HMM) described for various applications in \citet{e2011principles}. In a general setting, the HMM method connects a macro scale model with parameters dependent on micro scale variables to a model of this micro variables in order to simulate a physical process. Typically one has a macro scale model $F(U,D)$ where $U$ is the physical macro scale variable and $D$ the data needed in order for the macro scale model to be complete, a stress tensor for example. Paired with the macro scale model we have a micro scale model $f(u,d)=0$ and $d=d(U)$ where $d$ is the data needed to set up the micro scale model and is dependent on the macro scale state.
Typically the HMM process proceeds as follows.
\begin{enumerate}
\item Given the current state of the macro variables, initialise the micro variables using the needed micro model data $d=d(U)$.
\item Evolve the micro variables for some micro model time steps.
\item Through the appropriate method, calculate $D$, needed for the macro model.
\item Evolve the macro variables using the macro-solver.
\end{enumerate}
In this setting if an ensemble member is solution of either the macro or micro models disruption in their fidelity will naturally cause a problem with steps 1. and 3. through an inaccurate calculation of $d$ or $D$ and likely propagating such errors in the evolution steps. An example of a system like this can be found in Cloud-Resolving Convection Parameterization (CRCP) \citet{Grabowski2001}. There, a macro model solving inviscid moist equations is coupled with a micro model representing sub grid scale cloud physics.
We saw in Fig.~\ref{BGMsurf} and
\ref{KSMsurf} that there are regions of low analysis mean RMSE for relatively high values of $\alpha_J$, suggesting that the mean is smoothing out the added noise from jitter. In Fig.~\ref{wiggles} we show examples of the analysis mean, truth, and a typical ensemble member of the BGM model for a fixed inflation and three values of $\alpha_J$. The inflation chosen corresponds to the optimal value found for an ensemble size of 50. We show $\alpha_J=0$, the optimal $\alpha_J$ (in terms of lowest time averaged RMSE), and a larger $\alpha_J$ for which the ensemble mean still has low time averaged RMSE. The figure clearly shows that applying jitter to the ensemble members has the potential to disrupt them (see the waving profile of the displayed arbitrarily chosen ensemble member), especially if your scheme requires you to act on each ensemble member as we do here with dimension matching and return. Depending on the application, such as a model using the HMM framework, it may be better to sacrifice a small amount of analysis accuracy to preserve the fidelity of each ensemble member in terms of representing a valid solution to the underlying PDE. In other applications, that may not matter quite so much.
\begin{figure*}
\centering
{\bf HR Scheme}\par \medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HR_10_0_1.13.pdf}
\label{BGMEXJ_0HR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HR_10_0.02_1.13.pdf}
\label{BGMEXJ_0_02HR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HR_10_0.04_1.13.pdf}
\label{BGMEXJ_0_04HR}
\end{subfigure}
{\bf HRA Scheme}\par \medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HRA_10_0_1.07.pdf}
\label{BGMEXJ_0HRA}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HRA_10_0.02_1.07.pdf}
\label{BGMEXJ_0_02HRA}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HRA_10_0.04_1.07.pdf}
\label{BGMEXJ_0_04HRA}
\end{subfigure}
\caption{Examples of BGM analysis and an ensemble member for different inflation and jitter choices. Top HR and bottom HRA. The RMSE of the analysis mean can be low even when the ensemble members themselves represent unrealistic solutions to the PDE. }\label{wiggles}
\end{figure*}
To better quantify the effect that adding jitter and inflation may have on the PDE fidelity of the ensemble members we look at three different metrics. The average of the time averaged variance of the difference between the ensemble members and the truth at each node ($\sigma_{{\rm ens}}$), the kurtosis of the same difference ($k_{{\rm ens}}$), and the average of the time averaged RMSE errors of the ensemble members (RMSE$_{{\rm ens}}$). If ${ \bf d}_i^{\tau_j}=\left({ \bf u}_i^{\tau_j}-{ \bf u}_T^{\tau_j}\right) \in \mathbb{R}^M$ is the difference between the $i^{th}$ ensemble member and the truth at assimilation time $\tau_j$ then we can define these quantities as,
\begin{align}
\sigma_{ens}=\frac{1}{N^{\rm e}}\sum_{i=0}^{N^{\rm e}} \frac{1}{N_{an}}\sum_{\tau_1}^{\tau_n} \frac{1}{M} \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}-E\left[ d_{i_{k}}^{\tau_j}\right] \right)^2, \label{sigens} \\
k_{ens}=\frac{1}{N^{\rm e}}\sum_{i=0}^{N^{\rm e}} \frac{1}{N_{an}}\sum_{\tau_1}^{\tau_n} \frac{\frac{1}{M} \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}-E\left[ d_{i_{k}}^{\tau_j}\right] \right)^4}{\left(\frac{1}{M} \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}-E\left[ d_{i_{k}}^{\tau_j}\right] \right)^2 \right)^2}, \label{kurtens} \\
RMSE_{ens}=\frac{1}{N^{\rm e}}\sum_{i=0}^{N^{\rm e}} \frac{1}{N_{an}}\sum_{\tau_1}^{\tau_n} \frac{1}{M} \sqrt{ \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}\right)^2}, \label{rmseens}
\end{align}
where $N_{an}$ is the number of assimilation steps completed.
If the ensemble members have a low $\sigma_{ens}$ this would suggest they don't deviate from the mean error along the domain axis which can suggest that the shape of the curve is consistent with the true solution to the PDE and had not been overly distorted by the inflation or jitter. The kurtosis can give us a measure of how concentrated around the mean error the errors are. A low value for the kurtosis suggests a more uniform distribution with the normal distribution having a kurtosis of 3. Kurtosis above 3 would suggest either that the probability mass is concentrated around the mean and values far from the mean are rare, or that the probability mass is concentrated in the tails. In this particular case, a high $k_{ens}$ likely implies the ensemble members are not overly distorted by jitter and inflation with large error occurring infrequently along the domain. A low $k_{ens}$ signals that the ensemble members are distorted by jitter and inflation with larger deviations from the mean occurring more uniformly. However, one may have a large $k_{ens}$ with ensemble members that have high PDE fidelity but are far apart from each other and or far from the truth. Nevertheless, in this analysis we are looking at a {\it long time average} and expect the ensemble to converge around the true solution in time. Examples of ensemble members with low and high kurtosis at a specific time are shown in Fig.~\ref{KURTEX}. Ideally one would hope for each ensemble member to have a low variance, low RMSE and high kurtosis calculated from ${ \bf d}_i^{\tau_j}$. In Fig.~\ref{BGMextraFC},~\ref{BGMextraAN},~\ref{KSMextraFC} and~\ref{KSMextraAN} we show $\sigma_{ens}$, $k_{ens}$ and RMSE$_{ens}$ as a function of $\alpha$ and $\alpha_J$ for ensemble members before and after the update step. The lowest values for $\sigma_{ens}$ and RMSE$_{ens}$ are denoted by a red star while the largest value of $k_{ens}$ is denoted by a blue star. We calculate these metrics for the both the forecast ensemble, right before the update step, and the analysis ensemble after.
\begin{figure*}
\centering
{\bf Kurtosis Examples}\par \medskip
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{Figures/KURT_EX/EX_HRA_63_0_1.pdf}
\label{HRALOWKURT}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{Figures/KURT_EX/EX_HRA_50_0.1_1.6.pdf}
\label{HRAHIGHKURT}
\end{subfigure}
\caption{Examples of the three statistical measures we use to measure curve distortion.}\label{KURTEX}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Forecast Members (BGM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_VAR/ENS_HR_90.pdf}
\label{VARBGMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_KURT/ENS_HR_90.pdf}
\label{KURTBGMSHRURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_RMSE/ENS_HR_90.pdf}
\label{RMSEBGMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_VAR/ENS_HRA_90.pdf}
\label{VARBGMHRASURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_KURT/ENS_HRA_90.pdf}
\label{VARBGMSHRAURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_RMSE/ENS_HRA_90.pdf}
\label{RMSEBGMHRASURFFC}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and RMSE$_{ens}$ as a function of jitter and inflation for the BGM model using the forecast ensemble members, right before update, for HR (top row) and HRA (bottom row). It is notable that both $\sigma_{ens}$ and RMSE$_{ens}$ increase primarily as a function of jitter ($\alpha_J$) with less dependence on multiplicative inflation ($\alpha$) while the same is true for decreasing kurtosis evidenced by horizontal contours. This identifies the jitter as the primary source of ensemble member distortion. The red stars represent lowest values and blue stars highest values.}\label{BGMextraFC}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Analysis Members (BGM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_VAR/ENS_HR_90.pdf}
\label{VARBGMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_KURT/ENS_HR_90.pdf}
\label{KURTBGMSHRURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_RMSE/ENS_HR_90.pdf}
\label{RMSEBGMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_VAR/ENS_HRA_90.pdf}
\label{VARBGMHRASURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_KURT/ENS_HRA_90.pdf}
\label{VARBGMSHRAURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_RMSE/ENS_HRA_90.pdf}
\label{RMSEBGMHRASURFAN}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and RMSE$_{ens}$ as a function of jitter and inflation for the BGM model using the analysis ensemble members for HR (top row) and HRA (bottom row). It is notable that horizontal contours shown in the same analysis for the forecast members are now slanted with improvements in $\sigma_{ens}$ and RMSE$_{ens}$ for lower values of inflation ($\alpha$). This is due to the update step, however the Kurtosis remains relatively unchanged implying that the members are still distorted for larger values of $\alpha_J$ and only the scale of the errors has been reduced. The red stars represent lowest values and blue stars highest values.}\label{BGMextraAN}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Forecast Members (KSM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_VAR/ENS_HR_90.pdf}
\label{VARKSMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_KURT/ENS_HR_90.pdf}
\label{KURTKSMSHRURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_RMSE/ENS_HR_90.pdf}
\label{RMSEKSMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_VAR/ENS_HRA_90.pdf}
\label{VARKSMHRASURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_KURT/ENS_HRA_90.pdf}
\label{VARKSMSHRAURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_RMSE/ENS_HRA_90.pdf}
\label{RMSEKSMHRASURFFC}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and RMSE$_{ens}$ as a function of jitter and inflation for the KSM model using the forecast ensemble members, right before update, for HR (top row) and HRA (bottom row). It is notable that both $\sigma_{ens}$ and RMSE$_{ens}$ increase primarily as a function of jitter ($\alpha_J$) with less dependence on multiplicative inflation ($\alpha$) while the same is true for decreasing kurtosis evidenced by horizontal contours. This identifies the jitter as the primary source of ensemble member distortion. The red stars represent lowest values and blue stars highest values.}\label{KSMextraFC}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Analysis Members (KSM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_VAR/ENS_HR_90.pdf}
\label{VARKSMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_KURT/ENS_HR_90.pdf}
\label{KURTKSMSHRURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_RMSE/ENS_HR_90.pdf}
\label{RMSEKSMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_VAR/ENS_HRA_90.pdf}
\label{VARKSMHRASURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_KURT/ENS_HRA_90.pdf}
\label{VARKSMSHRAURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_RMSE/ENS_HRA_90.pdf}
\label{RMSEKSMHRASURFAN}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and RMSE$_{ens}$ as a function of jitter and inflation for the KSM model using the analysis ensemble members for HR (top row) and HRA (bottom row). It is notable that horizontal contours shown in the same analysis for the forecast members are now slanted with improvements in $\sigma_{ens}$ and RMSE$_{ens}$ for lower values of inflation ($\alpha$).
The red stars represent lowest values and blue stars highest values.}\label{KSMextraAN}
\end{figure*}
When comparing between the forecast and analysis metric surfaces for the BGM model there is a detectable change in structure for $\sigma_{ens}$ and RMSE$_{ens}$ before and after the update. Before the update, both metrics tend to increase with increasing $\alpha_J$ somewhat independently of $\alpha$ and after the update the metrics are significantly reduced for lower values of $\alpha$. This is evidenced by the relatively horizontal contours in the forecast surfaces, which changes after the update. However, the general structure of the kurtosis remains similar before and after the update, implying that the update is reducing the size of the error, but that the distortions remain among the ensemble members for larger values of $\alpha_J$. The analysis mean is anyway smoothed through averaging still providing low RMSE$_{ens}$. Comparing the Kurtosis surfaces between the HR and HRA schemes also again shows that some jitter is desirable in the HRA scheme as the larger $k_{ens}$ values are away from the x-axis. This may be caused by the ensemble members collapsing around a solution and then deviating from the truth in time due to model instability and over confidence in the model solutions. It is notable that the lowest values for $\sigma_{ens}$ and RMSE$_{ens}$ occur for almost the same value of $\alpha_J$ at which we have our maximum $k_{ens}$ for the analysis metric surfaces in the HRA case.
When making similar considerations between the forecast and analysis metric surfaces for the KSM model we see a similar change in structure between the $\sigma_{ens}$ and RMSE$_{ens}$ surfaces as that of the BGM case. However, the contours in the KSM forecast surfaces are less horizontal implying that multiplicative inflation alone can increase the average errors. This does not necessariliy mean that the solutions are of low fidelity given that the chaos exhibited by the KSM model can simply produce ensemble members which are further from the truth but still viable solutions of the underlying PDE. In fact, they do seem to be corrected at the analysis step. Interestingly the Kurtosis surface structure changes more for the HR scheme than for HRA between forecast and analysis. It is important to note though that there is a significant difference in the range of $k_{ens}$ between the BGM (1-15) and KSM (3-5) models. This is likely due to the presence of chaos in the KSM model naturally increasing the spread of the ensemble members causing some to have little distortion and thus more a more normal distribution of errors. The small range in $k_{ens}$ likely makes this a less informative measure for the KSM model.
How one would choose $\alpha$ and $\alpha_J$ would depend on the problem at hand, minimizing only the time averaged RMSE may be the desired outcome but if ensemble member fidelity is important considering other metrics such as those presented for the forecast ensemble may also be important. For example, with the BGM model there is a drop in $k_{ens}$ from 10 to 8 when going from $\alpha_J=0.01$ to $\alpha_J=0.02$, the optimal value suggested when using 50 ensemble members ({\it cf} Fig.~\ref{BGMsurf}), for the forecast ensemble with almost no trade off in the time averaged RMSE between the two values. Depending on the sensitivity another model component may have on the fidelity of the ensemble members a more careful choice of inflationary parameters may be warranted.
\section{Conclusions}\label{conclusions}
Adaptive mesh solvers have the potential to greatly improve model skill and predictions but present difficulties for traditional data assimilation methods such as the EnKF. We consider here the particular case of a non conservative adaptive moving mesh for which each member of an ensemble will have different numbers of nodes in different locations. The key steps in an EnKF scheme for models of this sort are {\it dimension matching} often involving interpolation with a sub step of paring state vector components should they be in different locations, and {\it dimension return}. Dimension return involves removing added points in the matching step, if they were, or if points were removed whether or not to add points back in. Building on the work presented in \citet{Aydodu2019npg} we develop an EnKF scheme for a non-conservative adaptive moving mesh solver in 1-d using an augmented state vector that includes the locations of the nodes, locations that are also updated in the analysis step. Dimension matching is done using the properties of the adaptive mesh scheme itself, via a partition of the domain with intervals of the same size as the proximity tolerance $\delta_1$ which guarantees each interval will have at most one node in it. In the HR scheme developed in \citet{Aydodu2019npg} component paring is done by shifting the nodes in each interval to the their nearest interval boundaries and then interpolating new points to any interval boundaries which are empty. In this way the HR method compares ensemble members on the same mesh updating only the physical values of the nodes. Dimension return is then done by deleting interpolated points and shifting the updated physical values back to the previous mesh. In contrast, the HRA method leaves ensemble member points where they are and interpolates new points to empty intervals with the location drawn from a normal distribution with variance $\delta_1 / 2$ with a check the location resides within that interval. Component paring is then done using like intervals. Next, the state vector is formed with the node locations appended and both physical values and nodes are updated. After the update the remeshing scheme is applied to enforce a valid mesh, and points in previously empty intervals are deleted.
We find that when updating the node locations ensemble collapse becomes a problem and some additive or multiplicative inflation may be necessary. This is less of an issue for the HR method due to some inherent stochasticity arising from the mapping procedures, although jitter and multiplicative inflation can improve RMSE values there as well. Given an initial mesh size, ensemble size, and observation error, the jitter and inflation can be optimised with twin model experiments. When this is done we find that the HRA method typically provides better performance in terms of the time average RMSE for both the function and the first derivative. When using additive inflation, such as the jitter as we have defined it, there is potential to distort ensemble members while still obtaining a good analysis mean. This could be problematic in some frameworks such as the HMM frame work discussed in section~\ref{results}. To quantify the severity of the distortions we calculate several metrics defined in Eqs.~\ref{sigens},~\ref{kurtens} and~\ref{rmseens}. From this analysis we can see that the addition of jitter is primarily responsible for distorting the ensemble members while multiplicative inflation has less of an effect. One would want to weigh what is more important, low RMSE of the analysis mean or preserving ensemble member fidelity and should be application specific.
We would also like to address the question of computational efficiency between these two approaches. When using the augmented state vector of the HRA scheme the size of the error covariance matrix is doubled compared to that of the HR scheme. For very high dimensional models this may be problematic, yet as can be seen in Fig.~\ref{optimalKSM} when updating the node locations the first spatial derivative RMSE is much improved and if that information is needed the extra computational cost may be worth it. It remains to be seen how a scheme like this plays out in 2-d or 3-d AMM models however, we speculate that the utility in the inclusion of the cross covariances between physical values and node locations may be far more significant in these higher dimensional cases. The complexity and range of types of patterns that can form in 2-d or 3-d is far grater than is possible in 1-d. This implies that far more information may be carried in the cross-covariances between physical values and node locations. Further, if the motivation for the use of an AMM scheme is to focus computational power in regions of strong gradients, updating those node locations in accordance with where observations of those gradients are large may be very advantageous. If the remeshing rules for the AMM model are based on strict considerations of node distances and mesh geometries a 2-d or 3-d analogue of the HR reference mesh should be attainable enabling the application of the HR or HRA schemes presented here. An example of such a model is the novel lagrangian sea ice model neXtSIM \citet{rampal2016nextsim}. neXtSIM uses a finite element method based on a triangular non-conservative adaptive mesh with strict rules on the distance between nodes and angles between edges and was the motivation behind our exploration in 1-d presented in this work. The authors are currently working on implementing the precursor of the current method \cite[{\it i.e.} without node updates]{Aydodu2019npg} in neXtSIM and shall investigate the joint physics and nodes updates based on this study soon afterwards.
\section{Acknowledgements}\label{acknowledgements}
The research in this work has been funded by the US Office of Naval Research grants Data Assimilation Development and Arctic Sea-Ice Changes (award A18-0960) and DASIM-II (award N00014-18-1-2493), A.C. has been funded by the UK Natural Environment Research Council (award NCEO02004).
\section{Conflict of interest}
{\label{644442}}
The authors attest to no conflicts of interest regarding this work.
\selectlanguage{english}
\section{Introduction}
Modern geophysical modeling typically makes use of numerical solvers to produce solutions to the underlying partial differential equations (PDE'S) which represent the physics of a system of interest. Numerical schemes require a discritization of the relevant spatial and temporal domains. Typically, the spatial discritization, mesh, remains fixed in time but this is not necessary nor always optimal. In many physical application meshes with change or adapt in time are desirable.
Adaptive meshes have gained popularity in recent years for use in numerical solutions to geophysical PDE's. Adaptive meshes can focus resolution in places of interest in order to make better use of available computational power \cite{huang2010adaptive} or may require the mesh to change as the system evolves to better represent the underlying physics \cite{weller2010bams}. Adaptive meshes are typically governed by a set of rules suitable to the specific problem being solved. In a fluid setting it may be beneficial to allow the nodes to advect with the flow to naturally concentrate nodes in locations of increased activity or to solve the equations in a Lagrangian frame. In a setting like this, it will almost certainly happen that some nodes will be too close together and some too far apart, requiring the mesh to adapt. We consider a mesh like this in this work with the specific details outlined in Section \ref{amm}.
As Adaptive meshes have gained popularity so has Data Assimilation. Data Assimilation (DA) is a process where best estimates from observational data and model output are combined to produce an estimate closer to the truth while leveraging knowledge of the observational and model errors. A survey of some DA methods can be found in \cite{budhiraja2018}. DA has become an integral tool in the geosciences and meteorology improving numerical weather prediction and as a tool for parameter estimation. A review of DA in the geosciences can be found in \cite{carrassi2018}.
One very popular DA methodology is that of ensemble methods \cite{evensen2009,houtekamer2016enkf}. These methods make used of estimated statistics from an ensemble of model runs at an analysis time step. When using an adaptive mesh there is the potential for nodes to be in different locations and in different numbers when the time comes to estimate the needed statistics. In this case one must decide how to compare ensemble members and how or whether one can gain information from the differences in the adaptive meshes at the analysis step. If the adaptave mesh is not-physically driven, perhaps one would want to avoid considering those differences. If the meshes are driven by model physics, as is the case we consider, there may be useful information in their individual structures.
This paper is organised as follows, first we present the adaptave mesh scheme and models we will investigate. Discuss and compare two EnKF schemes we have developed, one which leverages mesh structure in the update step and one that does not. We then present the results of the numerical experiments and follow up with an analysis of those results using some novel metrics to understand what the update schemes do to the structure of the meshes and ensemble members themselves.
\section{Model and Mesh}\label{modelandmesh}
\subsection{Adaptive Mesh} \label{amm}
In this work we are interested in adaptive meshes that evolve with the flow of a physical system and which are non-conservative. We will make use of the same 1-d adaptive mesh scheme as the one developed in \cite{Aydodu2019npg} which is motivated by the type of adaptive mesh of the Lagrangian sea ice model neXtsim \cite{rampal2016nextsim}.
The mesh itself is a 1-d mesh defined a domain $D=[0,L)$ with nodes $\{z_1, z_2, \dots, z_N\}\in D$. It is assumed that $0\leq z_i < z_i+1 < L$ and that the positions of the nodes satisfy criteria which define a valid mesh through two tolerance paramaters $\delta_1, \delta_2$. A valid mesh is one for which,
\begin{align}
\delta_1 \leq |z_{i+1}-z_i| \leq \delta_2 \quad \forall i \in \mathbb{N}: 1\leq i < N-1 \label{valid1}\\
\text{and} \quad \delta_1\leq |z_1+L-z_N| \leq \delta_2 \label{valid2}.
\end{align}
This criteria ensures that the mesh is periodic and that no two nodes are closer than $\delta_1$ or further apart than $\delta_2$. Further, $\delta_1,\delta_2$ are chosen so that $\delta_1/\delta_2\geq2$ and are both divisors of $L$.
The mesh points themselves evolve directly with the velocity $u$ as,
\begin{align}
\frac{dz_i}{dt}=u(t,z_i).
\label{zevolove}
\end{align}
Equation (\ref{zevolove}) together with one of the models described in section \ref{models} provide a coupled system of equations which can be solved alternately or simultaneously \cite{huang2010adaptive}.
Given that we may now treat the node locations as a function of time $Z_i=Z_i(t)$ it is clear that there will be instances when the criteria for a valid mesh given in equations (\ref{valid1}) and (\ref{valid2}) is violated. In this way we need a suitable remeshing scheme to enforce our criteria. For each $i$ if $|z_{i+1}-z_i|<\delta_1$, $Z_{i+1}$ is deleted. Alternately, if $|z_{i+1}-z{i}|<\delta_2$ a new point $z^*$ is inserted at the mid point between $z_{i+1}$ and $z_i$ and the points are re-indexed according to their order from left to right.
\subsection{Models and Observations} \label{models}
In this work we consider two models for use in our numerical experiments. The first is a diffusive form of Burgers' equation (BGM), \cite{burgers1948}, with periodic boundary conditions and chosen initial condition, given in equations (\ref{BGM}), (\ref{BGMBC}),and (\ref{BGMIC}) respectively,
\begin{align}
BGM:\quad \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial z}=\nu \frac{\partial^2u}{\partial z^2}, \quad z\in [0,1), \label{BGM} \\
BC: u(0,t)=u(1,t), \label{BGMBC} \\
IC: u(z,0)=\sin(2\pi z)+\frac{1}{2} \sin(\pi z), \label{BGMIC}
\end{align}
with viscosity, $\nu=0.08$. We note that Burgers equation has been used in several DA studies \cite{cohn1993dynamics, verlaan2001, pannekoucke2018parametric}. This model is of particular interest because of the steep gradients near the shock, a motivating reason to use an adaptive mesh.
We will also consider a a version of the Kuramoto-Shvashinsky (KSM) equation \cite{papageorgiou1991route} with periodic boundary conditions, and chosen inital condition defined in in equations (\ref{KSM}), (\ref{KSMBC}),and (\ref{KSMIC}) respectively,
\begin{align}
KSM: \quad \frac{\partial u}{\partial t}+\nu\frac{\partial u^4}{\partial z^4}+\frac{\partial u^2}{\partial z}+u\frac{\partial u^4}{\partial z}=0 \quad z\in[0,2\pi)\label{KSM} \\
BC: u(0,t)=u(2\pi,t),\label{KSMBC}\\
IC: u(z,0)=-sin(2\pi z).\label{KSMIC}
\end{align}
Here the viscosity $\nu=0.027$ is chosen so that we see chaotic behavior in the model. Both models are solved using central differences and an Eulerian time stepping scheme with time steps of $10^{-3}$ for BGM and $10^{-5}$ for KSM. The tolerances used in the remeshing scheme outlined in section \ref{amm} are $\delta_1=0.01$ , $\delta=0.02$ for BGM and $\delta_1=0.02\pi$, $\delta_2=0.04 \pi$ for KSM.
The observations are generated from high resolution nature runs for both models. Mean zero white noise is added to the observations and experiments carried out with differing variances. The observations are euilerian and taken on a regularly spaced grid on the interval $[0,L)$ at regular time intervals.
\section{EnKF for an Adaptive Moving Mesh (AMMEnKF)}
The ensemble Kalman filter relies on estimates of error statistics using an ensemble of model runs assumed to be Gaussian distributed. The error estimates themselves are calculated using the state vector formed from each ensemble member. In the case of an Eulerian solver with a fixed mesh, this calculation is easily carried out as the number of nodes and their locations are the same for each ensemble member and thus, the dimension of the state vector is the also the same for each ensemble member. In contrast for an AMM the mesh node locations for each ensemble member will almost certainly be in different locations at an assimilation time. Further, due to the re-meshing outlined in section \ref{amm}, will have different numbers of nodes as well. This makes estimation of the error statistics less direct and lends to a need for the development of modified versions of the EnKF suited to models with solvers like those we consider here. For a non-conservative adaptive mesh solver we see two additional steps to be necessary each with their own important considerations. The first key step needed before applying the EnKF we refer to as {\it dimension matching} needed to provide consistent estimations of ensemble statistics. One would need to decide to add or remove points from ensemble members to achieve the same number of components among state vectors. In addition a sub-step is that of {\it component paring}, that is, how to assign which points, which may be in different locations, are to be compared in the state vectors. The second key step comes after applying the update and we refer to it as {\it dimension return}. This would involve deciding whether or not to remove points which were added, if they were, or if points were removed, weather or not to add points back into the ensemble members. Both of these steps have the potential to disrupt the ensemble statistics and should be tailored to the model and meshing schemes.
One avenue toward an AMMEnKF involves the use of a reference mesh to which each ensemble member can be mapped and on which error statistics can be estimated. This has been explored in \cite{Du2016} with the reference (need to confirm). In \cite{Aydodu2019npg} use of a reference mesh was explored where the reference mesh itself is chosen based on the properties of the mesh adaptation scheme. In particular, two meshes are explored. The first is a high resolution (HR) mesh defined by the node proximity tolerance, $\delta_1$, which ensures {\it at most} one point from each ensemble member can be in any given interval of the partitioned domain. The second is a low resolution mesh (LR) defined by the node separation tolerance, $\delta_2$, which ensures each ensemble member has {\it at least} one point in any given interval of the partitioned domain. In both cases each ensemble member is mapped to the reference mesh before error statistics are calculated and then mapped back to their original meshes after the physical velocity values are updated in the analysis step. As a result the mesh locations are not updated during analysis. However, it is important to consider the fact that the node positions themselves are driven by the physical flow and as such can be considered time dependent state variables. In this work we consider updating the node locations making use of the HR partitioning of the interval domain for the same models considered in \cite{Aydodu2019npg}. The key difference between the previous work and the work presented here is that we now augment our state vector with the node locations and update them in the analysis step. Previous work for a conservative moving mesh was carried out in \cite{bonan2017}, there they also augment their state vector but avoid the issue of dimension matching. We are interested in exploring the use of the augmented state vector to leverage extra statistical information implied by the different meshes among the ensemble members. This is because, in this case, the mesh is connected to the physics and where nodes are clustered or spread apart says something about the system.
We will compare the results of using the augmented state vector on the high resolution mesh to that of using the reference mesh. We will, when needed, describe the methods in \cite{Aydodu2019npg} so that the reader may understand the relevant differences. In particular we focus on the HR method and refer to the augmented state vector as the HRA method. We outline the two approaches below followed by a more detailed discussion of the specifics.
{\bf Dimension Matching}
\begin{itemize}
\item{HR Scheme}
\begin{enumerate}
\item{Create evenly spaced partition of the domain for which at most one node from each ensemble member will be in each interval of the partition.}
\item{Shift each node to the nearest boundary of the interval it occupies.}
\item{Boundaries which remain empty have points interpolated to them from the nearest occupied neighbours.}
\item{Component paring is done naturally on the interval boundaries. Update the physical values }
\end{enumerate}
\item{HRA Scheme}
\begin{enumerate}
\item{Create evenly spaced partition of the domain for which at most one node from each ensemble member will be in each interval of the partition.}
\item{If an interval is empty, randomly assign a node location with in the interval and interpolate to that point.}
\item{Append the locations to the vector of physical values.}
\item{Component paring is done via the interval the components occupy. Update both the physical values and the node locations.}
\end{enumerate}
\end{itemize}
{ \bf Dimension Return}
\begin{itemize}
\item{HR Scheme}
\begin{enumerate}
\item{Remove values from interval boundaries which were interpolated to.}
\item{Shift the physical values from the interval boundaries back to the orignal node locations before the update}
\end{enumerate}
\item{HRA Scheme}
\begin{enumerate}
\item{Since node locations may change, reorder them in the state vector if needed.}
\item{Apply the re-meshing scheme to ensure a valid mesh.}
\item{Remove nodes which occupy previously unoccupied partition intervals.}
\end{enumerate}
\end{itemize}
\subsection{Matching Dimension}
\subsection{HR}\label{dimmatchHR}
In order to avoid the statistical consistency issues presented by having ensemble members with differing numbers of nodes at different locations, one can map each ensemble member to a reference mesh. The reference mesh can be defined on the physical domain $[0,L)$ into $M$ intervals of equal length $\Delta \gamma$,
\begin{align}
[0,L]=L_1\bigcup L_2 \bigcup \dots \bigcup L_m
\end{align}
where $L_i=[\gamma_i,\gamma_i+1)$. In this case $\gamma_1=0$, $\gamma_i=(i-1)\Delta \gamma$ for each i. Further $\Gamma_M=L-\Delta\gamma$ as $0$ and $L$ are identified on the periodic domain. The points $\gamma_i$ form the nodes of the reference grid.
The reference grid is chosen in one of two ways, to ensure that each ensemble member has {\it at most} one point in each interval, $\Delta \gamma=\delta_1$, or that each ensemble member has {\it at least}, $\Delta \gamma=\delta_2$, one point in each interval. The former is referred to as the high resolution mesh (HR) and the latter the low resolution mesh (LR).
Here we focus on the HR mesh as since we partition our physical domain in the same way. The mapping from an ensemble member to the HR mesh will take the state vector $\vec{x}=(u_1 \dots u_N, z_1 \dots z_N) \in \mathbb{R}^{2N}$ to the vector,
\begin{align}
\vec{x}=\left(\vec{\widetilde{u}},\vec{\gamma}\right)^T=(\widetilde{u}_1 \dots \widetilde{u}_M, \gamma_1 \dots \gamma_M)^T \in \mathbb{R}^{2M} \quad \text{with} \quad M\geq N.
\label{svHR}
\end{align}
Here $\widetilde{u}_i$ will be the physical value assigned to $\gamma_i$ through the introduction of a shifted mesh where $L_i\rightarrow \widetilde{L}_i=[\gamma_i-\delta_1/2,\gamma_i+\delta_1/2 )$ for $i=2,\dots M$. The first interval is is taken to be $\widetilde{L}_1=[0\delta_1 /2 )\bigcup[L-\delta_1 /2 )$ since we identify 0 and $L$. if there is a $z_k\in \widetilde{L_i}$, then set $\widetilde{u}_i=u_k$. If there is no such $z_k$ but $z_k\leq\gamma_i$ find $k$ such that $z_k\leq\gamma_i\leq z_{k+1}$ and set
\begin{align}
\widetilde{u}_i=\frac{u_k+u_{k+1}}{2}
\label{setutildehr}
\end{align}
if there is no such $z_k$, then set
\begin{align}
\widetilde{u}_i=\frac{u_1+u_{N}}{2}.
\label{setutildehrend}
\end{align}
This mapping is illustrated in Figure \ref{HRembed}. Once each ensemble member has been mapped to the fixed reference grid the standard stochastic EnKF may be applied.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/embed2.pdf}
\caption{The state vector is formed by interpolating velocity values to the center of any intervals which do not already have a mesh point in them. In this approach the locations of the mesh points that were evolved in the forecast do not change. Rather we only interpolate missing values to create "ghost" points we use during the update step.}\label{embeddings}
\end{figure*}
\subsubsection{HRA} \label{dimmatchHRA}
In the HRA setting, each ensemble member may have a different number of points in different locations. To choose which nodes will be compared, we partition the domain $D=\left[0,L\right)$ into $N$ subintervals ($L_i$) each of length $\delta_1$ so that $D=\bigcup_i L_i$. Since $\delta_1$ is the node proximity tolerance we are guaranteed that each subinterval will have {\it at most} one point in it. With this we can component match nodes which fall in the same subintervals. If an ensemble member does not have a point in a given subinterval we will insert one, a {\it ghost point}, based on the nearest neighbors.
In this approach we take the state vector of a $j^{th}$ ensemble member to be of the form
\begin{align}
\vec{x}=\left(\vec{u},\vec{z}\right)^T=\left(
u_1,u_2, \dots ,\widetilde{u_i},\dots, u_M, z_1, z_2 ,\dots,\widetilde{z_i},\dots,z_M\right)^T
\label{svHRA}
\end{align}
where $u_i$ or $\widetilde{u}_i$ would be the value of the velocity in the $i^{th}$ sub interval of the reference mesh. A value with no tilde would mean the ensemble member had a point in that interval while a tilde implies the member did not have a point in the $j^{th}$ interval and one was inserted based on a weighted average from it's nearest neighbours. The location of an interpolated point is drawn from $N\left(\frac{\gamma_i+\gamma_{i+1}}{2},\delta_1/2\right)$ with a check that the point drawn actually resides in the interval $L_i$, if not we draw again until it does. This is illustrated in Figure \ref{embeddings}. It is possible that we may have in invalid mesh in this process, however we don't enforce validity at this step as there will be many cases where no location in an empty interval can be chosen for which there is not a point with in $\delta_1$ near it. This is because the intervals themselves are of size $\delta_1$.
The physical value assigned to a ghost point $\widetilde{z}_i$ is calculated as:
\begin{align}
\widetilde{u}_i=\frac{b}{a+b}u_l+\frac{a}{a+b}u_r && a=\widetilde{z}_i-z_l, b=z_r-\widetilde{z}_i
\label{physicalinterp}
\end{align}
where $(z_l,u_l)$ and $(z_r,u_r)$ are the closest nodes to $\widetilde{z}_i$ to the left and right respectively and $u_l,u_r$ the corresponding physical values at those nodes. This is done from left to right which does allow for the possibility that a nearest left neighbor may have been a ghost point. However, in the case that $\delta_2=2\delta_1$ we are guaranteed each empty interval will have a non-empty interval to its left and right. This embedding process is illustrated in Figure \ref{embeddings}.
\subsection{Observation Operator}
For the HR method the observation operator takes the form
\begin{align}
h\left(\widetilde{\vec{x}}\right)=
\widetilde{u}_i+\frac{z_j^o-\gamma_i}{\gamma_{i+1}-\gamma_i}(\widetilde{u}_{i+1}-\widetilde{u}_i)
\label{obsopref}
\end{align}
Where $z_j^o$ is the observation location with $\gamma_i \leq z_j^o \leq \gamma_{i+1}$. In a similar way we may define the observation operator for the HRA method as,
\begin{align}
h(\vec{x})=u_i+\frac{z_j^0-z_i}{z_{i+1}-z_i}(u_{i+1}-u_i)
\label{obsop}
\end{align}
Where either $z_i,z_{i+1},u_i,u_{i+1}$ could have a tilde if they were inserted due to the ensemble member having no value in the $i^{th}$ interval.
This form of the observation operator means that we are not considering the location of the observation in the update, just the value of the velocity. The Kalman gain and innovation only have $z$ dependence through the error covariance estimates. The update is then performed using the stochastic EnKF.
\subsection{Analysis using the EnKF}
Once the dimensions of the state vectors of each ensemble member have been matched the stochastic EnKf can be applied in the usual way. We define the forecast ensemble matrix $E^f$ as:
\begin{align}
{\bf E}^f = \left[ \vec{x}_1^f, \dots , \vec{x}^f_{N^e} \right] \in \mathbb{R}^{2M \times N^e} ,
\end{align}
Where the forecast state vectors $\vec{x}_i^f$ takes the form as in equation (\ref{svHR}) for the fixed reference mesh case and equation (\ref{svHRA}) for the augmented case where we also update node locations.
Where $M$ is the number of subintervals, $L_i$, which partition the domain $D$ into subintervals of size $\delta_1$ and $N^e$ the number of ensemble members. Here the vectors ${\vec{x}_i^f}$ are the dimension matched state vectors taken to be the columns of ${\bf E}^f$.
The forecast anomaly matrix ${\bf X}^f$ takes the form
\begin{align}
{\bf X}^f = \frac{1}{\sqrt{N^e-1}}\left[ \vec{x}_1^f-\bar{\vec{x}}^f, \dots , \vec{x}^f_{N^e}-\bar{\vec{x}}^f \right]
\end{align}
where $\bar{\vec{x}}^f$ is the forecast ensemble mean defined as,
\begin{align}
\bar{\vec{x}}^f=\frac{1}{N_e}\sum_{i=1}^{N^e} \vec{x}_i^f
\end{align}
In each case we then employ the stochastic EnKF \cite{Burgers1998} for which the observations are treated as random variables so that each ensemble member is compared to a slightly different perturbation of the observation vector. That is, given an observation vector $\vec{y}$ we generate $N^e$ observations according to,
\begin{align}
\vec{y}_i=\vec{y}+\epsilon_i \quad 1\leq i\leq N^e \quad \epsilon_i \sim N(0,R)
\end{align}
where $R$ is the covariance of the assumed zero mean, white-in-time noise $\epsilon$. We can then calculate the normalized anomaly ensemble of observations,
\begin{align}
Y_o &= \frac{1}{\sqrt{N^e-1}} \left[ \vec{y}_1-\vec{y}, \dots , \vec{y}_{N^e}-y\right] \\
&=\frac{1}{\sqrt{N^e-1}}\left[ \epsilon_1, \dots \epsilon_2\right],
\end{align}
which in turn defines the ensemble observation error covariance matrix,
\begin{align}
R^e=Y_o\left(Y_0\right)^T.
\end{align}
We then define the observed ensemble-anomaly matrix using our observation operator $h$ as,
\begin{align}
Y=h(E^f)-h(\bar{E}^f).
\end{align}
This then defines the Kalman Gain matrix, $K$ to be,
\begin{align}
K=X^fY^T\left[\frac{1}{N^e-1}YY^T+R^e\right]^{-1}
\label{kgain}
\end{align}
which we use to individually update each ensemble member according to,
\begin{align}
\vec{x}^a_i=\vec{x}^f_i+K\left[\vec{y}_i-h(\vec{x}^f_i)\right] \quad 1\leq i \leq N^e
\end{align}
With the HRA method, however, there is the possibility that an ensemble member will have an invalid mesh after the update step. For this reason the re-meshing algorithm is applied to each ensemble member after updating.
In this work we also make use of covariance multiplicative scalar inflation \cite{anderson1999monte} in which the ensemble forecast anomaly matrix is inflated as,
\begin{align}
X^f\rightarrow \alpha X^f,
\end{align}
with $\alpha \geq1$, before $X^f$ is used in the analysis update. This parameter is one that can be tuned through numerical experimentation.
After updating each ensemble member the mean of each analysis can be used to obtain a best estimate of the physical state of the system.
\subsection{Dimension Return}
After the update is complete each ensemble analysis vector has its dimension returned to its pre-analysis value. In the HR case a backward map is used to return them to their original meshes before forecasting again. In the forward mapping step, the mapping indices associating the nodes in the adaptive moving mesh with nodes in the reference mesh are stored in an array. These are the indices resulting from the projections on to the HR reference mesh. This allows us to map the updated physical values $\widetilde{\vec{u}}^a$ back to the mesh that the ensemble member came into the update step with, that is, the values updated at $\gamma_i$ are shifted back to their previous node locations. From there, the forecast is run until the next assimilation time step. It is notable that this can have the effect of introducing some amount of noise in each ensemble member as physical values determined at one location are moved to another, illustrated in Figure \ref{backmap}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/backwardmap.pdf}
\caption{the back map}
\label{backmap}
\end{figure*}
In the HRA case, the physical values and nodes that were in intervals which were unoccupied by a point before the update are deleted for each ensemble member using the stored indices as in the HR case. As will be discussed in \ref{results}, we find that when using the HRA method, the spread in ensemble members can collapse quickly and adding white noise to the physical values, and not the node locations, can improve the update step. We refer to this noise as jitter and it can be added to each ensemble member after the update step. For a given ensemble member analysis vector $\vec{x}_i^a$ the jitter is added according to,
\begin{align}
\vec{x}_i^a=\vec{x}_i^a+\left( N(0,\sigma_j), \vec{0} \right)^T \quad \text{with} \quad N(0,\sigma_j), \vec{0} \in R^{M/2}\quad \text{and} \quad \sigma_j=\alpha_j \max_{u_i,u_j \in \vec{X}}|u_i-u_j|.
\label{jitter}
\end{align}
We will refer to $\alpha_j$ as the amount of jitter. We take $0\leq\alpha_j\leq 1$ so that we add a percentage of the maximum difference between the physical values of the ensemble member. This way the jitter is an adaptive form of additive inflation. We also experimented with this addition of noise in the HR method for comparison and found improvements in time averaged RMSE values for both schemes.
\section{Results}\label{results}
The primary difference we find between the HR and HRA approaches is in how the spread of the ensemble members evolves over time. The mapping to and from the reference mesh with the HR scheme induces stochasticity in the physical values when a value at one location is moved to another. This is not the case in the HRA method since both physical values and their locations are updated together. The spread when using the HR case tends to be higher than the HRA case and infact collapses quickly with the HRA method. This difference is shown in Figure \ref{INHERENT}. The collapse in ensemble spread can be detrimental when the model error causes the ensemble to diverge away from the truth. It is for this reason that we add jitter as defined in equation \ref{jitter}. Overall, we find improvements in RMSE values when adding jitter for both schemes with the HRA method typically preforming better or comparably to the HR method when using scheme specific optimized values of jitter and inflation.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/Inherent_HR_20_0_1.pdf}
\label{INHERENT_0HR}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/Inherent_HRA_20_0_1.pdf}
\label{INHERENT_0_11HR}
\end{subfigure}
\caption{Examples of the spread in the ensembles and an example of an ensemble member for the HR (left) and HRA (right) schemes. The forward and backward mapping of the HR scheme induces some inherent jitter increasing the spread. For the HRA case this does not happen and the ensemble members can collapse quickly. Also notable is the small oscillations in the ensemble member shown for the HR case caused by the mapping procedure.} \label{INHERENT}
\end{figure*}
We consider ensemble size, initial mesh size, and observation error to be largely dictated by the model rather than the DA scheme. Model complexity may limit mesh and ensemble sizes while observation error is instrument dependent. The main parameters that one has total control over are the jitter and inflation and are thus tuneable. In practice one may tune parameters like this with a twin model experiment. With this viewpoint in mind, we ran a sensitivity analysis for both schemes in which we optimize the jitter and inflation for various parameter settings and use optimal pairs to understand how the schemes compare and how that changes over the parameter settings. For these experiments we fix two of the three ``model dictated'' parameters mentioned above and vary the other. For each parameter setting, an optimal jitter and inflation are chosen for each scheme based on the lowest obtained time averaged RMSE error and then that error can be compared between schemes. This way we compare the best performance possible for each scheme for each of the ``model dictated'' parameters examined.
We show these comparisons in figures \ref{optimalBGM} and \ref{optimalKSM} respectively. We also show the average RMSE error for the first spatial derivative of the analysis average, $\partial_x \langle \widetilde{\vec{u}}^a \rangle$. We consider this quantity to determine how well each of the methodologies employed preserve derivative information. This is for two reasons, the first is that we are adding jitter to each ensemble member after the update which distorts the curve smoothness affecting local rates of change. The second is that the mapping scheme in the HR reference mesh method can create artificially sharp changes in function values when mapping to the refrence mesh and when the analysis vector is mapped back to the original ensemble member mesh when the original node location is sufficiently far away from a reference mesh location. These sharp changes over the domain, due to the jitter, HR mapping, or both, can disrupt local rates of change which has to potential to lead to the violation of conservation rules, such as incompressibility ($\nabla \cdot \vec{u}=0$), for example. While we make no direct study of conservation laws in this particular work, we evaluate the fidelity of the first derivative after the update step for each of these methods as a proxy for the potential violation of conservation laws. In these results the time averaged RMSE's for the derivatives are obtained using the parameters that optimize the time averaged RMSE of the solution analysis mean. Depending on the situation, one may run similar experiments and choose a jitter and inflation that best preserve the first derivative if high fidelity of it is needed. In this work for these models, there is not much difference in time averaged RMSE when using parameters that optimize the RMSE either the first derivative or the solution itself. The time averaged RMSE's here are calculated beginning from a time at which transient solutions have passed for both models.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HRA_VS_HR_FUNCTION.pdf}
\label{ENSoptimalfunctionBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HRA_VS_HR_FUNCTION.pdf}
\label{INIoptimalfunctionBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HRA_VS_HR_FUNCTION.pdf}
\label{OBSoptimalfunctionBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HRA_VS_HR_DERIVATIVE.pdf}
\label{ENSoptimalderivativeBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HRA_VS_HR_DERIVATIVE.pdf}
\label{INIoptimalderivativeBGM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HRA_VS_HR_DERIVATIVE.pdf}
\label{OBSoptimalderivativeBGM}
\end{subfigure}
\caption{Sensitivity analysis for the HR and HRA schemes with the BGM model. For each parameter shown along the x-axis, the optimal jitter and inflation are used to obtain a time averaged RMSE for the analysis. The time averaged RMSE values for the spatial derivatives correspond to the parameter which optimize analysis RMSE.}\label{optimalBGM}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HRA_VS_HR_FUNCTION.pdf}
\label{ENSoptimalfunctionKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HRA_VS_HR_FUNCTION.pdf}
\label{INIoptimalfunctionKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HRA_VS_HR_FUNCTION.pdf}
\label{OBSoptimalfunctionKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HRA_VS_HR_DERIVATIVE.pdf}
\label{ENSoptimalderivativeKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HRA_VS_HR_DERIVATIVE.pdf}
\label{INIoptimalderivativeKSM}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HRA_VS_HR_DERIVATIVE.pdf}
\label{OBSoptimalderivativeKSM}
\end{subfigure}
\caption{Sensitivity analysis for the HR and HRA schemes with the KSM model. For each parameter shown along the x-axis, the optimal jitter and inflation are used to obtain a time averaged RMSE for the analysis. The time averaged RMSE values for the spatial derivatives correspond to the parameter which optimize analysis RMSE.}\label{optimalKSM}
\end{figure*}
In Figures \ref{KSMsurf} and \ref{BGMsurf} we show examples of the time averaged RMSE surface as a function of inflation and jitter with the optimal values marked by a red star. The need for some jitter in the HRA method is highlighted by the fact that the time averaged RMSE error increases near the x-axis ($\alpha_j=0$) for both models. While this is also the case for the HR scheme with the BGM model it is less pronounced. For the HR method and the KSM model there is a region with $\alpha_j=0$ that the time averaged RMSE is remains close to the one obtained using optimal jitter and inflation, this is likely achievable due to the chaos in the KSM model naturally increasing the spread. The difference in the smoothness of the contour plots arises from the fact that we run the KSM model for a longer time than the BGM model which dissipates quickly due to the chosen viscosity term. The longer run provides a larger sample of RMSE values to average over producing a smoother surface.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HR_50.pdf}
\label{ENSKSMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HR_50.pdf}
\label{INIKSMSHRURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HR_0.798.pdf}
\label{OBSKSMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS/ENS_HRA_50.pdf}
\label{ENSKSMHRASURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_INI/INI_HRA_50.pdf}
\label{INIKSMSHRAURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_OBS/OBS_HRA_0.798.pdf}
\label{OBSKSMHRASURF}
\end{subfigure}
\caption{Contour plots for the jitter and inflation calibrations for the KSM model for the HR (top row) and HRA (bottom row) schemes for the ``model dictated'' parameters. The points in the plot represent the sample points used while the red star represents the jitter and inflation with the lowest time averaged RMSE. Of particular note is that the HRA scheme has its valley of low RMSE well away from the x-axis implying that some jitter is beneficial while the while this is not the case with the HR method due to the inherent jitter added during the dimension matching step.}\label{KSMsurf}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HR_50.pdf}
\label{ENSBGMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HR_50.pdf}
\label{INIBGMSHRURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HR_0.01.pdf}
\label{OBSBGMHRSURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS/ENS_HRA_50.pdf}
\label{ENSBGMHRASURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_INI/INI_HRA_50.pdf}
\label{INIBGMSHRAURF}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_OBS/OBS_HRA_0.01.pdf}
\label{OBSBGMHRASURF}
\end{subfigure}
\caption{Contour plots for the jitter and inflation calibrations for the BGM model for the HR (top row) and HRA (bottom row) schemes for the ``model dictated'' parameters. The points in the plot represent the sample points used while the red star represents the jitter and inflation with the lowest time averaged RMSE. Of particular note is that the HRA scheme has its valley of low RMSE well away from the x-axis implying that some jitter is beneficial while the while this is not the case with the HR method due to the inherent jitter added during the dimension matching step. }\label{BGMsurf}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HR_10_0_1.13.pdf}
\label{BGMEXJ_0HR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HR_10_0.02_1.13.pdf}
\label{BGMEXJ_0_02HR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HR_10_0.04_1.13.pdf}
\label{BGMEXJ_0_04HR}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HRA_10_0_1.07.pdf}
\label{BGMEXJ_0HRA}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HRA_10_0.02_1.07.pdf}
\label{BGMEXJ_0_02HRA}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_EXAMPLES/EX_HRA_10_0.04_1.07.pdf}
\label{BGMEXJ_0_04HRA}
\end{subfigure}
\caption{Examples of BGM analysis and an ensemble member for different inflation and jitter choices. Top HR and bottom HRA. The RMSE of the analysis mean can be low even when the ensemble members themselves represent unrealistic solutions to the PDE. }\label{wiggles}
\end{figure*}
As discussed earlier the addition of jitter can disrupt the the shape of the ensemble members while improving the analysis. This may be problematic if the ensemble members are used to feed information to another model component. In Figures \ref{KSMsurf} and \ref{BGMsurf} it is apparent that there are regions of low average RMSE but relatively high levels of jitter, in this case the mean is smoothing out the added noise from jitter. In Figure \ref{wiggles} we show examples of the analysis mean, truth, and a typical ensemble member of the BGM model for a fixed inflation and three values of jitter. The inflation chosen corresponds to the optimal value found for an ensemble size of 50. We show zero jitter, the optimal jitter (in terms of lowest time averaged RMSE), and a larger jitter which still has low time averaged RMSE. Artificially inflating a covariance matrix or adding stochasticty to the ensemble members has the potential to disrupt the ensemble members in a negative way, especially if your scheme requires you to disturb each ensemble member as we do here with dimension matching and return. Depending on the application it may be better to sacrifice a small amount of analysis average accuracy to preserve the physical representativeness of each ensemble member in other applications that may not matter.
\begin{figure*}
\centering
{\bf Kurtosis Examples}\par \medskip
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{Figures/KURT_EX/EX_HRA_63_0_1.pdf}
\label{HRALOWKURT}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{Figures/KURT_EX/EX_HRA_50_0.1_1.6.pdf}
\label{HRAHIGHKURT}
\end{subfigure}
\caption{Examples of the three statistical measures we use to measure curve distortion.}\label{KURTEX}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Forecast Members (BGM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_VAR/ENS_HR_90.pdf}
\label{VARBGMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_KURT/ENS_HR_90.pdf}
\label{KURTBGMSHRURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_RMSE/ENS_HR_90.pdf}
\label{RMSEBGMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_VAR/ENS_HRA_90.pdf}
\label{VARBGMHRASURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_KURT/ENS_HRA_90.pdf}
\label{VARBGMSHRAURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_FC_RMSE/ENS_HRA_90.pdf}
\label{RMSEBGMHRASURFFC}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and $RMSE_{ens}$ as a function of jitter and inflation for the BGM model using the forecast ensemble members, right before update, for HR(top row) and HRA(bottom row). It is notable that both $\sigma_{ens}$ and $RMSE_{ens}$ increase primarily as a function of jitter ($\alpha_j$) with less dependence on multiplicative inflation ($\alpha$) while the same is true for decreasing kurtosis evidenced by horizontal contours. This identifies the jitter as the primary source of ensemble member distortion. The red stars represent lowest values and blue stars highest values.}\label{BGMextraFC}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Analysis Members (BGM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_VAR/ENS_HR_90.pdf}
\label{VARBGMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_KURT/ENS_HR_90.pdf}
\label{KURTBGMSHRURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_RMSE/ENS_HR_90.pdf}
\label{RMSEBGMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_VAR/ENS_HRA_90.pdf}
\label{VARBGMHRASURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_KURT/ENS_HRA_90.pdf}
\label{VARBGMSHRAURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/BGM_ENS_IND_AN_RMSE/ENS_HRA_90.pdf}
\label{RMSEBGMHRASURFAN}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and $RMSE_{ens}$ as a function of jitter and inflation for the BGM model using the analysis ensemble members for HR(top row) and HRA(bottom row). It is notable that horizontal contours shown in the same analysis for the forecast members are now slanted with improvements in $\sigma_{ens}$ and $RMSE_{ens}$ for lower values of inflation ($\alpha$). This is due to the update step, however the Kurtosis remains relatively unchanged implying that the members are still distorted for larger values of $\alpha_j$ and only the scale of the errors has been reduced. The red stars represent lowest values and blue stars highest values.}\label{BGMextraAN}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Forecast Members (KSM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_VAR/ENS_HR_90.pdf}
\label{VARKSMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_KURT/ENS_HR_90.pdf}
\label{KURTKSMSHRURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_RMSE/ENS_HR_90.pdf}
\label{RMSEKSMHRSURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_VAR/ENS_HRA_90.pdf}
\label{VARKSMHRASURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_KURT/ENS_HRA_90.pdf}
\label{VARKSMSHRAURFFC}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_FC_RMSE/ENS_HRA_90.pdf}
\label{RMSEKSMHRASURFFC}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and $RMSE_{ens}$ as a function of jitter and inflation for the KSM model using the forecast ensemble members, right before update, for HR(top row) and HRA(bottom row). It is notable that both $\sigma_{ens}$ and $RMSE_{ens}$ increase primarily as a function of jitter ($\alpha_j$) with less dependence on multiplicative inflation ($\alpha$) while the same is true for decreasing kurtosis evidenced by horizontal contours. This identifies the jitter as the primary source of ensemble member distortion. The red stars represent lowest values and blue stars highest values.}\label{KSMextraFC}
\end{figure*}
\begin{figure*}
\centering
{\bf Extra Metrics Analysis Members (KSM)}\par\medskip
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_VAR/ENS_HR_90.pdf}
\label{VARKSMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_KURT/ENS_HR_90.pdf}
\label{KURTKSMSHRURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_RMSE/ENS_HR_90.pdf}
\label{RMSEKSMHRSURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_VAR/ENS_HRA_90.pdf}
\label{VARKSMHRASURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_KURT/ENS_HRA_90.pdf}
\label{VARKSMSHRAURFAN}
\end{subfigure}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{Figures/KSM_ENS_IND_AN_RMSE/ENS_HRA_90.pdf}
\label{RMSEKSMHRASURFAN}
\end{subfigure}
\caption{Surfaces for $\sigma_{ens}$, $K_{ens}$ and $RMSE_{ens}$ as a function of jitter and inflation for the KSM model using the analysis ensemble members for HR(top row) and HRA(bottom row). It is notable that horizontal contours shown in the same analysis for the forecast members are now slanted with improvements in $\sigma_{ens}$ and $RMSE_{ens}$ for lower values of inflation ($\alpha$). This is due to the update step, however the Kurtosis remains relatively unchanged implying that the members are still distorted for larger values of $\alpha_j$ and only the scale of the errors has been reduced. The red stars represent lowest values and blue stars highest values.}\label{KSMextraAN}
\end{figure*}
To better quantify the effect that adding jitter and inflation may have on the physical representativeness of the ensemble members we look at 3 other metrics. The average of the time averaged variance of the difference between the ensemble members ($\sigma_{ens}$) and the truth at each node, the kurtosis ($k_{ens}$) of the same difference, and the average of the time averaged RMSE errors ($RMSE_{ens}$) of the ensemble members. If $\vec{d}_i^{\tau_j}=\left(\vec{u}_i^{\tau_j}-\vec{u}_T^{\tau_j}\right) \in R^M$ is the difference between the $i^{th}$ ensemble member and the truth at assimilation time $\tau_j$ then we can define these quantities as,
\begin{align}
\sigma_{ens}=\frac{1}{N_{ens}}\sum_{i=0}^{N_{ens}} \frac{1}{N_{an}}\sum_{\tau_1}^{\tau_n} \frac{1}{M} \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}-E\left[ d_{i_{k}}^{\tau_j}\right] \right)^2, \label{sigens} \\
k_{ens}=\frac{1}{N_{ens}}\sum_{i=0}^{N_{ens}} \frac{1}{N_{an}}\sum_{\tau_1}^{\tau_n} \frac{\frac{1}{M} \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}-E\left[ d_{i_{k}}^{\tau_j}\right] \right)^4}{\left(\frac{1}{M} \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}-E\left[ d_{i_{k}}^{\tau_j}\right] \right)^2 \right)^2}, \label{kurtens} \\
RMSE_{ens}=\frac{1}{N_{ens}}\sum_{i=0}^{N_{ens}} \frac{1}{N_{an}}\sum_{\tau_1}^{\tau_n} \frac{1}{M} \sqrt{ \sum_{k=0}^M \left(d_{i_{k}}^{\tau_j}\right)^2}. \label{rmseens}
\end{align}
If the ensemble members have a low $\sigma_{ens}$ this would suggest they don't deviate from the mean error along the domain axis which can suggest that the shape of the curve is consistent with a solution to the PDE and had not been overly distorted by the inflation or jitter. The kurtosis gives us a measure of how concentrated around the mean the errors are. A low value for the kurtosis suggests a more uniform distribution with the normal distrubution having a kurtosis of 3. Kurtosis above 3 would suggest either that the probability mass is concentrated around the mean and values far from the mean are rare, or that the probability mass is concentrated in the tails. In this particular case a high $k_{ens}$ likely implies the ensemble members are not overly distorted by jitter and inflation with large error occurring infrequently along the domain. A low $k_{ens}$ signals that the ensemble members are distorted by jitter and inflation with larger deviations from the mean occurring more uniformly. Examples of ensemble members with low and high kurtosis are shown in Figure \ref{KURTEX}, ideally one would hope for each ensemble member to have a low variance, low RMSE and high kurtosis calculated from $\vec{d}_i^{\tau_j}$. In Figures \ref{BGMextraAN}, \ref{BGMextraFC}, \ref{KSMextraAN}, and \ref{KSMextraFC} we show $\sigma_{ens}$, $k_{ens}$ and $RMSE_{ens}$ as a function of $\alpha$ and $\alpha_j$ for ensemble members before and after the update step.
When comparing between the forecast and analysis surfaces there is a detectable change in structure for $\sigma_{ens}$ and $RMSE_{ens}$, before the update both metrics tend to increase with increasing $\alpha_j$ somewhat independently of $\alpha$ and after the update the metrics are significantly reduced for lower values of $\alpha$. This is evidenced by the horizontal contours in the forecast surfaces. It is also clear from the figures that multiplicative inflation does not distort the ensemble members alone. The general structure of the kurtosis remains the same however, implying that the update is reducing the size of the error, but that the distortions remain among the ensemble members. The analysis mean however is smoothed through averaging.
There is also a significant difference between the range of $K_{ens}$ between the BGM (1-15) and KSM (3-5) models. This is likely due to the presence of chaos in the KSM model naturally increasing the spread of the ensemble members causing some to have little distortion but relatively uniform error distribution across the domain.
\section{Conclusions}
Adaptive mesh solvers have the potential to greatly improve model skill and predictions but present difficulties for traditional data assimilation methods such as the EnKF. We consider here the particular case of a non conservative adaptive moving mesh for which each member of an ensemble will have different numbers of nodes in different locations. The key steps in an EnKF scheme for models of this sort are {\it dimension matching} often involving interpolation with a sub step of paring state vector components should they be in different locations, and {\it dimension return}. Dimension return involves removing added points int the matching step, if they were, or if points were removed whether or not to add points back in. Building on the work presented in \cite{Aydodu2019npg} we develop an EnKF scheme for a non-conservative adaptive moving mesh solver in 1-d using an augmented state vector that includes the locations of the nodes, locations that are also updated in the analysis step. Dimension matching is done using the properties of the adaptive mesh scheme itself, via a partition of the domain with intervals the same size as the proximity tolerance $\delta_1$ which guarantees each interval will have at most one node in it. In the HR scheme developed in \cite{Aydodu2019npg} component paring is done by shifting the nodes in each interval to the their nearest interval boundaries and then interpolating new points to any interval boundaries which are empty. In this way the HR method compares ensemble members on the same mesh updating only the physical values of the nodes. Dimension return is then done by deleting interpolated points and shifting the updated physical values back to the previous mesh. In contrast the HRA method leaves ensemble member points where they are and interpolates new points to empty intervals with the location drawn from a normal distribution with variance $\delta_1 / 2$ with a check the location resides within that interval. Component paring is then done using like intervals. Next, the state vector is formed with the node locations appended and both physical values and nodes are updated. After the update a the remeshing scheme is applied to enforce a valid mesh and points in previously empty intervals are deleted.
We find that when updating the node locations ensemble collapse becomes a problem and some additive or multiplicative inflation may be necessary. This is less of an issue for the reference mesh method due to some inherent stocasticity arising from the mapping procedures, although jitter and multiplicative inflation can improve RMSE values there as well. Given an initial mesh size, ensemble size and observation error the jitter and inflation can be optimised with twin model experiments, when this is done we find that the HRA method typically provides better performance interms of the time average RMSE for both the function and the first derivative. When using additive inflation, such as the jitter as we have defined it, there is potential to distort ensemble members while still obtaining a good analysis mean. This could be problematic if the ensemble members are used as input to some other model component, particularly if it is their derivative that is needed. {\color{red}Is there a nice citation where this is shown to be a problem? Would one ever do this?} To quantify the severity of the distortions we calculate several metrics defined in equations \ref{sigens}, \ref{kurtens} and \ref{rmseens}. From this analysis we can see that the addition of jitter is primarily responsible for distorting the ensemble members while multiplicative inflation does not. One would want to weigh what is more important, low RMSE of the analysis mean or preserving ensemble member shape and should be application specific.
We would also like to address the question of computational efficiency between these two approaches. When using the augmented state vector of the HRA scheme the size of the error covariance matrix is doubled compared to that of the HR scheme. For very high dimensional models this may be problematic, yet as can be seen in Figure \ref{optimalKSM} when updating the node locations the first spatial derivative RMSE is much improved and if that information is needed the extra computational cost may be worth it.
{\color{red}Wonder if I should say something about neXtSIM here?}
\selectlanguage{english}
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"redpajama_set_name": "RedPajamaGithub"
} | 3,207 |
{"url":"https:\/\/www.mathdoubts.com\/cofunction-identities\/","text":"Cofunction identities\n\nA mathematical relation of two trigonometric functions whose angles are complementary is called cofunction identity.\n\nList\n\nCo-function identities can be called as complementary angle identities and also called as trigonometric ratios of complementary angles. There are six trigonometric ratios of complementary angle identities in trigonometry.\n\nRemember, theta ($\\theta$) and $x$ represent angle of right triangle in degrees and radians respectively. You can use any one of them as formula in trigonometry problems.\n\nSin cofunction identity\n\nThe sine of complementary angle is equal to cosine of angle.\n\nIn Degrees\n\n$\\sin{(90^\\circ-\\theta)} \\,=\\, \\cos{\\theta}$\n\n$\\sin{\\Big(\\dfrac{\\pi}{2}-x\\Big)} \\,=\\, \\cos{x}$\n\nCos cofunction identity\n\nThe cosine of complementary angle is equal to sine of angle.\n\nIn Degrees\n\n$\\cos{(90^\\circ-\\theta)} \\,=\\, \\sin{\\theta}$\n\n$\\cos{\\Big(\\dfrac{\\pi}{2}-x\\Big)} \\,=\\, \\sin{x}$\n\nTan cofunction identity\n\nThe tangent of complementary angle is equal to cotangent of angle.\n\nIn Degrees\n\n$\\tan{(90^\\circ-\\theta)} \\,=\\, \\tan{\\theta}$\n\n$\\tan{\\Big(\\dfrac{\\pi}{2}-x\\Big)} \\,=\\, \\cot{x}$\n\nCot cofunction identity\n\nThe cotangent of complementary angle is equal to tangent of angle.\n\nIn Degrees\n\n$\\cot{(90^\\circ-\\theta)} \\,=\\, \\tan{\\theta}$\n\n$\\cot{\\Big(\\dfrac{\\pi}{2}-x\\Big)} \\,=\\, \\tan{x}$\n\nSec cofunction identity\n\nThe secant of complementary angle is equal to cosecant of angle.\n\nIn Degrees\n\n$\\sec{(90^\\circ-\\theta)} \\,=\\, \\csc{\\theta}$\n\n$\\sec{\\Big(\\dfrac{\\pi}{2}-x\\Big)} \\,=\\, \\csc{x}$\n\nCosec (or) Csc cofunction identity\n\nThe cosecant of complementary angle is equal to secant of angle.\n\nIn Degrees\n\n$\\csc{(90^\\circ-\\theta)} \\,=\\, \\sec{\\theta}$\n\n$\\csc{\\Big(\\dfrac{\\pi}{2}-x\\Big)} \\,=\\, \\sec{x}$","date":"2020-03-30 16:03:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9217779636383057, \"perplexity\": 3695.618857309747}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585370497171.9\/warc\/CC-MAIN-20200330150913-20200330180913-00480.warc.gz\"}"} | null | null |
March 1, 2013 (originally published by Booz & Company)
You Gotta Serve Somebody
Jeff Thull, author of Mastering the Complex Sale: How to Compete and Win When the Stakes Are High, introduces a passage that overturns negative stereotypes about sales from To Sell Is Human: The Surprising Truth about Moving Others, by Daniel H. Pink.
If you want to succeed in sales, set aside manipulative games and mesmerizing presentations. Instead, figure out how you can improve your customer's life. This is particularly important in the increasingly complex B2B markets, where customers face daunting buying decisions that can involve billions of dollars and where sales professionals struggle to differentiate themselves and their solutions from competitors.
Let's also set aside outdated sales stereotypes once and for all. I've always found that the most successful sales professionals are what Dan Pink would call servant sellers. In the excerpt that follows, Pink hits the nail on the head when he quotes Robert Greenleaf's vision of servant leadership—"do no harm...listen first...accept and empathize"—as a model for sales professionals. This model is well-supported by my firm's research: The thinking and behavior of top salespeople are a close match to those of the best doctors. They diagnose and prescribe while keeping the well-being of their clients foremost in their minds.
In the rush to close deals and make the numbers, it is too easy to forget that we need to address the hopes, fears, and aspirations of our customers. But today's customers are not saying, ''We need solutions!'' They are saying, ''We need help!'' If you reframe the way you sell in line with this reality, it will lead you to more valuable business relationships.
— Jeff Thull
An excerpt from chapter 9 of To Sell Is Human: The Surprising Truth about Moving Others
In 2008, [Wharton professor Adam Grant] carried out a fascinating study of a call center at a major U.S. university. Each night, employees made phone calls to alumni to raise money for the school. As is the habit of social psychologists, Grant randomly organized the fund-raisers into three groups. Then he arranged their work conditions to be identical—except for the five minutes prior to their shift.
For two consecutive nights, one group read stories from people who'd previously worked in the call center, explaining that the job had taught them useful sales skills.... This was the "personal benefit group." Another—the "purpose group"—read stories from university alumni who'd received scholarships funded by the money this call center had raised describing how those scholarships had helped them. The third collection of callers was the control group, who read stories that had nothing to do with either personal benefit or purpose. After the reading exercise, the workers hit the phones, admonished not to mention the stories they'd just read to the people they were trying to persuade to donate money.
A few weeks later, Grant looked at their sales numbers. The "personal benefit" and control groups secured about the same number of pledges and raised about the same amount of money as they had in the period before the story-reading exercise. But the people in the purpose group kicked into overdrive. They more than doubled "the number of weekly pledges that they earned and the amount of weekly donation money that they raised."
Sales trainers, take note. This five-minute reading exercise more than doubled production. The stories made the work personal; their contents made it purposeful. This is what it means to serve: improving another's life and, in turn, improving the world. That's the lifeblood of service and the final secret to moving others.
In 1970, an obscure sixty-six-year-old former mid-level AT&T executive named Robert Greenleaf wrote an essay that launched a movement. He titled it "Servant as Leader"—and in a few dozen earnest pages, he turned the reigning philosophies of business and political leadership upside down. Greenleaf argued that the most effective leaders weren't heroic, take-charge commanders but instead were quieter, humbler types whose animating purpose was to serve those nominally beneath them. Greenleaf called this notion "servant leadership" and explained that the order of those two words held the key to its meaning. "The servant-leader is servant first," he wrote. "Becoming a servant-leader begins with the natural feeling that one wants to serve, to serve first. Then conscious choice brings one to aspire to lead."
The very idea of leaders subordinating themselves to followers, of inverting the traditional pyramid, made many people uncomfortable. But Greenleaf's philosophy excited many more. Those who embraced it learned to "do no harm," to respond "to any problem by listening first," and "to accept and empathize" rather than reject. Over time, companies as diverse as Starbucks, TD Industries, Southwest Airlines, and Brooks Brothers integrated Greenleaf's ideas into their management practices. Business schools added Greenleaf to their reading lists and syllabi. Nonprofit organizations and religious institutions introduced his principles to their members.
What helped servant leadership take hold wasn't merely that many of those who tried it found it effective. It was also that the approach gave voice to their latent beliefs about other people and their deeper aspirations for themselves. Greenleaf's way of leading was more difficult, but it was also more transformative. As he wrote, "The best test, and the most difficult to administer, is this: Do those served grow as persons? Do they, while being served, become healthier, wiser, freer, more autonomous, more likely themselves to become servants?"
The time is ripe for the sales version of Greenleaf's philosophy. Call it servant selling. It begins with the idea that those who move others aren't manipulators but servants. They serve first and sell later. And the test—which, like Greenleaf's, is the best and the most difficult to administer—is this: If the person you're selling to agrees to buy, will his or her life improve? When your interaction is over, will the world be a better place than when you began?
Servant selling is the essence of moving others today. But in some sense, it has always been present in those who've granted sales its proper respect. For instance, Alfred Fuller [founder of The Fuller Brush Company]... said that at a critical point in his own career, he realized that his work was better—in all senses of the word—when he served first and sold next. He began thinking of himself as a civic reformer, a benefactor to families, and "a crusader against unsanitary kitchens and inadequately cleaned homes." It seemed a bit silly, he admitted. "But the successful seller must feel some commitment that his product offers mankind as much altruistic benefit as it yields the seller money." An effective seller isn't a "huckster, who is just out for profit," he said. The true "salesman is an idealist and an artist."
So, too, is the true person. Among the things that distinguish our species from others is our combination of idealism and artistry—our desire both to improve the world and to provide that world with something it didn't know it was missing. Moving others doesn't require that we neglect these nobler aspects of our nature. Today it demands that we embrace them. It begins and ends by remembering that to sell is human.
Reprinted with permission from Riverhead Books, a member of the Penguin Group (USA). Copyright 2012 by Daniel H. Pink.
Jeff Thull (jpthull@primeresource.com) is founder and CEO of Prime Resource Group, a sales consultancy. He is the author of Mastering the Complex Sale: How to Compete and Win When the Stakes Are High, 2nd Edition (Wiley, 2010), Exceptional Selling: How the Best Connect and Win in High Stakes Sales (Wiley, 2006), and The Prime Solution: Close the Value Gap, Increase Margins, and Win the Complex Sale (Dearborn, 2005).
To Sell Is Human: The Surprising Truth about Moving Others (Riverhead Books, 2012), by Daniel H. Pink
Daniel H. Pink (dhp@danpink.com) is also the author of A Whole New Mind: Moving from the Information Age to the Conceptual Age (Riverhead Books, 2005) and Free Agent Nation: The Future of Working for Yourself (Warner Books, 2002). Previously, he served as chief speechwriter for Vice President Al Gore.
Topics: sales, leadership, skills, leaders, execution
A brave new workless world
Pessimism dematerialized: Four reasons to be hopeful about the future
Before Murphy Brown and Liz Lemon, there was Emma McChesney | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,337 |
using System.Collections.Generic;
using BizHawk.Emulation.Common;
using BizHawk.Emulation.Cores.Components.M68000;
namespace BizHawk.Emulation.Cores.Consoles.Sega.gpgx
{
public partial class GPGX : IDisassemblable
{
public string Cpu
{
get
{
return "M68000";
}
set
{
}
}
public string PCRegisterName
{
get { return "M68K PC"; }
}
public IEnumerable<string> AvailableCpus
{
get { yield return "M68000"; }
}
public string Disassemble(MemoryDomain m, uint addr, out int length)
{
_disassemblerInstance.ReadWord = (a) => (short)m.PeekUshort(a, m.EndianType == MemoryDomain.Endian.Big);
_disassemblerInstance.ReadByte = (a) => (sbyte)m.PeekByte(a);
_disassemblerInstance.ReadLong = (a) => (int)m.PeekUint(a, m.EndianType == MemoryDomain.Endian.Big);
var info = _disassemblerInstance.Disassemble((int)addr);
length = info.Length;
return string.Format("{0:X4} {1,-7} {2}", info.RawBytes.Substring(0, 4), info.Mnemonic, info.Args);
}
// TODO: refactor MC6800's disassembler to be a static call
private MC68000 _disassemblerInstance = new MC68000();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,884 |
There's confusion surrounding the position of Owen Heary as Sligo Rovers manager.
The club has yet to make any statement on the future of their manager, despite reports that veteran player Joseph Ndo took charge of training yesterday.
Heary only took charge at the Showgrounds last October, when he signed a two-year contract. Former Shelbourne midfielder Stuart Byrne says that Sligo need to clear-up the confusion. | {
"redpajama_set_name": "RedPajamaC4"
} | 76 |
THE STATISTICIAN'S GUIDE TO UTOPIA: THE FUTURE OF GROWTH
by Morten Tønnessen
"... Abstract. In this article I paint a concise portrait of world economic and population history. Key factors include the world population and Gross Domestic Product (GDP). The role of technology in relation to the environmental impact of economic activity is represented by an Environmental Efficiency ..."
GDP. In conclusion, I describe under what circumstances it is conceivable that the growth economy can persist for at least 300 more years. Directions of inquiry are offered to three groups: Those who want to maintain the growth economy for as long as possible; those who want world population to stay
Bootstrap - inspired techniques in computation intelligence
by Robi Polikar, Prof Dr, Karl Heinrich Hofmann - Signal Processing Magazine, IEEE , 2007
"... [Ensemble of classifiers for incremental learning, data fusion, and missing feature analysis] This article is about the success story of a seemingly simple yet extremely powerful approach that has recently reached a celebrity status in statistical and engineering sciences. The hero of this story—boo ..."
—bootstrap resampling—is relatively young, but the story itself is a familiar one within the scientific community: a mathematician or a statistician conceives and formulates a theory that is first developed by fellow mathematicians and then brought to fame by other professionals, typically engineers, who point to many
Data Selection For Broadcast News Csr Evaluations
by William Fisher, Walter S. Liggett, Audrey Le, Jonathan G. Fiscus, David S. Pallett - Proc. of the Broadcast News Transcription and Understanding Workshop, February 811
"... Composition of the 1997 Hub-4 broadcast news test set is discussed. The composition is based on concurrent selection of a statistically-equivalent test set for a future evaluation, adjustment of the set to match the training data, and other considerations. This paper discusses both the principles in ..."
to be the focus of a CSR research effort. A statistician might conceive of this class of material as a po...
4 Trends in Statistical and Analytic Methodology: Implications for National Surveys THIS PAGE INTENTIONALLY LEFT BLANK"So What? " The Implications of New Analytic Methods for Designing NCES Surveys
by Robert F. Boruch, George Terhanian
"... This report was commissioned to address the question "How can advances in statistical analysis be used to improve the design of surveys? " The surveys of paramount interest are those sponsored by the National Center for Education Statistics (NCES). The "advances, " as initially conceived, include ne ..."
This report was commissioned to address the question "How can advances in statistical analysis be used to improve the design of surveys? " The surveys of paramount interest are those sponsored by the National Center for Education Statistics (NCES). The "advances, " as initially conceived, include
Invited Paper Confidentiality and Data Protection Through Disclosure Limitation: Evolving Principles and Technical Advances 1
by Stephen E. Fienberg
"... Confidentiality and privacy are widely conceived of as ethical matters and they impinge directly upon the work of statisticians and statistical agencies. But providing access to publicly-collected data is also an ethical matter and the goal of agencies should be to release the maximal amount of info ..."
Confidentiality and privacy are widely conceived of as ethical matters and they impinge directly upon the work of statisticians and statistical agencies. But providing access to publicly-collected data is also an ethical matter and the goal of agencies should be to release the maximal amount
by William Fisher Walter, Walter S. Liggett, Audrey Le, Jonathan G. Fiscus, David S. Pallett - Proc. of the Broadcast News Transcription and Understanding Workshop, February 811
SEVEN NewPerspectiveson(SomeOld)Problems of Frequentist Statistics I Frequentist Statistics as a Theory of Inductive Inference 1
by Deborah G. Mayo, David Cox
"... The philosophical foundations of statistics may be regarded as the study of the epistemological, conceptual, and logical problems revolving around the use and interpretation of statistical methods, broadly conceived. As with other domains of philosophy of science, work in statistical science progres ..."
The philosophical foundations of statistics may be regarded as the study of the epistemological, conceptual, and logical problems revolving around the use and interpretation of statistical methods, broadly conceived. As with other domains of philosophy of science, work in statistical science
December 2009Important Lessons from Studying the Chinese Economy
by Gregory C. Chow, Gregory C. Chow , 2009
"... to visit China and study the Chinese economy. After doing so for thirty years since and advising the government of Taiwan in the 1960s and the 1970s and the government of the People's Republic of China in the 1980s and the 1990s this is an opportune moment for me to summarize the important lessons t ..."
an economic relation using the official data the result confirmed the well-established economic theory. It would be a miracle if I had the power to make the Chinese official statisticians fabricate data to support my hypotheses. Even if I had had the power, most of the data had already been published
2nd Lehmann Symposium- Optimality IMS Lecture Notes- Mongraphs Series (2006) FREQUENTIST STATISTICS AS A THEORY OF INDUCTIVE INFERENCE
by G. Mayo, David R. Cox
"... After some general remarks about the interrelation between philosophical and statistical thinking, the discussion centres largely on significance tests. These are defined as the calculation of p-values rather than as formal procedures for "acceptance " and "rejection". A number of types of null hypo ..."
, broadly conceived. As with other domains of philosophy of science, work in statistical science progresses largely without worrying about "philosophical foundations". Nevertheless, even in statistical practice, debates about the different approaches to statistical analysis may influence and be influenced
"... A few years ago, Camus & Lima (2002) wrote an essay to stimulate ecologists to think about how we define and use a fundamental concept in ecology: the population. They concluded, concurring with Berryman (2002), that a population is "a group of individuals of the same species that live toge ..."
of ecological terms. The point I wish to stress here is that we ecologists tend to forget that when we use statistical tools to infer results from our sample to a population we work with what statisticians term "imaginary", "hypothetical " or "potential " popula-tions. As Zar (1999 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,752 |
In late 2003 Lason was named New Zealand's first equal Automotive Electrical Apprentice of the Year by MTA. His employer said that they had quite a few customers who would specifically ask for Jason due to his excellent knoweldge and was often mistaken for a qualified tradesperson due to his high work standard and ethics while still a Modern Apprentice.
It was focus and motivation that saw Michael gain top marks nationwide in his New Zealand Motor Industry Training Organisation (MITO) National Certificate in Panel Beating examination. A year later he was also first equal in the MITO A-Grade Panel Beating examination. In 2003 he received the Collision Repair Association's Golden Hammer Award. The Golden Hammer is the highest award in the panel beating trade and it is believed it was the first time that the award came to Gisborne.
One of the regions most skilled apprentices after taking part in the joinery section of the 2002 Waikato Regional SkillEX competition and winning first place in joinery.
Ari-Wayne has been awarded a gold medal from the New Zealand Baking Society as Apprentice of the Year for the second year. Hundreds of bakeries entered their apprentices in the competition that Ari-Wayne won.
Modern Apprentice mechanic, Michael Tu, ( Te Aitangi a Mahaki) has come a long way from his days at Waikohu College when he didn't really know what he wanted to do. A one-week overview course at Gisborne Development Inc in 1999, which Michael really enjoyed, led to his decision to enrol in the Rangatahi Maia Skill Enhancement automotive engineering course at GDI in January 2000. This was followed by a Pathway to apprenticeships course in January 2001 at Gisborne Development Inc and he has not looked back since. Doing the courses locally knocked off about two years from his apprenticeship and the tutors pushed Michael along and kept him motivated.
An integral part of the course content at Gisborne Development Inc is on the job training. Michael carried out work experience at several local automotive workshops with the positive outcome of full-time employment at Watts Motors, Gisborne in August 2001 run by Brian Watts, Owner/Manager. Modern Apprenticeships Co-ordinator Chris Hatwell, (also C.E.O. of Gisborne Development Inc) saw the potential for Michael to do well and has watched him mature from his early days in 1999 as a cheeky teenager to a hardworking young adult with definite goals and a desire to succeed in 2002.
Michael has continued to study and complete his units of learning and in addition to the goal of completing his Modern Automotive Apprenticeship modules as soon as possible, he is also studying a Honda NZ apprenticeship through correspondence. He also wants to gain experience in other facets of the organisation at Watts Motors and works in the truck shop when required and runs the parts departments on weekends, so his job is quite varied. He finds his career as a mechanic challenging and enjoys both driving the expensive cars as well as working on them whilst learning a skill that's basically universal no matter what country you live in.
When asked if he would encourage other young people to follow the Rangatahi Maia path, Michael's advice was, "It's really something for people to make up their own minds about, but my suggestion is to 'Go for it! It's much better than sitting around all day doing nothing."
Brent was encouraged to compete for the Joinery Manufacturers Customwood Award for Year 1 apprentices. After much deliberation Brent decided to make a replica of the Auckland Sky Tower - which, whilst being a challenge was certainly eye catching. He was awarded first place for his efforts at the New Zealand Joinery Association Annual Conference.
Steven was 1998 New Zealand Top Apprentice of the Year when he gained his National Certificate as a Craftsperson Joiner. Steven is now self employed and living in Wellington restoring old houses.
Christina in 1998 topped New Zealand when she gained her National Certificate in Signmaking. Christine is one of the few women in this trade. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,001 |
import Queue
import logging
import threading
import time
import subprocess
import os
from datetime import datetime, timedelta
from mirrors.libmirrors import t2s
class Singleton(type):
"""Singleton Class for RepoManager."""
_instances = {}
def __call__(cls, *args, **kwargs):
if cls not in cls._instances:
cls._instances[cls] = super(Singleton, cls).__call__(*args, **kwargs)
return cls._instances[cls]
class Repo(object):
def __init__(self, name, config):
"""A repo object which stores info about a single repo.
:param str name: Name of repo
:param config: running config options
:type config: ConfigParser.ConfigParser
"""
# ConfigParser Object which all of the repo info is stored under the repo name
self.config = config
# Name of the Repo
self.name = name
# inactive repos will not run
self.deactive = False
# Status of Repo Queue
self.queued = False
# Singleton of RepoManager
self.repo_manager = RepoManager()
# Contains rsync_thread
self.__sync = None
# Config Validation Section
if not self.config.has_option(self.name, 'source'):
raise RepoConfigError("No Source Defined".format(self.name), self.name)
if not self.config.has_option(self.name, 'destination'):
self.config.set(self.name, 'destination', './distro/')
directory = os.path.dirname(self.config.get(self.name, 'destination'))
if not os.path.exists(directory):
logging.info("Creating {0}".format(directory))
os.makedirs(directory)
if not self.config.has_option(self.name, 'rsync_args'):
raise RepoConfigError("No rsync_args Defined".format(self.name), self.name)
if not self.config.has_option(self.name, 'weight'):
self.config.set(self.name, 'weight', '0')
if self.config.has_option(self.name, 'deactive'):
self.deactive = self.config.getboolean(self.name, 'deactive')
else:
self.config.set(self.name, 'deactive', 'False')
if self.config.has_option(self.name, 'async_sleep') and self.config.has_option(self.name, 'hourly_sync'):
raise RepoConfigError("Both async_sleep and hourly_sync cannot be defined".format(self.name), self.name)
elif not self.config.has_option(self.name, 'async_sleep') and not self.config.has_option(self.name, 'hourly_sync'):
raise RepoConfigError("Either async_sleep or hourly_sync must be defined".format(self.name), self.name)
if not self.config.has_option(self.name, 'pre_command'):
self.config.set(self.name, 'pre_command', '')
if not self.config.has_option(self.name, 'post_command'):
self.config.set(self.name, 'post_command', '')
if not self.config.has_option(self.name, 'log_file'):
self.config.set(self.name, 'log_file', './log/{0}.log'.format(self.name))
logging.info("No log_file declared in {0}, defaulting to '{0}.log'".format(self.name))
# end config validation section
log_file = self.config.get(self.name, "log_file")
directory = os.path.dirname(log_file)
if not os.path.exists(directory):
logging.info("Creating {0}".format(directory))
try:
os.makedirs(directory)
except IOError:
logging.error("Failed to create {0}".format(directory))
try:
open(log_file, 'a').close()
logging.debug("{0} log file good for writing".format(self.name))
except IOError:
logging.error("Error opening {0} for writing".format(self.name))
if(self.deactive):
logging.info("{0} loaded successfully, but disabled".format(self.name))
else:
logging.info("{0} loaded successfully".format(self.name))
def is_alive(self):
"""Bool of syncing status."""
if self.__sync:
return bool(self.__sync.p)
return False
def running_time(self):
"""Total running time of active sync.
:rtype: int
:returns: An int of total syncing time elapsed
:rtype: None
:returns: None if not syncing
"""
if self.__sync:
if self.is_alive():
delta = datetime.now() - self.__sync.start_time
return delta - timedelta(microseconds=delta.microseconds)
def sleep_time(self):
"""Sleep duration of sleeping sync.
:rtype: int
:returns: A int of time elapsed since sleeping
:rtype: None
:returns: None if not in sleeping state
"""
if self.__sync:
if self.__sync.sleep_start:
delta = datetime.now() - self.__sync.sleep_start
return delta - timedelta(microseconds=delta.microseconds)
def time_remaining(self):
"""Return time left until sleep is over.
:rtype: int
:returns: A int of time remaining in sleep state
:rtype: None
:returns: None if not in sleeping state
"""
if self.__sync:
if self.__sync.sleep_start:
delta = timedelta(seconds=t2s(self.config.get(self.name, "async_sleep"))) - self.sleep_time()
return delta - timedelta(microseconds=delta.microseconds)
def terminate(self):
"""Send SIGTERM To the rsync process."""
if self.is_alive():
logging.info("Terminating {0}".format(self.name))
self.__sync.p.terminate()
def kill(self):
"""Send SIGKILL To the rsync process."""
if self.is_alive():
logging.info("KIlling {0}".format(self.name))
self.__sync.p.kill()
def __rebuild(self):
"""Destroy and recreate the rsync object and settings.
This will wipe all currently running rsync timers
"""
self.__sync = self.rsync_thread(self.name, self.config)
def start_sync(self):
"""Run an rsync against the repo source."""
self.__rebuild()
self.__sync.start()
class rsync_thread(threading.Thread):
"""Extended threading.Thread class to control rsync via subprocess.
:param str name: Name of repo
:param config: Running config options
:type config: Configparser.Configparser
"""
def __init__(self, name, config):
threading.Thread.__init__(self)
self.config = config
self.p = None
self.name = name
# Singleton of RepoManager
self.repo_manager = RepoManager()
self.start_time = None
self.finish_time = None
self.start_sleep = None
self.thread_timer = None
self.daemon = True
def run(self):
logging.debug("Opening {0} for writing".format(self.config.get(self.name, 'log_file')))
output_file = open(self.config.get(self.name, 'log_file'), 'a')
logging.debug("Running rsync with {0} {1} {2}".format(
self.config.get(self.name, "rsync_args"),
self.config.get(self.name, "source"),
self.config.get(self.name, "destination")))
self.start_time = datetime.now()
logging.info("Starting sync {0} at {1}".format(self.name, self.start_time))
self.p = subprocess.Popen("rsync {0} {1} {2}".format(
self.config.get(self.name, "rsync_args"),
self.config.get(self.name, "source"),
self.config.get(self.name, "destination")).split(),
shell=False,
stdout=output_file,
stderr=subprocess.STDOUT)
# bock until the subprocess is done
self.p.wait()
if self.config.get(self.name, "post_command"):
logging.debug("running post_cmd {0}".format(self.config.get(self.name, "post_command")))
self.post_cmd = subprocess.Popen("{0}".format(
self.config.get(self.name, "post_command")),
shell=True,
stdout=output_file,
stderr=subprocess.STDOUT)
self.post_cmd.wait()
logging.info("Done running post_command for {0}".format(self.name))
t = t2s(self.config.get(self.name, "async_sleep"))
self.thread_timer = threading.Timer(t, self.repo_manager.enqueue, [self.name])
self.thread_timer.start()
# Time that thread starts sleeping
self.sleep_start = datetime.now()
# clear out the current process when it finishes
self.p = None
# Remove state from running_syncs
self.repo_manager.running_syncs -= 1
self.finish_time = datetime.now()
logging.info("finished {0} at {1}, sleeping for {2}".format(self.name, self.finish_time, self.config.get(self.name, "async_sleep")))
logging.debug("closing {0}".format(self.config.get(self.name, 'log_file')))
output_file.close()
class RepoManager(object):
__metaclass__ = Singleton
def __init__(self, config):
"""Singleton manager of the repositories and threading.
:param config: Running config options
:type config: Configparser.Configparser
"""
# configparser object which all of the repomanager configs are stored under the GLOBAL Section
self.config = config
# priority queue for async processing
self.repo_queue = Queue.PriorityQueue(0)
# list of repo objects
self._repo_dict = dict()
if not self.config.has_section("GLOBAL"):
raise GlobalError("Config requires GLOBAL Section")
if not self.config.has_option('GLOBAL', 'async_processes'):
raise GlobalError("No async_processes value defined in GLOBAL")
if not self.config.has_option('GLOBAL', 'check_sleep'):
config.set("GLOBAL", 'check_sleep', '30')
# current running syncs: compared against max set in config
self.running_syncs = 0
self.async_thread = threading.Thread(name="async_control", target=self.__check_queue)
self.async_thread.daemon = True
self.async_thread.start()
def __check_queue(self):
"""Queue loop checker for async_control."""
while(True):
# Check for inactive repos
found = None
while not found:
repo = self.repo_queue.get()[1]
if not repo.deactive:
found = True
else:
# If inactive, toss aside
break
if self.running_syncs <= self.config.getint("GLOBAL", "async_processes"):
logging.debug("Acquired {0}".format(repo.name))
repo.queued = False
self.running_syncs += 1
logging.debug("Running Sync {0}, {1} slots available".format(repo.name, self.config.getint("GLOBAL", "async_processes")-self.running_syncs))
repo.start_sync()
else:
logging.debug("Requeuing {0}, no open threads".format(repo.name))
self.repo_queue.put([-11, repo])
time.sleep(30)
def get_repo(self, name):
"""Return repo object if exists.
:param str name: name of repo
:rtype: Repo
:returns: Repo Object
:rtype: None
:returns: None if no repo exists by passed in name
"""
if name in self._repo_dict:
return self._repo_dict[name]
def gen_repo(self):
"""Generator for repo_dict.
:rtype: Repo
:returns: Repo Object
"""
for name in self._repo_dict:
yield self._repo_dict[name]
def add_repo(self, name):
"""Create a repo for a section in the running config.
:param str name: Name of repo
:raises Repo.RepoConfigError: if no config exists for given repo name
"""
if self.get_repo(name):
raise RepoConfigError("Cannon create repo {0}, already created".format(name), name)
if self.config.has_section(name):
repo = Repo(name, self.config)
self._repo_dict[name] = repo
else:
raise RepoConfigError("Cannot create repo, section {0} does not exist".format(name), name)
def deactivate(self, name):
"""Deactivate repo from syncing.
:param str name: Name of repo
:raises Repo.RepoError: if no repo exists by given name
"""
if self.get_repo(name):
if self.get_repo(name).deactive:
# nothing to do, already deactive
return
self.get_repo(name).deactive = True
logging.info("Deactivating {0}".format(name))
else:
raise RepoError("No Repo Named {0}".format(name), name)
def activate(self, name):
"""Activate repo for syncing.
:param str name: Name of Repo
:raises Repo.RepoError: if no repo exists by given name
"""
if self.get_repo(name):
if not self.get_repo(name).deactive:
# nothing to do, already active
return
self.get_repo(name).deactive = False
self.enqueue(name)
logging.info("Activating {0}".format(name))
else:
raise RepoError("No Repo Named {0}".format(name), name)
def status(self, name):
"""Return status of Repo.
:param str name: Name of Repo
:rtype: str
:returns: str status of Repo
"""
if not self.get_repo(name):
raise RepoError("Repo {0} doesn't exist".format(name), name)
if self.get_repo(name).deactive:
return "{0} is deactive".format(name)
elif self.get_repo(name).queued:
return "{0} is queued".format(name)
elif self.get_repo(name).is_alive():
return "{0} is syncing, active for {1}".format(name, self.get_repo(name).running_time())
else:
return "{0} is sleeping, sync in {1}".format(name, self.get_repo(name).time_remaining())
def del_repo(self, name):
"""Delete repo object from dict.
:param str name: Name of repo
:raises Repo.RepoError: if no repo exists by passed in name.
"""
if self.get_repo(name):
del self._repo_dict[name]
else:
raise RepoError("Cannot delete repo, repo {0} does not exist".format(name))
def enqueue(self, name):
"""Add repo to the queue.
:param str name: Name of repo
:raises Repo.RepoError: if repo is already queued or doesn't exist
"""
if not self.get_repo(name):
raise RepoError("Repo {0} doesn't exist".format(name), name)
if self.get_repo(name).deactive:
raise RepoError("Failed to queue repo, {0} is deactive.".format(name), name)
if self.get_repo(name).queued:
raise RepoError("Failed to queue repo, {0} already queued.".format(name), name)
if self.get_repo(name).is_alive():
raise RepoError("Failed to queue Repo, {0} is syncing.".format(name), name)
self.repo_queue.put([self.config.get(name, "weight"), self.get_repo(name)])
self.get_repo(name).queued = True
class GlobalError(Exception):
def __init__(self, message):
"""Fatal GLOBAL Section Config Error."""
Exception.__init__(self, message)
self.message = message
class RepoConfigError(Exception):
def __init__(self, message, name):
"""Non-Fatal Repo config Error."""
Exception.__init__(self, message)
self.message = message
self.name = name
class RepoError(Exception):
def __init__(self, message, name):
"""Non-Fatal Repo Error."""
Exception.__init__(self, message)
self.message = message
self.name = name
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,384 |
Q: Approximating the solution to \begin{align*} \frac{d^2x}{dt^2} = \frac{-981x}{50\sqrt{1+4x^2}}, \; x(0)=-1, \; x'(0)=0 \end{align*} The Problem:
In an exploration of a ball rolling inside a parabola, I discovered and needed to solve or approximate the following differential equation:
\begin{align*}
\end{align*}
\begin{align*}
\frac{d^2x}{dt^2} = \frac{-981x}{50\sqrt{1+4x^2}}, \; x(0)=-1, \; x'(0)=0
\end{align*}
\begin{align*}
\end{align*}
where $x$ is a function of $t$.
\begin{align*}
\end{align*}
Attempts:
Trying to solve this analytically lead me to dead ends, so I figured the best way to solve this DE is numerically. Alternatively, I tried using a Taylor series centred about $x = 0$. I managed to find the first 15 non-zero terms.
\begin{align*}
\end{align*}
\begin{align*}
\end{align*}
However, the series appears to converge quite slowly and this took a lot of tedious, repetitive calculation. So, trying to obtain an accurate approximation of the solution this way feels futile. Are there any other ways of approximating this DE more efficiently and accurately, or maybe is it solvable analytically? Thanks.
A: You can turn it into two first-order DEs, the first is doable, and the second you might feed into your graphics calculator.
Let $v=\frac{dx}{dt}$, then $$\frac{d^2x}{dt^2}=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}$$
You integrate both sides to get $v=f(x)$. Then, recall $v=dx/dt$, so $$t=\int\frac{dx}{f(x)}$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,929 |
\section{Introduction :}
The quasiparticle quark-gluon plasma (qQGP) is a phenomenological model,
with few fitting parameters, widely used to describe the nonideal behaviour
of quark-gluon plasma (QGP). It was first proposed by Goloviznin and Satz
\cite{sz.1} and then by Peshier {\it et. al.}
\cite{pe.1} to explain the equation of state (EoS) of QGP from lattice gauge
theory (LGT) simulation of quantum chromodynamics (QCD) at finite temperature
\cite{l1.1}. The model, however, failed \cite{ba.1} to explain the more
accurate, recent lattice data \cite{l2.1}. Further, Gorenstein and Yang
\cite{go.1} pointed out that the model is thermodynamically inconsistent.
This thermodynamically inconsistency problem was remedied by them by
introducing a temperature dependent vacuum energy and
forced it to cancel the thermodynamically
inconsistent term, which was named as the reformulated statistical mechanics.
It is still not clear what is the physics or origin of this constraint which
was called as thermodynamic consistency check in Ref. \cite{go.1,pe.2,lh.1,s.1}
Here we show that the whole exercise is unnecessary and following the
standard statistical mechanics (SM), we propose a new qQGP model which
contains a single phenomenological parameter. Our model is thermodynamically
consistent and explains lattice data very well.
\section{Our model of qQGP:}
Let us start with the work of Peshier {\it et. al.} \cite{pe.1} on gluon
plasma. All thermodynamic quantities were derived from the pressure,
$P$, which was assumed as
\begin{equation}
\frac{P\,V}{T} = - \sum_{k=0}^\infty \ln (1 -
e^{ - \beta \epsilon_k})\,\, , \label{eq:p}
\end{equation}
where the right hand side is the logarithm of the grand partition function,
$Q_G(T)$, and $\epsilon_k$ is the single particle energy of quasi-gluon,
i.e, gluon with temperature dependent mass, given by,
\[ \epsilon_k = \sqrt{k^2 + m^2 (T)} \,\,, \]
where $k$ is momentum and $m$ is mass. $\beta$ is defined as $1/T$.
The expression for pressure
is similar to that of ideal gas with temperature dependent mass and
hence let us denote it as $P_{id}$.
We want to stress that this assumption itself is
the root cause of thermodynamic inconsistency and
hence the reformulation of SM by Gorenstein and Yang \cite{go.1}.
Generally, in grand canonical ensemble (GCE), energy ($E_r$) and number
of particles ($N_s$) fluctuate, but temperature ($T$) and
the chemical potential ($\mu$) are fixed. Hence, the average energy ($U$)
and average number of particles ($N$) are defined and may be related to
the grand partition function or q-potential,
\begin{equation}
q \equiv \ln Q_G =
\ln ( \sum_{s,r} e^{- \beta E_r - \alpha N_s})
= \mp \sum_{k=0}^\infty
\ln (1 \mp z\, e^{ - \beta \epsilon_k})
\,\,, \label{eq:q} \end{equation}
where $\mp$ for bosons and fermions and
$z \equiv e^{\mu /T} = e^{-\alpha}$ is called fugacity.
The average energy $U$ is defined as,
\begin{equation}
U \equiv <E_r> = \frac{\sum_{s,r}\,E_r \,e^{- \beta E_r - \alpha N_s}}{Q_G} =
- \frac{\partial }{\partial \beta} \ln Q_G =
\sum_k \frac{z \,\epsilon_k e^{- \beta \epsilon_k }}{1 \mp
z\, e^{- \beta \epsilon_k} }
\,\,. \label{eq:uo}
\end{equation}
Note that the partial differentiation with repect to $\beta$ above is just
a mathematical method to express $U$ in terms of sum over single particle
energy levels, $\epsilon_k$, making use of Eq. (\ref{eq:q}). While
differentiating, indirect dependence of $\beta = 1/T$ in the fugacity, $z$,
and mass, $m(T)$, must be ignored by definition.
Otherwise, we will not get back $<E_r>$.
Similarly, the average density $N$ is defined as,
\begin{equation}
N \equiv <N_s> = \frac{\sum_{s,r}\,N_s \,e^{- \beta E_r - \alpha N_s}}{Q_G} =
- \frac{\partial }{\partial \alpha} \ln Q_G =
z \frac{\partial }{\partial z} \ln Q_G =
\sum_k \frac{z \,e^{- \beta \epsilon_k }}{1 \mp
z\, e^{- \beta \epsilon_k} } \,\,,\label{eq:n}
\end{equation}
These (Eqs. (\ref{eq:uo}), (\ref{eq:n})) are the standard relations \cite{pa.1}
of $U$ and $N$ to the partition function,
which is valid even for quasiparticle with ($T$,$\mu$) dependent mass
by the definition of averages.
Here, for gluon plasma, we have $\mu = 0$ or $z=1$.
Next, pressure may be obtained by two methods. In method-I, one starts from
$U$ and using thermodynamic relation,
\begin{equation}
\varepsilon \equiv \frac{U}{V} = T \frac{\partial P}{\partial T}
- P \,\, , \label{eq:td}
\end{equation}
and on integration, one gets pressure which is the procedure that we follow here.
In method II, again following the standard text books on SM \cite{pa.1},
we can relate $P$ to q-potential as follows. The variation
in q-potential due to variations in it's dependence, namely
$T$, $\mu$ and volume $V$, specifying the macro-state of GCE system, is,
\begin{equation}
\delta q = \frac{1}{Q_G} \left[ \sum_{r,s} e^{- \beta (E_r - \mu N_s)}
\, \left( - E_r \,\delta \beta - \beta \,\delta E_r + N_s \,\delta (\beta \mu)
\,\right) \right] \,\,.
\end{equation}
Now, when compared with the text books results, we have an extra term coming
from $\delta E_r$ due to temperature dependent mass
and then using the definition of averages, we get,
\begin{equation}
\frac{P\,V}{T} = \mp \sum_{k=0}^\infty \ln (1 \mp z\,
e^{ - \beta \epsilon_k}) + \int d\beta\, \beta \,
\frac{\partial m}{\partial \beta}\,<\frac{\partial E_r}{\partial m}>
\,\,. \label{eq:pn}
\end{equation}
Therefore we see that $P$ is not just equal to $P_{id}$, but there is an
extra term. This extra term ensure thermodynamic consistency of the relation
as follows. From above $P$, on differentiating with respect to $T$ for
a system with $\mu=0$ or $z=1$,
\begin{equation}
\frac{\partial P}{\partial T} = \frac{P}{T} + \frac{\varepsilon}{T}
- \frac{1}{V}\,< \frac{\partial \epsilon_k}{\partial T} >
+ \frac{1}{V}\,< \frac{\partial E_r}{\partial T} >
\end{equation}
where the last two terms exactly cancels (following the procedure used
in deriving Eq, (\ref{eq:uo})) and hence the thermodynamic
relation, Eq. (\ref{eq:td}), is obeyed as expected.
Further more, this $P$ is also consistent with the $P$ obtained from $U$
through thermodynamic relations which may be shown as follows.
The Eq. (\ref{eq:pn}) may be simplified by evaluating
$<\frac{\partial E_r}{\partial m}>$ and taking continuum limit
($V \rightarrow \infty$), for a system with $\mu = 0$, as,
\begin{equation}
\frac{P}{T} = \mp \frac{g_f}{2 \pi^2} \int_0^{\infty} dk\,k^2\,
\ln(1 \mp \,e^{-\beta \epsilon_k}) + \int d\beta \,\beta
\frac{g_f}{2\pi^2}\,m\,\frac{dm}{d\beta}\,\int_0^{\infty} dk
\frac{k^2}{\epsilon_k \,( e^{\beta \epsilon_k} \mp 1)}
\,\,, \label{eq:pn1} \end{equation}
which on simplification, reduces to,
\[ \frac{P}{T} = \frac{g_f}{2\,\pi^2} \left[ T^3\, \sum_{l=1}^{\infty}
(\pm 1)^{l-1}\,\frac{1}{l^4}\, (\beta\,m\,l)^2\,K_2 (\beta\,m\,l)\, \right. \]
\begin{equation}
\left. +\, \int d\beta \frac{\beta}{m}\,\frac{\partial m}{\partial \beta}\,
\frac{1}{\beta^4} \sum_{l=1}^{\infty}
(\pm 1)^{l-1}\,\frac{1}{l^4}\, (\beta\,m\,l)^3\,K_1 (\beta\,m\,l)\,\right]
\,\,, \end{equation}
where $g_f$ is the internal degrees of freedom and $K_1$, $K_2$ are
modified Bessel functions. Using the recurrence relations of
Bessel functions and on integration by parts, above equation may
be further simplified to get,
\begin{equation}
\frac{P}{T} = \frac{P_0}{T_0} - \int_{\beta_0}^{\beta} \, d\beta\,
\varepsilon \,\,, \label{eq:td2} \end{equation}
where $\varepsilon$ is the energy density and $P_0$ is the pressure
at some temperature $T_0$ or $\beta_0$. This equation is nothing
but the thermodynamic relation, Eq. (\ref{eq:td}).
Therefore, Gorenstein and Yang's starting argument that above two
methods give different $m(T)$ does not exist now by using our
derived expression for $P$, instead of the assumption \cite{pe.1,go.1}.
\section{Question of vacuum energy $B(T)$ :}
After the reformulation of SM by
Gorenstein and Yang, almost all study in qQGP is based on the
thermodynamic consistency relation, related to vacuum energy $B(T)$. Different authors call and interpret $B(T)$ in a different way, like vacuum energy, background field or bag pressure. But, by definition of quasiparticle, whole thermal energy
is used to excite these quasiparticles. So quasiparticles are excitations
above the ground state or vacuum state which may not depend on
temperature or chemical potential. This is our assumption. As noted earlier, we also don't have any thermodynamic inconsistency in our model.
In fact, when we redo our derivation of pressure
with vacuum or zero point energy in single particle energy,
like in Ref. \cite{go.1}, Eq. (\ref{eq:pn1}) is modified as,
\begin{equation}
P = P_{id} - B(T) + T\,\left( \int_{T_0}^T \frac{d\tau}{\tau}
\left[ \frac{g_f}{2\pi^2}\,m\,\frac{dm}{d\tau}\,\int_0^{\infty} dk
\frac{k^2}{\epsilon_k \,(e^{\epsilon_k/\tau} \mp 1)}
+ \frac{\partial B}{\partial \tau} \right] \right) \,\,, \label{eq:pn2}
\end{equation}
and the energy density,
\begin{equation}
\varepsilon = \varepsilon_{id} + B(T) \,\,.
\end{equation}
where $\varepsilon_{id}$ is the expression for $\varepsilon$ similar to
ideal gas. Again it is easy to show that above $P$ and $\varepsilon$
obey thermodynamic relation Eq. (\ref{eq:td}).
The thermodynamic consistency relation \cite{go.1}, used in other qQGP
models, is nothing but a restrictive condition that the terms inside
the square bracket in Eq. (\ref{eq:pn2}) is zero. At present it is
not clear what is the physical origin of this constraint. Note that
without this constraint, so called thermodynamic consistency relation,
our system is thermodynamically consistent even with the zero-point energy contribution, $B(T)$. One may model $B(T)$ based on other effects of strongly interacting QCD system, like hadronic states, resonances and may be relevent at the
transition point. In our study of gluon plasma here, we neglect all
these effects and consider only the thermal properties of gluons. Hence
we take $B(T) = 0$ and we get a very good fit to lattice results except at very close to the transition temperature, i.e, for $T<1.2 T_c$.
\section{EoS of gluon plasma:}
As an example, let us apply our model to gluon plasma which is a QCD plasma
without quarks. We first calculate
the energy density, expressed in terms of
$e(T) \equiv \varepsilon / \varepsilon_s$,
and then obtain $P$ from thermodynamic relation, Eq. (\ref{eq:td}). So we have,
from Eq. (\ref{eq:uo}) after some algebra,
\begin{equation}
e(T) = \frac{15}{\pi ^4}
\sum_{l=1}^\infty \frac{1}{l^4}
\left[ (\frac{m_g\,l}{T})^3 K_1 (\frac{m_g\,l}{T})
+ 3\, (\frac{m_g\,l}{T})^2 K_2 (\frac{m_g\,l}{T}) \right]
\end{equation}
where $\varepsilon_s$ is the Stefan-Boltzman gas limit of QGP,
$m_g$ is the temperature dependent mass and $K_1$, $K_2$ are
modified Bessel functions.
The results are plotted in Fig. 1, for two different mass
terms, $m_g^2(T) = \omega_p^2 = g^2(T)\,T^2\,/3$ (our model)
and $m_g^2(T) = g^2(T)\,T^2\,/2$ (other qQGP models).
$g^2(T)$ is related to the two-loop order running coupling constant, given by,
\begin{equation} \alpha_s (T) = \frac{6 \pi}{(33-2 n_f) \ln (T/\Lambda_T)}
\left( 1 - \frac{3 (153 - 19 n_f)}{(33 - 2 n_f)^2}
\frac{\ln (2 \ln (T/\Lambda_T))}{\ln (T/\Lambda_T)}
\right) \label{eq:ls} \;, \end{equation}
where $\Lambda_T$ is a parameter related to QCD scale parameter. This
choice of $\alpha_s (T)$ is an approximate expression of the running coupling
constant used in lattice simulations \cite{l2.1}.
Then the pressure is obtained from the
thermodynamic relation Eq. (\ref{eq:td}) or Eq. (\ref{eq:td2}).
Since we have only one parameter to adjust, we don't get good fit for
the generally used second choice of quasi-gluon mass. The best fitted
parameter is $\Lambda_T /T_c = 0.3$. But a remarkably good
fit may be obtained for our choice of gluon mass which is motivated from the
fact that the quasi-gluons are the thermal excitations of plasma waves
with mass equal to the plasma frequency \cite{me.1}. The value of the fitted
parameter is $\Lambda_T /T_c = 0.65$.
Let us now compare our results with the results from other qQGP models, for example Ref. \cite{pe.2}, where $B(T)$ is not zero, but is determined by thermodynamic consistency relation. From the Fig. 2, we see that only at large $T/T_c$ both the results almost match, but differ near to $T/T_c = 1$. We used the same $\alpha_s(T)$ with $\Lambda_T/T_c = .65$ for both the cases.
Further, results from our model with $B(T) = 0$
fits well the lattice data. A very good fit to lattice data was
also obtained in Ref. \cite{pe.2}, but with a different expression for
$\alpha_s(T)$, having two free parameters, and an additional parameter related to degrees of freedom.
\section{Conclusions:}
Here we have pointed out the basic reason for the thermodynamic inconsistency
of the extensively studied quasiparticle QGP models \cite{pe.1} and it's consequence of the reformulation of statistical mechanics \cite{go.1}.
To revise it, we have proposed a new qQGP model which follows from the standard SM and has no thermodynamic inconsistency. When we extend our formalism to a system with temperature dependent vacuum energy, again, we get a thermodynamically consistent general model and we obtained other widely studied qQGP models as a special case of our model under certain restrictive condition which was called as
thermodynamic consistency relation in Ref. \cite{go.1,pe.2,lh.1,s.1}.
As an example, we studied the gluon plasma
using our model. A remarkable good fit to the
LGT data was obtained by adjusting just one parameter and without
the temperature dependent vacuum energy $B(T)$. Whereas we know that the
other qQGP models has 3 or more parameters. Further extension of our
model to flavored QGP without and with masses, and also without and with
chemical potential, fit remarkably well
the lattice results and were reported in \cite{ba.2,ba.3}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 36 |
Q: Accessing an array element that does not exist in C I was writing some C code today and I noticed something strange. First, I defined an array with a certain size. Then, I indexed it in a for loop and accidentally indexed it past the size that was allocated for it. I believe that this should have thrown an error, but I didn't get a compiler error or an error when I ran the executable.
The code went something like this:
#include <stdio.h>
#include "functions.h"
int main()
{
unsigned char arr1[5] = {1, 2, 3, 4, 5};
int N=100;
int arr2[N];
int i;
for(i=0;i<N;++i)
{
arr2[i] = i;
//Some function defined in a file functions.c
func1(arr2[i],arr1[i]);
}
return 1;
}
How does this not throw an error when I try to index, say, arr1[5]?
Not that it is important to the question, but I realized that I needed to do a nested for loop. Like this:
#include <stdio.h>
#include "functions.h"
int main()
{
unsigned char arr1[5] = {1, 2, 3, 4, 5};
int N=100;
int arr2[N];
int i;
int j;
for(j=0;j<5;++i)
{
for(i=0;i<N;++i)
{
arr2[i] = i;
//Some function defined in a file functions.c
func1(arr2[i],arr1[j]);
}
}
return 1;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,489 |
That has such great memories for me from camp and the different tunes we used to sing it as a grace before meals.
The greeting time near the beginning of service was the 'musical chairs' type, in which everyone moves all over church to say hi to everyone. I got a hug from an older gentleman named Jay, who seemed nice. Also there was a prayer time open to the congregation, in which we took turns giving Pastor Abby prayer requests. Then she led us in prayer over the requests. This again was very like the Methodist Churches I've been to.
The church is small. The building isn't too big and those attending church don't fill the building more than one third. They told me it was down to about five families at one point, in danger of closing. But it didn't and has grown some since. And it is fairly active in terms of projects geared towards helping the needy, especially considering its size. I've been to churches with size/attendance issues before and I've seen how tough that can be, so I hope this church can continue and be where it needs to be. As for me, the nature of my project says I must go on to the next church on my list. I do however think I will be stopping at this church too sometimes with canned goods for their food bank affiliate. They are conveniently located right next to a grocery store.
Overall Feelings: I like this place, and would be willing to go back again. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,392 |
{"url":"http:\/\/math.stackexchange.com\/questions\/274906\/let-h-be-a-subgroup-of-g-such-that-for-all-x-in-g-x2-in-h\/274923","text":"# Let $H$ be a subgroup of $G$ such that for all $x \\in G$, $x^2 \\in H$\n\nLetting $H$ be a subgroup of $G$ such that for every $x\\in G$, $x^{2}\\in H$. So, which of the following is true?\n\na. $H$ is a normal subgroup containing $G'$.\n\nb. $H$ is a normal abelian subgroup.\n\nc. $H=G$.\n\nd. $H$ is a maximal subgroup.\n\nc need not be true. For example, consider $H=2\\Bbb Z$ and $G=\\Bbb Z$. I want to know which of the a, b or d is true?\n\nThank you.\n\n-\nFor $(d)$, do you mean the subgroup $\\{0,4\\}=H\\leq\\Bbb Z\/8\\Bbb Z$? If so, then it is not a counterexample since $1+1=2\\notin H$. Or do you mean something else by your notation? \u2013\u00a0 Clayton Jan 10 '13 at 3:16\n@ Clayton Thank you. The question is edited. \u2013\u00a0 aliakbar Jan 10 '13 at 3:20\nwhat does $G'$ denote? \u2013\u00a0 rondo9 Jan 10 '13 at 3:29\nThe commutator subgroup of $G$ \u2013\u00a0 Clayton Jan 10 '13 at 3:31\n\nHints:\n\n(1) In any group $\\,G\\,$ , the subgroup $\\,G^n:=\\langle\\,x^n\\;;\\;x\\in G\\,\\,,\\,n\\in\\Bbb N\\,\\rangle\\,$ is normal (in fact, it is a fully invariang subgroup)\n\n(2) Any group $\\,G\\,$ for which $\\,x^2=1\\,\\,\\,\\forall\\,x\\in G\\,$ is abelian\n\n(3) In your case, $\\,H\\triangleleft G\\,$ and $\\,G\/H\\,$ is abelian.\n\n(4) $\\,\\forall\\,N\\triangleleft G\\,$ in any group $\\,G\\,$ , $\\,G\/N\\,$ is abelian iff $\\,G'\\leq N\\,$\n\nCan you take it from here?\n\n-\nNote that in $\\mathbb Q_8$, the subgroup $H=\\{1,-1\\}$ is as you defined above but it is not maximal at all. Nice suggestions +1 \u2013\u00a0 B. S. Jan 10 '13 at 10:08\n@Babak I do not understand that how $H$ is normal \u2013\u00a0 aliakbar Jan 10 '13 at 11:40\n@aliakbar: Google the Quaternion subgroups to see that $H$ is a char subgroup so is normal. \u2013\u00a0 B. S. Jan 10 '13 at 11:42\n@aliakbar, look: $$\\forall\\,x,g\\in G\\;,\\;\\;g^{-1}x^ng=\\left(g^{-1}xg\\right)^n$$ and this shows that any generator of $\\,G^n\\,$ remains a generator of $\\,G^n\\,$ after being conjugated (an element of $\\,G\\,$ to the $\\,n-$th power) , so $\\,G^n\\,$ is normal as it is generated by a normal set. In the case of $\\,\\{-1,1\\}\\leq Q_8\\,$ , you can try to prove the beautiful lemma: if a group has one unique element $\\,t\\,$ of order two, then $\\,\\{1,t\\}\\leq Z(G)\\,$ which, in particular, means that this subgroup's normal. \u2013\u00a0 DonAntonio Jan 10 '13 at 12:56\n@ DonAntonio I still do not understand why $H$ is a normal subgroup. I would be most grateful if you would explain \u2013\u00a0 aliakbar Jan 23 '13 at 15:19\nI'm not entirely sure about $(a)$ and $(b)$, let me think about it for a while.\nFor $(d)$, $H$ need not be maximal, for we can consider a group of order $8$, where all elements have order $2$ except for the identity. Then $H=\\{e,a\\}$ is a subgroup, every element not in $H$ has order $2$, i.e., $g^2=e\\in H$, but $J=\\{e,a,b,ab\\}$ is a subgroup which contains $H$, hence $H$ is not maximal.\n$(a)$ is correct. It suffices to prove that $[a,b] \\in H$ for all $a,b \\in G$. It is an easy job by some commutator relations.","date":"2014-03-10 07:35:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9236211180686951, \"perplexity\": 207.7730372787985}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1394010693428\/warc\/CC-MAIN-20140305091133-00010-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
//#include https://ajax.googleapis.com/ajax/libs/prototype/1.7.2.0/prototype.js
//#include https://code.jquery.com/ui/1.11.0/jquery-ui.min.js
//NOTE: $() is reserved for Prototype, jQuery must use jQuery() format
//#include js/framework/Service.js
var ResourceProvider = Class.create({
initialize: function() {
this.eventBus = new EventBus();
this.images = {};
this.numImagesLoading = 0;
this.sprites = {};
this.numSpritesLoading = 0;
this.jsonFiles = {};
this.numJsonFilesLoading = 0;
this.dynamicRes = {};
this.baseURL = "";
this.verbose = false;
Service.add("rp", this);
},
isLoading: function() {
if( this.numImagesLoading > 0 ) return true;
if( this.numSpritesLoading > 0 ) return true;
if( this.numJsonFilesLoading > 0 ) return true;
return false;
},
_didLoad: function(fileName, resource) {
this.eventBus.dispatch({"evtName":fileName, "status":"loadComplete", "res":resource});
this.eventBus.clearListeners(fileName);
},
/**
* Loads the image if it's not already loaded
* @return:Image
*/
getImage: function(fileName, fnOnLoad) {
if(!this.images[fileName] || !this.images[fileName].isLoaded)
{
this.loadImage(fileName, fnOnLoad);
return;
}
if(fnOnLoad) fnOnLoad({"evtName":fileName, "status":"loadComplete", "res":this.images[fileName]});
return this.images[fileName];
},
loadImage: function(fileName, fnOnLoad) {
var RP = this;
if(this.images[fileName]) {
if(this.images[fileName].isLoaded) {
//already loaded
if(fnOnLoad) fnOnLoad({"evtName":fileName, "status":"loadComplete", "res":this.images[fileName]});
} else {
//pending load
if(fnOnLoad) RP.eventBus.addListener(fileName, fnOnLoad);
}
return;
}
if(this.verbose) console.log("load image " + fileName);
var img = new Image();
img.isLoaded = false;
img.src = this.baseURL + fileName;
this.images[fileName] = img;
if(fnOnLoad) RP.eventBus.addListener(fileName, fnOnLoad);
img.onload = function() {
//console.log("image loaded: " + fileName);
RP.images[fileName].isLoaded = true;
RP.numImagesLoading--;
RP._didLoad(fileName, RP.images[fileName]);
}
if(img.complete)
{
console.warn("image already complete, abort loading")
}else {
this.numImagesLoading++;
}
},
/**
* Loads the sprite if its not already loaded
* @return:Sprite
*/
getSprite: function(fileName, fnOnLoad) {
if(!this.sprites[fileName] || !this.sprites[fileName].isLoaded)
{
this.loadSprite(fileName, fnOnLoad);
return;
}
if(fnOnLoad) fnOnLoad({"evtName":fileName, "status":"loadComplete", "res":this.sprites[fileName]});
return this.sprites[fileName];
},
loadSprite: function(fileName, fnOnLoad) {
var RP = this;
if(this.sprites[fileName]) {
if(this.sprites[fileName].isLoaded) {
//already loaded
if(fnOnLoad) fnOnLoad({"evtName":fileName, "status":"loadComplete", "res":this.sprites[fileName]});
} else {
//pending load
if(fnOnLoad) RP.eventBus.addListener(fileName, fnOnLoad);
}
return;
}
if(this.verbose) console.log("load sprite " + fileName);
var sprite = new Sprite(fileName);
if(fnOnLoad) RP.eventBus.addListener(fileName, fnOnLoad);
jQuery.getJSON(this.baseURL + fileName, function(data) {
if(this.verbose) console.log("sprite loaded: " + fileName);
RP.sprites[fileName]._load(data, function(){
RP.numSpritesLoading--;
RP._didLoad(fileName, RP.sprites[fileName]);
});
});
this.sprites[fileName] = sprite;
this.numSpritesLoading++;
},
getJson: function(fileName, fnOnLoad) {
if(!this.jsonFiles[fileName] || !this.jsonFiles[fileName].isLoaded)
{
this.loadJson(fileName, fnOnLoad);
return;
}
if(fnOnLoad) fnOnLoad({"evtName":fileName, "status":"loadComplete", "res":this.jsonFiles[fileName].data});
return this.jsonFiles[fileName].data;
},
loadJson: function(fileName, fnOnLoad) {
var RP = this;
if(this.jsonFiles[fileName]) {
if(this.jsonFiles[fileName].isLoaded) {
//already loaded
if(fnOnLoad) fnOnLoad({"evtName":fileName, "status":"loadComplete", "res":this.jsonFiles[fileName].data});
} else {
//pending load
if(fnOnLoad) RP.eventBus.addListener(fileName, fnOnLoad);
}
return;
}
if(this.verbose) console.log("load json: " + fileName);
if(fnOnLoad) RP.eventBus.addListener(fileName, fnOnLoad);
jQuery.getJSON(this.baseURL + fileName, function(data) {
if(this.verbose) console.log("json loaded: " + fileName);
RP.jsonFiles[fileName].data = data;
RP.jsonFiles[fileName].isLoaded = true;
RP.numJsonFilesLoading--;
RP._didLoad(fileName, RP.jsonFiles[fileName].data);
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if(this.verbose) console.log("json being loaded: "+fileName);
this.jsonFiles[fileName] = { file:fileName, data:null, isLoaded:false };
this.numJsonFilesLoading++;
},
setResource: function(fileName, res) {
if(this.dynamicRes.hasOwnProperty(fileName)) {
console.warn("tried to set already existing resource " + fileName);
return;
}
this.dynamicRes[fileName] = res;
},
getResource: function(fileName) {
return this.dynamicRes[fileName];
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}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,603 |
package RegoForm_MemberPasswordReminder;
require Exporter;
@ISA = qw(Exporter);
@EXPORT = qw(
HandlePasswordReminder
sendPwdReminder
);
@EXPORT_OK = qw(
HandlePasswordReminder
sendPwdReminder
);
use lib "..","../..";
use Defs;
use Utils;
use TemplateEmail;
use RegoForm_Common;
use strict;
use Data::Dumper;
sub HandlePasswordReminder {
my $self = shift;
my $emailaddress = $self->{'RunParams'}{'emailaddress'} || '';
my $natNum = $self->{'RunParams'}{'natnumber'} || '';
my $surname = $self->{'RunParams'}{'surname'} || '';
my $resultHTML = '';
if($emailaddress) {
my $countSent = sendPwdReminder($self->{'Data'}, $self->AssocID(), $emailaddress);
if ($countSent) {
$resultHTML = qq[
<div class="OKmsg">Your password reminder has been sent</div><br>
<p>Your password has been emailed to the address you provided.</p>
<p>Please be patient, it may take a few minutes to receive it. Remember to check your SPAM folder if you have not received it.</p>
<p>Click your browser's back button to return to the login page</p>
];
}
else {
$resultHTML = qq[<div class="warningmsg">No matches found for this email address<br/ > Please contact your organization for username and password to process your registration online.</div>];
print STDERR Dumper($self->{'Data'});
$emailaddress='';
}
}
if (
$self->{'SystemConfig'}{'AllowStupidPwdReminder'}
and ($natNum or $surname)
) {
$resultHTML = showPwdReminder($self->{'Data'}, $self->AssocID(), $natNum, $surname);
if ($resultHTML) {
$resultHTML = qq[
<div class="sectionheader">Username and Password Reminder</div>
<div class="OKmsg">Your Details have been found</div>
$resultHTML
<p>Click your browser's back button to return to the login page</p>
];
}
}
if (! $emailaddress and ! $natNum and ! $surname) {
my $hiddenfields = $self->stringifyCarryField();
my $body = qq[
<form method="POST" action="$self->{'Data'}{target}">
<input type="hidden" name="a" value="PWD">
<input type="hidden" name="rfp" value="vt">
$hiddenfields
<div class="sectionheader">Username or Password Reminder via Email</div>
$resultHTML
<p>
Please enter your email address below<br>
<b>Email Address</b> <input type="text" value="" name="emailaddress">
</p>
<p>
When you click <b>Send me my Username and Password</b> you will receive
an email with all usernames and passwords that are assigned to this emails
address.<br>
Please make sure that you have allowed incoming email from
sportingpulse.com, and check your junk mail for your password reminder.
</p>
<input type="submit" name="submit" value="Send me my Username and Password">
</form>
];
if ($self->{'SystemConfig'}{'AllowStupidPwdReminder'}) {
my $hiddenfields = $self->stringifyCarryField();
$body .= qq[
<form method="POST" action="$self->{'Data'}{target}">
<input type="hidden" name="a" value="PWD">
<input type="hidden" name="rfp" value="vt">
$hiddenfields
<div class="sectionheader">Username or Password Reminder</div>
<p>Please enter your National Number and Surname below</p>
<table>
<tr>
<td><b>National Number</b></td>
<td><input type="text" value="" name="natnumber"></td>
</tr>
<tr>
<td><b>Surname</b></td>
<td><input type="text" value="" name="surname"></td>
</tr>
<table>
<p>
When you click <b>Show me my Username and Password</b> you will be shown
the username and password which match this National Number and Surname.
</p>
<input type="submit" name="submit" value="Show me my Username and Password">
</form>
];
}
$resultHTML = $body;
}
return $resultHTML || '';
}
sub showPwdReminder {
my ($Data, $assocID, $natNum, $surname) = @_;
my $tNatNum= $natNum;
my $tSurname= $surname;
my $st = qq[
SELECT
CONCAT(M.strFirstname, " " , M.strSurname) as MemberName,
M.strEmail,
ATH.strUsername,
ATH.strPassword,
ATH.intAuthID,
M.intMemberID,
MA.intAssocID
FROM tblMember as M
LEFT JOIN tblMember_Associations as MA ON (
M.intMemberID = MA.intMemberID
AND MA.intAssocID = ?
AND MA.intRecStatus <> ?
)
LEFT JOIN tblAuth as ATH ON (
ATH.intID = M.intMemberID
AND ATH.intLevel = ?
)
WHERE
M.strNationalNum = ?
AND M.strSurname = ?
AND M.intStatus <> ?
AND M.intRealmID = ?
LIMIT 1
];
my $query = $Data->{'db'}->prepare($st);
$query->execute(
$assocID,
$Defs::RECSTATUS_DELETED,
$Defs::LEVEL_MEMBER,
$natNum,
$surname,
$Defs::MEMBERSTATUS_DELETED,
$Data->{'Realm'},
);
my $body = '';
my $assocname= '';
my $dref=$query->fetchrow_hashref();
if ($dref->{intMemberID}) {
if (! $dref->{intAuthID}) {
my $strPassword = generateRandomPassword();
my $st_ins = qq[
INSERT INTO tblAuth (
strUsername,
strPassword,
intLevel,
intID,
dtCreated
)
VALUES (
?,
?,
$Defs::LEVEL_MEMBER,
?,
SYSDATE()
)
];
my $q = $Data->{'db'}->prepare($st_ins);
$q->execute(
$dref->{intMemberID},
$strPassword,
$dref->{intMemberID},
);
$dref->{strUsername} = $dref->{intMemberID};
$dref->{strPassword} = $strPassword;
}
my $username = "1".$dref->{strUsername};
if ($dref->{intAssocID} == $assocID) {
$body = qq[
<table>
<tr><td><b>National Number</b></td><td>$tNatNum</td></tr>
<tr><td><b>Surname</b></td><td>$tSurname</td></tr>
<tr><td><b>Username</b></td><td><b>$username</b></td></tr>
<tr><td><b>Password</b></td><td><b>$dref->{strPassword}</b></td></tr>
</table>
];
}
}
$body = qq[Record not found] if ! $body;
return $body;
}
sub sendPwdReminder {
my ($Data, $assocID, $email) = @_;
my $tEmail = $email;
return 0 if !$email;
my $st = qq[
SELECT
CONCAT(M.strFirstname, " " , M.strSurname) as MemberName,
M.strEmail,
ATH.strUsername,
ATH.strPassword,
ATH.intAuthID,
M.intMemberID,
A.strName
FROM
tblMember as M
INNER JOIN tblMember_Associations as MA
ON (M.intMemberID = MA.intMemberID AND MA.intAssocID = ?)
INNER JOIN tblAssoc as A ON (A.intAssocID = MA.intAssocID)
LEFT JOIN tblAuth as ATH
ON (ATH.intID = M.intMemberID and ATH.intLevel = ?)
WHERE
(M.strEmail = ? OR M.strP1Email = ? OR M.strP2Email = ?)
AND M.intStatus <> ?
AND MA.intRecStatus <> ?
AND M.intDeRegister <> 1
];
my $query = $Data->{'db'}->prepare($st);
$query->execute(
$assocID,
$Defs::LEVEL_MEMBER,
$email,
$email,
$email,
$Defs::MEMBERSTATUS_DELETED,
$Defs::RECSTATUS_DELETED,
);
my $body = '';
my $count=0;
my $assocname= '';
my @members = ();
while (my $dref=$query->fetchrow_hashref()) {
$assocname = $dref->{strName};
$count++;
if (! $dref->{intAuthID}) {
my $strPassword = generateRandomPassword();
my $st_ins= qq[
INSERT INTO tblAuth (
strUsername,
strPassword,
intLevel,
intID,
dtCreated
)
VALUES (
?,
?,
1,
?,
SYSDATE()
)
];
$Data->{db}->do(
$st_ins,
undef,
$dref->{intMemberID},
$strPassword,
$dref->{intMemberID},
);
$dref->{strUsername} = $dref->{intMemberID};
$dref->{strPassword} = $strPassword;
}
my $username = "1".$dref->{strUsername};
push @members, {
MemberName => $dref->{'MemberName'} || '',
Username => $username || '',
Password => $dref->{'strPassword'} || '',
};
}
my %TemplateData = (
AssocName => $assocname,
UserList => \@members,
UsePassport => $Data->{'SystemConfig'}{'usePassportInRegos'},
);
my $emailresponse = 0;
if($tEmail and @members) {
$emailresponse = sendTemplateEmail (
$Data,
'regoform/password_reminder.templ',
\%TemplateData,
$tEmail,
"Password reminder for $assocname",
"$Defs::null_email_name <$Defs::null_email>",
$Data->{'SystemConfig'}{'regoFormPwd_CC'} || '',
'',
);
}
return $emailresponse || 0;
}
1;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 39 |
Q: Configuring simplesamlphp for IIS instead of Apache (as a service provider) A project I am working on has migrated from an Apache LAMP stack to and Azure enrivonrment, which is based on IIS. I'm trying to install SAML2.0 support and am attempting to use simplesamlphp as I did on the previous set up.
However, working through the installation advice I'm running up against section 6 "congfiguring Apache" where it requires you to "Find the Apache configuration file for the virtual hosts where you want to run SimpleSAMLphp." (page is here).
Is it possible to do the equivalent on IIS? I can't find any instructions about simplesamlphp. Skipping this step and doing everything else the installation / setup page of the app is unresponsive.
If simplesamlphp isn't a good option of IIS is there another package anyone can recommend?
A:
Is it possible to do the equivalent on IIS? I can't find any
instructions about simplesamlphp.
SimpleSamlPHP for IIS works great, but I couldn't find any good documentation either. Just do the following where you should be configuring Apache:
*
*Enable FastCGI in IIS
*Create a virtual directory in IIS to the www-folder in the simplesamlphp folder.
*Add read & execute permissions to the application pool running on the server
Rightclick the new virtual directory in IIS -> edit permissions -> security -> edit -> add.
Use the the computer as location, not the domain.
Search for IIS APPPOOL\DefaultAppPool, or replace DefaultAppPool with the application pool used in IIS.
Continue with configuration of SimpleSamlPHP from adding baseurlpath to config.php
Good luck!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,156 |
Q: How to get the Spotify refresh token in iOS SDK In the web api for /authorize a refresh and access token are returned. How can I access/ receive a refresh token similar to what is returned in /authorize?
Something like SPTAuth.defaultInstance().refreshToken?
A: You need to create a URL scheme for your app. Something like:
appName://SpotifyAuthentication
Then when you register your dev account with Spotify, you need to enter that as the redirect URI. When you make the request on the device (GET https://accounts.spotify.com/authorize?client_id=.....&response_type=code& redirect_uri=appName%3A%2F%2FSpotifyAuthentication&.....), it will call this URI automatically and will call: application:openURL:options: in AppDelegate.
The URL query string will contain your auth token. IE: appName://SpotifyAuthentication?authToken=someToken.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 274 |
\section{Introduction}
The purpose of this paper is to reconsider our understanding of
the mechanism of spontaneous chiral symmetry breaking (S$\chi$SB)
in QCD as reflected by the (possibly small) size of $\bar q q$ condensates
and to interpret its relation to the Zweig rule (ZR) violation in the scalar
channel and to the quark mass ratio $r =2 m_s/(m_u + m_d)$. New precise
experimental results which are expected to come soon \cite{daphne,mainz,exper}
will merely concern
low-energy $\pi \pi$ scattering which is governed by the chiral dynamics
of $u$- and $d$-quarks with the $s$-quark playing
at most the role of a ``sea-side spectator''. From this point of view the
existence of a relationship between low-energy $\pi \pi$ observables and the
quark mass ratio $r =m_s/m$ ($m=m_u=m_d$) is by no means obvious. A related
question concerns determinations of $\mathrm{SU}(2)\times \mathrm{SU}(2)$ low-energy constants
from $\mathrm{SU}(3)\times \mathrm{SU}(3)$ observables: for example, the low energy constants
$l_4$ and $l_3$, which are essential to assess the prediction of standard chiral
perturbation theory (S$\chi$PT) \cite{Gasser-Leutwyler} for
$\pi \pi$ s-wave scattering lengths \cite{pipistd}, are usually inferred using the
experimental values of $F_K/F_{\pi}$ and of the $K$- and $\eta$-meson
masses respectively.
The theoretical counterpart of these and similar phenomenological questions
concerns the dependence of various order parameters of S$\chi$SB on the number
$N_f$ of massless quark flavours. This $N_f$-dependence is an effect of
light quark loops and it is generally expected to be rather weak: it is
suppressed in the large-$N_c$ limit and it violates the Zweig rule which
are both considered as a good approximation to the real QCD (with a possible
exception of the anomalous $0^{-+}$ channel). On the other hand, more recent
investigations suggest that $N_f$-dependent light-quark loop effects
could sometimes be rather important. First, Nature does not seem to
respect the large-$N_c$ predictions in the scalar channel
which is not dominated by ideally mixed $\bar qq$ nonets, as required by the
Zweig rule \cite{Pennington}. Next, some recent lattice simulations with
dynamical fermions \cite{lattflav1,lattflav2}
observe a rather strong $N_f$-dependence of S$\chi$SB signals.
Finally, a new method of estimating the variation of the chiral condensate
between $N_f=2$ and $N_f=3$ from the
data has recently been proposed \cite{Moussallam}.
Using as input experimental informations
on ZR violation in the scalar channel at low and medium energies,
a large variation of $\langle\bar q q\rangle$ has been found.
For sufficiently large $N_f/N_c$,
the existence of chiral phase transitions is generally expected on the
basis of the behaviour of the perturbative QCD
$\beta$-function \cite{Banks-Zaks}. Various approaches have been proposed to
study the transitions arising when $N_f$ increases. Investigations about the
QCD conformal window, where the theory is asymptotically free in the
ultraviolet, but is governed by a non-trivial fixed point in the infrared,
suggest a restoration of chiral symmetry for $N_f\sim 10$ at $N_c=3$
\cite{conform}. The analysis of gap equations in the same conformal window
puts a slightly different bound, with a critical $N_f$ above 12 \cite{gapeq}.
On the other hand, the instanton liquid model indicates that instantons do not
significantly contribute to Chiral Symmetry Breaking for $N_f > 6$
\cite{instanton}.
We would like to stress that particular properties of
vector-like gauge theories such as QCD allow a new type of
non-perturbative interpretation of S$\chi$SB and of its $N_f$-dependence,
suggesting a natural link between a suppression of $\langle \bar q q\rangle$
and an enhancement of ZR violation in the scalar channel for not too
large $N_f/N_c$. No reference is made to concepts and methods such as gap
equation, mean field approximation, or the critical effective
coupling, which in confining non-Abelian gauge theories lack a truly
non-perturbative gauge-invariant definition.
Instead, QCD may be formulated in Euclidean space-time and the mechanism of
S$\chi$SB can be related to the dynamics of the lowest modes of the
hermitean Dirac operator \cite{Banks-Casher,Vafa-Witten,Leutwyler-Smilga}
\begin{equation} \label{dirac}
H[G] = \gamma_{\mu} (\partial_{\mu} + iG_{\mu}) = H^{\dagger}[G],
\end{equation}
averaged over gauge field configurations $G_{\mu}(x)$. In particular,
the role of increasing $N_f$ naturally appears as a consequence of the
paramagnetic effect of light quark loops: it tends to suppress the order
parameters that are dominated by the small eigenvalues of $H$. Whether this
suppression is strong enough to imply a phase transition for not too large
$N_f$ is a dynamical question which at present we are not able to answer
analytically. At least one can arrive at a theoretically coherent framework
suggesting a semi-quantitative understanding of various aspects of S$\chi$SB
and of their possible interconnection.
\section{$N_f$-dependence of infrared-dominated order parameters}
The theory will be considered in a four-dimensional
Euclidean box $L \times L \times L \times L$ with (anti)periodic
boundary conditions modulo a gauge transformation (i.e. on a torus).
We are interested in gauge invariant correlation functions which break the
chiral symmetry of the vacuum, i.e. in vacuum expectations of operators
which do not contain the singlet representation of the chiral symmetry group.
All external momenta are set to 0 and all quarks are first taken to be
massive. At the end we consider the limit in which the first $N_f$ lightest
quarks become massless keeping the remaining quark masses fixed.
One randomly chooses a gauge field configuration
$G_{\mu}(x)$ and performs the Grassmann integral over quark fields.
The result can be formally expressed in terms of eigenvalues and
eigenfunctions of the hermitean Dirac operator (\ref{dirac}):
\begin{equation} \label{eigen}
H[G] \phi_n = \lambda_n[G] \phi_n, \qquad\int dx\ \phi_n^{\dagger}(x) \phi_k(x)
=\delta_{nk}.
\end{equation}
Since $H[G]$ anticommutes with $\gamma_5 =\gamma_5^{\dagger}$, the spectrum
is symmetric around $0$. In addition, for gauge field configurations with
non-vanishing winding number $\nu \in Z$, the spectrum contains $|\nu|$
topological zero modes. Positive eigenvalues will be ranged
in the ascending order and numerated by a positive integer, defining
$\lambda_{-n}=-\lambda_{n}$. Notice that the eigenvalues $\lambda_{n}$ and
the wave functions
$\phi_{n}(x)$ are entirely given by the gauge field configuration
$G_{\mu}(x)$;
in particular, they are independent of the quark flavour and of the quark
masses\footnote{The wave functions $\phi_{n}$ live in the spin $\times$ colour
space and they transform as the fundamental representation of
$\mathrm{SU}(N_c)$ and as a 4-dimensional
$\mathrm{O}(4)$ spinor respectively.}.
The dependence on quark flavour and masses arises from the propagator
\begin{equation} \label{prop}
S_{j}(x,y|G) =\sum_{n}\frac{\phi_{n}(x)\phi_{n}^{\dagger}(y)}
{m_j - i\lambda_{n}[G]} \end{equation}
and from the fermionic determinant $\det(M-iH)=\prod_{j}\Delta(m_j|G)$, where
the single flavour determinant can be expressed as
\begin{equation}\label{det}
\Delta(m|G) =m^{|\nu|} \prod _{n > 0}\frac{m^2 + \lambda_{n}^2[G]}{m^2 +
\omega_{n}^2}.\end{equation}
The $G$-independent numbers $\omega_{n}$, which essentially coincide with the
free eigenvalues, provide a convenient overall
normalization of (\ref{det}) and will be specified shortly. An integral over
fermion fields of any product of bilinear quark currents can be expressed
in terms of the propagator (\ref{prop}) and of the determinant (\ref{det}).
The next step then consists in taking an average over all gluon
configurations. Before
we comment on this last point, let us concentrate on the simplest example
of an order parameter of S$\chi$SB.
The chiral condensate $\langle\bar u u\rangle$, where $u$ stands for
the lightest quark,
will be considered in the $\mathrm{SU}(N_f)\times \mathrm{SU}(N_f)$ chiral limit
\begin{equation} \label{lim}
m_1 = m_2 = \ldots = m_{N_f} = m \rightarrow 0, \qquad m_j \neq 0,\ j>N_f
\end{equation}
and it will be denoted as $-\Sigma(N_f)$. One has
\begin{equation} \label{cond}
\Sigma (N_f) = \lim \frac{1}{L^4} \ll \int\ dx
{\mathrm{Tr}}\ S(x,x|G)\gg _{N_f} = \lim \frac{1}{L^4} \ll
\sum_{n} \frac{m}{m^2 + \lambda_{n}^2}\gg _{N_f}.
\end{equation}
Hereafter, $\lim$ denotes the $\mathrm{SU}(N_f)\times \mathrm{SU}(N_f)$ limit (\ref{lim})
preceeded by the large volume
limit. The symbol $\ll \gg _{N_f}$ represents the normalized
($\ll\!1\!\gg_{N_f}=1$) average over gauge-field configurations weighted by
the fermionic determinant,
\begin{equation} \label{average}
\ll \Gamma \gg _{N_f} = Z^{-1} \int d\mu[G]\ \Gamma\ \Delta^{N_f}(m|G)
\prod _{j>N_f}\Delta (m_j|G) \exp\{-S[G]\},
\end{equation}
where $S[G]$ stands for the Yang-Mills action.
Since every gauge-field configuration $G_{\mu}(x)$ can be globally
characterized by the corresponding set of Dirac eigenvalues ${\lambda}$ and
eigenvectors ${\phi}$, the functional integral (\ref{average}) may be viewed
as an average over all possible Dirac spectra. The probability distribution
of Dirac eigenvalues should be, in principle, calculable from the theory
itself. In practice, it requires a non-perturbative regularization and
renormalization of the gluonic average (\ref{average}) which (as in
perturbation theory) may depend on the observable $\Gamma$. Whilst
this problem is hard to solve in general, in the particular case of chiral
order parameters such as (\ref{cond}), the formal structure of
Eq.~(\ref{average}) suggests some possibly interesting properties of
S$\chi$SB even before an analytic solution becomes available.
The fact that $\langle\bar u u\rangle$ is an order parameter of S$\chi$SB is
reflected by the vanishing of the $m \to 0$ limit of Eq.~(\ref{cond})
taken at finite volume. In order to get a non-trivial result, the large
volume limit has to be taken first and the spectrum of the Dirac operator
has to become sufficiently dense around the origin. Actually, the average
distribution of the smallest Dirac eigenvalues is all what matters: if in
Eq.~(\ref{cond}) one cuts the infrared end of the spectrum and sticks to
$|\lambda_n|>\Lambda$, the result would be zero, no matter how small
$\Lambda$ is. For the same reason, the ultraviolet divergences of the sum
$\sum_{n}$
in Eq.~(\ref{cond}) become irrelevant in the chiral limit. It turns out
that only eigenvalues which in the average accumulate like $1/L^4$ contribute
to the chiral condensate $\langle\bar q q\rangle$
\cite{Banks-Casher,Vafa-Witten,Leutwyler-Smilga}.
A similar discussion applies to other
order parameters of S$\chi$SB, in particular to the coupling $F_{\pi}$ of
Goldstone bosons to the axial current, whose square can be expressed as
the two-point correlator of left-handed and right-handed currents
$\langle L_{\mu}R_{\nu}\rangle$ at zero momentum transfer.
Denoting by $F^2 (N_f)$ the $\mathrm{SU}(N_f)\times \mathrm{SU}(N_f)$
chiral limit (\ref{lim}) of $F^2 _{\pi}$, one has
\cite{Stern}
\begin{equation} \label{conduct}
F^2 (N_f) = \lim \frac{1}{L^4} \ll \sum _{k,n} \frac {m}{m^2 + \lambda ^2 _{k}}
\frac {m}{m^2 + \lambda ^2 _{n}} J_{kn} \gg _{N_f},
\end{equation}
where $\lim$ has the same meaning as in Eq.~(\ref{cond}) and
\begin{equation} \label{mob}
J_{kn} = \frac{1}{4} \sum_{\mu} \left|\int dx\ \phi^{\dagger} _{k} (x)
\gamma _{\mu} \phi _{n}(x)\right|^2.
\end{equation}
Due to the Goldstone theorem, $F^{2}(N_f) \ne 0$ is both sufficient and
necessary for the chiral symmetry $\mathrm{SU}(N_f) \times \mathrm{SU}(N_f)$
to be spontaneously broken. Again, $F^{2}(N_f)$ merely receives
contributions
from the lowest Dirac eigenvalues but it is less infrared sensitive
than $\langle\bar u u\rangle$: the eigenvalues behaving in the average as
$1/L^2$ could now be sufficient to produce a non-zero value of
(\ref{conduct}), since there are two factors
$m/(m^2 + \lambda^2)$ for a single inverse power of volume \cite{Stern}.
Eq.~(\ref{average}) suggests that for $m \to 0$ the effect of the fermionic
determinant and the $N_f$-dependence will be stronger for observables
$\Gamma$ which are dominated by the lowest Dirac eigenvalues. This observation
follows from the rigorously proven inequality~\cite{Vafa-Witten}
\begin{equation} \label{VW}
|\lambda _{n}[G]| < C \frac{n^{1/d}}{L} \equiv \omega _{n},
\end{equation}
where $d$ is the space-time dimension ($d=4$ in our case) and $C$ is a constant
independent of the gauge field configuration $G_{\mu}(x)$, of the integer $n$
and of the volume $V=L^d$. (In general, $C$ depends on the shape of the space
time manifold, once the volume has been fixed.)
The existence of the uniform upper bound (\ref{VW}) reflects
the paramagnetic response of the Euclidean Dirac spectrum to an external
gauge field \cite{paramagnet}.
It allows to split the single flavour determinant (\ref{det})
into infrared and ultraviolet parts \cite{Duncan-Eichten}:
one chooses a cutoff $\Lambda$ and
defines an integer $K$ such that $\omega _{K} = \Lambda$. The determinant is
then written
\begin{equation} \label{split}
\Delta(m|G) = m^{|\nu|} \Delta_\mathrm{IR}(m|G) \Delta_\mathrm{UV}(m|G),
\end{equation}
where $\Delta_\mathrm{IR}$ involves the first $K$ non-zero eigenvalues and is
bounded by 1 as a consequence of the inequality (\ref{VW}),
\begin{equation} \label{IR}
\Delta_\mathrm{IR}(m|G) = \prod \limits_{k=1}^{K} \frac{m^2 + \lambda_{k}^{2}[G]}
{m^2 +\omega _{k}^2} < 1. \end{equation}
One may expect that in the case of chiral order parameters such as
$\langle\bar u u\rangle$
(\ref{cond}) or $F^2$ (\ref{conduct}) which are entirely dominated by the
infrared extremity of the Dirac spectrum, the $\Delta_\mathrm{IR}$ part of the
determinant will describe the bulk of the effect of light quark loops in
Eq.~(\ref{average}). This effect should be paramagnetic,
\begin{equation} \label{para}
\Sigma (N_f + 1) < \Sigma (N_f),\qquad F^2 (N_f + 1) < F^2 (N_f),
\end{equation}
indicating the suppression of chiral order parameters with increasing $N_f$.
How strong is this suppression depends on how sensitive is the observable
$\Gamma$ in Eq.~(\ref{average}) to the smallest Dirac eigenvalues for large
volumes and small quark masses. For this reason one may expect a stronger
suppression in the case of $\langle \bar q q\rangle$ than for $F_{\pi}$.
On the other hand, for observables that are not dominated by the
infrared extremity of the Dirac spectrum, the sensitivity to the
determinant and to $N_f$ can remain marginal, as expected in the large-$N_c$
limit.
\section{Why the scalar channel does not obey large-$N_c$ predictions}
We now turn to the connection between flavour dependence of order
parameters of S$\chi$SB and the observed violation of the Zweig rule in
the scalar channel. As before, we consider first $N_f$ light flavors of
common mass $m \to 0$ and denote by $s$ the ($N_{f}+1$)-th quark, whose mass
$m_s$ is non-zero, but still considered as light compared to the scale of the
theory. For $N_{f}=2$, this corresponds to the situation in real QCD.
$\Sigma(N_f)$ is a function of $m_s$, and its derivative reads
\begin{eqnarray} \label{der}
\frac{\partial}{\partial m_s} \Sigma(N_f) &=&
\lim_{m\to 0} \int\ dx \langle \bar u u(x) \bar s s(0)\rangle^{c}
\ \equiv\ \Pi_{Z}(m_s) \nonumber \\
&=&\lim \frac{1}{L^4} \ll \left(\sum_{k} \frac{m}{m^2 + \lambda_{k}^2}\right)
\left(\sum_{n} \frac{m_s}{m_{s}^2 + \lambda_{n}^2}\right)\gg _{N_f}^c ,
\end{eqnarray}
where the notations are as before and the superscript $c$ stands for the
connected part. Since $\Sigma(N_f) \to \Sigma(N_{f}+1)$
for $m_s \to 0$, one can write
\begin{equation} \label{diff}
\Sigma(N_f) = \Sigma(N_{f}+1) + \int_{0}^{m_s}\!\!d\mu\ \Pi_{Z}(\mu)=
\Sigma(N_{f}+1) + m_s Z_\mathrm{eff}^S(m_s) + O(m_{s}^2 \log m_s).
\end{equation}
In general, and for $N_f \sim 2-3$, the difference
$\Delta(N_f) = \Sigma(N_f) -\Sigma(N_{f}+1)$ is expected to be negligible
compared to $\Sigma(N_f)$ for two independent reasons: first, it is chirally
suppressed due to the smallness of $m_s$ relative to the condensate
$\Sigma(N_{f}+1)$, at least provided the latter is of the standard ``normal''
size. Second, the connected correlator (\ref{der}) of scalar quark densities
of different flavours is suppressed in the large-$N_c$ limit relative to
$\Sigma(N_f)$. An important $N_f$-dependence of the condensate would imply
that \emph{both} these arguments fail. We argue that this should naturally
be expected close to the critical point $n_{\mathrm{crit}}(N_c)$
at which $\Sigma(N_f)$ vanishes. Suppose that for a given value of $N_c$
(e.g. $N_c=3$), we have $N_{f} + 1 <n_{\mathrm{crit}}(N_c)$, so that
$\Sigma(N_{f}+1)$ is still non-zero but
already small. Then (for the actual value of $m_s$) the condensate term
need not dominate the expansion (\ref{diff}) in powers of $m_s$, not because
the chiral expansion breaks down but due to the suppression of
$\Sigma(N_{f}+1)$. The generalized chiral perturbation theory (G$\chi$PT) is
precisely designed to cope with such a situation
\cite{gchipt,pionpion1,pionpion2}.
The suppression of the
condensate means that near the critical point, the average
density of states \cite{Banks-Casher}
\begin{equation} \label{density}
\frac{1}{L^4} \ll \rho (\lambda)\gg _{N_f}, \qquad \rho (\lambda) =
\sum_{n}\delta(\lambda -\lambda_{n}[G]),
\end{equation}
drops for $\lambda \sim m$. Indeed, Eq.~(\ref{cond}) can be rewritten as
\begin{equation} \label{BC}
\Sigma(N_f) = 2 \lim \int_{0}^{\infty} \frac{du}{1+u^2} \frac{1}{L^4}
\ll \rho (mu)\gg_{N_f}\!.
\end{equation}
It is then natural to expect that the proximity of
a phase transition will further manifest itself by an increase of
fluctuations of the density of states and/or by an enhancement of the
density-density correlation
$L^{-4}\ll\!\rho(\lambda)\rho(\lambda^{\prime})\!\gg^{c}_{N_f}$ for
$\lambda \sim \lambda^{\prime} \sim m$. This quantity determines the
variation of the condensate, see Eqs.~(\ref{der}) and (\ref{diff}):
\begin{equation}
\frac{\partial}{\partial m_s} \Sigma(N_f) =
4 \lim \int_{0}^{\infty} \frac{du}{1+u^2}
\frac{dv}{1+v^2} \frac{1}{L^4} \ll \rho(mu) \rho(m_{s}v)\gg^{c}_{N_f}\!.
\end{equation}
Far away from the critical point, the correlation of small Dirac eigenvalues
should not be important, as predicted by the large-$N_c$ limit.
Since this limit leads to a suppression of quark loops,
any infrared-dominated order parameter becomes independent of $N_f$.
Therefore, the large-$N_c$ limit prevents $\Sigma$ from vanishing, since
the critical number of flavours $n_\mathrm{crit}(N_c)$ moves away to
infinity as $N_c\to\infty$. This asymptotic behaviour is
also supported by perturbative calculations,
in which $N_f$ and $N_c$ usually arise through their ratio.
Large-$N_c$ expansion is therefore expected to converge slowly for $N_f$
fixed just below the critical point $n_{\mathrm{crit}}(N_c)$, and to yield
irreparably false results for $N_f$ above it.
This argument merely concerns the vacuum channel $0^{++}$. The variation of
any chiral order parameter between $N_f$ and $N_f+1$ is given by a
correlation function which violates the Zweig rule precisely in that
channel, cf Eq.~(\ref{diff}): a strong variation would imply the existence
of $J^{PC}=0^{++}$, SU$_V(N_f)$-singlet states strongly coupled both to the
first $N_f$ light quarks and to the \emph{scalar} density $\bar s s $
of the $(N_f+1)$-th quark. The proximity of a phase transition could then
explain in a natural way why the spectrum of $0^{++}$ states is not
dominated by ideally mixed scalar mesons, presenting significant
discrepancies with large-$N_c$ predictions\footnote{
Connections with scenarios invoking trace anomaly and light
dilatons~\cite{dilaton} remain
to be seen. We thank H.~Leutwyler for bringing our attention to this
question.}.
In the following we concentrate on the actual case $N_f = 2-3$,
having in mind the possibility that (for $N_c=3$)
the real world might already be close to the critical point. It has recently
been pointed out \cite{Moussallam} that the correlation function
(\ref{der}) satisfies a
well convergent sum rule which allows a phenomenological estimate of
the $N_f$-dependence of the condensate between $N_f=2$ and $N_f=3$,
\begin{equation} \label{sumrule}
\Pi_{Z}(m_s) = \frac{1}{\pi} \int_{0}^{\infty}\frac{dt}{t}\ \sigma(t),
\end{equation}
where the spectral function $\sigma(p^2)$ (defined in Minkowski
space-time) collects ZR violating contributions which couple both
to $\bar uu$ and $\bar ss$
\begin{equation} \label{spectral}
\sigma(p^2) = \frac{1}{2}\sum_{n} (2\pi)^{4} \delta^{(4)}(p - P_{n})\langle 0|\bar
uu|n\rangle \langle n|\bar ss|0\rangle.
\end{equation}
Since we take $m_u = m_d=m$ and isospin symmetry cannot be spontaneously
broken \cite{restbreak}, only isoscalar states $|n\rangle$ contribute in
Eq.~(\ref{spectral}) and $\langle 0 | \bar u u | n \rangle = \langle 0 |
\bar d d | n \rangle$.
Under the plausible assumption that the dominant contribution comes from
two-particle states ($|\pi\pi\rangle, |K \bar K\rangle, \ldots $),
the spectral function
(\ref{spectral}) can be reconstructed from the corresponding multi-channel
T-matrix, which contains the experimental information on the size of Zweig
rule violation in the $0^{++}$ channel, such as the effect of the $f_0(980)$
resonance. Using further $\chi$PT to normalize the
solution of the multi-channel Muskhelishvili--Omn\`es equation, one can obtain
the corresponding form factors $\langle 0|\bar q q |n\rangle$. Within
the standard version of $\chi$PT, a strong $N_f$-dependence of the condensate
has been found \cite{Moussallam}: $\Sigma(3)/\Sigma(2) = 1-0.54\pm 0.27$.
We shall further comment on this result shortly.
\section{$\chi$PT considerations}
The case of a large Zweig-rule violation leading to a substantial difference
between the $N_f=2$ and $N_f=3$ condensates has never been fully
included into the $\chi$PT analysis before. This leads us to
carefully reconsider the G$\chi$PT relation between the actual size of the
condensate(s), the quark mass ratio $r=m_s/m$, and some low
energy observables. What matters is the renormalization group invariant product
$m\Sigma$ in physical units of $F^{2}_{\pi} M^{2}_{\pi}$, i.e.
the Gell-Mann--Oakes--Renner ratio(s) \cite{gor}
\begin{equation} \label{GOR}
X(N_f) = \frac{2m \Sigma(N_f)}{F^{2}_{\pi} M^{2}_{\pi}}.
\end{equation}
The standard chiral expansion presumes that both $X(2)$ and $X(3)$ are close to
1 which is, a priori, hardly compatible with an important flavour dependence
of the condensate. On the other hand,
in G$\chi$PT, the GOR ratio (\ref{GOR}) can be significantly below 1,
leaving enough space for a large
$N_f$-dependence. It is worth stressing that an important ZR violation
would not affect the quantitative relation between $X(2)$ and low energy
$\pi\pi$ observables provided it is systematically based on
$\mathrm{SU}(2)\times \mathrm{SU}(2)$ G$\chi$PT. The
reason is not a ``bad convergence'' of the expansion in the strange quark
mass. The ZR violation effects do not manifest themselves through a
specific low-energy constants in the $N_f=2$ effective Lagrangian, whereas
in the three-flavour $\mathcal{L}_\mathrm{eff}$
they show up as extra (ZR violating) low energy
constants $L_{6}(\mu)$ and $L_{4}(\mu)$ which are a priori unknown and
have never been determined experimentally\footnote{The reason of this
difference resides in group theory: while the ZR violating correlator
$\langle \bar uu(x) \bar dd(0)\rangle$ does not break the chiral symmetry
$\mathrm{SU}(2)\times \mathrm{SU}(2)$, the two-point function
$\langle \bar uu(x) \bar ss(0)\rangle$ is an order parameter for both $N_f=2$
and $N_f=3$ chiral symmetry.}. Hence, $X(2)$ remains accessible to experiment
via low-energy $\pi\pi$ phases \cite{pionpion1,pionpion2}, azimuthal
asymmetries in the decay $\tau \to 3 \pi \nu_{\tau}$ \cite{tau3pi},
or in the reaction $\gamma+\gamma \to 3 \pi$ \cite{gammagamma}.
The question remains how this information is related to the size of $X(3)$,
to the ZR violation in the $0^{++}$ channel and to the quark
mass ratio $r=m_s/m$.
A partial answer to these questions can be obtained from a new look
at old $\mathrm{SU}(3)\times \mathrm{SU}(3)$ G$\chi$PT expansions of
kaon and pion masses (see Ref.~\cite{gchipt} and App.~A of
Ref.~\cite{pionpion2}),
rewritten as
\begin{eqnarray}
F^2_\pi M^2_\pi &=& 2m\left[\Sigma(3)+(m_s+2m) Z_\mathrm{eff}^S(m_s)\right]
+4m^2 A_\mathrm{eff} + F^2_\pi \delta M^2_\pi,
\label{expan1}\\
F^2_K M^2_K &=& (m+m_s)\left[\Sigma(3)+(m_s+2m) Z_\mathrm{eff}^S(m_s)\right]
\nonumber\\
&&\qquad +(m+m_s)^2 A_\mathrm{eff} + F^2_K \delta M^2_K.\label{expan2}
\end{eqnarray}
These formulae collect in a scale independent manner all contributions
linear {\em and} quadratic in quark masses. They are useful
to the extent that $\delta M^2_{P} \ll M^2_{P}$, which is certainly a much
weaker requirement than the assumption which is at the basis of S$\chi$PT.
The constants $Z_\mathrm{eff}^S$ and $A_\mathrm{eff}$
are related to the quark-mass independent constants of the $O(p^2)$ G$\chi$PT
Lagrangian, $Z_0^S$ and $A_0$, renormalized at a scale $\mu$:
\begin{eqnarray}
Z_\mathrm{eff}^S(m_s)&=&2F^2(3) Z_0^S(\mu)-\frac{B_0^2}{16\pi^2}
\left\{\log\frac{\bar{M}_K^2}{\mu^2}+
\frac{2}{9}\log\frac{\bar{M}_\eta^2}{\mu^2}\right\},
\label{eff1}\\
A_\mathrm{eff}&=&F^2(3) A_0(\mu) \nonumber\\
&& \quad -\frac{B_0^2}{32\pi^2}
\left\{\log\frac{\bar{M}_K^2}{\mu^2}+
\frac{2}{3}\log\frac{\bar{M}_\eta^2}{\mu^2}
+3\log\frac{M_K^2}{\bar M_K^2}
+\frac{10}{9}\log\frac{M_\eta^2}{\bar M_\eta^2}
\right\},\label{eff2}
\end{eqnarray}
where we use the notations
$B_0=\Sigma(3)/F^2(3)$, $\bar M^2_{K,\eta}=\lim_{m\to 0} M^2_{K,\eta}$.
Both expressions (\ref{eff1}) and (\ref{eff2}) are scale independent.
The connection with the S$\chi$PT $O(p^4)$ constants is
$F^2(3)Z_0^S(\mu)=16 B_0^2L_6(\mu)$ and $F^2(3)A_0(\mu)=16 B_0^2L_8(\mu)$
respectively. The constant $Z_{\mathrm{eff}}^S(m_s)$,
independent of $m_u=m_d=m$, is the same as in Eq.~(\ref{diff}), taken for
$N_f=2$. It is convenient to split the scale independent remainders
$\delta M^2_{\pi,K}$ into two scale independent parts,
$\delta M^2 = \delta_{(1)}M^2 + \delta_{(2)}M^2$:
\begin{eqnarray}
\delta_{(1)} M_\pi^2&=&\frac{4m^2 B_0^2}{32\pi^2 F_\pi^2}\nonumber\\
&&\qquad\times \left\{3\log\frac{M_K^2}{M_\pi^2}
+\log\frac{M_\eta^2}{M_K^2}
+\frac{m_s}{m}\left[\log\frac{\bar M_K^2}{M_K^2}
+\frac{2}{9}\log\frac{\bar M_\eta^2}{M_\eta^2}\right]
\right\},\label{logs1}\\
\delta_{(1)} M_K^2&=&\frac{m(m+m_s) B_0^2}{32\pi^2 F_K^2}\nonumber\\
&&\qquad\times \left\{3\log\frac{M_K^2}{M_\pi^2}
+\log\frac{M_\eta^2}{M_K^2}
+2\log\frac{\bar M_K^2}{M_K^2}
+\frac{4}{9}\log\frac{\bar M_\eta^2}{M_\eta^2}
\right\}.\label{logs2}
\end{eqnarray}
Substituting Eqs.~(\ref{logs1}) and (\ref{logs2}) in Eqs.~(\ref{expan1}) and
(\ref{expan2}),
one recovers the full $O(p^4)$ standard expansion \cite{Gasser-Leutwyler}.
In this case, $\delta_{(2)}M^2$ consists of
$O(p^6)$ (two-loop) and higher standard contributions \cite{Bijnens}.
In the G$\chi$PT reading, $\delta_{(2)}M^2$ are $O(m_\mathrm{quark}^3)$ as
well, but now, they consist
of the tree $\mathcal{L}_{(0,3)}$ component of
$\mathcal{L}_\mathrm{eff}$, of the remaining
scale independent part of the one-loop $O(p^4)$ contributions not included in
Eqs.~(\ref{logs1}) and (\ref{logs2}), and of higher order terms.
As a result we expect $\delta M^{2}_{\pi,K}/M^{2}_{\pi,K}$ to be at most
3-4 per cent in the whole
range $0< X(3) < 1$ (this statement will be made more quantitative in the
final result). The control of the accuracy, independently of the size of
$X(3)$, is considerably simplified expanding the product $F_P^2 M_P^2$ rather
than $M_P^2$ and $F_P^2$ separately. It avoids uncertainties related to the
low energy constant $\xi$ ($L_5$) and to its ZR violating counterpart
$\tilde \xi$ ($L_4$). Further advantages of this way of
ordering the expansion of Goldstone boson masses will appear shortly.
A simple algebra allows one to infer from Eqs.~(\ref{expan1}) and
(\ref{expan2}) a relation between $X(3)$, the ZR violating constant
$Z_\mathrm{eff}^S$ and the quark mass ratio $r = m_s/m $:
\begin{equation} \label{x3}
X(3) + \frac{2m(m_{s}+2m)}{F^{2}_{\pi}M^{2}_{\pi}} Z_\mathrm{eff}^S = 1 - \tilde
\epsilon(r) + \delta X(3),
\end{equation}
where $\delta X(3)$ is a simple combination\footnote{
Hereafter, all manipulations with Eqs.~(\ref{expan1}) and~(\ref{expan2}) are
algebraically exact, making no use of expansions in $\delta M_{\pi,K}^2$ or
in quark masses.} of $\delta M^{2}_{\pi,K}$ and
\begin{equation} \label{epsilon}
\tilde \epsilon(r) = 2 \frac{\tilde r_{2}-r}{r^{2}-1},\qquad
\tilde r_{2}=2\frac{F^{2}_{K}M^{2}_{K}}{F^{2}_{\pi}M^{2}_{\pi}} - 1 \sim 39.
\end{equation}
This information can now be combined with the general Eq.~(\ref{diff})
(considered here for $N_f=2$). The latter can be recovered considering the
limit $\Sigma(2) = \lim_{m \to 0}(F^{2}_{\pi}M^{2}_{\pi})/2m$ of Eq.
(\ref{expan1}) keeping $m_s$ fixed:
\begin{equation} \label{x2}
X(2) = X(3) + \frac{2mm_s}{F^{2}_{\pi}M^{2}_{\pi}} Z_\mathrm{eff}^{S}
+\delta X(2).
\end{equation}
Eliminating $Z_\mathrm{eff}^{S}$ from Eqs.~(\ref{x3}) and (\ref{x2}), one arrives
at a simple relation between the $N_f=2$ and $N_f=3$ GOR ratios (\ref{GOR})
and the quark mass ratio $r$
\begin{equation}\label{rel}
X(2) = [1 - \tilde \epsilon(r)] \frac{r}{r+2} + \frac{2}{r+2} X(3) + \Delta.
\end{equation}
Before we show that the remainder $\Delta$ is small and well under control,
the simple meaning of Eq.~(\ref{rel}) should be stressed. The
three-flavour GOR ratio should be in the interval $0\le X(3) \le X(2)$
because of the vacuum stability and of the paramagnetic inequality
(\ref{para}). Since, furthermore,
$r>\tilde r_{1}=2 F_{K}M_{K}/F_{\pi}M_{\pi}-1 \sim 8$,
the second term on the r.h.s.
of Eq.~(\ref{rel}) represents a small correction all over the interval of $r$:
the quark mass ratio $m_s/m$ is merely given by the two-flavour GOR ratio
$X(2)$ which is more easily accessible experimentally, whereas $X(3)$
represents the fine tuning of the above relation. It might be, for instance,
conceivable that $X(2)$ would be close to 1, the quark-mass ratio $r$ close
to its standard value (or even larger) and yet $X(3) \sim 0$, implying
a very steep decrease of $X(N_f)$ to the critical point and a huge amount
of ZR violation. This is a possibility which has never been
considered before.
We finally discuss the uncertainty in Eqs.~(\ref{x3}), (\ref{x2}) and
(\ref{rel}). One has
\begin{equation} \label{err3}
\delta X(3) = \left[\tilde \epsilon(r) +\frac{2}{r-1}\right]
\frac{\delta M^{2}_K}{M^{2}_K}
- \frac{r+1}{r-1} \frac{\delta M^{2}_{\pi}}{M^{2}_{\pi}},
\end{equation}
whereas $\delta X(2)$ can be read off from the expansion of
$F^{2}_{\pi}M^{2}_{\pi}$ inspecting linear terms in the limit $m \to 0$
with $m_s$ fixed (including the whole $O(p^4)$ G$\chi$PT order, see
Eq.~(5.46) in \cite{pionpion2}). As a result,
$\delta X(2) =\delta_{(2)}M^{2}_{\pi}/M^{2}_{\pi} +\ldots $, where the
dots stand for contributions for which the dimensional estimate gives
$O[m^{2}m_s/(M^{2}_{\pi}\Lambda_{H})] \sim \pm 0.4/r^2$. Hence,
this uncertainty is at the per mille level. $\Delta$ is a simple linear
combination
$\Delta = \delta X(2) + \delta X(3)\cdot r/(r+2)$ and it may be represented
as $\Delta = \Delta_{1} \pm \epsilon$. $\Delta_{1}$ represents the
contributions of chiral logs to (\ref{err3}) arising from
$\delta_{(1)}M^{2}_{\pi,K}$. The contribution of $\delta_{(2)}M^{2}_{\pi}$ to
$\Delta$ almost exactly cancels between $\delta X(3)$ and $\delta X(2)$ and
the remaining uncertainty $\epsilon$ is dominated by $\delta_{(2)}M^{2}_K$.
The dimensional estimate gives $\epsilon = 0.025$ for $r=10$, $\epsilon=0.006$
for $r=20$ and even much smaller values for higher $r$. In
Fig.~\ref{x2vsr}, the correlation between $X(2)$ and $r$ is plotted,
including $\Delta_{1}$: Eq.~(\ref{x2}) is considered either with a maximal ZR
violation $[X(3)=0]$ or no violation at all $[X(3)=X(2)]$. We check the
fine tuning role of $X(3)$ in the significant correlation between $r$ and
$X(2)$. If the two-flavour GOR ratio is close to 1, the quark mass ratio
is not very accurately determined, but it is much more restricted for smaller
$X(2)$.
\EPSFIGURE{x2x3.eps,width=10cm,angle=-90}{Correlation between the
Gell-Mann--Oakes--Renner ratio for two massless flavours $X(2)$ and the
quark mass ratio $r=m_s/m$. The uncertainty due to $\epsilon$ is not taken
into account (see text). The upper curve respects strictly the Zweig rule,
which is maximally violated on the lower curve.
\label{x2vsr}}
Further insight can be obtained combining the general perturbative
expressions displayed above with Moussallam's sum rule (\ref{sumrule}).
Differentiating Eq.~(\ref{diff}) with respect to $m_s$ and using
Eq.~(\ref{eff1}) yields the identity
\begin{equation}\label{delta}
X(2)-X(3)=\frac{2mm_s}{F^2_\pi M^2_\pi}
\left[\Pi_Z(m_s) + \frac{B^2_0}{16\pi^2}
\left(\bar\lambda_K+\frac{2}{9}\bar\lambda_\eta\right)\right] + \Delta X,
\label{diffgor}
\end{equation}
where $\bar\lambda_P=m_s\cdot\partial(\log \bar{M}_P^2)/\partial m_s$, and
\begin{equation}
\Delta X=\frac{2m}{M^2_\pi F^2_\pi}
\left(1-m_s \frac{\partial}{\partial m_s}\right)
\lim_{m\to 0} \frac{F^2_\pi \delta_{(2)}M^2_\pi}{2m}.
\end{equation}
The identity (\ref{diffgor}) is a variant of Eq.~(\ref{x2}), in which the
ZR violating constant $Z_\mathrm{eff}^S$ has not been eliminated
but reexpressed using the sum rule (\ref{sumrule}) taken at $m=0$. $\Delta X$
differs from $\delta X(2)$ by insertion of
$(1- m_s \partial/\partial m_s)$. They are both expected of the same
order of magnitude and of opposite sign. In particular, $\Delta X$ receives
contributions starting at the two-loop order in S$\chi$PT.
Following the same dimensional estimate as in the discussion of the
uncertainty in Eq.~(\ref{rel}), one expects
$|\Delta X | < 0.05$ for $r \sim 10$, further decreasing for larger $r$.
Eq.~(\ref{delta}) provides a general framework for the discussion of the
variation of the condensate $X(N_f)$
between $N_f=2$ and $N_f=3$ in terms of the sum rule (\ref{sumrule}),
independently of any prejudice about the size of $X(3)$. It is in particular
valid in S$\chi$PT, in which the deviation of $X(3)$ from $1$ is considered
as a small perturbation. In the latter case,
$\bar\lambda_K =\bar\lambda_{\eta} = 1$ at the leading order, and
$2m m_sB_{0}^2$ can be replaced by $M^{2}_{\pi} ( M^{2}_{K} - M^{2}_{\pi}/2)$.
In this way one recovers the S$\chi$PT-based analysis of
Ref.~\cite{Moussallam}.
The second term on the right hand side of Eq.~(\ref{delta}) then takes the value
0.21, whereas the evaluation of the sum rule (\ref{sumrule}),
as discussed in Ref.~\cite{Moussallam}, corresponds to the final result
$0.38 < X(2) - X(3) < 0.73$, which is compatible with the conclusion
expressed in Ref.~\cite{Moussallam} in terms of the ratio $X(3)/X(2)$.
Notice that $\Pi_{Z}(m_s) >0$ for not too large $m_s$, because the two-point
function (\ref{der}) exhibits
a positive logarithmic increase as $m_s \to 0$. Consequently, in
S$\chi$PT, the bound $X(2)-X(3)>0.21$ must hold up to two-loop corrections.
A too large difference $X(2)-X(3)$ could hardly be reconciled with the
premises of S$\chi$PT which require both GOR ratios to be reasonably
close to 1. Consequently, a new analysis of the sum rule (\ref{delta})
within G$\chi$PT would be highly desirable.
\section{Summary and concluding remarks}
{\bf 1.} Order parameters of SB$\chi$S which are dominated by the
infrared extremity
of the Euclidean Dirac spectrum ($\Sigma=-\langle \bar{u}u\rangle$,
$F_\pi^2$, \ldots) could be sensitive to the paramagnetic effect of light
quark loops : in the chiral limit, the fermionic determinant reduces the
statistical weight of the lowest Dirac eigenvalues and gradually suppresses
these order parameters as the number of massless flavours $N_f$ increases.
This might result into a rich chiral phase structure as a function of $N_f$
and $N_c$.
{\bf 2.} For $N_f$ approaching the critical point $n_\mathrm{crit}(N_c)$, the
average density of low Dirac eigenvalues should drop and its fluctuations
should increase. This would naturally lead to a significant reduction of
$\langle \bar{q}q \rangle$, i.e. to a rapid decrease of the
Gell-Mann--Oakes--Renner (GOR) ratio $X(N_f)$ (\ref{GOR}), and to an
enhancement of
the Zweig rule (ZR) violation in the scalar channel as compared to the
large-$N_c$ predictions.
{\bf 3.} In the real world, where
$m_u\sim m_d \ll m_s \ll \Lambda_H \sim 1\ \mathrm{GeV}$, $X(2)$ and $X(3)$
are of a direct phenomenological interest ($1>X(2)>X(3)>0$). If the critical
point (for $N_c$=3) is far away from $N_f=3$, it is natural to expect
$X(3)\sim X(2) \sim 1$ and a less important ZR violation in the
$0^{++}$ channel. If, on the other hand, the real world is close to a phase
transition, $X(N_f)$ should quickly fall towards the critical point,
leading to a large ZR violating difference $X(2)-X(3)$. This would
force a value of $X(3)$ significantly below 1, leaving open the
question whether $X(2)$ already feels the influence of the critical point
or still remains close to 1.
{\bf 4.} $X(2)$ can be extracted from precise low-energy $\pi\pi$-scattering
experiments, independently of the ZR violation and of the size of
$X(3)$. Furthermore, $X(2)$ is closely related to the quark mass ratio
$r=2m_s/(m_u+m_d)$, and this relation is only marginally affected by $X(3)$.
On the other hand, even if $X(2)\sim 1$, and S$\chi$PT were a reliable
expansion scheme in the two-flavour sector, its accurate predictions for
s-wave scattering lengths \cite{Gasser-Leutwyler,pipistd} would be obstructed by
important ZR
violation already at the one-loop level: sofar, a reliable determination of
the $\mathrm{SU}(2)\times\mathrm{SU}(2)$ low-energy $O(p^4)$ constants
$l_3$ and $l_4$ from independent experimental data requires (among other
things) the knowledge of the $\mathrm{SU}(3)\times\mathrm{SU}(3)$ ZR
violating constants $L_6$ and $L_4$. The determination of the two-flavour GOR
ratio $X(2)$ remains the central goal of ongoing $\pi\pi$ scattering
experiments \cite{daphne,mainz,exper} and of related proposals
\cite{tau3pi,gammagamma}. If $X(2)$ turned out to be close to 1,
these experiments could be interpreted as a first measurement of the
low-energy constants $l_3$ and $l_4$.
{\bf 5.} Whatever the experimental output for $X(2)$ will be, additional
information will be necessary to pin down $X(3)$ and to settle the
theoretical issue of a nearby phase transition. The sum rule analysis of
Ref.~\cite{Moussallam} could be extended, but a more direct access to the ZR
violation in the $0^{++}$ channel and to the three-flavour condensate would
be desirable despite its difficulty.
{\bf 6.} The theoretical question of what happens in the phase in which we
likely do not live [$N_f>n_\mathrm{crit}(N_c)$], is at present hard to
answer unambiguously. The first chiral transition should merely affect
observables that are particularly sensitive to the lowest modes of the Dirac
operator. The $\rho$-meson mass, the string tension and the characteristics
of confinement in general need not be affected at all. Among chiral order
parameters, $\langle \bar{q}q\rangle$ exhibits the strongest infrared
sensitivity and is expected to vanish first. Whether this implies a complete
or partial restoration of chiral symmetry \cite{Appelquist}
is a question that involves the hard problem of non-perturbative regularization
and renormalization of the chiral symmetry-breaking sector of QCD. Within the
cut-off dependent bare theory, the vanishing of $\langle \bar{q}q\rangle$ implies
$F^2_\pi=0$, and consequently the full restoration of chiral symmetry
\cite{Shifman}. The validity of this argument in the full theory remains to
be clarified. The answer likely resides in a non-perturbative study of the fixed
points of renormalization group flows.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,377 |
Q: How do i find $(1+i)^{100}?$ How do I find $(1+i)^{100}$ without expanding $(1+i)$ 100 times?
Is there a quicker way to do this?
The hint was to find the modulus and argument of $1+i$ which I've got as $\sqrt{2}$ and $\pi/4$ but I'm not sure what to do from here.
A: Hint:
$1+i=\sqrt{2} e^{i\pi/4}$
Therefore, $(1+i)^{100}= [\sqrt{2} e^{i\pi/4}]^{100}=\sqrt{2}^{100}e^{i100\pi/4}=[{{2^{0.5}}}]^{100}e^{i(24\pi+\pi)}=2^{50}e^{i\pi}=2^{50}(-1)=\boxed{-2^{50}}$
Your job: how did we get $1+i=\sqrt{2} e^{i\pi/4}$?
To do this kind of problem in general, look at the polar form of a complex number:
http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Forms.aspx
A: You can use De Moivre's formula, $1+i=\sqrt{2}(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4})$, so $(1+i)^{100}=(\sqrt{2})^{100}(\cos 25\pi+i \sin 25\pi)$
A: $$(1+i)^2=2i\qquad \Longrightarrow\qquad (1+i)^4=-4.$$
A: Hint: $$(1+i)=\sqrt{2}e^{\frac{i\pi}{4}}$$
A: The ( at least a ) key is that the powers are periodic (argument-wise) .One way of doing Complex multiplication is by doing it geometrically, you multiply two numbers by squaring their respective moduli and by adding their respective arguments. Notice that $1+i$ lies on the line $y=x$, so that its argument is $\pi/4$. What happens if you go around $\frac {2k\pi}{\pi/4}$ times? The key is that $\pi/4$ is "commensurate" to $2\pi$, meaning it divides exactly (an integer number of times) into $2 \pi$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,191 |
{"url":"http:\/\/timescalewiki.org\/index.php\/Delta_derivative_of_squaring_function","text":"# Delta derivative of squaring function\n\nLet $\\mathbb{T}$ be a time scale. The following formula holds for $f \\colon \\mathbb{T} \\rightarrow \\mathbb{R}$ defined by $f(t)=t^2$: $$f^{\\Delta}(t)=t+\\sigma(t),$$ where $f^{\\Delta}$ denotes the delta derivative and $\\sigma$ denotes the forward jump.","date":"2019-01-20 06:52:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9997343420982361, \"perplexity\": 189.6277324248464}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583700734.43\/warc\/CC-MAIN-20190120062400-20190120084400-00630.warc.gz\"}"} | null | null |
Q: Need to find the correct PHP string I want one else if to return result for more than one value.
This is returning the information as 0534 holds.
I have some more values that i want to return. But not sure how to do this?
Is not possible to use stristr for this?
} else if( stristr( $product->prodno, '0534' ) ) {
$enlargements []= $productarray;
}
Grateful for any help i can get.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,990 |
\section{Introduction}\label{sec:}
Cognitive Decision Support System (DSS) is an integrated part of various complex systems that are designed based on the automated cognition concept, which is based on the perception-action cycle. Examples of DSS include
unmanned aircraft systems \cite{hou2007intelligent, hou2014intelligent}, biometric-enabled security checkpoints \cite{yanushkevich2019cognitive}, supply chain management \cite{ojha2018bayesian}, and automated interviewing \cite{yanushkevich2019cognitive}.
Performance of cognitive DSS is evaluated in various dimensions:
\begin{itemize}
\item [$-$]Technical, e.g. false accept and false reject rate
\cite{lai2017bridging},
\item [$-$]Social, e.g. public acceptance \cite{andreou2017identity},
\item [$-$]Psychological, e.g. efficiency of human-machine interactions \cite{hu2018computational,hugenberg2013towards,montibeller2015cognitive}, and
\item [$-$]Privacy and security domain, e.g. vulnerability of personal data
\cite{andreou2017identity,merler2019diversity,yanushkevich2019cognitive}.
\end{itemize}
Taxonomy and regulators such as \emph{risk, trust}, and \emph{bias} are useful in the performance evaluation of complex dynamical systems (Fig. \ref{fig:Risk-Trust_Bias_Regulators}). For example, trust to the artificial intelligence (AI) interview assistant addresses the so-called AI (as well as cognitive) biases \cite{yanushkevich2019cognitive};
risk and trust to DSS is related to various kinds of biases, e.g. in human identification based on face biometrics \cite{das2018mitigating,grother2019face} and social profiles \cite{andreou2017identity}. This paper focuses on risk and related assessment of an \emph{ensemble of biases} observed in a cognitive DSS, in particular, in a cognitive security checkpoint. Specifically, we are interested in the impact of biases on the DSS's performance.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.35]{fig/System_Perfomance.png}
\vspace{-4mm}
\end{center}
\caption{Illustration of the problem:
risk, trust, and bias are system regulators of technical, social, psychological, and security performance at various decision-making levels throughout various stages of human-machine and machine-machine interactions of cognitive DSS.
}\label{fig:Risk-Trust_Bias_Regulators}
\end{figure}
Many kinds of biases in biometric-based DSS were identified and studied. However, \textbf{dealing with an ensemble of biases is an open problem}. This is the focus of our interest. In our study, we model the DSS as a complex multi-state dynamic system \cite{yanushkevich2019cognitive,yanushkevich2019cognitive2}. Each state is characterized by a particular kind of bias. \textbf{The research question is how to fuse these biases in order to obtain the combined risk of biases and related trust estimators.}
Some preliminary results have been reported in the related areas.
In \cite{yao2013multi}, a trust bias interpreted as a prior knowledge has been incorporated into a non-probabilistic trust inference procedure. Three kinds of biases were distinguished: 1) a global bias, 2) a trustor bias, and 3) a trustee bias. A probabilistic network that is suited for approximate probabilistic confidence reasoning for trust inference problems has been developed in \cite{kuter2007sunny}. The DSS such as semi-automated cognitive checkpoint deals with \textbf{multiple (ensemble) of biases of different nature} from the following sources:
\begin{footnotesize}
\begin{center}
\begin{parbox}[h]{0.95\linewidth} {
\vspace{-2mm}
\begin{center}
\begin{eqnarray*}
\textbf{\texttt{Bias}}
\equiv \left\{
\begin{tabular}{ll}
\texttt{\small Human Perception} & \cite{hugenberg2013towards,montibeller2015cognitive} ; \\
\texttt{\small Artificial Intelligence} &\cite{yanushkevich2019cognitive} ;\\
\texttt{\small Biometrics} &\cite{das2018mitigating,grother2019face,merler2019diversity}; \\
\texttt{\small Identity Management} & \cite{andreou2017identity,yanushkevich2019cognitive}
\end{tabular}
\right.
\end{eqnarray*}
\end{center}}
\end{parbox}
\end{center}
\end{footnotesize}
A phenomenon of \emph{own-race bias} is well-known in psychology, the tendency to have better recognition for faces of one's racial ingroup rather than for racial outgroup faces \cite{hugenberg2013towards}, was recently confirmed in face recognition experiments \cite{das2018mitigating,grother2019face,merler2019diversity}. The source of additional cognitive biases is human-human interactions. The AI biases are introduced by intelligent support of human-machine interactions \cite{yanushkevich2019cognitive}. Finally, identity management bias was analyzed in multiple social profiles \cite{andreou2017identity}.
While all these biases are different, they are probabilistic in nature because the evidence and information gathered to make a decision is always incomplete, often inconclusive, frequently ambiguous, commonly dissonant, conflicting, and
has various degrees of believability. Trust and risk contribute to subject acceptance and rejection, respectively. Trust deviation impacts bias and vice versa. For example, by declining trust, one can increase bias. Trust is characterized by risks taken by a trustor in the presence of uncertainty.
\textbf{The goal of this paper} is to develop a bias-specific projection of the DSS such as cognitive security checkpoints. The performance of the security checkpoint is a good indicator of technological progress. The checkpoint is a complex system that includes various sensors, biometrics and data processing, pattern recognition and decision-making components. It is also an indicator of public acceptance of the privacy transformation. \textbf{The contributions} of our study include: 1) a systematic view on the risk-trust-bias impact on the cognitive DSS performance, and 2) an approach to the ensemble of bias assessment and operation.
This paper is organized as follows. After providing a background (Section \ref{sec:Basics}), {our approach is introduced in Section \ref{sec:Taxonomical-projection}. The details are highlighted in Section \ref{sec:experiment} via an experiment.} Section \ref{sec:Summary} concludes the paper.
\section{Background}\label{sec:Basics}
Bias is an application-specific phenomenon in the real-world:
\begin{itemize}
\item [$-$]In statistics, bias refers to the tendency of a measurement process to systematically over- or under-estimate the value of a population parameter.
\item [$-$]In psychology, bias is a well-defined phenomenon:
e.g., a customer bias is a deviation of the forecast from the true expectation.
\item [$-$]In biometric systems, biases occur, for example, in the form of a systematic difference between facial recognition of individuals belonging to different demographic groups
(performance bias), a systematic decision-making mistake (prediction bias), and a systematic mis-classification (classification bias).
\item [$-$]In Artificial Intelligence (AI), any machine learning-based assessment has a bias.
The key question is how to teach an AI assistant to act within the ethical and legal guidelines. Fairness detection can be referred to as AI bias.
\end{itemize}
Identifying and mitigating bias is essential to build {trust} between humans and machines. The machines can learn and assess the related {risks}. Risk of the biased AI judgment is a focus of interest in all AI applications, e.g. in medicine \cite{gates2018technology}.
Various characteristics are needed for the performance evaluation of the DSS. For example, the commonly used false reject rate and false accept rate in decision-making are affected by the level of the individual's satisfactory and social acceptance of profiling technologies. Other research challenges include identifying biases in human biometric and behavior recognition, in confidence estimation, AI trustworthiness, and belief inference in machine reasoning.
Each of these extensions is considered over specific operational landscapes. For example, bias cannot be propagated through system states, but risk and trust of bias can be represented by probability distributions and quantitatively assessed as probabilities.
{Risk} is a function of (a) an adverse impact, or magnitude of the harm, that would arise if the circumstance or event occurs; and (b) a likelihood of occurrence, \texttt{Risk$=F$(Impact, Probability)} \cite{nist2017security}.
\section{Bias taxonomy and operational landscape}\label{sec:Taxonomical-projection}
There are three phases of bias analysis in the DSS:
1) Bias identification, e.g, what kind of biases are manifested in a system?
2) Bias assessment, e.g. a unified metric for different kinds of biases, such as the risk of bias; and
3) Bias operation, e.g. fusion of risks of biases.
Recent studies are focused on bias identification such as demographic bias in facial recognition \cite{das2018mitigating,grother2019face,merler2019diversity} and AI bias \cite{gates2018technology}. This is a starting point in cognitive DSS development. Given the DSS multi-state model, the key requirements to the identified bias are assessment metric that provides bias operations such as propagation, adjustment, fusion, and prediction.
The aforementioned aspects of bias analysis are introduced in Fig. \ref{fig:Taxonomical_projection} for the particular case of the DSS, cognitive security checkpoint. The traveler's identity management process is implemented as a process that goes through multiple states \cite{yanushkevich2019cognitive,yanushkevich2019cognitive2}. Each state is characterized by a specific bias such as bias in face recognition when using surveillance and ID check, the bias of ID source reliability, etc. Statistics of these biases are being used for machine learning. The biases are mostly represented by the tailed probability distributions. A unified metric of bias that we consider in this study is the risk of bias and the related trust in the technology that is biased.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig/Biases.png}
\caption{Taxonomical projection of the cognitive security checkpoint in terms of biases: biases are propagated between states, and at each state, their status is analyzed, adjusted, fused, and predicted.}
\label{fig:Taxonomical_projection}
\end{center}
\end{figure}
Given the risk of a particular bias, various operations can be available with this risk value.
For example, the \emph{forward propagation} reflects a process that relates the causes to their respective effects. The bias \emph{backward propagation} reflects the bias assessment that traces the effects to their respective causes. Forward and backward bias propagation provide a systematic evaluation of the biases in traveler profiling or risk assessment. These biases are caused by various threats, hazards, and concerns, and are useful in deriving the cost-effective measures for lowering these risks to an acceptable level. We suggest that various techniques from related areas can be adopted for this purpose, in particular, causal relationships of risk factors and their propagation \cite{feng2014security}, risk propagation in supply network \cite{garvey2015analytical,ojha2018bayesian}, as well as trust prediction and belief propagation \cite{zhang2014trust}.
\section{Motivational experiment}\label{sec:experiment}
Fig. \ref{fig:Biases} illustrates various operations on biases. Bias is represented in metrics that must be compatible with metrics of risk and trust. The latter is represented by probability distributions of measures of the AI decision performance, decision reliability, confidence, credibility, and trustability. These values, in terms of posterior probabilities, can be inferred based on the prior probabilities and the current observation or conditions. The mechanism we apply for such inference is Bayesian, also called belief networks.
The motivations of an experiment are as follows:
1) Explain the details of the ensemble bias assessment, and
2) Introduce the essentials of reasoning mechanism.
Our motivational experiment addresses multiple biases in a cognitive security checkpoint (Fig. \ref{fig:Taxonomical_projection}) such as ``ID Reliability Bias'', ``ID Validation Bias'', and ``Trustworthiness Bias''.
Among various candidates of biases, we will consider ``Face Recognition Bias''.
Few results on demographic bias in facial recognition have been recently reported, in particular, in \cite{das2018mitigating,grother2019face,merler2019diversity}.
Our motivational experiment aims at highlighting the practical details of assessing an ensemble of biases.
\subsection{Causal model}
The causal network shown in Fig. \ref{fig:Biases} describes how the quality of facial recognition can be compromised by various facial attributes that are ``biased'' based on the year-of-birth (YOB) $Y$, gender $G$, ethnicity $E$, mustache $H$, beard $B$, and glasses $S$. The parent nodes to the ``Correctness'' node represents the bias attributed to face recognition. The ``Correctness'' node presents the probability of the neural network in predicting a positive (genuine subject) or negative (imposter) identity, whereas the ``Match'' node determines whether the positive or negative prediction matches the ground truth label.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.35\textwidth,interpolate]{fig/bn.png}
\caption{ A simplified causal network of biases for facial recognition. Risk is derived based on the results of the ``Match'', and Trust is affected by the Operator's Bias }
\label{fig:Biases}
\end{center}
\end{figure}
\subsection{Formalization}
The risk of the decision is evaluated based on the results of the ``Match'' node. The Risk affects the perceived Trust, which is also affected by the Operator's Bias (such as trained operator, inexperienced operator, perceived AI trustworthiness, and other human factors). The Operator then can use their judgment to adjust the parameters of the matcher in order to improve the decision outcome, thus implementing the ``Perception-Action'' cycle of the cognitive DSS. The adjustment action, in turn, will affect the Operator's trust in the DSS decision. It should be noted that the trust assessment would require the quantitative representation of the human factor (Operator) bias, which was not evaluated in this experiment. Therefore, we will focus on the risk assessment in the experiment described below.
Risk is estimated as \texttt{Risk$=F$(Impact, Probability)} which relates to the error rates of the system, specifically the false non-match rate (FNMR) and false match rate (FMR). At a high level of abstraction (e.g. ignoring metric and dependencies), risk of a particular bias $\texttt{Risk}_\text{\slshape Bias}$ is defined as follows:
\vspace{-4mm}
\begin{small}
\begin{eqnarray} \label{eq:riskb}
\texttt{Risk}_\text{\slshape Bias}&=& \underbrace{\texttt{Impact}_{\text{\slshape FMR}}}_\text{Cost of a FMR} \times \underbrace{\texttt{Error}_{\text{\slshape FMR}}}_{\text{FMR}} \nonumber\\
&+&\underbrace{\texttt{Impact}_{\text{\slshape FNMR}}}_\text{Cost of a FNMR} \times \underbrace{\texttt{Error}_{\text{\slshape FNMR}}}_{\text{FNMR}}
\end{eqnarray}
\end{small}
\vspace{-3mm}
For example, given the scenario of a security checkpoint, the FMR is related to a wrongly granted access, while the FNMR contributes to travelers' inconvenience. The impact of the FMR is a breach of security which, given this scenario, should have a high impact. The impact of the FNMR is a negative user experience which is rather of a low impact. Using this scenario, we assign a 10:1 ratio, that is $\texttt{Impact}_{FMR}=10$ and $\texttt{Impact}_{FNMR}=1$. Given the Year-of-Birth (YOB) attribute, the risk for individuals born in the 1930s is computed as follows: $10 \times 0.0208 + 1 \times 0.0012 = \fbox{0.2092}$.
The ensemble risk bias in identifying (matching) a particular individual $\texttt{Risk}_{\text{\slshape Bias}}\texttt{(Ensemble)}$ is assessed as the sum of the risk biases according to his/her attributes:
\vspace{-3mm}
\begin{small}
\begin{eqnarray} \label{eq:risk}
\texttt{Risk}_{\text{\slshape Bias}}\text{(\slshape Ensemble)}&=& \sum_{\text{\slshape Bias}=1}^{N} \texttt{Risk}_\text{\slshape Bias}
\end{eqnarray}
\end{small}
where $\text{\slshape Bias}$ represents one of the attributes, $Y,G,E,H,B,$ and $S$ in Fig. \ref{fig:Biases}, $N=6$.
\subsection{Experimental setup}
For this experiment, we demonstrate biases using the FERET face database. The database was collected between August 1993 and July 1996 and contains a total of 14,126 images, including 1199 individual subjects \cite{phillips1998feret}. The FERET database was chosen because of its detailed meta-data information which includes information such as gender, year-of-birth, ethnicity, and pose.
The typical performance of face recognition includes accuracy, FNMR, and FMR and are defined as $\mathrm{Accuracy} = \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{FP}+\mathrm{FN}+\mathrm{TP}+\mathrm{TN}}$, $\mathrm{FNMR} = \frac{\mathrm{FN}}{\mathrm{FN}+\mathrm{TP}}$, and $\mathrm{FMR} = \frac{\mathrm{FP}}{\mathrm{FP}+\mathrm{TN}}$, where TP are true positives (correct prediction identity), TN are true negatives (correct prediction of imposter), FP are false positives (incorrect prediction of identity), FN are false negatives (incorrect prediction of imposter).
The features used for face identification is extracting using a pre-trained Resnet50 convolutional neural network. The features are then passed through a fully-connected layer and softmax activation to perform identification. The model is trained using stochastic gradient descent with a learning rate $lr=0.0001$ and a momentum of 0.9 for 100 epochs.
\subsection{Results}
The overall identification accuracy, FNMR, and FMR for the FERET dataset are reported in Table \ref{tab:recognition}. The results are separated based on the biases such as gender, year-of-birth, ethnicity, and facial attributes (glasses, beard, and mustache). The performance measures are evaluated based on the Rank-1 results. Given a baseline accuracy of $91.63\%$, \textbf{positive biases are highlighted in green while negative biases are highlighted in red}. For example, observing the attribute year-of-birth, a significant bias is shown as the year-of-birth increases, specifically significant bias against younger individuals. \textbf{An accuracy difference of $\mathbf{17.65\%}$ between those born in the 1920s and the 1980s}. The FNMR and FMR for each bias are computed based on the bias nodes of the causal network shown in Fig. \ref{fig:Biases}.
\begin{table}[!htb]
\centering
\caption{Performance of the face identification process in terms of the accuracy, FNMR, and FMR (FERET dataset).}\label{tab:recognition}
\begin{tabular}{@{}c|ccc@{}}
Method & Accuracy & FNMR & FMR \\
\hline
Baseline & 0.9163 & 0.0941 & 0.0001 \\
\hline
Female &\ccar{0.8908 }& 0.1233 & 0.0003 \\
Male &\ccag{0.9308 }& 0.0749 & 0.0001 \\
\hline
1920s &\ccag{1.0000 }& 0.0000 & 0.0000 \\
1930s & 0.9697 & 0.0208 & 0.0012 \\
1940s & 0.9667 & 0.0443 & 0.0004 \\
1950s & 0.9513 & 0.0625 & 0.0003 \\
1960s & 0.9122 & 0.1020 & 0.0004 \\
1970s & 0.8873 & 0.1124 & 0.0003 \\
1980s &\ccar{ 0.8235 }& 0.1875 & 0.0105 \\
\hline
Asian & 0.9424 & 0.0664 & 0.0003 \\
Black-or-African-American & 0.9195 & 0.0984 & 0.0014 \\
Hispanic &\ccar{0.8254 }& 0.1981 & 0.0033 \\
Native-American &\ccag{1.0000 }& 0.0000 & 0.0000 \\
Other &\ccag{1.0000 }& 0.0000 & 0.0000 \\
Pacific-Islander &\ccag{1.0000 }& 0.0000 & 0.0000 \\
White & 0.9119 & 0.0965 & 0.0002 \\
\hline
No Glasses &\ccag{0.9214 }& 0.0923 & 0.0001 \\
Glasses &\ccar{0.8859 }& 0.1028 & 0.0009 \\
\hline
No Beard &\ccar{0.9134 }& 0.0976 & 0.0001 \\
Beard &\ccag{0.9615 }& 0.0357 & 0.0007 \\
\hline
No Mustache &\ccar{0.9144 }& 0.0947 & 0.0001 \\
Mustache &\ccag{0.9328 }& 0.0891 & 0.0008 \\
\end{tabular}
\end{table}
Table \ref{tab:subjects} illustrates the effect of biases on the accuracy of the chosen classifier in identifying 11 randomly selected subjects with their associated attributes. The top 3 identity predictions (with their respective score values) are shown in the last three columns. \textbf{The first 8 subjects are samples of correctly identified subjects, while the last 3 subjects are incorrectly identified subjects when using only the top/rank-1 prediction.} The highlighted results indicate a match to the ground truth label. The score values indicate that if we choose to use a threshold-based method of classification, it is not possible to decisively separate the genuine and imposter scores.
Only the top 5 scores of each prediction are used for creating the genuine and imposter scores. Scores, representing the similarity between faces, are generated from the penultimate layer of the neural network. Similar/identical faces generate large scores while unrelated faces produce low/negative scores. Through softmax normalization, the sets of scores are converted into probabilities. Due to the exponential nature of softmax normalization, lower scores, when converted to probabilities, are effectively reduced to zero, and thus, should not influence the decision-making process.
\begin{table*}[!htb]
\centering
\caption{Bias phenomenon of the facial recognition process: attributes for random Subjects (FERET dataset).}\label{tab:subjects}
\begin{scriptsize}
\begin{tabular}{r|ccc|ccc|rrr}
\multirow{2}{*}{Subject} & Year of & \multirow{2}{*}{Gender} & \multirow{2}{*}{Ethnicity} & \multicolumn{3}{c|}{Facial Features} & \multicolumn{3}{c}{Top Subject Predictions} \\
& Birth & & & Glasses & Beard & Mustache & First ( Score ) & Second ( Score ) & Third ( Score ) \\
\hline
15 & 1970s & Male & Asian & False & False & False & \ccag{ 15 }( 11.8175 ) & 913 ( 7.9727 ) & 612 ( 7.5015 ) \\
189 & 1970s & Female & Asian & False & False & False & \ccag{ 189 }( 15.7484 ) & 434 ( 8.5322 ) & 566 ( 6.9052 ) \\
143 & 1970s & Female & Hispanic & False & False & False & \ccag{ 143 }( 12.1884 ) & 402 ( 6.6154 ) & 763 ( 5.4235 ) \\
295 & 1980s & Female & Black-or-African-American & False & False & False & \ccag{ 295 }( 14.4057 ) & 364 ( 6.6878 ) & 808 ( 6.2970 ) \\
561 & 1940s & Male & Asian & False & False & False & \ccag{ 561 }( 10.4680 ) & 187 ( 6.6516 ) & 695 ( 6.0315 ) \\
917 & 1940s & Female & Hispanic & False & False & False & \ccag{ 917 }( 10.5805 ) & 251 ( 6.8186 ) & 972 ( 6.5261 ) \\
948 & 1950s & Male & White & False & False & True & \ccag{ 948 }( 10.7142 ) & 330 ( 5.4853 ) & 962 ( 5.1746 ) \\
684 & 1930s & Male & Asian & False & False & False & \ccag{ 684 }( 7.7132 ) & 501 ( 6.3049 ) & 775 ( 5.7674 ) \\
\hline
0 & 1940s & Male & White & True & False & False & 457 ( 8.0854 ) & \ccag{ 0 }( 6.7735 ) & 348 ( 6.3287 ) \\
774 & 1950s & Female & White & False & False & False & 704 ( 9.4579 ) & 801 ( 7.3339 ) & 942 ( 7.0282 ) \\
255 & 1960s & Female & White & False & False & False & 924 ( 8.8947 ) & \ccag{ 255 }( 7.0531 ) & 270 ( 6.9886 ) \\
\end{tabular}
\end{scriptsize}
\end{table*}
\subsection{Reasoning on risk of bias}
The model for reasoning upon the probability distribution of matching scores and the chosen biases is a probabilistic graphical model called Bayesian, or belief, network is shown in Fig. \ref{fig:Biases}. This is a causal network with the assigned Conditional Probability Tables (CPTs). Bayesian inference is applied to the CPTs to update the probability given evidence of various biases.
Example of calculation the prior probabilities for the parent nodes in the belief network (Fig. \ref{fig:Biases}) is as follows:
\begin{footnotesize}
\begin{center}
\begin{tabular}{cc}
\begin{tabular}{c|cc}
& \multicolumn{2}{c}{Gender} \\
$G$ & Male & Female \\
\hline
\(\Pr(G)\) & 0.6383 & 0.3617 \\
\end{tabular} &
\begin{tabular}{c|cc}
& \multicolumn{2}{c}{Wear Glasses} \\
$S$ & True & False\\
\hline
\(\Pr(S)\) & 0.1425 & 0.8575
\end{tabular}
\end{tabular}
\end{center}
\end{footnotesize}
These prior probabilities are vital for calculating the posterior probabilities representing the impact of biases on the resulting matching rate. It should be noted that for better dealing with uncertainty and conflicts, more advanced metrics rather than point probabilities as in Bayesian networks can be chosen to be applied in the causal network. These metrics of uncertainty include interval probabilities, fuzzy probabilities, as well as Demster-Shafer and Dezert-Smaradache extensions \cite{yanushkevich2019cognitive2}.
Any bias scenario can be represented by a causal network, e.g. Bayesian network.
Using the reported performance from Table \ref{tab:recognition} for risk estimation and an impact factor of 1 for both FNMR and FMR, the baseline risk of bias is computed according to Equation (\ref{eq:risk}) as
$1\cdot 0.0941+1\cdot 0.0001=\fbox{0.0942}$.
Let the attribute such as gender be known. Then the risk of bias is evaluated as follows:
\vspace{-5mm}
\begin{small}
\begin{eqnarray*}
\texttt{Risk}_\text{\slshape Bias}(\text{\slshape Female})&=&
\texttt{Impact}_{\text{\slshape FMR}} \times \texttt{Error}_{\text{\slshape FMR}}^{\text{\slshape Female}}\\
&+&{\texttt{Impact}_{\text{\slshape FNMR}}} \times \texttt{Error}_{\text{\slshape FNMR}}^{\text{\slshape Female}}\\
&=&1\times 0.1233+1\times 0.0003=\fbox{0.1236};\\
\texttt{Risk}_\text{\slshape Bias}(\text{\slshape Male})&=&
\texttt{Impact}_{\text{\slshape FMR}} \times \texttt{Error}_{\text{\slshape FMR}}^{\text{\slshape Male}}\\
&+&{\texttt{Impact}_{\text{\slshape FNMR}}} \times \texttt{Error}_{\text{\slshape FNMR}}^{\text{\slshape Male}}\\
&=&
1\times 0.0749+1\times 0.0001=\fbox{0.0750}
\end{eqnarray*}
\end{small}
\vspace{-5mm}
The risk associated with females (0.1236) is higher than males (0.0750) due to the recognition model being gender-biased. This can be reasoned due to a larger representation of males in the dataset, as evident in the prior node for gender distribution.
To evaluate the perceived trust of the Operators in the matcher decision, the Operator's bias shall be taken into account. As well, based on the assessed risk, the Operator can take action in adjusting the parameters of the matcher such as the decision threshold, since the Operator wishes to improve the trustworthiness of the decision. Another cycle of inference can be performed, while this time the risk is interpreted as evidence of the existing bias. This ``Perception-Action'' cycle of the cognitive DSS will lead to continuous re-assessment of the risk and trust in the DSS decision.
\section{Summary and conclusion}\label{sec:Summary}
Our study addresses the probabilistic reasoning as a mechanism for assessing an ensemble of biases aiming at their further propagation, adjustment, fusion, and prediction. In real-world cognitive DSS scenarios, each risk of bias in the ensemble is introduced by a probability distribution function (pdf) rather than CPTs as considered in the motivational example. The future research avenue here would be to perform a fusion of pdf(s) using the standard copula technique that shall result in the joint pdf. Marginalization will result in conditional pdf(s) that are assigned to the nodes of the causal network. The followed up reasoning mechanism will allow for inferences of risks and trust in the causal model.
\section*{Acknowledgments}
\begin{small}
This Project was partially supported by
Natural Sciences and Engineering Research Council of Canada (NSERC) through grant ``Biometric-Enabled Identity Management and Risk Assessment for Smart Cities'', and the Department of National Defence's Innovation for Defence Excellence and Security (IDEaS) program, Canada.
\end{small}
{\small
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,426 |
Produced by Jeroen Hellingman and the Online Distributed
Proofreading Team at http://www.pgdp.net/ for Project
Gutenberg (This book was produced from scanned images of
public domain material from the Google Print project.)
HOUSEHOLD STORIES
FROM THE LAND OF HOFER
OR,
POPULAR MYTHS OF TIROL,
INCLUDING THE
ROSE-GARDEN OF KING LAREYN.
BY
THE AUTHOR OF "PATRANAS; OR, SPANISH
STORIES," &c.
WITH ILLUSTRATIONS BY T. GREEN.
LONDON:
GRIFFITH AND FARRAN,
SUCCESSORS TO NEWBERY AND HARRIS,
CORNER OF ST. PAUL'S CHURCHYARD.
MDCCCLXXI.
LONDON: GILBERT AND RIVINGTON, PRINTERS, ST. JOHN'S SQUARE.
CONTENTS.
PAGE
Introduction 1
Norg Myths 13
1. The Eggshells 14
2. The Reckoning Day 15
3. Fritzl and the Tarnkappe 22
4. The Rose-garden of King Lareyn 26
Myths of North and South Tirol.
The Nickel of the Roehrerbuechel 73
The Wilder Jaeger and the Baroness 110
The Grave Prince and the Beneficent Cat 131
Klein-Else 158
Prince Radpot 188
The Three Black Dogs 207
Ottilia and the Death's Head 217
The Two Caskets 229
The Prudent Counsellor 235
The Geeseherds 253
St. Peter's Three Loaves 265
Myths of Waelsch-Tirol.
The two Cousins of St. Peter 273
Luxehale's Wives (including the Legends of the
Marmolata) 278
Zovanin Senza Paura; or, the Boy who went out to
discover Fear 335
The Dove-Maiden 356
Myths of Vorarlberg.
Kriselda 386
The Golden Pears 394
How the Poorest became the Richest 403
HOUSEHOLD STORIES
FROM THE LAND OF HOFER.
INTRODUCTION.
"Blessed are the people of whom history is silent; for
history occupies itself more with the doings of fools than of
the wise; with storms than with tranquil days: it immortalizes
the butcher and the tyrant, and consigns to oblivion
the innocent and peaceful."--Cibrario.
Something of the deep, strong attachment to their native mountains
which is innate in the children of the Alps steals over me when I
think of my pleasant journeyings in Tirol [1].
Though it is a little, out-of-the-way country whose cry is seldom heard
in the newspapers, though it exercises little influence in political
complications, the character of its people is one which, next after
that of our own, has a claim to our esteem and admiration. Hardy,
patient, and persevering; patriotic and loyal to a fault; honest and
hospitable to a proverb--they carry the observance of their religion
into the minutest practice of every-day life; and there underlies
all these more solid qualities a tender, poetical, romantic spirit
which throws a soft halo round their ceaseless toil, and invests
their heroic struggles for independence with a bright glow of chivalry.
Surrounded from their earliest years with living pictures of Nature's
choicest forms and colouring, they need no popular fiction to cultivate
their imagination, no schools of design to educate their taste.
Shut out from the world's ambitions by their pathless Alps, they have
learned to see before them two aims alone,--to maintain the integrity
and the sanctity of their humble homes on earth, and to obtain one
day a place in that better Home above, to which the uplifted fingers
of their sun-bathed mountain-peaks ever gloriously point.
The paramount claims on their hearts' allegiance of the hearth and
the altar are inseparably interwoven in their social code, and their
creed scarcely knows of a distinction between Nature and Nature's God.
At their mother's knee they have learnt, every one, to prattle
of their Father in heaven with as complete a realization of His
existence as of that of their father on earth. Just as they receive
their toil-won food and raiment as an earnest of the paternal care
of the one, the change of the seasons, the sunshine and the rain,
betoken to them as certainly that of the Other. They scarcely trace
any line of demarcation between the natural and the supernatural;
and earth and sky are not for them the veil which hides Divinity, but
the very temple and shrine of the Godhead dwelling among His creatures.
Going forth in this simple faith through the pure, unfogged atmosphere
which surrounds them, it is scarce to be wondered at if they can trace
the guiding footprints and the unerring hand of Providence where for
others are only chances and coincidences. Or that--like the faint
outline of wished-for land revealing itself to the trained eye of
the sailor, where the landsman sees but a hopeless expanse of sky and
ocean--they should recognize a personal will and individuality in the
powers which are the messengers to them of the good pleasure of Heaven,
in the germination of fruit and grain, in the multiplication of their
flocks and herds; or of the envious malice of the Evil One, in the
wind and the lightning, the torrent and the avalanche, destroying
the work of their hands.
It is necessary to bear this well in mind, or we shall not appreciate
the delights which their fantastic tales have in store for us. We
must learn to realize that this way of viewing things has created a
nomenclature, almost a language, of its own.
When the boisterous blast sweeps through their valleys, scattering the
scent of the wild game, and driving them far out of their reach, they
say it is the Wilder Jaeger [2], the Beatrik [3], or the Nachtvolk
[4], on his chase. Their restless energies, pent up within the
shelter of their rattling walls and casements, invest him with a
retinue of pitiless followers and fiery-eyed hounds--while the fate
of some who have ventured out while he is said to be abroad, blown
over precipices or lost in crevasses, is expressed by the fancy that
his train is closed by a number of empty pairs of shoes, which run
away with those who come within his influence.
When the bright beams of sun and moon enliven their landscape,
or fructify their seed, or guide their midnight way, they fable of
them as beautiful maidens with all sorts of fanciful names derived
from associations as old as the world: Perahta, brightness, daughter
of Dagha, the daylight--hence, also, Perchtl and Berchtl. In other
localities, Holda or Hulda; in others again, they are known as Angane,
and Enguane, the Saligen Fraueelein, Nornen, Zarger Fraueelein, and
Weissen Fraueelein. They say they smile on the overburdened peasant,
beguile his labour by singing to him, show him visions of beautiful
landscapes, bestow wonderful gifts--loaves which never diminish,
bowls and skittles, charcoal and corn of pure gold; to the husbandman
they give counsels in his farming; to the good housewife an unfailing
store--bobbins of linen thread which all her weaving never exhausts;
they help the youth or the maiden to obtain the return of the love they
have longed for, and have some succour in store for every weary soul.
Such helpers the people recognize of the masculine gender, also, in
the so-called Noergl, Pechmannl, Puetzl, Wiehtmaennlein, Kaesermaennlein,
and Salvanel; for possibly, they say, not all the angels who rebelled
with Lucifer may have been cast into the outer darkness. There may
have been some not so evilly disposed themselves, but talked over
and led astray by others; and such, arrested in their descent by a
merciful reprieve, may have been only banished to the desolate and
stony places of the earth, to tops of barren mountains and fruitless
trees. Such as these might be expected to entertain a friendly
feeling for the human beings who inhabit the regions which gave them
shelter, and to be ready to do them a good turn when it lay in their
way--lift weights, and carry burdens for them up the steep heights,
and protect their wild game. And, also, it is not inconsistent with
their nature to love to play them a mischievous trick full oft--make
off with the provision of loaves prepared for the mowers; sit, while
remaining invisible, on their sledges and increase their difficulty
and confusion in crossing the mountain-paths lost in snow; entice
them into the woods with beseeching voices, and then leave them to
wander in perplexity; overturn the farm-maids' creaming-pans; roll
the Senner's cheeses down the mountain sides.
Worse tricks than these are those of the Wilder Mann. When the soil
is sterile and ungrateful; when any of the wonted promises of nature
are unfulfilled; when the axe of the lonely woodman rebounds from
the stubborn trunk and wounds him; when the foot of the practised
mountain-climber fails him on the crisp snow, or the treacherously
sun-parched heather; when a wild and lawless wight (for such there are
even in Tirol, though fewer, perhaps, than elsewhere) illtreats the
girl who has gone forth to tend her father's flock upon the mountains,
trusting in her own innocence and Heaven's help for her protection--it
is always the Wilder Mann--in some places called the Wilder Joergel, in
others, the Lorg, the Salvang or Gannes, the Klaubaut or Rastalmann,
in Vorarlberg, Fengg, Schraettlig, Doggi, and Habergass--to whose
account the misfortune or the misdeed is laid. His female counterpart
is called Trude and Stampa, and the Langtuettin.
The mineral riches of the country, and the miners occupied in
searching for them, are told of as of hidden treasure sought after or
revealed, as the case may be, by the Bergweib and the Bergmannlein,
or Erdmannlein, the Venedigermannlein and the Hahnenkekerle, the
stories of whose strength and generosity, cupidity and spite, are
endless; while the mountain echoes are the voices of sprites playfully
imitating the sounds of human life.
If the mountains and the forests are thus treated, neither are the
lakes and torrents without their share of personification, and many
are the legends in which the uses and beauties of the beneficent
element are interpreted to be the smiles and the helpful acts of the
Wasserfrauelein, while the mischances which occur at the water's edge
are ascribed to the Stromkarl and the Brueckengeist.
The sudden convulsions of nature to which their soil has been subject
from age to age are all charged with retribution for the sins of the
people, like the overthrow of Sodom and Gomorrah and the cities of
the plain. Castles and forest possessions of wicked rich men are sunk
beneath the waters of lakes so that their foundations may never again
be set up, and their place be no more found; while a curse pursues
those who attempt to dig out the ill-gotten treasure. Villages
are recorded to have been swallowed by the earth or buried by the
snow-storms when their people have neglected the commandments of God.
This literal adaptation of the admonitions of Holy Writ receives among
this people another development in traditions of instances where good
deeds done to the poor have been believed to have been actually done
to visions of our Lord and of His Saints. Then again, their devout
belief in both the irresistible justice and the ineffable love of
God convinces them that there must be a place on earth where souls
too soiled for heaven, yet not given over to utter reprobation, may
wander till the final day of rest. And thus every shepherd, as he keeps
his lonely watch upon the Alpine pastures, expects that he may meet
the feurige Sennin who broke the Sunday rest; or the Tscheier Friedl
who was cruel to the cattle in his charge; or the buessender Hirt who
stole the widow's kine; or the Markegger who removed his neighbour's
landmark; or the Pungga-Mannl who swore a false oath; or the feuriger
Verraether who betrayed the mountain pass to the Roman legions.
On the other hand, the heroes and types of the Christian faith are
thought of as taking a perpetual interest in the welfare of their
struggling brethren: St. Nothburga and St. Isidore watch over
the husbandman, and St. Urban over the vinedresser; St. Martin
over the mower; St. Martha, St. Sebastian, and St. Rocchus, the
drei Pestschutzheiligen [5], are expected to be as potent in their
intercession now as when at their prayers, when on earth, plagues
were stayed. St. Anthony and St. Florian similarly protect against
fire. St. Vigilius, the evangelizer of the country and martyr to
his zeal, is still believed to guard its jealously-preserved unity
of faith. In return, they receive special veneration: the ordinary
dealings of life are regulated by the recurrence of their festivals,
and the memory of sacred mysteries is kept in perpetual honour by
setting up their tokens in every homestead and every house, in every
vineyard and in every field, on every bridge and by every wayside.
It is not surprising that a people so minded have tales to tell
of wonderful events which seem to have befallen them, and which
take the record of their lives out of the prosaic monotony which
rules our own,--tales always bearing a wholesome moral lesson,
always showing trust in Providence and faith in the World Unseen,
and always told with the charming simplicity which only a logically
grounded expectation that events should turn out even so--and no
invention or imagination--can give.
A selection of these tales I have put into English dress in the
following pages. Though some few of them may be found to bear analogy
with similar tales of other German nations, the distinctive qualities
of the Tirolese, and the peculiar nature of the scenery amid which
they have been conceived, will be found to have stamped them with a
character entirely their own.
I think that what I have said is sufficient to give, to such of my
readers as did not possess it already, the key to their application,
and I need not now append to each a tedious interpretation of the
fantastic personages and scenes I shall have to introduce.
It remains only to say a word as to their distribution. The present
principality of Tirol is composed of four provinces. North Tirol,
South Tirol, Waelsch or Italian Tirol, and Vorarlberg. North and South
Tirol have been for long so closely united that, like their language
and customs, their mythology has become so intimately intermingled that
it scarcely comes within the scope of a work like the present to point
out their few divergences or local peculiarities. But those of Waelsch
Tirol and Vorarlberg each maintain a much more distinctive character,
and I have accordingly marked with a separate heading those which I
have gathered thence.
"The Rose-garden of King Lareyn," however, is not the peculiar property
of any one province. Though the three places which claim to occupy its
site are all placed in South Tirol, this pretty myth is the common
property of the whole country--its chief popular epic--and has even
passed into the folklore of other parts of Germany also. It is beside
the purpose of the present little work to enter into the controversy
which has been raised concerning the authorship. There can be little
doubt, however, that it was originally the utterance of some unknown
minstrel putting into rough-and-ready rhyme one of the floating myths
which symbolized the conflict of the heroes with the powers of evil,
so popular in the middle ages. Then poets of more pretensions wrote
out, and, as they wrote out, improved the song. Thus there are several
different manuscripts of it extant, of between two and three thousand
lines each, but not of equal value, for later scribes, in trying
to improve, overlaid the simple energy of its diction with a feeble
attempt at ornament which only served to damage its force. The name of
the Norg-king who is the subject of it, is in these spelt variously,
as Lareyn, Luarin, Luarine, &c.; the modern orthography is Laurin. The
spelling I have adopted is that of the Chronicon of Aventinus.
I have thought it well to precede the story by some account of the
Norg folk and some samples of their legends, that the reader may not
come wholly unacquainted with their traditional character to the tale
of the discomfiture of the last Norg-king.
NORG MYTHS.
The Norgen were a mighty folk in olden time in Tirol. In their
span-high bodies resided a power which no child of man, were he ever
so stalwart and well-limbed, could resist. But they were also for
the most part a peaceable race, and more inclined to assist than to
obstruct the industrious inhabitants of the country in their labours;
so long as they were treated with respect and deference they seldom
interfered with any one. Then they were generally scrupulously
honourable, and strict keepers of their word. A service rendered
one of them was sure to be repaid a hundredfold. An injury brought
a corresponding retribution, and scorn, contempt, or ridicule roused
their utmost vengeance; while some there were who entertained a true
spirit of mischief, and indulged in wanton tricks which showed their
character was not altogether free from malice.
They were most often to be met in lonely paths and unfrequented
fastnesses of nature, but a solitary Noerglein could also occasionally
stray within the haunts of men, at times asking hospitality at their
hands, and at others getting into the bedrooms at night, and teasing
the children in their sleep, hence the common proverb--
"Shut the door closely to,
Or the Norg will come through [6]."
And at other times, again, they would take part in the field and
household labours, as if they found it sport. The name of Norg
was chiefly appropriated to them in South Tirol; in Vorarlberg the
analogous cobbold went by the name of Rutschifenggen. Every locality,
every valley, every hamlet, and almost every farm, had its own familiar
dwarf whose doings were handed down as household words.
Thus it is told that there was once a countrywoman, who lived in a
lonely Meierhof [7] of the Passeierthal, standing over her stove,
preparing a pancake for her husband's dinner, and as he was a great
eater she used an immense number of eggs--three dozen and more--in
his pancake: as fast as she broke the eggs into the pan, she threw
the shells behind her. Three Norgs came by as she was so occupied
and amused themselves with playing with them and arranging them into
all kinds of patterns. The Meierin [8] was a grumpy sort of woman,
and instead of finding pleasure in the glee of the little people,
grew cross with them, and scattered the dirty black ashes among the
egg-shells they had arranged so prettily. Offended at this ill-natured
treatment, the Norgs took their departure, but first laid the thread
of the good wife's spinning-bobbin as a snare across the floor,
and then stationed themselves outside the window to see what happened.
Presently the husband called to know if the pancake was not ready, and
the Meierin, running to satisfy him, with both hands engaged in holding
the dish of the enormous pancake, caught her feet in the thread, and
fell flat on the ground with her face in the dish, while the three
Norgs completed her vexation by setting up a loud laugh in chorus.
Here is another story of their doings, in which they play a different
part. There was a storm in the valley of Matsch, and a storm in the
valley of Matsch is often a terrible matter. This was one of the
worst: the pitiless flood streamed down the heights, and threatened
to overflow the banks of the Hochseen [9]; the wind from the glacier
howled dismally over the mountain-sides; the people closed their
doors and shutters against the blast, and listened to the roar of the
elements, trembling with the thought that every moment might come the
signal of the inundation which should carry them and their habitations
away in its torrent. In the solidest and most important house of
the straggling village, which bears the same name as the valley,
was gathered the family of the richest man of the place, who had no
reason to share these fears, but with singing and lively conversation
chased away the dismal influence of the lugubrious sounds without.
Suddenly, between the angry gusts of wind, a doleful voice was heard
piteously praying for help. One of the party opened the casement,
and looked out, but with more of curiosity than interest, and then
quickly closing it again, came back into the room with a laugh to
describe the ludicrous figure he had seen. It was a little mannikin
with a beard big enough for a full-grown man, his clothes drenched
with the rain, and slung over his shoulder a tiny bundle tied in a
handkerchief, which yet seemed to bow him down with its weight. The
description provoked a chorus of laughter, and the wretched little
Norg--for it was a Norg--would have been no more thought of but that
his wail became more irritating than that of the wind, and at last
the master of the house got up and shouted to him to go on, for it
was useless to stand droning there, he was not going to open his house
at that time of night, or to such a ridiculous object. But though he
banged the window to as closely as possible after delivering himself
of this speech, the little man's menacing couplet yet reached his ear--
"The reckoning day
Is not far away [10]."
Nevertheless the Norg begged no more, but endeavoured to pass on his
way. He could not get far: the torrents of rain had obliterated the
path which led from the rising ground on which this house was built,
to the next, and it was scarcely safe to descend in the dark with the
loose stones rattling away under the feet. Fortunately a glimmering
light betrayed a low hut built into the <DW72>. It looked so poor and
humble, that the Norg felt ashamed to ask aught of its inhabitants,
who could scarcely have had enough for their own needs; but when he
saw how utterly forlorn was his position, he sat down on a stone,
and wept. Notwithstanding that the poor little Norg had such a
hoarse voice that it was more like that of a wild animal than a man,
there was a compassionate little maid within who perceived it was a
voice of distress, and put her head out to ask who was there. "Poor
old man!" she cried; "come inside and dry yourself, and let me give
you something warm." But before he could answer he heard a weak voice
within, "Beware, Theresl, of the wolves--remember we are in 'Matsch der
Woelfe Heimath [11].'" "Never fear, mother dear," replied the maiden,
"this is no wolf, but a very distressed little old man, who does not
look as if he could harm any one; and besides we are now in June--the
wolves don't threaten us in the summer," and she opened the door,
and let in the little man.
By the time she had dried his clothes and fed him with some warm
soup, the worst of the storm had abated, and he was able to go on his
way. The maiden offered him shelter for the night, but he declared
he must reach home before midnight, and prepared to depart. Before
he left he asked her what there was she most desired. "Oh, that my
mother be restored to health!" answered Theresa; "I desire nothing
more than that!"
The Norg walked to the bedside, and informed himself of the
nature of the sick mother's illness. "Your mother shall be
cured," said the little man; "but you must come to me to-morrow
at midnight to the Noergelspitz;" and as the girl started at the
impossibility of the feat, he continued, "You have only to make
your way as far as the Wetterkreuz, and there call three times
'Kruzinegele! Kruzinegele! Kruzinegele!' and I will be at your side,
and take you up the rest of the way." And he took his departure,
singing,--
"Morgen oder Heut
Kommt die Zahlzeit."
The next night Theresa courageously set out on her way, and climbed
as far as the Wetterkreuz--and it was lucky she had to go no farther,
for here she sank down quite exhausted. She had not lain there many
seconds when she saw a procession of little men just like Kruzinegele,
with a litter and torches, who carried her up till they came to
a door in the rock, which opened at their approach. This led to a
magnificent crystal hall glittering with gold and gems, and on a
gold throne sat Kruzinegele himself, with his fair daughter by his
side. When the litter was brought to the steps of the throne, he
came down courteously, and renewed his thanks for her hospitality,
but she could not find a word to say, in her astonishment at seeing
him so changed. Meantime he sent his daughter to fetch the herbs
which were to cure the poor mother, and gave them to her, telling
her how to administer them. "You see," he added,--
"Morgen oder Heut
Kommt die Zahlzeit;
and your rich neighbour will find it so too." Then he told the little
men to carry her home, and they laid her in the litter, and bore
her away; and she remembered nothing more till she found herself
comfortably in bed, with the rising sun kissing her cheeks. But the
appearance of every thing was as much changed as Kruzinegele himself
had been! The walls that used to bulge, and reek with mildew and
damp, were straight and smooth; glass casements replaced the ricketty
shutters; nice white curtains tempered the sunshine; the scanty and
broken furniture was replaced by new. But what she valued above all,
in her hand were the herbs which were to make her mother's healing
drink! Their decoction was her first occupation; and by the next day
they had restored her mother to health, and joy once more reigned in
the cottage, thanks to the Norg!
It had been the rich churl's custom, equally with the other villagers,
to take his cattle on to the mountain pastures to graze for the
beginning of the summer season am Johanni [12]. His grazing ground
was just the highest pasture of the Noergelspitz. The festival now
soon arrived, and the picturesque processions of cattle with their
herds went lowing forth as usual, to enjoy their summer feed.
When the Norg's enemy, however, arrived at his destination, instead
of the emerald <DW72>s he was wont to find, with their rich yield of
marbel and maim [13], all ready prepared by St. Martin's care [14] for
the delight of his cows and sheep, all was stony and desolate! Three
days they spent wandering about in search of a few blades to browse,
but even this was denied them--nor ever again did the Noergelspitz
bring forth any thing but ice and snow!
Of the sleek droves which had started, the envy of all beholders, few
beasts lived to return; the prosperity of the once flourishing Hof
had fled, and before many years were out its proprietor was obliged
to leave it, a ruined man. Theresa had in the meantime married a
thrifty peasant, whose industry enabled him to be the purchaser of
the abandoned Hof, which he soon stocked to the full extent of former
days. Ofttimes a curious grey-bearded little stranger would drop in
at night to share their comfortable meal, and before he went away he
would always sing his couplet--
"Morgen oder Heut
Kommt die Zahlzeit."
Such occasional apparitions of the strange visitants excited the
curiosity of the inhabitants of the earth to the utmost, and many a
weird story was told of frightful injury happening to those who had
striven to penetrate their retreat, and for a long period none had
any success in the enterprise.
It happened one day, however, that a daring hunter who had been led
far from his usual track, and far from the country with which he was
familiar, by the pursuit of a gemsbock, found himself at the entrance
of a low-arched cavern. As night was about to fall and the sky wore
a threatening aspect, he was glad to creep within this shelter till
the light of morning should enable him to find his way home once more.
He had not proceeded far within the dim corridor, when he perceived
that in proportion as he got farther from the light of day the cave
became brighter instead of darker! Eagerly seeking the cause of this
phenomenon, he perceived that the walls were all encrusted with gold
and precious stones, which emitted constant sparkles of light. He
thereby recognized at once that he had reached an approach to one of
the resorts of the Mountain-folk, as the Norgs were also called from
having their habitation in the hearts of the mountains.
To avoid the fate of those who had ventured within the mysterious
precincts, he was about to make good his escape, when he felt something
soft under his feet. It proved to be a red hood or cap, dropped there
by one of the Mountain-folk, a veritable Tarnkappe which had the
property of making the wearer invisible to men, and also enabled him
to command admission to any part of the subterranean settlement. He
had scarcely placed it on his head when one of the little men of the
mountain came running up to look for his lost cap. Fritzl the hunter
was much too cunning to give up the advantage of its possession, but
with great good humour he told the dwarf he reckoned it too great an
advantage to have the opportunity of visiting his beautiful territory
to give it up for nothing; but he assured him he should have no reason
to regret having given him admission. The dwarf could not choose but
obey, and the Jaeger enjoyed the singular privilege of surveying all
the hidden treasures of the underground world.
Beautiful are the glories of the mountain world as seen by mortal
eyes--gorgeous its colours when illumined by the southern sun,
but all this is as barren darkness compared with the glories hidden
within its stony recesses. Here, the sky overhead was all of diamonds
and sapphires and carbuncles, and their light sparkled with tenfold
glory and beauty to the light of the sun and moon and stars; the
trees were of living gold and silver, and the flowers and fruit of
precious stones; the grass all of crystal and emerald; there was
no cold or heat, no perplexing change of season, but one perpetual
spring spread its balmy air around; lakes there were all of opal and
mother-o'-pearl, and gorgeously plumed swans perpetually crossing
them served the inhabitants in lieu of boats.
The Jaeger's delight and admiration at all these sights won the sympathy
and regard of his guide, and by degrees he grew more communicative,
and explained to him the whole economy of their mode of life. He showed
him how they were divided into three distinct classes: those wearing
red caps, who were gay and good-natured, and filled with goodwill
towards mankind also, notwithstanding many wild pranks; those with
brown caps, whose mischief was mingled with malice rather than fun,
but who yet would suffer themselves to be propitiated; and those
with black caps, always gloomy and morose, who boded evil wherever
they went. His guide advised him to have nothing to say to these,
but with some of the red and brown he was admitted to converse: he
found them pleasant and sociable, and ready enough to communicate
their ideas. Some asked him questions, too, about various matters
which seemed to have puzzled them in their peregrinations on earth,
while others, who had never been outside their own habitations,
had other inquiries to make--but some there were also who had no
curiosity on the subject, but rather contempt; and one thing that
amused the Jaeger in them was their incapacity to conceive many of
the things he had to tell them, and particularly to understand what
he could mean when he talked about death.
Chiefly to keep the spiteful freaks of the black-caps in check
there was a guard of warrior dwarfs, whose array was shown to our
Jaeger. Formidable they must have been, for the armour of each was made
out of one diamond, and they wore helmets and greaves and shields all
of diamonds, and while they were thus impervious to every attack, their
swords were of diamond too, and resistless therefore in their thrusts.
The Jaeger could not restrain his raptures at their gorgeous show, as
the colours of the gems around were reflected in this shining armour.
The dwarf had nothing left to show after this, but then stood and
sighed over the glories of the past. "And what were the glories of the
past?" inquired the Jaeger, with intense interest. The dwarf watched his
interlocutor closely, and satisfied himself that his interest was not
feigned. Then he paused long, as hesitating whether to unburden himself
to a stranger of the sad thoughts which crowded into and oppressed
his mind. A few words of sympathy, however, decided him at last
"Yes, we still have some power and some riches left, and some of
our ancient strength, but we have lost our kings, the kernel of
our strength. It is true, we are able to surprise you with isolated
exhibitions of riches and power, but, on the whole, your people has got
the better of ours; and since your heroes of old destroyed the last of
our royal race, we have been a doomed, disorganized, dwindling race,
fast disappearing from our ancient fastnesses."
"And how happened it that our people got the better of yours? How
did our heroes destroy your royal race? I pray you tell me."
The dwarf led the Jaeger into a delicious alcove of the opal rock, whose
pure, pale lustre seemed more in accordance with his melancholy mood
than the garish brilliancies that had hitherto surrounded them. They
laid them down on the bank, and the dwarf thus recounted the story.
THE ROSE-GARDEN OF LAREYN, THE LAST NORG-KING.
The lineage of our kings had endured for countless generations,
he said, and had always enjoyed the undeviating homage of our people.
In our kings were bound up our life and our strength; they were the
fountain of our light and the guardians of our power. The royal race
was a race apart which had never mingled with the race of the governed,
yet which had never failed or been found wanting. But Adelgar cast his
eyes on Hoerele, one of the Norginnen of the common herd, and raised
her to share his throne. The union not only was unblessed--what was
worse, all the rest of the royal stock died out, and all the noble
princes of his first marriage died away one after the other [15];
and when Hoerele at last came to die herself, there was only one left.
This was Lareyn, the last of his race. Adelgar looked around him
with tears, for there was none left to whom he could marry his son,
and he had experienced in himself the ill effects of departing from
the ancient tradition which forbade him from mingling his race with
the race of the governed, and he bewailed his folly.
But Lareyn bethought him of a remedy; he determined to go out into
the outer world, and choose him a wife among the daughters of its
inhabitants, and bring her to reign over the mountain people and
continue the royal stock. In a supreme council of the elders of the
kingdom it was decided to approve what he proposed. But Adelgar only
consented with much reluctance, and accompanied his permission with
many conditions and counsels, the chief of which were that Lareyn and
his suite should every one go forth clothed in a Tarnhaut [16], and
that he should exercise his choice in a far distant country where the
ways of the dwarfs were not known, and where, whatever might befall,
no friend of the bride could think of coming to his palace to seek
her, for the old king rightly judged that the Christian folk would
not willingly give a daughter of theirs to the Norgs.
Lareyn promised his father to attend to his injunctions, and
gave orders to prepare a thousand suits of diamond armour for his
body-guard, and five hundred suits of silk attire for his pages, who
were to bear the gifts with which he meant to captivate the maiden
of his choice, and Tarnhauts to cover them all--and, above all,
the presents themselves of jewels and priceless goldsmith's works,
at which the Norgen were very expert.
While all these things were being got ready Adelgar died, and Lareyn
succeeded to the crown. However much he desired to adhere to his
father's injunctions, he was forced to decide that under the altered
circumstances it could not be well for him to journey to a distance
from his kingdom, and to leave it long without a head. He determined,
therefore, to search the neighbourhood for a maiden that should please
him. In the meantime he made use of his newly acquired power to prepare
a dwelling to receive her which should correspond with the magnificence
of his presents, and by its dazzling lustre should make her forget all
that she might be inclined to regret in her earlier home. The highest
title of honour was now promised to whoso of his subjects could point
out to him an unexplored mine of beauty and riches. This was found in
a vault all of crystal, which no foot of dwarf had ever trod. Lareyn
was beside himself with gladness when he saw this; he ordered a
hundred thousand dwarfs immediately to set to work and form of it
a residence for his bride; to divide it into chambers for her use,
with walls and columns encrusted with gold; to engrave the crystal
with pleasing devices; and to furnish it with all that was meet for
her service. Thus arose the great Krystallburg [17] ever famous in the
lays of the Norgs, and which the cleverest and richest of the children
of men might have envied. That so glorious a palace might be provided
with a garden worthy of it, hundreds of thousands of other dwarfs were
employed to lay out the choicest beds and bowers that ever were seen,
all planted with roses of surpassing beauty, whose scent filled the air
for miles round, so that, wherever you might be, you should know by
the fragrant exhalation where to find the Rosengarten of King Lareyn
[18]. Engrossed with these congenial preparations, Lareyn forgot all
his prudence and moderation: that they might be completed with all
possible expedition the whole working community of the dwarfs was
drawn off from their ordinary occupations; the cultivation of the
land was neglected, and a famine threatened. Lareyn then would go
out and make a raid on the crops of the children of earth, and take
possession of whatever was required for the needs of his own people,
without regard for the outcry raised against him, knowing that,
strong in his supernatural strength, he had no retaliation to fear.
While thus he pursued his ravages every where with indiscriminating
fury, he one day came upon the arativo [19] of a poor widow whose
only son was her one support. The golden grain had been gathered into
her modest barn just as Lareyn and his marauders came by; swift,
like a flock of locusts, they had seized the treasure. The widow
sobbed, and her stalwart son fought against them in vain; Lareyn was
inexorable. At another time the good-nature of his Norg blood would
have prompted him at least to repay what he had appropriated in the
gold and precious stones of which he had such abundant store, but now
he thought of nothing but the prompt fulfilment of his darling design;
and he passed on his way unheeding the widow's curse.
At last the Krystallburg was complete, and the Rosengarten budding
ready to burst into a bloom of beauty. To so fair a garden he would
have no other fence but a girdle of silk, only he gave it for further
defence a law whereby any who should violate that bound should forfeit
his left foot and his right hand.
Lareyn looked round, and his heart was content. He felt satisfied now
that he had wherewithal to make any daughter of earth forget her own
home and her father's people, how delightful soever might have been
the place of her previous sojourn.
Donning his Tarnhaut, he went forth with his followers marshalled
behind him, all equally hidden from human sight.
He wandered from castle to castle, from Edelsitz [20] to Edelsitz,
from palace to palace, but nowhere found he the bride of his heart,
till he came to the residence of the Duke of Styria. Here, in a
garden almost as lovely as his own Rose-garden, he found a number of
noble knights assembled, and their ladies, all of surpassing beauty,
taking their pleasure on the greensward amid the flowers.
Lareyn had never seen so much beauty and gallantry, and he lingered
long with his attendant wights running from one to another, and
scanning the attractions of each, as a bee hovers from flower to
flower, gathering the honey from their lips. Each maiden was so
perfect, that he would have been content with any one of them, but
each was so guarded by her cavalier that he saw no way of approaching
her; at last, driven to despair, he wandered away under the shade of
a lonesome grove.
Here, under a leafy lime [21], his eye met a form of loveliness which
surpassed the loveliness of all the dames he had heretofore seen
put together, and he felt thankful now that he had not been able to
possess himself of any of them, for then he had never seen her who
now lay before him in all the bloom of her virgin perfection. Lareyn,
accustomed to associate his conceptions of beauty with a dazzling
blaze of gold and jewels, found an entirely new source of admiration
in the simple attire of the Styrian princess, for it was Simild,
daughter of Biterolf, Duke of Styria, who lay before him, seeking
rest amid the midday heat, draped only in virgin white, with wreathed
lilies for her single ornament.
Lareyn stood absorbed for some time in contemplation of her perfect
image. Then, hearing the voices of her companions drawing near, quickly
he flung a Tarnhaut over her, so that they trooped by, searching
for her, and passed on--seeing her not--to seek her farther. Then
he beckoned to the bearers of a litter he had prepared in readiness
to approach, into which her sylph-like form was soon laid; and over
hill and dale he carried her towards the Rosengarten.
They had got some way before Simild woke. Lareyn rode by her side,
watching for her eyes to open, and the moment she gave signs
of consciousness he made a sign for the cortege to halt. Quick as
thought a refection was laid out on the greensward, while a band of
Norg musicians performed the most delicious melody.
Simild, enraptured with the new sights and sounds, gazed around,
wondering where she was and what all the little creatures could be who
hopped around ministering to her with so much thoughtfulness. Lareyn
hastened to soothe her, but fancying that some of the Norgs were
wanting in some of their due services to her, he rated them in such
a positive tone of command that Simild began to perceive that he was
the master of this regiment of ministrants, and hence she inferred
that by some mysterious means she had fallen into his power; but what
those means could be she was at a loss to conceive.
Lareyn now displayed his presents, and in presenting them poured
forth the most enthusiastic praise of her beauty. Simild's vanity
and curiosity were both won; yet the strangeness of the situation,
the sudden separation from her friends, her ignorance of what might
be going to befall her, roused all her fears, and she continued to
repeat in answer to all his protestations of admiration that she
could listen to nothing from him till he had restored her to her home.
"This is the one thing, sweet princess, that I cannot do at your
bidding," he replied. "Whatever else you desire me to do shall be
instantly executed. And it is hardly possible for you to exhaust my
capacity of serving you."
Then he went on to describe the magnificence and riches of his kingdom,
and all the glories over which, as his bride, she would be called to
reign, till her curiosity was so deeply excited, and her opposition
to his carrying her farther grew so faint, that he lost no time in
taking advantage of her mood to pursue the journey.
In the meantime the greatest consternation had fallen on all the
friends of Simild. The maidens whose duty it was to wait on her sought
her every where, and not finding her they were afraid to appear before
her father. The knights and nobles who had been in her company were
distracted, feeling the duty upon them to restore her, and not knowing
which way to begin. The old Duke Biterolf shut himself up within the
palace and wept, objecting to see any one, for his heart was oppressed
with sorrow; and he refused to be comforted till his child should be
restored to him.
But Dietlieb, Simild's brother, a stout young sword [22], when he
had exhausted every counsel that occurred to him for discovering his
sister's retreat, determined to ride to Gardenna on the Garda-See,
the castle where resided Hildebrand [23] the Sage, renowned for wisdom,
and prudence, and useful counsel.
When Hildebrand the Sage saw him come riding yet a long way off, he
said to those who stood beside him on the battlements, "See Dietlieb
the Styrian, how he rides! His heart is full of indignation. Up,
my men, there is work for us; some one has done him a great wrong,
and us it behoves to stand by him, and see him righted."
Ute, Hildebrand's wife, and her daughters prepared a warm welcome for
the prince, as was due; and the heroes gathered round Hildebrand held
out their hands to him as to one whose integrity and valour claimed
their respect. Hildebrand himself led him to his chamber, and left
to no maiden the task of helping him off with his armour [24], but
with his own hand lifted off his helmet and laid by his good shield.
Then they placed refreshing wine from the cool cellar in the rock
before him, and a banquet of many dishes, as became so worthy a
guest. When the tables had been removed [25], Hildebrand invited his
young guest to detail the cause which had brought him. Dietlieb, who
was burning to tell the story of his mishap, poured out the details
of his sister's misadventure, without omitting the smallest incident
which could serve Hildebrand to form an opinion as to the remedy to
be adopted.
The event was so strange that Hildebrand himself could not venture
all at once to divine the nature of the injury. But he forbore also
to express his perplexity, lest the bold young Styrian should be
discouraged. Without therefore expounding exactly what his views
were, but determining to ponder the matter more deeply by the way,
the advice he propounded in the first instance was, that they should
all repair forthwith to seek the aid of Berndietrich [26].
The counsel was received with joyful acclamation; and loud was the
clanging as every one ran to don his chain-armour, for all were glad
to be called to deeds of high emprise, and such they deemed were in
store for them if Dietrich von Bern was to be their leader.
Ute and her daughters, to whom their courage and mettle was well known,
greeted them as they went forth with no sinking hearts, but gave them
augury of good success.
As they journeyed along, they came to a broad heath, which they were
about to pass over with their train, when up sprang a man of forlorn
aspect, who cried after Hildebrand, and asked his aid.
Hildebrand, seeing him in such sorry plight, turned aside out of
compassion, to ask what had befallen him. It was no other than the
peasant--the widow's son--whom Lareyn had so deeply wronged, and,
seeing the heroes go forth in such brave array, he besought their aid
against the oppressor of his mother. Some of them laughed at his wild
mien and uncouth gestures, but Hildebrand the Sage took him apart,
and lost not a word of his story of how the Norg-king lived in the
heart of the mountains, of how he came out with his mighty little men,
and ravaged all the face of the country, contrary to all the habits of
his former life, and of how it was all because his own labourers were
engaged in preparing the most magnificent palace for the reception
of a daughter of earth, whom he meant to make his bride.
Hildebrand now felt he knew all, and with the help of the poor
countryman, the widow's son, would be able to conduct the heroes
into his retreat, inflict condign punishment, and release the captive
princess.
How, with purely natural means, to overcome the resistless strength
of the Norgs did not indeed make itself apparent; this was matter
for further consideration, and sufficed to engross his thoughts for
the rest of the journey. Of one thing he was satisfied--that he was
right in claiming the intervention of Berndietrich, whose traffic
with the supernatural powers [27] made him, of all the wigands [28],
alone capable of conducting such an expedition.
Hildebrand and his companions were received by Theodoric with hearty
welcome and hospitable care and cheer. As they sat at table, all
the heroes together vied with each other in lauding the prowess of
Theodoric, till they had pronounced him the bravest sword of which
the whole world could boast.
This was the time for Hildebrand. "No!" he cried, as he upsprang,
and by his determined manner arrested the attention of all the
wigands. "No, I say! there is one mightier than he; there is one with
whom he has never yet ventured to measure his strength----"
"Who? Name him!" shouted Theodoric, rising to his feet, and glaring
round him with defiant fury, only kept in check by his regard for
Hildebrand.
"I speak of Lareyn, the Dwarf-king, the dweller in the depths of the
mountains of Tirol," replied Hildebrand, in a voice of firm assurance.
"The Dwarf-king!" exclaimed Theodoric, with incredulity and contempt;
and he sat down again.
"As long as the Dwarf-king is suffered to live in his mountain
stronghold, and to ravage the lands of the peaceful peasants, I call
no man who knows of him a hero. But him who overcomes this little
one--him I will call a hero indeed, above all others!"
"If your Dwarf-king were so formidable, Meister Hildebrand," replied
Theodoric, "you would have told me of him before now, I ween. How
has he raised your wonderment just at this time?"
"Because just at this time his insolence has increased. He has built
a palace surpassing all palaces in magnificence, which he calls his
Krystallburg, and has surrounded it with a garden of beauty, which
he calls his Rosengarten, fenced round only with a silken girdle,
but of whomsoever crosses that boundary he forfeits the left foot
and the right hand."
The report of this boast was enough to decide Theodoric, the impetuous
sword. "If it is thus he vaunts him," he cried, "he shall know that
there is one will dare brave his decree, and destroy the garden his
ferocity guards after the manner you describe."
With that up he rose, and called for his Velsungen [29], for his
armour he never put off, and he called for his helmet and his horse;
and before another had time to frame his purpose, he had started,
without parley and without guide.
Only Wittich the Wigand, his boon companion, who loved to share his
rash ventures, and was familiar with his moods, could bestir himself
to follow before he was too far gone to be overtaken.
To Tirol they rode by day and by night, without slacking rein, for
their anger brooked no reprieve. They slacked not their speed for
dell or mountain, and they rode forty miles through the dense forest;
but every where as they went along they tested the air, as it was
wafted past them, to see if they could discern the perfumes of the
Rosengarten. At last, as they toiled up the mountain side, a majestic
sight was suddenly opened to their view. The white shining rock of the
living mountain was cut and fashioned into every pleasing device of
turret and tower, diamonds and rubies were the windows, and the dome
was of pure gold set with precious stones. "We have far to ride yet,"
said Wittich the Wigand, as he scanned the lordly place. "And yet the
perfume of the Rose-garden reaches even hither," said the Bernaere
[30]. "Then we know we are on the right track," answered Wittich;
so they put spurs to their horses, and rode forward with good heart.
They had pushed on thus many a mile when the blooming Rosengarten
itself came in sight, entrancing their senses with its beauty and
its odours.
"What was that?" asked Theodoric, who always rode ahead, as some
light obstruction made his mount swerve for a moment.
"Why, you have burst the silken girdle of King Lareyn's
Rosengarten!" said Wittich the Wigand; "so now we have incurred the
vengeance of the little man."
"Ah!" said Theodoric, as he gazed around, "let us not harm this
pleasant place; sweetly are the flowers disposed, and in the fragrant
hours of evening and of morning it must be well to be here: let us
destroy naught!"
"Nay!" said honest Wittich, "came we not forth to destroy this
devil's-work, and to reduce the pride of the boasting Norg-king who
spares none? Shall we return, and leave our work undone? I have no
such mind; nothing will I leave of what we see before us now."
He dismounted to carry his threat into effect; and Theodoric, not
to be behindhand, or to incur suspicion of fearing the Norg-king,
dismounted too. Then with one consent they hewed down and rooted up
the fair plants, till the whole garden was a wilderness, and they
lay them down upon the grass to rest.
As they lay, there appeared before them, coming at full speed,
as on swift wings, a knightly form clad in shining armour, so that
Wittich cried, "See, Lord Dietrich, who comes to visit us--surely it
is St. Michael, leader of the heavenly hosts!"
"I see no St. Michael," answered Theodoric, sullenly. "It is one of
no heavenly build, albeit he bears him so bravely. We may rue that
we have loosed our helmets and shields, for methinks he regards us
with no loving eye."
While they spoke the rider had advanced over a good space of the
way, and they could discern the manner of his bearing. His horse
was lithe as a roe-buck upon the wild mountain heights, and its
housings of cloth-of-gold gave back the rays of the golden sun; the
bridle was studded with precious stones, and embroidered with cunning
workmanship of gold, moreover it was held in a commanding hand. The
saddle was dazzling with rubies, and so were the stirrups no less;
but the armour was most dazzling of all, and all hardened in dragon's
blood [31]. His sword of adamant could cut through steel and gold;
the handle was one carbuncle, which darted rays of light. Over his
breast-armour he wore a tight tunic of cloth-of-gold, with his arms
embroidered in glowing colours. His helmet was of burnished gold and
topazes mingling their yellow light, and between them many a carbuncle
which by night gave the light of day, and from within it there sang
pleasant voices of birds--nightingales, bulfinches, and larks, with
softened voices, as if they lived, and breathed their song upon the
branches of their native trees.
His shield was likewise of gold, and recorded many a deed of prowess
of him who bore it; on it was painted a leopard, too, with head erect,
as though preparing to spring upon his prey. In his right hand was a
spear, and from its point floated a small red banner [32], on which
appeared two swift greyhounds intent on following the wild game. But
more imposing than all this display of gold and art was the rider's
majestic mien, which was as of one used to know no law but his own
will, and to be obeyed by all who approached him; and yet, with all
this, he was only a span high! For it was King Lareyn, and he wore
tightly buckled his girdle of twelve-men's-strength.
Theodoric would gladly have laughed his little figure to scorn, but
when he caught the fire of his eye he was fain to acknowledge he was
no puny antagonist in fierce intention, whatever was his height.
Nor did Lareyn spare angry words when he had come up with the knights,
and saw what they had done; there was no epithet of scorn in his
vocabulary that he did not pour out upon them. He told them their
lives were forfeit for the mischief they had wreaked upon his roses,
and they could only redeem them by the surrender of their left foot
and right hand.
Theodoric was not slow to pay back his vituperation in corresponding
measure. He bid him remember what a little, wee mannikin he was; that
his was not the right tone in which he ought to talk to princes. Had
he ventured to ask a money-compensation, it would have been impertinent
enough, but what he had asked was a ludicrous pretention.
"Money!" shouted the Norg, in no way disconcerted; "I have more gold at
command than any three of you together. You call yourselves princes,
do you? You have done no princely deed to-day; you have incurred
the common penalty which I have decreed for all alike who trespass
on my Rose-garden--so no more words: hand over your horses, armour,
and clothing [33], together with the left foot and right hand of each."
"Herr Dietrich!" interposed Wittich, "is it possible you have patience
to listen to the insolent railing which this little mite pours out
in his folly? Say but the word, and I will punish him once and for
all. It needs but to take him and his mount by the leg, with one
grasp of my strong hand, and knock their heads against the stone wall,
that they may lie as dead as the roses we have already strewn around!"
"God is exhaustless in His wonders!" replied the Bernaere; "for aught
we know, He has laid up within this mite's body all the strength of
which he is so forward to boast: or by some magic craft he may have
possessed himself of might commensurate with the riches which we can
see plainly enough he has at command. If it comes to fighting, we
will bear ourselves like men; but take my advice, and be not rash,
for very much I doubt if we shall leave these mountains of Tirol
alive this day."
"Now, prince of lineage high! if I knew not your prowess before this
day," cried Wittich, beside himself with indignation, "I had said you
were afraid of his sword, which a mouse might wield! Shall a Christian
knight shrink before any pagan hound? But a thousand such wights as
this could be overmatched by you; and without arms you could smite
them down, and hang them all on the trees!"
"Your ideas of your powers are not weak," interposed King Lareyn; "you
talk of one of you being a match for a thousand such as I: come on,
and let us see how you will bear you against one 'tiny antagonist'!"
Wittich's impatience knew no bounds at the challenge; without
exchanging another word with Theodoric he sprang into the saddle, and
Lareyn, who had chafed at being spoken of as an unworthy adversary,
now drew himself up, proud to find Wittich did not scorn to meet
him mounted.
They rode out opposite each other on the greensward with their lances
poised, and then dashed the one at the other like two falcons on
the wing. Wittich, not at all wanting in the science of handling his
lance, made sure to have hit Lareyn, but the spell that surrounded
him protected him against the thrust, while his lance struck Wittich's
throat where the helm was braced, and sent him backwards off the saddle
on to the ground with great force. As he fell among the clover he vowed
that no other lance had ever so offended him, for never before had
victory appeared so easy. Hastily he sprang to his feet, to wipe away
the shame by seeming indifference; but Lareyn stood before him in the
long grass with his sword ready to take the forfeit he demanded, the
left foot and the right hand,--and would have taken it, but Theodoric
deemed it time to interfere; he said he should have reckoned it a
shame on him could it have been said of him that he had stood by while
a companion was made to pay so hard a penalty for so small a harm.
"What is a shame to you is no affair of mine," cried Lareyn in return;
"but instead of defending your companion, it behoves you to defend
yourself, for, as you had your part in the destruction of the garden,
I demand my forfeit equally of you, and your left foot and right hand
I must have. Stand on guard then! for I am a match for twelve such
as you."
The words stung Theodoric to the quick. But with what celerity soever
he vaulted into the saddle, the moment had sufficed for Lareyn to
bind fast Wittich to a tree, and gain his stirrups in time to confront
his foe.
"I perceive you are the Bernaere," he said, "by your shield and helm;
and never have I poised lance so gladly against any foe, and never
have had such satisfaction in triumph as I shall when I have you bound
by the side of your companion, and when the great Dietrich von Bern
shall lie in the bonds of the little Norg!"
"Dwarf! waste not words," cried Theodoric, in a terrible voice, his
eyes flashing fire; his spear trembling in his hand with the fury that
burnt within him. Before the foes could meet, however, and not a whit
too soon, Hildebrand appeared upon the scene, having found his way,
with the bold Wolfhart who never shrank from any fight, and Dietlieb
the Steieraere [34], by the guidance of the injured widow's son.
Hastily Hildebrand reached Theodoric's ear: "Fight him not in that
way," he said; "he has ever the advantage with the lance, and if he
hurled you from the saddle, where would be your princely honour? Never
could you again reign in your Hall of Verona. Dismount, and meet him
on foot upon the grass, and watch for what further may be suggested
to us."
Theodoric gladly accepted the counsel of the sage, and, standing
once more on the ground, called to the Norg to meet him there. Lareyn
refused not, but met him with many a valiant thrust, which the wigand
parried, and returned too, as best he might, with Hildebrand's counsel,
till the little man complained of the interference, without which,
he swore, Dietrich had been bound even as Wittich. But Theodoric bid
him not talk, but fight, and with that planted him a blow between
the eyes which shut out the light of the sun. Hildebrand, meantime,
released Wittich, as it behoved while one fought in his defence. But
Lareyn, finding he could gain no signal advantage against the hero,
drew his Tarnhaut from his pocket, and, slipping it over his head,
became invisible to his antagonist. Now it was a weird running
hither and thither, as the deft Norg paid out his cunning blows,
and the bold wigand in vain sought him, that he might return them;
now his blow fell on the stone wall, and now on a tree, while the
Norg's mocking laughter echoed at each mistake.
"One counsel only I see," cried good Hildebrand, distressed to see
his prince so hard thrust; "call to him to drop his sword, and wrestle
with you; so shall you reach him, and at least know where he stands."
The hero followed the counsel of his master, nor did the Norg
refuse. True, Theodoric now could at least feel his unseen foe, but
he felt him to his cost, for it was impossible to stand against his
strength, nor was it long before the dwarf forced the hero down upon
his knees on the grass. Great was the wigand's distress, for never
had antagonist so dealt with him before.
"Dietrich! beloved lord," cried Hildebrand, "list to my word. One
way of safety there is: wrench from him his girdle--his girdle which
gives him twelve men's strength!"
Gladly Theodoric heard the counsel, nor was he long in finding with
his hand the girdle; by it he raised King Lareyn from the ground,
and dashed him down again, till the girdle burst and fell beneath
their feet. Hildebrand quickly caught it up, lest the dwarf should
again possess himself of it; but Lareyn gave a cry of despair which
might have been heard o'er mountain and forest three days' journey
off! Then, with doleful voice, he said,--
"Dietrich von Bern! if you are the noble sword for which men hold you,
you will be now content, and will give me my life; while I will be
your tributary, and mighty are the gifts I have to offer you."
"No!" replied Theodoric; "your haughtiness and pretensions have been
too gross. I pardon not such as you so easily; we must have another
trial, in which you must yield up your worthless life."
"I have no power in fighting against such as he now, without my
girdle," mused the Norg; "my only chance of safety lies in getting
one of the heroes who is equal to him to fight for my cause in my
place. So he made up to Dietlieb the Steieraere, and conjured him,
as he was his brother-in-law, to help him in his need--even as he
loved his sister's honour."
"True!" replied Dietlieb; "since you confess honestly that you have my
sister, it is meet that I should be your champion; and I will deliver
you or die." With that he went to Theodoric, and prayed him earnestly
four times, by his regard for knightly honour, for woman's worth, for
friendship, and for virtue--four things which, at receiving his sword,
every knight bound himself to honour, that he would spare Lareyn. But
Theodoric was not to be moved, and each time only swore the harder
that he would fight it out to the last; that Lareyn had offended him
too deeply, and that he could not be suffered to live. When Dietlieb
found the ambassage he had undertaken unsuccessful, and that he would
have to own his failure, he grew impatient and wroth, and riding his
horse up to Theodoric, he proclaimed in a loud voice,--
"Be it known, Prince Dietrich, highly praised, that I declare King
Lareyn, great in power and riches, shall not be bound your prisoner,
nor his life taken; that I appear here to answer for him with brotherly
service, and that either he shall be let go scot free, or in my person
only shall the death-blow be dealt out for him."
Theodoric, unwilling to enter a feud of life and death with one of
his own allies, and yet too proud to refuse the challenge, answered
him nothing. But Dietlieb took the Norg and hid him away in safety
in the long grass out of Theodoric's sight, and then returned ready
to confront him. Theodoric, finding he was determined in his attack,
called for his horse, and bound on his helmet, his shield he took in
his hand, and hung his sword to his girdle.
"Think not I spare you more than another, Lord Theodoric, when I have
found the cause I ought to defend," cried Dietlieb, and his flashing
eye told that he would fight his fight to the end.
Theodoric still said no word, but his anger was the more desperate.
Thus minded, they rode at each other, and the lance of each hurled the
other from his horse upon the grass. Up each sprang again, and drew
his trenchant sword; the one struck, and the other pierced, till the
grass all around, as high as their spurs, was dyed as red as the roses
they had destroyed anon. Then Theodoric dealt such a mighty stroke
on Dietlieb's helmet that the fire flashed again, and he thought,
"Now have I conquered him and Lareyn at one blow." But Dietlieb,
recovering from the momentary shock, struck Theodoric's shield with
such force that he dashed it from his grasp; you might have heard
the clash a mile off!
When the bold Theodoric found he had his shield no longer, he took his
sword in both his hands, and gave the wigand such a mighty Schirmschlag
[35] that he felled him to the ground.
"Now then, foolish man!" he cried, in scorn, "do you still hold out
for Lareyn?"
Dietlieb sprang to his feet once more with a start which made his
armour ring again, and, for an answer, ran at Theodoric, and tried to
repeat his stroke; but Theodoric was more difficult to bring down,
and answered his attack by striking him on the rim of his shield so
forcibly that he loosed the band by which he held it.
Meantime, Hildebrand had been occupied stirring up the other wigands
to part the combatants, and at this moment Wittich and Wolfhart came
up to Dietlieb and seized him, and with main force dragged him off
the field; while Hildebrand reasoned with Theodoric about the merit
and friendship of Dietlieb, and the advantage of compromise now that
he had done enough to prove his superiority in the fight. Theodoric,
who ever gave weight to Hildebrand's reasoning, agreed to be friends
again with Dietlieb, and to leave Lareyn his life and liberty, only
exacting homage and tribute of him. To these terms Dietlieb also
agreed, and all entered the bonds of good friendship.
Lareyn, who had watched the combat and listened to the treaty of
peace from his hiding-place in the long grass, gave in his adhesion,
promising to pay tribute of all his wealth.
"And now, good brother-in-law," he said, addressing Dietlieb, "or
brother-in-law that-is-to-be,--for Simild has not yet given her consent
to be my wife--let us talk a little about your lovely sister. You
are doubtless burning to know how I became possessed of her, and I
no less to tell." Then he told him how he had found her under the
linden-tree, and had enveloped her in the Tarnhaut and carried her
away unseen by mortal eye; and of how all Norgdom was subject to her,
of how he had laid an empire of boundless wealth at her feet, and how,
if she preferred reigning on earth, he was able to buy a vast kingdom
to endow her with. Then he noticed that the day was declining, and they
far from shelter, and bade them all welcome to his underground home,
promising them good cheer and merry pastime. Dietlieb, anxious to
see his dear sister again, accepted the offer, and the other wigands
agreed to follow him. Stern Hildebrand the Sage would have preferred
camping in the open air, but Theodoric told him it would be a shame on
his name before all heroes if, having been so near the Norg kingdom,
of which all had heard, he should have feared to make acquaintance
with its economy and government. All the others were of his mind, but
Hildebrand reminded Theodoric, that as he whom all were ready to obey
had counselled incurring the danger, he made himself responsible for
all their lives. "He who gave us prudence will guard our lives and
honour," said the prince; and without further parley they rode on,
after Lareyn's guidance.
On they rode, through thick forest and narrow mountain-path, till,
as it grew dark, they came to a golden door in the rock. It opened at
Lareyn's approach, and the moment they had passed within they found
themselves surrounded by a light above the light of day from the
shining stones that glittered around. Trumpets sounded to herald their
entrance. As they advanced through the sparkling trees friendly birds
warbled a sweet welcome; and as they neared the hall soft melodies
of lutes and harps enchanted their ear. All around them the Norgs
disported themselves, ready to render any service the wayfarers might
require. Refreshment was all ready, as if they had been expected;
and when the wigands had done justice to the spread, they were led
each to his apartment to take their rest, which they well needed.
In the morning Lareyn prayed them to stay and enjoy the wonders
of his kingdom and taste his hospitality, whereupon new debate
arose. Theodoric was disposed to trust him; and Dietlieb desirous
to keep friends with him for the sake of his sister; while Wolfhart
was ready for any sort of adventure; but Wittich, who had tasted the
effects of Lareyn's guile and strength, used all his persuasion to
induce the others to return, and prudent Hildebrand deemed it the
wiser part. At last, however, Wolfhart said, scornfully to Wittich,
that if he was afraid to stay he could go back; he had no need to
spoil their pleasure. After that Wittich said no more, but by his
sullen looks he showed he disapproved the venture.
Lareyn, seeing them doubtful, came up, and with much concern bid them
have no hesitation or fear, for all they saw was at their service--they
had but to command. To which Theodoric made answer that such words
were princely indeed, and if his deeds accorded therewith he never
would have reason to rue the league he had made with them.
Then with delight Lareyn led them through the riches of his
possessions. So much heaped-up gold, so many precious stones, such
elaborate handiwork none of Theodoric's band had ever seen before;
and the place rang with their exclamations of wonder.
But all this was nothing to the cunning feats of the Norgs, who,
at a sign from Lareyn, displayed their various talents before the
astonished eyes of the heroes. Some there were who lifted great stones
bigger than themselves, and threw them as far as the eye could reach,
then by swiftness attained the goal before the stone they threw! Others
rooted up great pine-trees, and broke them across as sticks. Others
did feats of tilting and horsemanship, and others danced and leapt
till the knights were lost in wonder at their agility and strength.
Lareyn now called his guests in to dine; and all manner of costly
dishes were set before them, arranged with greater care and taste
than Theodoric was used to in his own palace, while sweet-voiced
minstrels sang, and nimble Norgs danced. In the midst of the repast,
Simild, summoned by Lareyn, entered the hall, attended by a train of
five hundred choicely-robed Norginnen; her own attire a very wonder
of art. It was all of silk and down, and set off with ornaments of
jewellery beyond compare with any on earth; stones there were of value
enough to ransom three kingdoms; and in her coronet one which lighted
up the hall with its radiance--meet crown of her own loveliness! At
Lareyn's courteously worded request she gave all the guests a joyful
welcome, with a word of praise from her rosy lips for each, for their
fame of knightly deeds. But when she saw Dietlieb her joy knew no
bounds; they embraced each other with the heartfelt joy of those who
have been long and cruelly separated.
"Tell me, sister mine," said Dietlieb, anxiously turning to account
the brief opportunity her embrace gave him of whispering into her ear,
"is it of your own will that you are here, in this strange mountain
dwelling? is this Lareyn dear to you? and do you desire to dwell with
him? Or has his artifice been hateful to you? Say, shall I rid you
of his presence?"
"Brother, it is your help I need to decide this thing," replied the
maid. "Against Lareyn's mildness I have no word to say: gift upon gift
has been heaped upon me; with honour after honour have I been endowed;
and every wish of mine is fulfilled ere it is born. But when I think
of Him of whom all our pleasures are the bounty, I feel no pleasure in
pleasures so bestowed. This pagan folk holds Christ, our dear Lord, in
hate--and when I think of Him, I long to be again in Christendom [36]."
"Yes, Simild, sister dear, in Christendom is your place, not here; and
since such is your mind, cost what it will, I will set you free from
the Norg-king's power," was Dietlieb's answer; and there was no time
for more, for Lareyn called them back to the fresh-dressed banquet.
"Come, new allies but trusted friends!" cried the dwarf, "come, and
let us be merry, and pledge our troth in the ruby bowl! Lay aside
your heavy arms and armour, your sword and shield. Let us be light
and free as brothers together."
As he spoke a whole host of waiting-men appeared, who helped the
knights off with their armour, and brought them robes of rich stuffs
and costly work. The guests suffered them to do their will, for
they were lost in admiration at the choice banquet; at the table,
all of ivory inlaid with devices of birds and game so lifelike they
seemed to skim across the board; at the vessels of silver and gold
and crystal of untiring variety of design; and, above all, at the
order and harmony with which all was directed.
Cool wine from cellars under earth was now served round [37]; then
various dishes in constant succession, each rarer than the last;
and then again sounded soft, clear voices to the accompaniment of the
harmonious strings. And again and again the tankards were filled up
with Lautertrank, Moras [38], and wine.
At last the tables were drawn away, and at the same time Simild and
her maids withdrew; but many an hour more the guests sat while the
music and the singing continued to charm them. But lest even this
should weary, King Lareyn, as if determined there should be no end
to the change of pastimes with which he had undertaken to amuse his
guests, sent to fetch a certain conjuror who dwelt in the heart of a
high mountain, and whose arts surpassed any thing that had been done
before. The magician came at his bidding, and exhibited surprising
evidences of his craft, till at last the king said,--
"You are a cunning man, no doubt, but there is one exhibition of
your power you have never been able to give me, and I shall think
nothing of your art till you can satisfy me. In this country within
the mountains, these jewels fixed in vault, wall, and sky, weary one
with their perpetual glare. Make them to move as the luminaries of
earth, so that we may have calm, peaceful night for repose."
"True, O king! I have never before been able to accomplish this
desire," replied the magician; "but now I have acquired this art
also, and waited for a fitting occasion to make the first display of
the same."
"No occasion can be more fitting than the present," answered Lareyn,
"when by its inauguration you shall celebrate the visit of my honoured
guests, and also by its achievement afford them that rest from the
glare of day to which they are accustomed in their own nights."
"I desire but to obey," replied the magician; and forthwith he threw on
to the fire that burnt on a black stone before him, a powder which no
sooner touched the flame than a pale blue smoke arose with pleasing
scent, and, curling through the hall, presently extinguished the
brilliant shining of every countless jewel with which the walls and
roof were set.
"Now, if you are master of your art," continued the king, "let us
have light once more."
The magician, wrapt in his incantation, spoke not, but dropped another
powder on the flame, which at once sent up a wreathing fume of rainbow
hues, carrying back to every precious stone its lustre.
"Wondrous!" "Brave artist!" "Wondrous show indeed!" were the
exclamations which broke spontaneously from every lip.
"Now let it be dark again," said the king; and the magician quenched
the sparkling light as before.
"Now light," he cried; and so alternated until the sight was no longer
new. Now, it was dark, and this time Lareyn called no more for light,
nor spoke, and the silence was long; till the heroes grew anxious, and
Wittich turned to where Wolfhart had sat, and said, "I like not this:
who knows but that while we can see naught the Norgs may fall upon us
and destroy us?" But Wolfhart answered not, for a stupor had fallen
upon him that the fumes had been gifted to convey; and Wittich, too,
felt their influence before he could utter another word; so it was
with Hildebrand the Sage no less. Theodoric only had time to answer,
"Such treachery were not princely; and if Lareyn means harm to us,
he may be sure he will rue this day," and then sleep fell upon him
as on the others.
Dietlieb had already left the hall, thinking under cover of the
darkness to find his sister, but being met by a page had been conducted
to his apartment, and knew nothing of what had befallen the others.
Lareyn, meanwhile, sought out Simild in anxious mood. "Ever lovely
virgin!" he exclaimed, "support me with your prudent counsel in this
strait. I have already told you how your people have avenged on me
that I have loved you; how they have laid low my silken fence and
golden gates, and wasted my choice garden of roses. Good reprisals
I had thought to have taken, and had I been left man to man against
them I had overcome them all; but Hildebrand the Sage interposed
his advice: it was thus the Bernaere had the advantage over me, and
had it not been for your brother Dietlieb's stout defence, he had
even taken my life. But in all the other four beside him there is no
good, and in one way or another I had found means to rid me of them,
but for Dietlieb's sake, who would be as ready to oppose me in their
defence as he opposed Dietrich in mine. So, fair lady mine, say how
shall I end this affair?"
"If you would follow my advice," replied Simild, "be not rash;
and, above all, use no treachery; keep to the pact of peace that
you have sworn; and be sure the Christian knights will not go back
from their plighted word. But in place of the little girdle of
twelve-men's-strength that they took from you, here is a ring of
equal power which your seven magicians welded for me: with that you
will feel all your old consciousness of strength and dignity. But,
by all you hold dear, let the wigands go forth with honour!"
Lareyn was not slow to own that the counsel was good, and spoke as
if he would have followed it. But when he put on the ring, and found
himself endowed once more with twelve men's strength, he could not
forbear taking his sweet revenge for his yesterday's defeat and danger.
First, he had sevenfold bolts put on Dietlieb's door, that he might
not be able to come forth and aid his brethren; and then he sent
and called for one of the giants, who were always true allies to the
dwarfs, and entreated him to carry the heroes to a deep dungeon below
the roots of the mountains, where they should be bound, and shut out
from the light of day, and never again be able to do him harm.
The feat pleased the giant well; and, having bound a cord round the
waist of each of the sleeping heroes, slung the four over his shoulder
as if they had been no heavier than sparrows, and carried them to the
dungeon below the roots of the mountains, whither Lareyn led the way,
now skipping, now dancing, now singing, now laughing in high glee,
to think how well he had succeeded in ridding him of his foes--but
forgetting all about Simild's advice, and his promise to her.
It was not till next morning that the heroes woke; and then all was
cold and dark around them, and they knew they were no longer in the
hall of the banquet, for the iron chains and stanchions, the chill,
and must, and damp, and slime, told them they were in a dungeon
under earth.
Loudly they all exclaimed against the deceit with which they had
been caught, and loudly they all swore to find means to punish the
treacherous captor. But Theodoric's anger was greater than the anger
of them all; and the fiery breath [39] glowed so hot within him that
it scorched away the bonds with which he was bound!
Once more, then, his hands at least were free, and his companions
gave him joy; but his feet were still held to the rock by chains
of hard steel, the links as thick as a man's arm. Nevertheless, his
indignation was so great that when he beat them with his fists they
were obliged to yield, as they had been made of egg-shell; and when he
had broken his own chains he set to work and released the others also.
Great was their joy and thankfulness; but heaviness came down on them
again when they saw themselves closed in by the cruel rock, and all
their armour and weapons of defence locked up far away from them in
the Norg's castle. Another day they lay there in despair, and another,
for wise Hildebrand saw no way of passing through the rock [40].
Meantime Simild had grown uneasy at the silence that reigned in
the palace; there was no more sound of revel and festivity, and of
entertaining guests. She was no more sent for to entertain them, and
Lareyn hid himself from her, and avoided her. In dire fear she hunted
out the right key of her brother's apartment, and having covered
the glowing carbuncle in her coronet, which lighted up every place,
crept along silently till she had reached him.
"Sister mine!" exclaimed Dietlieb, "what does this mean? why am I
held fast by seven locks? and why do no tidings of my companions
reach me? Oh! had I but my sword and shield, I would release them
from the hands of Lareyn, and of how many Norgs soever he may have
at his command! or at least I would not survive to bear the shame of
living while they are in I know not what plight."
"Dietlieb, be guided by me," replied the maiden: "we must deliver them
out of the dire dungeon in which Lareyn has treacherously confined
them, but also we must have your life and honour safe. Take this ring
upon your hand, for against him who wears it none can prevail; and
then go and deliver your companions." With that she took him along
to where his armour lay concealed; and having girt him with it, she
said many a fervent blessing [41] over him, to preserve him from harm.
Endowed with the strength the ring gave him, Dietlieb was able to
load himself with the arms and armour of all the four heroes; and at
its command a way was made in the rock, through which he passed it in
to them. As each piece fell upon the hard floor, the clang re-echoed
through the far-off mountains.
Lareyn heard the noise, and knew what had befallen, so he sounded on
his horn the note that was known far and wide through all the lands
of the Norgs; and at the call three hundred thousand dwarfs appeared
swarming over the whole face of the country.
"To me, my men! to me!" cried Lareyn, as they drew near. "Before you
stands he who has essayed to release our enemies whom I and the giant
had bound under the roots of the mountains. He has given them back
their strong armour and their weapons of war, and if they get loose
and come among us, great havoc will they make of us, therefore smite
him down and destroy him!"
The dwarfs rushed on Dietlieb at the bidding of the king; but
Lareyn would not engage him himself, because he had fought for his
release. Dietlieb, young and strong, stood planted against a vault
of the rock, and as the mannikins approached him, he showered his
blows upon them, and sent them sprawling, till the dead and mangled
were piled up knee-deep around him.
The heroes heard the sound of the battle in their prison, and they
longed to take their part in the fray; but they saw no means of
breaking through the rock to reach him, till Hildebrand bethought him
that he had yet with him the girdle he had picked up when Theodoric
tore it from the Norg-king's body. This he now handed to the hero.
Theodoric took it, and spoke not for joy, but with its strength tore
down the living rock round the opening Dietlieb's ring had made,
and burst his way to stand beside the brave young Steieraere. This
done, scorning the girdle's strength, he cast it back to Hildebrand,
trusting in his good sword alone.
"Now, treacherous dwarf, come on!" he cried. "No knightly troth has
bound you, but against us, your guests and allies, you have acted
as one who has no right to live! Come here, and let me give you the
guerdon you have earned!"
Lareyn refused not; and the two fought with fury terrible to
behold. And yet Theodoric prevailed not. Then Hildebrand discerned
the ring of twelve-men's-strength on Lareyn's hand, where it was
not before, and knew it was a talisman, so he called to Theodoric,
and said,--
"Dietrich, my prince, seize yonder ring upon the Norg-king's hand! so
shall his strength be no more increased by the powers of his magic."
Theodoric, ever prone to be guided by the advice of the Sage, directed
a mighty blow upon the ring, so that the hoop must fain give way;
and the dwarf's power went from him.
"Now all your hosts, and all your arts, and all your gold shall profit
you nothing more!" So cried the Bernaere; "but condign penalties you
must suffer for your crime. My prisoner you are, nor is there any
can deliver you more."
The Norgs, grieving for their king's loss, trooped round Theodoric
and attacked him on every side; but he swang his good sword Velsungen
around, and at every sweep a hundred Norg's heads fell pattering at
his feet. Suddenly a little dwarf came running out from the mountain
rock, and seizing Lareyn's horn, blew on it notes which wandered wild
through all the forest-trees.
Five giants lived in the forest, and when they heard those notes they
knew the Norgs were in dire distress. With swift strides they came;
their helmets flashed like lightning over the tops of the pines;
and each brought his sword and pike of trenchant steel. The little
dwarf saw his brethren mown down like grass before the scythe,
and again sent forth his far-sounding notes of distress. The giants
heard it, and marched over hill and dale, till they came before the
mountain-side. Again the little dwarf sent out his appeal, and the
giants burst their way through the mountain; but albeit they came
with such speed, twelve thousand Norgs were meantime lost to King
Lareyn by Velsungen's strokes.
Dietlieb and Hildebrand, Wittich and Wolfhart mowed down their
harvest too.
Now they had to prepare for another kind of attack, for in fearful
array the five giants came down upon them, brandishing their clubs of
steel. But neither could these stand before the swords of the heroes,
and each several one laid low his adversary.
When the Norgs saw that their king was bound, and their best
fighting-men destroyed, and the giants themselves without breath,
they knew they could stand no longer before the wigands, but each
turned him and fled for refuge to the mountains.
The heroes then, seeing no more left to slay, went into the
banquet-hall, where only Simild stood, for all the Norgs had hidden
themselves in fear.
"Welcome, noble brother! and welcome, bold swords all!" cried the maid;
"you have delivered us from this treacherous king. Now you will go
home to your own land with glory and honour, and take me with you."
The heroes returned her greeting, and rejoiced in her praise; then
they piled up the treasure on to waggons, all they could carry, and
in triumph they made their way to earth, and Lareyn with them, bound.
First they directed their steps to Styria, till they came to the
spreading linden-tree whence Simild had first been taken; for there
sat Duke Biterolf, her father, bewailing his bereavement, and around
him trooped her maidens lamenting their companion.
All was restored to joy and gladness now that Simild was at home
again. They passed seven days in high festival, the heroes all
together; and many a time they had to tell the tale of their bold
deeds, and the wonders of the mountain-world. And the minstrels sang
to the merits of the conquerors, while the merry bowl passed round
and round.
At last Theodoric rose and thanked Biterolf for his hospitality,
who thanked him in return right heartily for the help he had lent
his son. With that Theodoric took his leave, and along with him went
Hildebrand the Sage, and Wittich the Wigand, and the strong Wolfhart,
and King Lareyn too, of whom Theodoric made his court-fool in his
palace at Verona.
THE NICKEL [42] OF THE ROeHRERBUeCHEL.
From the fourteenth to the sixteenth, in some few places down
to the seventeenth, centuries the mountains of Tirol were in many
localities profitably worked in the search after the precious metals;
many families were enriched; and the skill of the Tirolese miners
passed into a proverb throughout Europe. When the veins lying near the
surface had been worked out, the difficulty of bringing the machinery
required for deeper workings into use, in a country whose soil has
nowhere three square miles of plain, rendered the further pursuit so
expensive that it was in great measure abandoned, though some iron and
copper is still got out. There are many old shafts entirely deserted,
and their long and intricate passages into the bowels of the earth
not only afford curious places of excursion to the tourist, but are
replete with fantastic memories of their earlier destination.
One of the most remarkable of these is the so-called Roehrerbuechel,
which is situated between Kitzbuehl and St. Johann, and not far from
the latter place. It was one of the most productive and one of the
latest worked, and it boasted of having the deepest shaft that had
ever been sunk in Europe; but for above a hundred and fifty years it
has been taub [43], that is, deaf, to the sound of the pick and the
hammer and the voices of the Knappen [44].
I have given you my way of accounting for the cessation of the
mining-works. The people have another explanation. They say that the
Bergmaennlein, or little men of the mountains--the dwarfs who were the
presiding guardians of these mineral treasures--were so disgusted
with the avarice with which the people seized upon their stores,
that they refused to lend them their help any more, and that without
their guidance the miners were no longer able to carry on their search
aright, and the gnomes took themselves off to other countries.
One of these little men of the mountains, however, there was in
the Roehrerbuechle who loved his ancient house too well to go forth
to seek another; he still lingered about the mile-long clefts and
passages which once had been rich with ore, and often the peasants
heard him bewailing, and singing melancholy ditties, over his lonely
fate. They even thought he came out sometimes to watch them sadly
in their companionship of labour, and peeped through their windows
at them in their cosy cottages, while it was cold and dark where he
stood without: and many there were who took an interest in the Nickel
of the Roehrerbuechel.
The Goigner Joessl [45] had been mowing the grassy <DW72> near the
opening of the Roehrerbuechl; he was just putting up his implements to
carry home after his day's toil, when he espied the orphan Aennerl
[46] coming towards him. Her dark eyes had met his before that day,
and he never met her glance without a thrill of joy.
"I have been over to Oberndorf for a day's work," said orphan Aennerl,
"and as I came back I thought I would turn aside this way, and see
how you were getting on; and then we can go home together."
"So we will," answered Joessl; "but we're both tired, and the sun
isn't gone yet--let's sit down and have a bit of talk before we go."
Orphan Aennerl was nothing loath; and they sat and talked of the
events of the day, and their companions, and their work, and the
weather, and the prospects of the morrow. But both seemed to feel
there was something else to be said, and they sat on, as not knowing
how to begin.
At last Joessl removed his pointed hat from his head and laid it by his
side, and took out and replaced the jaunty feathers which testified
his prowess in the holiday sport [47], and finally cleared his throat
to say, softly,--
"Is this not happiness, Aennerl?--what can we want more in this
world? True, we work hard all day, but is not our toil repaid when
we sit together thus, while the warm evening sun shines round us,
and the blue heaven above and the green fields below smile on us,
and we are together? Aennerl, shall we not be always happy together?"
They were the very words that orphan Aennerl had so often longed to
hear her Joessl say. Something like them she had repeated to herself
again and again, and wondered if the happy day would ever come when
she should hear them from his own lips. Had he said them to her any day
of her whole life before, how warmly would she have responded to them!
To-day, however, it was different. The rich peasant's wife for whom
the poor orphan worked had been harsh to her that day, and for the
first time envious thoughts had found entrance into her mind, and
discontent at her lowly lot.
So, instead of assenting warmly, she only said,--
"Of course it's very nice, Joessl, but then it's only for a little
bit, you see. The hard toil lasts all day, nevertheless. Now to have
a Hof [48] of one's own, like the one I work upon at Oberndorf, with
plenty of cattle, and corn, and servants to work for you, that's what
I should call being happy! Sitting together in the sunset is all very
well, but we might have that besides."
The good, hard-working, thrifty, God-fearing Joessl looked aghast to
hear his Aennerl speak so. Beyond his day's wage honestly worked for,
and the feather in his Trutzhut bravely contended for, and his beloved
Aennerl wooed with tenderness and constancy--he had not a thought or
a wish in the wide world. Hitherto her views had been the counterpart
of his; now, for the first time, he perceived there was something
had come between them, and he felt disappointed and estranged.
"If that's your view, Aennerl girl, it isn't the Goigner Joessl that
will be able to make you happy," replied the youth at length, coldly;
"your best chance would be with the Nickel of the Roehrerbuechel," he
added, almost bitterly, as one who would say, Your case is desperate;
you have no chance at all.
"What was that?" said Aennerl, suddenly starting. "Who can be working
so late? Don't you hear a pick go 'click, clack'? Who can it be?"
"No one is working at this hour," replied Joessl, in no mood to be
pleased at the interruption. But as he spoke the bells of the villages
around toned forth the Ave-Maria. Both folded their hands devoutly
for the evening prayer; and in the still silence that ensued he could
not deny that he heard the sound of the pick vigorously at work,
and that, as it seemed, under the ground directly beneath their feet.
"It is the Bergmaennlein--it must be the Bergmaennlein
himself!" exclaimed Aennerl, with excitement.
"Nonsense! what silly tales are you thinking of?" replied Joessl,
inwardly reproaching himself for the light words he had just spoken
suggesting the invocation of a superstition with which his honest,
devout nature felt no sympathy; and, without letting the excited girl
exert herself to catch the strange sounds further, he led her home.
Aennerl's curiosity was roused, however, and was not to be so easily
laid to rest.
The next evening Joessl's work lay in a different direction, but no
sooner had the hour of the evening rest arrived than he started on the
road to Oberndorf, to see if he could meet his Aennerl coming home. But
there was no Aennerl on the path; and he turned homewards with a heavy
heart, fearing lest he had offended her, and that she was shunning him.
But Aennerl, whom the desire of being rich had overcome with all the
force of a new passion, had been more absorbed on that last memorable
evening by the idea of having heard the Bergmaennlein at work amid the
riches of the mines than with--what would have been so terrible a grief
at any other time--having offended her faithful Joessl. Accordingly,
on the next evening, instead of being on the look-out for Joessl to
walk home with her, her one thought had been to find out the same
place on the bank where they had sat--not with loving affection to
recall the happy words she had heard there, but to listen for the
sound of the Bergmaennl's axe, and perhaps follow it out; and then--and
then--who could tell what might befall? Perhaps she might be able to
obtain some chips of those vast wealth-stores unperceived; perhaps
the Bergmaennl's heart might be opened to her--who could say but,
in some mode or other, it might be the way to fortune?
She was not long in tracing out the spot, for she had marked the angle
which the well-known outline of the mighty Sonnengebirg bore to the
jagged "comb [49]" of the Kitzbichler-Horn, and for a nearer token,
there lay, just before her, the crushed wild-flower which her Joessl
had twisted and torn in his nervousness as he had brought himself to
speak to her for the first time of their future. But she thought not
of all that at that time; she was only concerned to find the spot,
and to listen for the stroke of the Nickel's pickaxe. "Hush!" that
was it again, sure enough! She lingers not on that happy bank; she
stops not to pick up one of those wild-flower tokens: 'click, clack,'
goes the axe, and that is the sound to guide her steps. The village
bells sound the Ave-Maria, but the sacred notes arrest her not--the
evening prayer is forgotten in the thirst for gold.
But Joessl heard the holy sound as he was retracing his steps
mournfully from his fruitless search after her, having missed her by
but a minute's interval. He heard it as he was passing a little old,
old wayside chapel, which you may yet see, with a lordly pine-tree
overshadowing it, and which records the melancholy fate of some Knappen
who perished in the underground workings. Joessl, who has no fear on the
steep mountain-side, and loves to hang dangerously between earth and
sky when he is out after the chamois, shudders when he thinks of those
long, dark, mysterious passages where the miners worked underground,
and, as he kneels on the stone step of that wayside memorial, obedient
to the village-bell, involuntarily applies his prayer to all those
who have to penetrate those strange recesses: "Be with them; help them
now and in the hour of death. Amen." If you had told him his Aennerl
was included in that prayer he would not have believed you then.
Meantime Aennerl had found her way to the opening of the old mine. It
has a lateral shaft through which you may walk some distance--a
very long way it seemed to Aennerl, now breathless and trembling,
but the nearing sounds of the Bergmaennl's tool kept up her courage,
and determined her not to give in till she had attained the goal.
On she went, groping her way with fear and trembling, and expecting
every moment to come upon some terrible sight. But, far from this,
in proportion as she got deeper into the intricate passages of the
Roehrerbuechel, the way, instead of getting darker, grew lighter and
lighter. A pale, clear, rosy light played on the sides of the working,
which, now that she looked at them close, she found to her astonishment
were not made of rough, yellow clay, as she had thought hitherto,
but of pure, sparkling gold, and encrusted with gems!
It was no longer fear that palsied her, it was a fascination of
delight at finding herself in the midst of those riches she coveted,
but the near approach of which brought back misgivings of the danger
of their possession of which she had so often heard, though without
ever previously feeling an application to herself in the warning.
Her curiosity far too strongly stimulated to yield to the counsel of
her conscience to turn and flee the temptation, she walked stealthily
on and on, till the faint, rosy light grew into a red, radiant glow,
which, as she reached its focus, quite dazzled her senses.
She now found herself in a broad and lofty clearing, into which the
long narrow passage she had so long been timorously pursuing ran,
and in the sides of which she saw the openings of many other similar
ramifications. The walls, which arched it in overhead and closed
it from the daylight, were of gold and silver curiously intermixed,
burnished resplendently, and their brilliance so overcame her that
it was some minutes before she could recover her sight to examine
more particularly the details of this magnificent abode.
Then she discovered that all this blaze of light came from one huge
carbuncle [50], and that carbuncle was set in the breast-bib of the
leathern apron worn by a dwarf, the clang of whose pickaxe had lured
her to the uncanny spot.
The dwarf was much too busily and too noisily engaged to notice
Aennerl's footsteps, so she had plenty of leisure to examine him. He
was a little awkward-shaped fellow, nearly as broad as he was long,
with brawny muscular arms which enabled him to wield his pick with
tremendous effect. He seemed, however, to be wielding it merely
for exercise or sport, for there did not appear to be any particular
advantage to be gained from his work, which only consisted in chipping
up a huge block of gold, and there were heaps on heaps of such chips
already lying about. Though his muscles displayed so much strength,
however, his face gave you the idea of a miserable, worn-out old man;
his cheeks were wrinkled and furrowed and bronzed; and the matted
hair of his head and beard was snowy white. As he worked he sang,
in dull, low, unmelodious chant, to which his pick beat time,--
"The weary Bergmaennl, old and grey,
Sits alone in a cleft of the earth for aye,
With never a friend to say, 'Good day.'
For a thousand years, and ten thousand more,
He has guarded earth's precious silver store,
Keeping count of her treasures of golden ore
By the light of the bright Karfunkelstein [51],
The only light of the Bergmaennlein
But never a friend to say, 'Good day,'
As he sits in a cleft of the earth for aye,
Has the lonely Bergmaennl, old and grey!"
He had poured out his ditty many times over while Aennerl stood
gazing at the strange and gorgeous scene. The ugly, misshapen,
miserable old man seemed altogether out of place amid the glories of
the wonderful treasure-house; and the glittering treasures themselves
in turn seemed misplaced in this remote subterranean retreat. Aennerl
was quite puzzled how to make it all out. It was the Nickel of the
Roehrerbuechel who was before her, she had no doubt of that, for he
was exactly what the tradition of the people had always described
him, and she had heard his ungainly form described before she could
speak; so familiar he seemed, indeed, from those many descriptions,
that it took away great part of the fear natural to finding herself
in so novel a situation.
At last the dwarf suddenly stopped his labour, and, as if in very
weariness, flung the tool he had been using far from him, so that
it fell upon a heap of gold chips near which Aennerl was standing,
scattering them in all directions. One of the sharp bits of ore hit
her rudely on the chin, and, anxious as she was to escape observation,
she could not suppress a little cry of pain.
Old and withered and haggard as he seemed, the Cobbold's eye glittered
with a light only second to that of the Karfunkelstein itself at the
sound of a maiden's voice, and quickly he turned to seize her. Aennerl
turned and fled, but the Nickel, throwing his leathern apron over
the shining stone on his breast, plunged the whole place in darkness,
and Aennerl soon lost her footing among the unevennesses of the way
and lay helpless on the ground. Her pursuer, to whom every winding
had been as familiar as the way to his pocket these thousand years,
was by her side in a trice, still singing, as he came along,--
"But never a friend to say, 'Good day,'
As he sits in a cleft of the earth for aye,
Has the lonely Bergmaennl, old and grey!"
The self-pitying words, and the melancholy tone in which they were
uttered, changed most of Aennerl's alarm into compassion; and when the
dwarf uncovered the carbuncle again, and the bright, warm red light
played once more around them, and showed up the masses of gold after
which she had so longed, she began to feel almost at home, so that
when the dwarf asked her who she was and what had brought her there,
she answered him quite naturally, and told him all her story.
"To tell you the truth," said the Cobbold, when she had finished,
"I am pretty well tired of having all this to myself. I was very
angry at one time, it is true, with the way in which your fellows
went to work destroying and carrying every thing away, and leaving
nothing for those that are to come after, and I was determined to
put a stop to it. I am not here to look after one generation, or two,
or three, but for the whole lot of you in all the ages of the world,
and I must keep things in some order. But now they have given this
place up and left me alone, I confess I feel not a little sorry. I
used to like to listen to their busy noises, and their songs, and
the tramp of their feet. So, if you've a mind to make up for it,
and come and sit with me for a bit now and then, and sing to me some
of the lively songs you have in your world up there, I don't say I
won't give you a lapful of gold now and then."
A lapful of gold! what peasant girl would mind sitting for a bit now
and then, and singing to a poor lonely old fellow, to be rewarded
with a lapful of gold? Certainly not Aennerl! Too delighted to speak,
she only beamed assent with her dark, flashing eyes, and clapped her
hands and laughed for joy.
"It's many a day since these walls have echoed a sound like that," said
the dwarf, with deep feeling, and as Aennerl's smile rested on him, it
seemed to wipe away some of the rough dark wrinkles that furrowed his
cheeks and relax the tension of his knit brows. "And yet there's more
worth in those echoes than in all the metallic riches which resound to
them! Yes, my lass, only come and see the poor old Bergmaennl sometimes,
and cheer him a bit, and you shall have what you will of his."
With that he led her gently back into the great vault where she had
first seen him working, and, stirring up a heap with his foot, said,--
"There, lass, there's the Bergmaennl's store; take what you will--it is
not the Bergmaennl that would say nay to a comely wench like you. Why,
if I were younger, and a better-looking fellow, it would not be my
lapfuls of gold I should offer you, it would be the whole lot of
it--and myself to boot! No, no, I shouldn't let you go from me again:
such a pretty bird does not come on to the snare to be let fly again,
I promise you! But I'm old and grey, and my hoary beard is no match for
your dainty cheeks. But take what you will, take what you will--only
come and cheer up the poor old Bergmaennl a bit sometimes."
Aennerl had not wanted to be told twice. Already she had filled her
large pouch and her apron and her kerchief with all the alacrity
of greed. So much occupied was she with stowing away the greatest
possible amount of the spoil, that she scarcely remembered to thank
the Bergmaennl, who, however, found pleasure enough in observing the
rapturous gestures her good fortune elicited.
"You'll come again?" said the Cobbold, as he saw her turn to go when
she had settled her burden in such a way that its weight should least
impede her walking.
"Oh, yes, never fear, I'll come again! When shall I come?"
"Oh, when you will! Let's see, to-day's Saturday, isn't it? Well,
next Saturday, if you like."
"Till next Saturday, then, good-bye!" said Aennerl, panting only to
turn her gold to account; and so full was she of calculation of what
she would do with it, that she never noticed the poor old dwarf was
coming behind her to light her, and singing, as he went,--
"The weary Bergmaennl, old and grey,
Sits alone in a cleft of the earth for aye,
With never a friend to say, 'Good day.'
For a thousand years, and ten thousand more,
He has guarded earth's precious silver store,
Keeping count of her treasures of golden ore
By the light of the bright Karfunkelstein,
The only light of the Bergmaennlein.
But never a friend to say, 'Good day,'
As he sits in a cleft of the earth for aye,
Has the lonely Bergmaennl, old and grey!"
Aennerl had no time for pity; she was wholly absorbed in the
calculation of the grand things she could now buy, the fine dresses
she would be able to wear, and in rehearsing the harsh speeches of
command with which she would let fling at the girls whom she would
take into her service, and who yesterday were the companions-in-labour
of orphan Aennerl.
The village was all wrapt in silence and sleep as Aennerl got back
with her treasure.
"So late, and so laden! poor child!" said the parish priest, as he
came out of a large old house into the lane, and met her. "I have
been commending to God the soul of our worthy neighbour Bartl. He was
open-handed in his charity, and the poor will miss a friend; he gave us
a good example while he lived--Aennerl, my child, bet' fuer ihn [52]."
Aennerl scarcely returned his greeting, nor found one word of sorrow
to lament the loss of the good old Bartl; for one thought had taken
possession of her mind at first hearing of his death. Old Bartl had
a fine homestead, and one in which all was in good order; but Bartl
was alone in the world, there was no heir to enter on his goods: it
was well known that he had left all to the hospital, and the place
would be sold. What a chance for Aennerl! There was no homestead in
the whole Gebiet [53] in such good order, or so well worth having,
as the Hof of old Bartl.
Aennerl already reckoned it as hers, and in the meantime kept an eye
open for any chances of good stock that might come into the market.
Nor were chances wanting. The illness which had carried old Bartl to
the grave had been caught at the bedside of the Wilder Juergl [54]. A
fine young man he had been indeed, but the villagers had not called him
"Wild" without reason; and because he had loved all sorts of games,
and a gossip in the tavern, and a dance with the village maids more
than work, all he had was in confusion. He always said he was young,
and he would set all straight by-and-by, there was plenty of time. But
death cut him off, young as he was; and his widow found herself next
morning alone in the world, with three sturdy boys to provide for,
all too young to earn a crust, and all Juergl's debts to meet into the
bargain. There was no help for it: the three fine cows which were the
envy of the village, and which had been her portion at her father's
death, only six months before, must be sold.
Aennerl was the purchaser. Once conscience reproached her with a
memory of the days long gone by, when she and that young widow were
playmates, when orphan Aennerl had been taken home from her mother's
grave by that same widow's father, and the two children had grown up
in confidence and affection with each other. "Suppose I left her the
cows and the money too?" mused Aennerl--but only for a moment. No;
had they been any other cows, it might have been different--but just
those three which all the village praised! one which had carried
off the prize and the garland of roses at the last cow-fight [55],
and the others were only next in rank. That was a purchase not to be
thrown away. Still she was dissatisfied with herself, and inclined
to sift her own mind further, when she was distracted by the approach
of loud tramping steps, as of one carrying a burden.
It was the Langer Peterl; and a goodly burden he bore, indeed--a
burden which was sure to gather round him all the people of Reith,
or any other place through which he might pass.
Aennerl laughed and clapped her hands. "Oh, Peterl, you come erwuenscht
[56]!" she exclaimed. "Show me what you have got to sell--show me all
your pretty things! I want an entirely new rig-out. Make haste! show
me the best--the very best--you have brought."
"Show you the best, indeed!" said the Langer Peterl, scarcely
slackening his pace, and not removing the pipe from his mouth; for
hitherto he had only known the orphan Aennerl by her not being one of
his customers. "Show you the best, indeed, that what you can't buy you
may amuse yourself with a sight of! And when you've soiled it all with
your greasy fingers, who'll buy it, d'you suppose? A likely matter,
indeed! Show you the best! ha! ha! ha! you don't come over me like
that, though you have got a pair of dark eyes which look through into
a fellow's marrow!"
"Nonsense, Peterl!" replied Aennerl, too delighted with the thought
of the finery in prospect even to resent the taunt; "I don't want
to look at it merely--not I, I can tell you! I want to buy it--buy
it all up--and pay you your own price! Here, look here, does this
please you?" and she showed him a store of gold such as in all his
travels he had never seen before.
"Oh, if that's your game," said the long Peter, with an entirely
changed manner, "pick and choose, my lady, pick and choose! Here are
silks and satins and laces, of which I've sold the dittos to real
ladies and countesses; there are----"
"Oh, show me the dittos of what real ladies and countesses have
bought!" exclaimed Aennerl, with a scream of delight; and the pedlar,
who was not much more scrupulous than others of his craft, made
haste to display his gaudiest wares, taking care not to own that
it was seldom enough his pack was lightened by the purchases of a
"real lady." To have heard him you would have thought his dealings
were only with the highest of the land.
But it needed only to say, "This is what my lady the Countess of
Langtaufers wears," "This is what my lady the Baronin Schroffenstein
bought of me," for Aennerl to buy it at the highest price the Long
Peter's easy conscience could let him extort; and, indeed, had he not
felt a certain commercial necessity for reserving something to keep up
his connexion with his ordinary customers on the rest of his line of
route, orphan Aennerl would have bought up all that was offered her
under these pretences, and without stopping to consider whether the
materials or colours were well assorted, or whether such titles as
those with which the pedlar dazzled her understanding existed at all!
The next day was a village festival in Reith. And the quiet people of
Reith thought the orphan Aennerl had gone fairly mad when they saw at
church the extravagant figure she cut in her newly-acquired finery;
for, in her hurry to display it, she had in one way and another piled
her whole stock of purchases on her person at once. A showy skirt
embroidered with large flowers of many colours, and trimmed with
deep lace, was looped up with bright blue ribbons and rosettes over a
petticoat of violet satin, beneath which another of a brilliant green
was to be seen. Beneath this again, you might have descried a pair of
scarlet stockings; and on her shining shoes a pair of many-
rosettes and shoe-buckles. The black tight-fitting bodice of the local
costume was replaced by a kind of scarlet hussar's jacket trimmed with
fur, fastened at the throat and waist with brooches which must have
been originally designed for a stage-queen. From her ears dangled
earrings of Brobdignagian dimensions; and on her head was a hat and
feathers as unlike the little hat worn by all in Reith as one piece
of head-gear could well be to another.
Of course, it did not befit a lady so decked to take the lowly seat
which had served the orphan Aennerl; before the Divine office began
she had seated herself in the most conspicuous place in the church,
so that no one lost the benefit of the exhibition; and it may well
be believed that the congregation had no sooner poured out of the
sacred building than the appearance of the orphan Aennerl was the
one theme of a general and noisy conversation.
For some it was a source of envy; for some, of ridicule; for some
unsophisticated minds, of simple admiration. But the wiser heads kept
silence, or said, in tones of sympathy, "The orphan Aennerl isn't
the girl the Goigner Joessl took her for."
Joessl had been to church in his own village of Goign, and had therefore
been spared the sight, as well as the comments it had elicited. But
as he came towards Reith to take his Aennerl for the holiday walk,
he noticed many strange bits of hinting in the greetings he received,
which puzzled him so, that, instead of going straight on to Aennerl,
he sat down on the churchyard wall, pondering what it could all
mean. "I wish you joy of your orphan Aennerl!" one had said. "Goigner
Joessl, Goigner Joessl, take my advice, and shun the threshold of orphan
Aennerl!" were the words of another, and he was an old man and a sage
friend too. "Beware, Goigner Joessl, beware!" seemed written on every
face he had met--what could it all mean? He wandered forward uncertain,
and then back again, then on again, till he could bear it no longer,
and he determined to go down to the Wirthshaus beim Stangl, and ask
his mates to their face what they all meant.
Before he came in sight of the door, however, he changed his
resolution. Through the open window he heard noisy talk, and noisiest
of all was the voice of the Langer Peterl. Honest Joessl had an
invincible antipathy to the wheedling, the gossip, the bluster, and
the evil tongue of the Langer Peterl, and he never trusted himself to
join his company, for he knew a meeting with him always led to words.
Determining to wait till he was gone, he walked about outside, and
as there is always a train of waggons waiting at the Wirthshaus am
Stangl while the wayworn carters refresh themselves, he could easily
remain unperceived.
Thus, however, he became unintentionally the hearer of all he desired
to know--much more than he desired, I should say.
"I tell you, she,--Aennerl would have bought my whole pack if I'd
have let her!" vociferated the Langer Peterl; "and I might have
saved myself all further tramping, but that I wouldn't disappoint my
pretty Ursal, and Trausl, and Moidl, and Marie," he added, in a tone
of righteousness.
"Buy it, man! you don't mean buy it! She got it out of you one way
or another, but you don't mean she bought it, in the sense of paying
for it?"
"Yes, I do. I say, she paid for it in pure gold!"
"No, that won't do!" said other voices; "where could she get gold
from?"
"Oh, that's not my affair," replied the pedlar, "where she got it
from! It wouldn't do for a poor pedlar to ask where his customers
get their money from--ha! ha! ha! I'm not such a fool as that! I
know the girl couldn't have it rightfully, as well as you do, but it
wouldn't do for me to refuse all the money I suspect is not honestly
come by--ha! ha! I should then drive a sorry trade indeed!"
Joessl's first impulse had been to fly at the Langer Peterl, and, as
he would have expressed it, thrust the lie down his throat; but then,
he reflected; where had the girl got the money from? what could he
say? To dispute it without having means of disproving it was only
opening wider the sore; and while he stood dejected and uncertain
the conversation went on more animated than before.
"I agree with you!" cried, between two whiffs of smoke, an idle
Bursch, on whom since the death of the Wilder Juergl that nickname had
descended by common consent. "What right have we to be prying into
our neighbour's business? If the girl's got money, why should any
one say she hasn't a right to it? She's an uncommon fine girl, I say,
and looks a long way better than she did before in her beggarly rags;
and a girl that can afford to dress like that is not to be despised,
I say."
That the speaker had only received the cognomen of Wild after the
Wilder Juergl was only in that he was younger; he had earned the
right to it in a tenfold degree. None of the steady lads of either
Goign or Reith or Elmau, or any other place in the neighbourhood,
would make a friend of him, and that is why he now sat apart from
the others smoking in a corner.
To be praised and defended by the Wilder Karl was a worse compliment
than to be suspected by the steadier ones. The words therefore threw
the assembly into some embarrassment for a moment, till the Kleiner
Friedl [57], a sworn friend of Joessl, thinking he ought to strike a
blow in his defence somewhere, cried out, in a menacing tone,--
"Very well played, Wilder Karl! but I see your game. You think because
the girl's got money she's a good chance for you. You think her
flaunting way will estrange steady Goigner Joessl, and then you think
you may step in between them--and a sorry figure she'd cut two days
after you'd had the handling of her! She wouldn't have much finery
left then, I'll warrant! The Langer Peterl there would have it all
back at half-price, and that half-price would all be in the pocket
of our honest Wirth am Stangl. But it's in vain; whatever she is,
she'll be true to the Goigner Joessl, I'll warrant--and as for you,
she wouldn't look at you!"
Wilder Karl rose to his feet, and glared at the Kleiner Friedl with
a glance of fury. "I wager you every thing you and I have in the
world, that I'll make her dance every dance with me at the Jause
[58] this very night!" and he shook his fist with a confident air,
for he had a smooth tongue and a comely face, and Aennerl would not
have been the first girl these had won over.
"That you won't," said the Wirth, coming to Friedl's rescue, who was
but a young boy, and had felt rather dismayed at the proposed wager,
"for I'm not sure, till all this is cleared up, that I should admit her
to the dance. But the difficulty will not arise, for Aennerl herself
told my daughter Moidl that now she could wear a lady's clothes it
would be impossible for her to come any more to the village dance."
Strengthened by the support of the Wirth [59], the Kleiner Friedl
felt quite strong again; and he could not forbear exclaiming, "There,
I told you there was no chance for you, Wilder Karl!"
But Wilder Karl, furious at the disappointing news of the Wirth,
and maddened by the invective of the Kleiner Friedl, rushed at the
boy head-over-heels, bent on mischief.
But Wilder Karl, though a bully and a braggart, inspired no respect,
because no feather adorned his hat, and that showed he was no champion
of any manly pursuit. So the whole room was on the side of Kleiner
Friedl; and the bully having been turned out, and the subject of
conversation pretty well exhausted, the Goigner Joessl turned slowly
home.
Now I don't say that he was right here. He was an excellent young
man, endowed in an especial degree with Tirolese virtues. His parents
had never had a moment's uneasiness about him; no one in the whole
village was more regular or devout at church; in the field none more
hard-working or trustworthy; at the village games and dances none
acquitted himself better; and had a note of danger to his country
sounded in his time I am sure he would have been foremost to take
his place among its living ramparts, and that none would have borne
out the old tradition of steadfastness more manfully than he.
But of course he had his faults too. And one of his faults was the
fault of many good people,--the fault of expecting to find every one
as good as themselves, of being harsh and unforgiving, of sulking
and pining instead of having an open explanation.
Now, mind you, I think it would have been much better if Joessl had,
after hearing the conversation I have just narrated, gone straight on
to Aennerl's, and had it all out with her, had heard from her own lips
the truth of the matter about which all Reith and Goign were talking,
and judged her out of her own mouth, giving her, if he could not
approve her conduct, advice by which she might mend it in the future.
But this was not his way. He had thought his Aennerl a model, almost
a divinity. He had always treated her as such, talked to her as such,
loved her as such. It was clear now, however, that in some way or
other she had done wrong. Instead of getting to the bottom of it, and
trying to set it straight, he gave himself up to his disappointment
and went home and sulked, and refused to be comforted.
Aennerl, meantime, knew nothing of all this. She had had a great desire
to be a lady and no longer a servant; and having plenty of money,
and plenty of fine clothes, she thought this made her a lady, and had
no idea but that every one acknowledged the fact. I don't think she
exactly wished that all the village should be envious of her, but at
all events she wished that she should enjoy all the prerogatives of
ladyhood, and this, she imagined, was one. Then she had no parents
to teach her better, and Joessl, who might have been her teacher,
had forsaken her.
But it was all too new and too exciting for her to feel any misgivings
yet. She amused herself with turning over all her fine things, and
fancied herself very happy.
In another day or two the Hof of good neighbour Bartl was put up for
sale, and another visit to the Bergmaennlein enabled her to become the
purchaser. She thus became the most important proprietor in Reith;
but she was so little used to importance that she did not at all
perceive that the people treated her very differently from the former
proprietor of the Hof.
Before him every hat was doffed with alacritous esteem due to his
age and worth. But poor Aennerl hardly received so much as the old
greeting, which in the days of companionship in poverty had always
been the token of good fellowship with her, as with every one.
It was long before any suspicion that she was mistrusted reached the
mind of Aennerl. In the meantime she enjoyed her new condition to
the full. Weekly visits to the Roehrerbuechel enabled her to purchase
every thing she desired; and when the villagers held back from her,
she ascribed their diffidence to the awe they felt for her wealth.
In time, however, the novelty began to wear off. She grew tired at
last of giving orders to her farm-servants, and watching her sleek
cattle, and counting her stores of grain. That Joessl had not been
to see her, she never ascribed to any thing but his respect for her
altered condition; and she felt that she could not demean herself by
being united to a lad who worked for day-wages.
Still grandeur began to tire, and her isolation made her proud,
and angry, and cross; and then people shunned her still more, and
upon that she grew more vexed and angry. But, worse than this, she
got even so used to her riches that she quite forgot all about the
Nickel to whom she owed them. Her farm was so well stocked that it
produced more than her wildest fancies required; she had no need to
go back to the Roehrerbuechel to ask for more gold, and she had grown
too selfish to visit it out of compassion to the dwarf.
The Bergmaennlein upon this grew disappointed; but his disappointment
was of a different kind from Joessl's. He was not content to sit apart
and sulk; he was determined to have his revenge.
One bleak October night, when the wind was rolling fiercely down
from the mountains, there was a sudden and fearful cry of "Fire!" in
the village of Reith. The alarm-bells repeated the cry aloud and
afar. The good people rose in haste, and ran into the lane with that
ready proffer of mutual help which distinguishes the mountain-folk.
The whole sky was illumined, the fierce wind rolled the flames and
the smoke hither and thither. It was Aennerl's Hof which was the
scene of the devastation. The fire licked up the trees, and the farm,
and the rooftree before their eyes. So swift and unnatural was the
conflagration that the people were paralyzed in their endeavour to
help. One ran for ladders, another for buckets; but before any help
could be obtained the whole homestead was but one vast bonfire. Then,
madly rushing to the top of the high pointed roof, might be seen the
figure of Aennerl clothed only in her white night-dress, and shrieking
fearfully, "Save me! save me!" Every moment the roof threatened to fall
in, and the agonized beholders watched her and sent up loud prayers,
but were powerless to save.
Suddenly, on the road from Goign a figure was seen hasting along. It
was the Goigner Joessl. Would he be in time? The crowd was silent now,
even their prayers were said in silence, for every one gasped for
breath, and the voice failed. A trunk of an old branchless tree yet
bent over the burning ruins. Joessl had climbed that trunk and was
making a ladder of his body by which to rescue Aennerl all frantic
from the roof. Will he reach her? Will his arm be long enough? Will
he fall into the flame? Will he be overpowered by the smoke? See! he
holds on bravely. The smoke rolls above his head, the flames dart
out their fierce fangs beneath him! He holds on bravely still. He
calls to Aennerl. She is fascinated with terror, and hears him
not. "Aennerl! Aennerl!" once more, and his voice reached her, and
with it a sting of reproach for her scornful conduct drives her to
hide her face from his in shame.
"Aennerl! Aennerl!" yet once again; and he wakes her, as from a dream,
to a life like that of the past the frenzy had obliterated. She forgets
where she is; but the voice of Joessl sounded to her as it sounded in
the years gone by, and she obeys it mechanically. She comes within
reach--and he seizes her! But the flames are higher now, and the
smoke denser and more blinding. "Jesus Maria! where are they? They
have fallen into the flames at last! Jesus, erbarme Dich ihrer [60]!"
"Hoch! Hoch! Hoch [61]!" shouts the crowd, a minute later. "They
are saved, Gott sei dank, they are saved!" and a jubilant cry rings
through the valley which the hills take up and echo far and wide.
On the edge of the crowd, apart, stands a little misshapen old man
with grey, matted hair and beard, whom no one knows, but who has
watched every phase of the catastrophe with thrilling emotion.
It was he who first raised the cry that they had fallen into the
flames; and the people sickened as they heard it, for he spoke it in
joy, and not in anguish. In the gladness of the deliverance they have
forgotten the old man, but now he shouted once more, as he dashed his
hood over his head in a tone of disappointed fury, "I did it! and I
will have my revenge yet!"
"No; let there be peace," said Joessl, who had deposited Aennerl
in safe hands, and now came forth to deal one more stroke for her;
"let there be peace, old man, and let bygones be bygones."
"Never!" said the Cobbold; "I have said I will have my revenge,
and I will have it!"
"But," argued Joessl, "have you not had your revenge? All you gave
her you have had taken away--she is as she was before: can you not
leave her so?"
"No!" thundered the dwarf; "I will have the life of her before
I've done."
"Never!" in his turn shouted Joessl; and he placed himself in front
of the elf.
"Oh, don't be afraid," replied the dwarf, with a cold sneer, "I'm not
going after her. I've only to wait a bit, and she'll come after me."
Joessl was inclined to let him go, but remembering the instability
of woman, he thought it better to make an end of the tempter there
and then.
"Will you promise me, that if I let you return to your hole in peace,
you will do her no harm should she visit you there again?"
"I promise you that I will serve her to the most frightful of
deaths--that's what I promise you!" retorted the enraged gnome.
"Then your blood be on your own head!" said Joessl, and, with his
large hunting-knife drawn in his hand, he placed himself in a menacing
attitude before the now alarmed dwarf.
Joessl was a determined, powerful youth, not to be trifled with. The
gnome trusted to the strength of his muscles, and fled with all
his speed; but Joessl, who was a cunning runner too, maintained his
place close behind him. The dwarf, finding himself so hotly followed,
began to lose his head, and no longer felt so clearly as at first the
direction he had to take to reach the Roehrerbuechel. Joessl continued
to drive him before him, puzzling him on the zigzags of the path
till he had completely lost the instinct of his way of safety. Then,
forcing him on as before to the edge of the precipice, he closed upon
him where there was no escape.
Yes, one escape there was--it was in the floods of the Brandenburger
Ache, which roared and boiled away some hundred feet below! Rather
than fall ignominiously by the hand of a child of man, the gnome
dashed himself, with a fierce shout, down the abyss. And that was
the last that was ever seen of the Nickel of the Roehrerbuechel.
Aennerl was now poorer than ever in this world's goods, but she was
rich in one deep and wholesome lesson--that it is not glittering wealth
which brings true happiness. The smiles of honest friends, and the
love of a true heart, and the testimony of an approving conscience are
not to be bartered away for all the gold in all the mines of the earth.
Wilder Karl laughed with his two or three boon companions, and said,
with a burst of contempt, "I've no doubt that fool of a Goigner Joessl
will marry the orphan Aennerl now that she hasn't a penny to bless
herself with!"
And the Wilder Karl judged right. Aennerl scarcely dared hope that he
could love her still, and she went forth humbly to her work day by
day, neither looking to the right hand nor the left, accepting all
the hardships and humiliations of her lot as a worthy punishment of
her folly and vanity.
But one evening as she came home from her toil, the Goigner Joessl
came behind her, and he said softly in her ear, "Do you love me still,
Aennerl?"
"Love you still, Joessl!" cried the girl; "you have thrice given me
life--first when I was a poor, heartbroken orphan, and you made me
feel there was still some one to live for in the world; and then a
second time, in that dreadful fire, when hell seemed to have risen
up out of the earth to punish me before the time; and now again this
third time, when I began to think my folly had sickened you for good
and all! Don't ask me that, Joessl, for you must know I love you more
than my life! If I dared, there is one question I should ask you,
Can you still love me? but I have no right to ask that."
"I must answer you in your own words, Aennerl," replied Joessl:
"you must know that I love you more than my life!"
"You must, you must--you have shown it!" exclaimed Aennerl. They
had reached the bank near the Roehrerbuechel where we first saw them;
the rosy light of the sunset, and the scent of the wild flowers,
was around them just as on that night.
"Yes," said Aennerl, after a pause, as if it were just then that
Joessl had said the words [62]--"yes, Joessl, this is happiness; we want
nothing more in this world than the warm sun, and the blue sky--and
to be together! Yes, Joessl, we shall always be happy together."
They walked on together; as they reached the memorial of the dead
miners the village bells rang the Ave. And as they knelt down,
how heartfelt was Joessl's gratitude that the prayer he had uttered
at that spot once before had been so mercifully answered, and his
Aennerl restored to him for ever!
THE WILDER JAeGER AND THE BARONESS.
There was a rich and powerful baron who owned a broad patrimony
in South Tirol, Baron di Valle. He was not only one of the richest
and most powerful, he was also one of the happiest, for he had the
prettiest and most sensible woman of Tirol for his bride. The brief
days were all too short for the pleasure they found in each other's
society, and they were scarcely ever apart the whole day through.
Once, however, the Baron went on a hunting party through a part of
the country which was too rough for the Baroness to follow him. The
day was splendid, the scent good, and the Baron full of enthusiasm
for his favourite sport; but what egged him on more than all these,
was the sight of a strange bold hunter who bestrode a gigantic mount,
and who dashed through brake and briar, and over hill and rock,
as if no obstacle could arrest him. Baron di Valle, who thought he
was the boldest hunter of the whole country-side, was quite mad to
see himself outdone; nor could he suffer this to be. Determined to
outstrip his rival, he spurred his horse on, so that he might but
pass him somewhere; but the Wilder Jaeger, for it was he, always kept
on ahead, and though the brave Baron kept close to his heels, he was
never able to pass him by.
They had long outstripped all the rest. But all this time the Baron
had taken no note of whither he went; now he found himself in the midst
of a thick forest of tall fir-trees, with their lower branches cut off
because they were planted so thick and close together that there was no
room between them, and their tops were intergrown so that they formed
one compact mass, excluding the very air and the light of day. The Wild
Hunter stopped his mad career before this barrier, and then, turning,
pretended for the first time to be aware of the Baron's presence.
"What do you want here?" he exclaimed, fiercely, his rolling eyes
glaring like fire. "How dare you invade my domain!" and with that he
blew a mighty blast on his hunting-horn, at sound of which a whole
troop of wild huntsmen, habited like himself, and with similar fiery
eyes, appeared suddenly, surrounding the Baron.
"Stand back!" cried the Baron, in a commanding tone, as the wild
huntsmen dismounted and prepared to seize him.
"No one commands here but I," said the Wilder Jaeger. And then he added,
addressing his men, "Seize him, and carry him off!"
"Hold!" said the Count, but speaking more humbly than before, for he
saw he must yield something to the necessity of the case; "I suppose
there is some ransom upon which you will let me off? I have wronged
you in nothing, and meant no offence. I admired your brave riding,
and I thought what one brave man might do, another might."
"Since you take that tone," said the Wilder Jaeger, "I will do what
I can for you. I will let you ransom yourself at one price. You must
know, that it is not you that I want at all; I only lured you here as
a means of getting hold of the Baroness, and had you been uppish and
violent, I should have kept you in chains for the rest of your life,
while I married her. But as you know how to keep a civil tongue in
your head, I will show you that I can appreciate courtesy. So now
I give you permission to return, to be yourself the bearer to your
wife of my conditions.
"Tell her, then, that I have won her for my own, and she belongs
irrevocably to me; it is useless that she attempt to escape, for you
see that my people are countless, and violence is of no avail against
me. But I am a good sort of fellow, and as I love her, I don't want to
do any thing to alarm her, so long as she shows no foolish resistance."
"But the ransom? You spoke of a ransom just now," interposed the Baron,
hastily; "what, about that?"
"All in good time," replied the Wilder Jaeger--"give a fellow time
to speak. The only mode of ransom is this--let the Baroness guess
my name. I give her three guesses of three words each, and an
interval of a month. But if she doesn't succeed, remember, she is
mine! this day month I appear and claim her. If, in the meantime,
she thinks she has made the guess, and wants to satisfy herself as to
its correctness"--and he laughed a ghastly laugh of scorn, as if to
impress the Baron with the hopelessness of the idea--"she has only
to come to the ilex grove on the border of this forest which marks
the frontier between your territory and mine. If she stands there,
beside the centre tree, and blows this horn--see what a pretty little
gold horn it is, that I have had studded with diamonds and rubies--just
fit for her pretty little fingers!" he added, with a grin of scorn--"at
sound of her voice I shall be with her on the instant."
The Baron was not one to have tolerated such talk from any human
being soever, but he felt the necessity of vanquishing his temper this
time--a more difficult matter ordinarily than vanquishing a foe--for a
dearer life than his own was at stake; and if he could not altogether
save the Baroness from the power of the Wilder Jaeger, he could take
counsel with her as to the means of finding out the hidden name,
and at least spend with her the last days that he could call her his.
Accordingly he took the horn, and stuck it in his belt without a
word. And indeed no word would have availed him, for the whole troop
of the wild huntsmen had vanished as it came, and he was left alone.
There was no difficulty in finding his way back by the path by which
he had come, for it was plainly marked by the havoc of the surrounding
vegetation the wild chase had cost. And though he now put spurs to
his steed that he might reach home without losing an hour more than
he could help of the companionship of his beloved wife, he now for
the first time apprehended how swiftly he had come, for, riding the
utmost of mortal speed, it took him three days to get back to the ilex
grove which marked the boundary of his own territory. Hence it was
still half a day's journey to reach his castle. But while he was yet
a great way off his loving wife came out to meet him, full of joy at
his approach, for since the rest of the hunt had come home without him
she had done nothing but watch from the highest turret of the castle,
that she might catch the first sight of him returning; her thirsty
eyes had not been slow to discern his figure as he hastened home.
Great was her amazement, however, to find that, instead of returning
her greeting with his wonted delight, he turned his head away, as if
he dared not look at her, and wept. She rode beside him all the way
home, but he still kept silence, for he could not bear to render her
sorrowful with the message of which he was the bearer. But he could
conceal nothing from her loving solicitude, and soon he had told
her all.
Being a woman of prudent counsel and strong trust in God, she was much
less cast down, however, than he had expected. Though bewildered at
first, and seeing no way out of the difficulty, she yet declared she
was sure some way of escape would be opened to her, it only remained
to consider where they should find it. And never a word of angry
recrimination did she utter to remind him that it was his mad vanity
had brought them to this plight.
The Baron felt the full force of this forbearance, for he did
nothing but reproach himself with his folly. But the fresh proof of
her amiability only occasioned another pang at the thought of the
approaching separation.
Still no good counsel came to mind, and the Baroness herself began
almost to lose heart. The Baron had abandoned the hunt and all his
sports, and sat gloomily in the ancient seat of his ancestors. The
Baroness sat among the flowers of her oriel window, her embroidery
in her hand; but her mind was far away over the tops of the dark
green trees, looking for some bright thought to bring deliverance
to her from above. Every morning and evening they knelt together in
the chapel of the castle, and prayed that a spirit of prudence and
counsel might be given them.
Ten days had passed, and no good thought had come. The Baron reclined
gloomily in the ancient seat of his ancestors, and the lady sat
among the flowers of the oriel window gazing over the tops of the
high dark fir-trees, full of hope that some wise counsel would be
given her. Suddenly she rose and clapped her hands, and her ringing
laugh brought the Baron bounding to her side.
"I have found it, Heinrich!" she exclaimed; "I am sure I have found
the name! Doesn't the Wilder Jaeger live among the tall fir-trees?"
"Yes; among the tall fir-trees is his dwelling."
"And didn't he speak of three names?"
"Yes; he said your guess must include three names."
"Then I have it, Heinrich! What more natural than that he should be
called from the names of the trees which form his palace? As I was
gazing over the tops of the high dark trees the words came into my
mind, 'Tree, Fir, Pine'--those will be the three words. Come, and let
us go out to the ilex grove, and be free to belong to each other as
of old!"
She was so lively that the Baron caught some spark of her hopefulness,
but he was too far sunk in despondency to enter into her joy all
at once. Nevertheless, it was not a moment when, if ever, he would
have thwarted her, so he ordered the horses to be saddled, for it was
still early morning. And they rode together to the ilex grove which
was the boundary of the Wilder Jaeger's domain; the Baroness striving
every minute by some sprightly speech to distract the Baron, and the
Baron utterly incapable of rousing himself from his gloomy fears.
The Baroness was the first to reach the grove; in fact, she had ridden
on a good way in advance, that she might have it out with the Wilder
Jaeger before her husband came, so that she might greet him on his
arrival with the news that she was free.
Merrily she sounded the jewelled horn, and before its sound had died
away the Wilder Jaeger was at her side. He no longer looked dusty,
wild, and fierce, as during the Baron's mad chase. He seemed a man
of noble presence, carefully dressed in a green hunting-suit, with a
powerful bow in his hand, and a beautiful boy to hold his arrows. In
his belt was a jewelled hunting-knife of exquisite workmanship, and
to a cord across his shoulder hung a golden horn of similar pattern
to that he had sent the Baroness, and, moreover, as a further act of
gallantry, he wore a scarf of red and white, the favourite colours
of the Baroness. A jewelled cap shaded the sun from his brow, which
a red and white plume gracefully crested.
The Baroness looked astonished to find she had nothing more formidable
to meet, and felt that had she not already been the wife of the
Baron di Valle, she would not have found it so great a calamity to
be obliged to marry the Wilder Jaeger.
The Wilder Jaeger was not slow to perceive that the impression he
had produced was good, and bowing towards her with courtly mien,
paid her a respectful salutation, and immediately added,--
"Your eyes are so clever, fair Baroness, that I very much fear you
are going to pronounce my name, and rob me of the happiness I had so
nearly bought! Spare me, therefore, lovely lady--say not the word! but
come with me into the shady pine-forest, where you shall have every
thing heart can desire--the noblest palace, the widest domain, and
unlimited command; retainers without number, pleasures without alloy,
and every wish gratified without condition!"
He approached her as he spoke. His eyes sparkled no longer with the
angry fury which had thrilled the Baron, but with a mild fire of
tenderness and devotion. Nothing more attractive and winning than his
whole appearance and manner could be conceived, and for a moment the
Baroness had almost forgotten the less accomplished--but, oh! more
sincere--passion of her Heinrich.
It was only for a moment. The weakness passed, she instantly drew
herself up with dignity, and stepped back against the friendly ilex.
"It was not to hear such words I came," she said, "but to pronounce
those which are to free me from ever having to listen to such
protestations again----"
"Oh, say them not! say them not!" said the Wilder Jaeger, throwing
himself at her feet. "Any thing but that! Name any wish by fulfilling
which I can win your favour; name any difficult task by accomplishing
which I can prove myself worthy of your love----"
"My love," said the Baroness, striving to speak coldly, "is another's
already; you see, there is none to be won from me. But interrupt me
no more. I have guessed your name, to discover which was to be the
price of my freedom. It is----"
The Wilder Jaeger clasped her feet in despair, entreating her not
to pronounce it, but she went on, with a clear, confident voice,
to utter the words,--
"Tree! Fir! Pine!"
The Wilder Jaeger looked up as if he did not quite understand what
she meant.
"Now, let go your hold, and let me pass, for I am free!" she said,
resolutely.
"'Free,' say you?" said the handsome Cobbold, with
astonishment. "Free? did you mean you thought that was my unknown
name?"
"Yes," replied the lady, in a voice of conviction.
"Oh, dear, it is nothing like it!" he answered, with glee, and yet
not without a delicate regard for her disappointment. "No, that is
not it; nor is it likely you should ever arrive at it. So days of
happiness are before us yet." He had no need to kneel to her longer,
but it was joy to him to be at her feet.
"Dare not to speak so before me!" replied the Baroness, trying to
tear herself away. "I know of no happiness, except with Heinrich; and
I am persuaded that, though I have failed this time, it will yet be
given to me to find the word which shall restore me to him completely."
The Baron arrived as she finished speaking; and though he saw by the
sorrowful look which now had possession of her bright face, and the
triumphant mien of the Cobbold, that she had failed, and that she was
still under the Wilder Jaeger's spell, he was so incensed to find him
in such an attitude that he drew his sword, and would have closed with
him then and there, but the Wilder Jaeger blew one note upon his horn,
and in an instant he was surrounded, as before, by his myrmidons,
who unarmed him and held him bound upon the ground, while the Cobbold
himself approached to seize the hand of the Baroness. A fiery fury
took possession of him, and sparks darted from his eyes which fell
smouldering among the twigs of fir. Powerless to defend his wife
by force, the Baron once more mastered his anger, and reminded his
adversary courteously of his promise to leave them at peace for the
interval of a month.
"I am always ready to answer you in whatever tone you elect to
adopt," said the Wilder Jaeger, rising, and leaving the side of the
Baroness. "You see, it is useless to attempt force against me; but when
you behave with due consideration, so will I." At a sign from him the
sprites loosed the Baron's bonds, gave him back his sword, and held
his stirrup with the most respectful care, while he mounted his horse.
"Depart, then, unharmed," said the Wilder Jaeger, "since you set so
much store on prolonging your suspense. I should say, it was wiser
to make the best of a bad bargain and submit to your fate at once,
with grace. However, I have given my word and won't go back from it. I
restrain my power over you till the full end of the month; and, what
is more, I not only give the lady three guesses, but as many as she
likes. For," he added, with a cynical leer, "she is as little likely
to guess it in thirty as in three; while every time that she chooses
to essay the thing, it gives me the happiness of seeing her." And he
turned away with a peal of wild laughter which made the lady shudder.
The sprites vanished as they had come; and the Baron and his wife
rode sadly home, without the courage to exchange a word.
If the Baroness had for a moment been won by the comely presence
and devoted admiration of the Wilder Jaeger, she had now seen enough
reason to fear his treacherous humour, and to dread her impending
fate as much as at the first.
They spent the rest of the evening in prayers and tears in the chapel
of the castle, and the next evening, and the next; and the days flowed
by as before, but more sadly, and with even less of hope. The Baroness
scarcely now dared raise her eyes so high as the tops of the tall
dark trees; they fell abroad over the beautiful landscape stretched
out beneath them, and the good gifts of God cropping up out of the
ground; and she thought how beautiful was that nature to which she
must so soon say adieu!
Thus ten days passed without a gleam of expectation. Suddenly she
rose and clapped her hands; and her silvery laugh brought her husband
bounding to her side.
"I have it this time, Heinrich!" she said.
And the Baron listened anxiously, but trusted himself never to speak.
"Said you not that the Wilder Jaeger's domain was entirely among the
tall dark trees?"
"So it seemed to me it was," responded her husband.
"But I certainly discerned through the forest patches of ripe golden
grain. Saw you them not too?"
"The first time I rode too fast to notice them, but I do think on
this last journey I saw such here and there by the wayside."
"No doubt," continued the lady, "it is hence he takes his name;
these small patches of golden grain are more worth than all the
vast forests. Order the horses, for I have guessed his name! It came
to my mind just now, as I looked over the harvest-fields stretched
out yonder.
"Wheat, Oats, Maize--that will be his name!"
The Baron knew her counsel had often proved right when he least
expected, and even disputed it, and though he was now too desponding
to expect success, he was likewise least inclined to dispute her
word. So he ordered the horses round, for it was yet early morning,
and they rode to the ilex grove.
The Baroness, whose hope seemed to rise as she got nearer the goal
of the journey, was full of spirit and cheerfulness, and, finding it
impossible to work up the Baron to the same expectation as herself,
rode on to accomplish her work ere he arrived.
One note of the jewelled horn brought the Wilder Jaeger to her presence.
As she had failed before, he had less fear of her success this time,
and he was proportionately less subservient and submissive.
"So you think you are come to give me my dismissal, beautiful
Baroness? But you have no reason to repulse me so--be assured I mean
it well with you; and though there is no limit to my power over you,
I shall never treat you otherwise than with honour," he said, with a
little scornful laugh which suited his fine features exactly, and made
him look handsomer than before. And as he spoke so, his haughty tone,
not unmixed with warmth and admiration, thrilled her with the notion
that, after all, if it were not for her troth plighted to the Baron,
it would not be so very dreadful to owe obedience to one who knew
how to command so gracefully.
But it was only for a moment. The weakness passed, she drew herself
up with dignity, and, retreating against the support of the friendly
ilex, said,--
"Silence! and remember your promise to leave me at peace till the
fatal month is out. I cannot listen to you. And now for your name----"
The Cobbold bowed, with a half-mocking, half-respectful inclination,
as if forcing himself to listen out of courtesy, but secure that she
would not guess right.
"Wheat! Oats! Maize!" said the Baroness, with a positive air.
The Cobbold stared comically, as if doubting whether she was in
earnest; and at last, as if to relieve her out of politeness,
he replied,--
"Oh, dear no, that's not at all like it!"
The Baroness hung her head in despair; then, drawing herself up again,
she said,--
"How do I know you are not deceiving me? You say this is not your name,
and I have to believe you--but suppose I maintain that it is it?"
"You are not fair, beautiful Baroness," replied the Wilder Jaeger,
with a charming dignity. "I have never deceived you, nor ever would
I deceive so noble a lady! what I have promised, I have kept; but in
this case I have no means of deceiving you--great as is my power, that
is one thing beyond it. Could a mortal, indeed, discover and pronounce
my name in my presence, I could not stand before him an instant. But
this it is not given to mortals to know, and that is why I proposed
this difficulty to you. Should I have paid you so bad a compliment,"
he added, with his cynical laugh, "as to render it possible that I
should lose so great a prize?"
The Baron rode up while he was saying this, and shrank dumb with
despair at the cruel words and the positive tone in which they were
uttered.
Without condescending to exchange a word with the Cobbold this time,
he lifted his wife on to her palfrey and rode away with her in silence.
It was now all over. His despondency even gained the Baroness, and she
ceased to rack her brain with the hope of finding the inconceivable
name. Her eye not only dared not raise itself to the tall dark
trees--it had not even power to range over the landscape. With her
head sunk upon her breast, she sat silently among her flowers in her
oriel window, nor cared even to look at them. Only in the morning
and the evening they knelt together in the chapel of the castle,
and prayed that the calamity might pass away yet.
The days went by, and now the last but one had come; and the Baroness
trembled, for her imagination pictured the Cobbold coming to carry her
away. But her courage did not forsake her even now, and she proposed
to go out into the forest to meet her fate, as more noble than waiting
for it to overtake her.
The Baron, too dispirited to discuss any matter, and indifferent to
every thing, now that all he cared for was to be taken from him, gave
a listless consent. The next morning, having prayed and wept together
in the castle chapel, they set out on their mournful pilgrimage,
the young wife led as a lamb to the sacrifice.
The flowers bloomed beneath their feet, and the sun shone warm
overhead, the birds sang blithe and gay--all nature was bright and
fresh; but with heavy hearts they passed through the midst, nor found
a thought but for their own great sorrow. As they came to the borders
of the forest, however, the Baroness discerned the cry as of one in
distress. Forgetting for the moment her own agony, her compassionate
heart was at once moved, and she begged her husband to turn aside
with her, and find out the poor wretch who pleaded so piteously. In
a little time they had followed up the sound, and they found one of
the Wilder Jaeger's men tied in front of a lately lighted fire. In a
few minutes more the heaped-up wood would have been all in flames,
and then the luckless wight must have been slowly roasted! At a word
from the Baroness, the Baron cut his bonds; and then they inquired
what was the occasion of his punishment. "Oh, it don't want much to
get a punishment out of the Wilder Jaeger!" was the answer.
"Is he so very severe, then?" asked the Baroness, her cheek blanching
with fear.
"At times, yes; it depends how the fancy takes him--if he is out of
humour he spares no one. If he were not so violent and arbitrary,
I would do you a good turn for that you have done me; but I dare not,
his anger is too fearful."
The more he descanted on the Wilder Jaeger's barbarity, the more the
Baroness prayed that he would tell her the word that would save her;
but he dared not, and all her instance was in vain. "And yet there
might be a means," he said, for he was desirous of doing a service
to his deliverers.
"Oh, speak! tell us what we can do--no matter what it is, we will do
it!" answered both at once.
"Well, if you happen to overhear it, I shan't have told you, and yet
it will serve your turn just as well;" and with that he walked on
close in front of them, singing carelessly as he went.
"How are we to 'overhear' it, Heinrich?" said the Baroness, after
a bit.
"He seems to have forgotten us," replied the Baron, in despair. "I have
been expecting him every minute to turn round and give us a hint of
how he meant to help us; but it is just like every one you do a favour
to--when they have got what they want, they forget all about you."
They walked on in silence; and the fellow kept on close in front of
them, singing as before, and always the same verse.
At last the Baroness got wearied with hearing the same thing over
and over again, and she began repeating the words over to herself,
mechanically. She could not make them out at all at first, for he
had a rough, abrupt articulation, but by dint of perseverance in an
occupation which served as a distraction to her agony, she at last
made it out, word by word:--
"The Wild Huntsman's betrothed (though he is not tamed)
To a lady fair
Driven to despair.
If she only knew he's Burzinigala named!"
"'Burzinigala named!' exclaimed the Baroness, with the ringing laugh
of former days, and clapping her hands merrily.
"I have it all right this time, you may depend, Heinrich!" and she
laughed again.
The Baron was too delighted for words--he embraced his wife in his
joy; and they walked on with a very different mien from what they
wore before. The first joy over, they turned to thank their helper;
but he had already disappeared, climbing over the tops of the trees
to get out of sight of the Wilder Jaeger's eye for as long as might be.
There was no more lingering now, they hasted on, anxious only to
proclaim their triumph.
The ilex grove was soon reached, and the jewelled horn quickly produced
the Wilder Jaeger.
To-day he was habited with greater care even than on the former
occasions, and there was also still more assurance in his manner,
and still more forwardness to flatter.
"Well, lady fair," he said, with a mocking air, "do you deem you have
guessed my name this time?"
"Really, it is so difficult," replied the lady, "that how can you
think I can hope to succeed? Besides, why should I wish to do what
would deprive me of so charming a companion?"
The Wilder Jaeger in his turn was perturbed. Nothing could have made
him happier than to hear such words from her lips, could he have deemed
them sincere; but there was an irony in her tone and a playfulness in
her countenance which showed that her heart was not in her words. Yet
he felt convinced she could not discover his name; and so he knew not
what to think, and scarcely what to say. And the Baroness, delighting
in his confusion, continued teasing him, like a cat with a mouse.
After a good deal of this bantering, in which the Wilder Jaeger got
quite bewildered, the Baroness rose majestically.
"Have we not had enough talking?" she said, with emphasis; "when are
you going to take me home--Sir Burzinigala?"
It would be impossible to describe the effect of this word. He rose
from the earth with one bound. The beauty, the calmness, the commanding
air, which had at one time charmed the Baroness, had all fled. Wild,
savage, and furious as he had first appeared and tenfold more, he now
showed; and the sparks flew from his eyes on all around. Through the
thick tops of the trees he passed, they hardly knew how; and soon the
only trace of him left was that of the sparks that smouldered on the
dry heath.
It only remained for the Baron and Baroness to return home, locked
in each other's arms. And they continued loving each other more than
ever before to the end of their days.
THE GRAVE PRINCE AND THE BENEFICENT CAT.
There once was a king in Tirol who had three sons. The eldest was grave
and thoughtful beyond his years; but he seldom spoke to any one, took
no pleasure in pastimes, and lived apart from those of his age. The
other two were clever and merry, always forward at any game, or at any
piece of fun, and passed all their time in merry-making and enjoyment.
Now though the eldest son was, by his character, more adapted to make
a wise and prudent sovereign, yet the two younger brothers, by their
lively, engaging manner, had made themselves much more popular in
the country; they were also the favourites of their father, but the
eldest was the darling of his mother.
The king was old and stricken in years, and would gladly have given
up the cares of government, and passed his declining years in peace,
but he could not make up his mind to which of the brothers he should
delegate his authority. The queen was persuaded of the excellent
capacity of her eldest son; but the two younger were always saying
he was half mad, and not fit to govern, and as they had the people
on their side, he greatly feared lest the kingdom should be involved
in civil war, so he always put off making any arrangement.
One day, however, an ancient counsellor observed to him, that if he
really feared that there would be a dispute about the succession,
it was much better to have it decided now while he was alive to act
as umpire, than that it should befall when they would be left to
wrangle with no one to make peace between them.
The king found the counsel good, and decided to retire from the
government, and to proclaim his eldest son king in his stead. When the
two younger sons, however, heard what he intended to do, they came
to him and urged their old charge, that their elder brother was not
fit to govern, and entreated the king to halve the kingdom between
them. But the king, anxious as he was to gratify them, yet feared to
displease the queen by committing so great an injustice against her
eldest son; and thus they were no further advanced than before.
Then the old counsellor who had offered his advice before spoke again,
and suggested that some task should be set for the three, and that
whoever succeeded in that should be king beyond dispute.
The three sons all swore to abide by this decision; and the king found
the counsel good. But now the difficulty arose, what should he set
them to do? for they had insisted so much on the weak intellect of
the eldest, that the queen feared lest, after all, he should fail in
the trial, and her care for him be defeated. She knew he had never
practised himself in feats of strength, or in the pursuit of arms,
so it was useless proposing such as these for the test, but she
persuaded him to set them something much simpler.
So, having called an assembly of all the people, he proclaimed
aloud that the three brothers should travel for a year and a day,
and whichever of them should bring him back the finest drinking-horn,
he should be the king--the three sons swearing to abide by his award.
The two younger brothers set out with a great retinue; and, as they
did not apprehend much difficulty in surpassing their brother in
whatever they might undertake, they spent the greater part of the year
allowed them in amusing themselves, secure in bringing back the best,
whatever they might bring.
The eldest set out alone through the forest. In his lonely wanderings
he had often observed a strangely beautiful castle on a far-off
mountain, concerning which he could find no record in any of his
books, nor could he learn that any one living knew any thing about
it. He now resolved to make his way thither, persuaded that if he was
to find something surpassing the work of human hands, it was like to
be in this enchanted castle.
Though it was so high-placed, the way was much easier than he thought,
and he was not more than five months getting there; so that he had
ample time for exploring its precincts, and yet get back within the
appointed date. He had, indeed, to traverse dark forests and steep
rocky paths, but when he got near the castle all these difficulties
ceased. Here there were only easy <DW72>s of greensward, diapered
by sparkling flowers; broad-leaved trees throwing delicious shade;
and rills that meandered with a pleasant music. Delicious bowers
and arcades of foliage of sweet-scented plants invited to repose;
and every where luscious fruits hung temptingly within reach. Birds
sang on every branch with a soft, dreamy melody which soothed, and
disturbed not the lightest slumber.
The prince thought it would have been delightful to pass the remainder
of his days there, but he remembered that it was an important mission
with which he was entrusted, and he passed on.
A broad flight of marble steps led from these amenities up to the
palace, and every now and then a thousand little jets were turned on,
to pour their tiny floods over them, and cool them for the tread of
those who entered.
And yet no one was near, no one to enjoy all this magnificence! The
prince entered the hall, but no one came to meet him; he passed through
the long corridors--all were deserted; he entered one apartment after
another--still no one. At last he came to one charming boudoir all
hung with pink satin, and lace, and beautiful flowers. On a pink
satin sofa covered with lace sat a large Cat with soft grey fur,
and soft grey eyes--the first living thing he had met!
As he entered, the Cat rose to meet him, walking on her hind-paws,
and, holding out her right front-paw in the most gracious manner,
asked him, in a sweet, clear voice, if there was any thing she could
do for him. Then, as if the effort was too great, she let herself
down on all fours, and rubbed her soft grey head against his boots.
Finding her so friendly, he was going to take her up in his arms:
this she would not allow, however, but sprang with an agile bound on
to a ledge above his head. "And now tell me," said she, "what is it
you want me to do for you?"
"Really, Lady Purrer, you are so kind, you confuse me! But, to tell
you the truth, I fear--"
"You fear that a poor puss can't be of any use," interposed the
Cat, smartly, "and that your requirements are much above her feeble
comprehension. But never mind, tell me all the same; there is little
fear but that I can help you, and if I can't, the telling me will do
you no harm."
"Quite the contrary," replied the prince, "it will be a great pleasure
to have only your sympathy, for I am in great distress." Her voice
was so sweet and kind, that he quite forgot it was only a Cat he was
talking to.
"Poor prince!" said the Cat, soothingly; "tell me all about it,
then. But stop, I'll tell you first what I think. I'm sure you are not
appreciated at home. I saw it in your look when you first came in. You
don't look bright and enterprising, as you ought to look. You look
as if you lived too much alone. Oh, you would be twice as handsome
if you only looked a little more lively and energetic--" and then
she stopped short, and sneezed a great many times, as if she feared
she had said what was not quite proper, and some other sound would
efface that of her words.
"There is a great deal of truth in what you say," replied the prince;
"they don't care much about me at home--at least my mother does,
but my father and brothers don't. And I do live too much alone--but
it's not my fault: it's a bad way of mine, and I don't know how to
get out of it."
"You want some one to pet you, and spoil you, and make you very happy;
and then you would be pleased to go into the society of others, because
then you could say to yourself, I'll show them that there's some one
understands me and makes a fuss about me--" and she stopped short,
as before.
"But who should care to spoil and pet me?" cried the prince,
despondingly, and too much interested in her words to see any reason
why she should be confused at what she had said.
"Why, a nice little wife, to be sure!" replied the Cat.
"A wife!" exclaimed the prince; "oh yes, my father's grey-bearded
counsellors will find me some damsel whom it is necessary I should
marry for the peace of the kingdom; and to her I shall be tied, and,
be she an idiot or a shrew, I shall have no voice in the matter."
"But do you mean to say," retorted the Cat, in a more excited voice,
"that if you found a nice little princess--I don't say any one they
could with justice object to, but a real princess--who cared very
much for you, and made you very happy, very happy indeed, so that you
determined to marry her, that you wouldn't be man enough to say to your
father and all his counsellors, 'Here is the princess I mean to make
my wife; I feel Heaven intended her for me. I am sure she will be the
joy of my people, as she is mine, and no other shall share my throne'?"
"Wouldn't I," exclaimed the prince, with energy, starting to his feet,
and placing his hand instinctively on his sword, his eye flashing
and the colour mounting in his cheek.
"Ah! if you always looked like that! Now, you are handsome
indeed!" exclaimed the Cat, enthusiastically, and purred away. "But,"
she added, immediately after, "all this time you haven't told me what
it was you came for."
"Ah!" said the prince, despondingly, at finding himself thus recalled
to the prosaic realities of his melancholy life from that brief
dream of happiness. "No; because you have been talking to me of more
interesting things" (the Cat purred audibly); and then he told her
what it was had really brought him there.
"You see, your mother understands your character better than all
the rest," said the Cat. "She knew you could be trusted to prove
your superiority over your brothers, though the others hope you
may fail. However, fail you won't this time, for I can give you a
drinking-horn which neither your brothers nor any one else on earth
can match!"
With that she sprang lightly on to the soft carpet, and ran out of
the room, beckoning to him to follow her. She led him through a long
suite of rooms till they came to a large dining-hall all panelled
with oak and filled with dark carved-oak furniture. In the centre of
one end of this hall, high up in the panelling, was an inlaid safe or
tabernacle curiously wrought. Puss gave one of her agile springs on to
the top of this cabinet, and, having opened its folding-doors gently
with her paw, disclosed to view a drinking-horn such as the prince had
never seen. It was a white semi-transparent horn, but close-grained,
like ivory, and all finely carved with designs of curious invention;
the dresses of the figures were all made of precious stones cunningly
let in, and they sparkled with a vivid lustre, like so many lamps. Then
it had a rim, stand, and handle of massive gold exquisitely chased,
and adorned with rows of pearls and diamonds.
"Kind Lady Purrer," exclaimed the prince, "you are right, there is
no doubt of my success! But how can I ever sufficiently thank you
for what you have done for me? for I owe all to you."
"And a little to your own discernment too," said the Cat, archly. "And
now, always look as much alive and as bright as you do now, and you
will see people will think better of you."
"But when shall I see you again, most sweet counsellor? May I come
back and see you again?" pleaded the prince, and he tried to stroke
her sleek fur as she rubbed her soft grey head, purring, against
his boots. The stroking, however, she would by no means allow, but
springing again on to the top of the cabinet, she said,--
"Oh, yes; it will not be long before you will have to come back to me,
I know. But go, now; you have spent more time here than you think,
and you have only just enough left to get back within the year."
The prince turned to obey her; and the Cat jumped down, and ran by his
side, purring. When he got out into the grounds again, she followed
him, climbing from tree to tree; and when he came to the boundary-wall
she ran all along on the coping. But here at last they had to part,
to her great regret, and for many a lonely mile he still heard her
low and plaintive mew.
It was true, he must have spent more time in her pleasant company than
he had thought, for when he reached home he found the day of trial
had arrived; the streets were deserted, and all the people gathered
in the palace to see the drinking-horns his brothers had brought,
and talking loudly of their magnificence. He passed through their
midst without being recognized, for the people knew him so little;
and thus he heard them speak of his younger brothers:--
"What bright faces they have! and what a merry laugh! it does the
heart good to hear them," said one.
"I wonder how the kingdom will be divided, and which half will be to
which of them," said another.
"For my part, I don't care to the lot of which I fall, for both are
excellent good fellows," replied a third.
And thus they had clearly settled in their own mind that his brothers
had carried the day, and they didn't even trouble themselves to
think what he would bring, or whether he would come back at all. It
was the same thing all the way along. The words were varied, but the
same idea prevailed every where, that the younger brothers had made
good their claim; there was no question at all of the eldest. The
prince's face was growing moody again; but just then one good woman,
wiping the soap-suds from her hands as she turned from her washing at
the river to join the throng, exclaimed, as she heard some neighbours
talking thus, "Hoity toity! it's all very well with you and your
laughing princes--a grave one for me, say I! Laughing may lead a man
to throw away his money, but it won't teach him to feed the poor,
or govern a kingdom. Wait till the Grave Prince comes back! I'll
warrant he'll bring the bravest drinking-horn!"
A chorus of mocking laughter greeted her defence of him.
"He bring the bravest drinking-horn!" said one.
"Don't believe he knows what a drinking-horn is for--or drink
either!" said another.
"No; his brothers understand that best, at all events. I like a man
who can drink his glass."
"And I like one who doesn't drink it, whether he can or not; but
keeps his head clear for his business," said the good wife who had
defended him before.
And as there were a good many who were too fond of the bottle in the
crowd, the laugh raised at him was turned against them.
He had one defender, then, in all that mass of people, but all the
rest judged him incapable, and without trial! He was too disheartened,
to make his way into the great hall where the success of his brothers
was being proclaimed, but instead trod sadly and secretly up to his
mother's chamber.
The queen was too distressed at the absence of her favourite son to
take part in the jocular scene below, and was seated, full of anxiety,
at her window, watching.
"What do you here, my son?" she exclaimed, when he entered; "you have
but one short half-hour more, and the time will be expired. The sun is
already gone down, and the time once past, whatever you have brought,
it will avail you not! Haste, my son, to the council-hall!"
"It is useless, mother; all are against me!" cried the prince; and
he laid the beautiful flagon on the table, and sank upon a chair.
In the mean time it had grown dark, but the queen, impelled by her
curiosity to know what success her son had had, pulled off the wrapper
that enclosed the drinking-horn, and instantly the apartment was
brilliantly lighted by the light of the precious stones with which
it was studded!
"My son, this is a priceless work! This is worth a kingdom! Nothing
your brothers can have brought can compare with this--haste, then,
my son!" and she led him along.
It was dark in the council-hall too; but when the queen had dragged
her son up to the throne where the king sat, she uncovered the flagon,
and the sparkling stones sent their radiance into every part.
Then there was one shout of praise. The drinking-horns of the younger
brothers, which had anon been so highly extolled, were no more thought
of, and every one owned that the Grave Prince had won the trial.
The king declared it was too late for any more business that night,
the proclamation of the new sovereign would be made the next morning;
and in the meantime they all retired to rest, the Grave Prince with
some new sensations of satisfaction and hope, and the queen assured
of the triumph of her son.
But in the silent night, when all were wrapt in slumber, and the king
could not sleep for the anxiety and perplexity which beset him as to
his successor, the two young brothers came to him and complained that
they had been circumvented. The Grave Prince had always shown himself
so gloomy and unenergetic, it was impossible they could conceive he
was going to distinguish himself, so they had taken no trouble to
beat him; but if their father would but allow another trial, they
would undertake he should not have the advantage of them again.
So the next day, instead of proclaiming the new sovereign, the king
announced that he had determined there should be a fresh trial
of skill; and whichever of the princes should bring him the best
hunting-whip, that day year, should have the crown.
The princes set off next day on their travels once more, the eldest
son of course directing his towards the castle of the Beneficent Cat.
This time he had not to traverse a file of deserted halls before
meeting her; she sat looking out for him on the coping of the wall
where he had left her mewing so piteously when he last parted from her.
"I told you it would not be long before you would have to come back
to me," she said, as he approached. "What can I do for you this time?"
"My brothers are discontented at being beaten with your beautiful
beaker," replied the prince, gallantly, "and they have demanded another
trial: this time my father sends us in quest of a hunting-whip."
"A hunting-whip?" echoed the Cat; "that is lucky, for I can suit you
with one neither they nor any one else on this earth can surpass!" and
she frisked merrily along the path before him till they came to the
stables; then she took him into a room where all manner of saddles,
and horse-gear, and hunting-horns were stored. But on a high ledge,
at the very top of the room, was a dusty hunting-whip of the most
unpretending appearance. With one of her bold springs she reached
the ledge, and jumped down again with this whip in her mouth.
"It is not much to look at, I own," she said, as she observed the
perplexed look with which the prince surveyed the present; "but
its excellent qualities are its recommendation. You have but to
crack this whip, and your horse will take any thing you put him at,
be it a river half a mile wide, or a tree fifty feet high. There
are plenty of horses in the stable, saddle any of them you like,
and make experience of it for yourself."
The prince did as she bid him; and at sound of the enchanted whip
his mount leapt with equal ease over hills and valleys.
"This is a whip indeed!" exclaimed the prince, his face flushed with
the unwonted exercise, and his heart beating high at the idea of
being the bearer of such a prize.
"Ah, that's how I like to see you!" said the friendly puss; "I like
to see you like that. Now you are handsome indeed!" and she scampered
away, as if coyly ashamed of what she had said.
It was not long before she returned; and then she invited the prince
into the next room, where an elegant dinner was laid out, of which
the Cat did the honours very demurely. A high divan was arranged
at the top of the table, on which she reclined, and ate and lapped
alternately out of the plates ready before her, while invisible
attendants served the viands and filled the glasses.
When they had finished their meal, they went out to repose in the
flowery bowers; and when the heat of the day was past, the Beneficent
Cat reminded her guest that he must be thinking of going home, if he
would not that his brothers should supplant him.
"Must I go so soon, sweet Lady Purrer?" replied the prince. "I know
not how to part from you; it seems I should be happy if I were always
with you. I have never felt so happy any where before!"
"You are very gallant, prince," responded the Cat, "and you have no
idea how well it becomes you to look as you do now; but the affairs
of your kingdom must be your first thought. You must first secure
your succession--and then we must look out for the nice little wife
we talked of last time."
"Ah," sighed the Grave Prince, "don't talk of that--that is not
for me! No one beautiful enough for me to care about will ever care
for me!"
"Not if you look desponding and gloomy, like that," replied the
Cat. "Do you know, you look quite like another being when you look
so gloomy; and yet you can be so handsome when you look bright and
hopeful! But now," she proceeded, laying her soft paw on his arm
to arrest the futile justification which rose to his lips, "before
you go, I have something very important to tell you. You will now go
back, and with the hunting-whip I have given you, you are safe to win
the trial which is to establish your right to the kingdom. But there
will be yet another trial exacted of you, and you will have to come
back again to me. What you are to do then, I must tell you now, for
it requires great prudence and courage, and one principal thing is,
that you don't say a word to me all the time. Can you promise that?"
"Well, that is hard indeed," said the prince; "but still, if you
command it, I think I can promise to obey, for the sake of pleasing
you."
"Then the next thing is harder. Do you think you can do whatever
I command?"
"Oh yes, I am sure I can promise that!" replied the prince, warmly.
"Mind, whatever I command, then--however hard, or however dreadful
it may be?"
"Yes, any thing--however hard, or however dreadful!"
"But will you swear it?"
"I see you doubt my courage," said the prince, half offended. "You
take me for a fool, like the rest. But no wonder; I know I look like
a fool!"
"Now don't look gloomy again! you were so handsome just now when
you said so firmly you would do 'any thing.' Will you gratify me
by swearing?"
"You doubt my courage."
"No; I don't doubt your courage. But I know how terrible a thing I
have to command you; and I know how many others have failed before
you. Now will you not swear, but to please me?"
"Yes; I swear," said the prince, energetically, "to do whatever it
may be that you tell me to do."
"Now, remember, you have undertaken it solemnly. This is what you
must do. When you come in, you will find me sitting on the kitchen
stove; you must then seize me by my two hind-paws, and dash me upon
the hearthstone till there is nothing left of me in your hands,
but the fur!"
"Oh dear! I can never do that!" exclaimed the prince, in great
embarrassment.
"But you have sworn to do whatever I told you!" replied the Cat.
"Well, but I thought you were going to order me to do something
rational, something noble and manly, requiring courage and
strength--not a horrible act like this."
"If it is the thing that has to be done, it does not matter what it
is. Besides, it does require courage, great courage; and that is why
I would not tell you first what it was, because others have failed
when they knew what it was."
"And you expect me to have less feeling and affection for you than
they?"
"No; but I expect more sense and judgment of you. I expect you to
understand and believe that if I say it has to be done, it is really
for the best, and that you will trust to me that it is right. And I
expect that you will respect your promise, which was made without
limit or exception. But now, go; you have no time to lose, if you
want to reach home with the hunting-whip in time for the trial."
He rose to leave; and she followed him down the path, purring by
his side. And after she had taken leave of him at the boundary-wall,
he heard her mewing sad adieus as he went on for many a weary mile.
When the prince reached the council-hall, he found, as before,
that his brothers were there first, and that every one seemed to
have decided that they had won the day--in fact no one showed any
curiosity to know what he would bring. As he had beaten them by his
lustrous jewels before, they had fancied he would bring something of
the same sort again; so, to conquer him on his own ground, they had
sought out and found two handles of hunting-whips mounted with jewels
as sparkling as those of his drinking-horn. When they saw him come
in with the shabby old whip the Beneficent Cat had given him, they
laughed outright in his face; and the king, in a fit of indignation,
ordered him to leave the hall for venturing to insult him by bringing
such a present. Some laughed him to scorn, and some abused him; but no
one would listen to a word he had to say. At last the tumult was so
great that it reached the queen's ears; and when she had learnt what
was the matter, she insisted that he should have a hearing allowed
him. When silence had been proclaimed the Grave Prince said,--
"It is true, my whip is not so splendid as that of my brothers, but
jewels are out of place on a hunting-whip, it seems to me; the handle
is wanted to be smooth, so that the hand may take a firm grip of it,
rather than to be covered with those points and unevennesses. The
merit of my whip is not in the handle, it is in the lash, which has
such excellent qualities, that you have but to crack it, and your
horse will immediately take you over any obstruction there may be in
your way--be it a house or a mountain, or what you will. If you will
allow me, I will give you proof of its powers."
Then they all adjourned to the terrace in front of the council-hall,
where was a fine avenue of lofty cypresses; and the queen ordered
a horse to be brought round from the stables. The people had never
seen the prince on horseback before; and when they saw him looking so
gallant, and noble, and determined, they could not forbear cheering
him, till his younger brothers began to fear that his real worth
would soon be found out, and their malice exposed.
Then the prince cracked his whip--and away went the horse over the tops
of the high trees, seeming to scrape the clouds as he passed. All
the people were lost in admiration, no one had ever seen such a
sight before; and while they were wondering whether it was possible
he could have reached the ground in safety from such a height, there
was a murmur in the air, and they saw him coming back again over the
tree-tops. With no more apparent effort than if he had merely taken
a hedge, he came softly to the ground; and then, kneeling gracefully
before his father on one knee, without a word of boasting or reproach,
he laid the clever whip at his feet.
The king raised him up, and said, aloud to the people, none could deny
that it was this whip that had won the trial, but that as it was now
late, he must leave the ceremony of proclaiming his successor till
the morrow.
All went home for the night, and the old king also went to bed; but he
could not sleep for anxiety, thinking of the anger and dissatisfaction
of his younger sons. And presently, in the silent hour, they came
to him, and said that he must allow them another trial; that it
was impossible they could conceive he meant them to bring him a
fantastical whip of that sort, or of course they would have brought
one which could do much better things. They thought it was the beauty
of the workmanship they had to look to, and so they had provided for
nothing else. They urged their suit so persistently, that the king,
who was now very old and weak, agreed to let them have their way.
Accordingly, next morning he had it proclaimed that the three princes
were to make one trial more; and that whichever brought back the most
beautiful and virtuous princess for his wife should have the crown.
The three princes set out again early the next morning; the two younger
ones providing themselves with jewels and riches, and many precious
things for presents; the eldest taking nothing, but walking off alone
towards the enchanted castle with a heavy heart. "It is all up with
me now," he said to himself, "after all! Why couldn't my father have
been satisfied when I had beaten them twice? Now I have to kill the
Beneficent Cat--the only being that ever assisted me; and then I shall
have no one to help me at all! They will come back with two beautiful
princesses, and I shall come back looking like a fool, because no
princess will ever come with me--and they will take my kingdom, and
laugh at me into the bargain! If it was not for my mother, I would
never come back at all; but it would break her heart if I stayed away,
and she is the only one of them who understands me and cares for me."
As he got nearer the castle, he grew more and more sad. "Why did she
make me swear? If it hadn't been for that, I could still have escaped
doing it; but now I cannot break my oath;" and he trudged on.
The gardens looked more lovely than ever. The scent of the flowers
seemed sweeter, and the melody of the birds more soothing. All was
full of harmony--and he who had never harmed a fly must cruelly use
the soft and beautiful Cat who had so befriended him!
He passed through the apartments where puss had purred round him so
happily--the dining-room where they had had their pleasant repast
together--the boudoir where she had given him such wise counsel.
At last he came to the kitchen; and there, sure enough, was the Cat
cosily curled round, her soft grey head buried in her long grey fur.
An energy and daring he had never known before seemed suddenly to
possess him. He took care not to speak, for she had particularly
recommended silence; but, approaching her on tiptoe, seized her rapidly
by her hind-paws before she had time to wake from her pleasant slumber,
and dashed her several times upon the hearth, scarcely knowing what
he did in his horror, till he perceived that he had nothing left in
his hand but the soft, limp, grey fur.
He sank upon the ground in tears, and commenced laying it out tenderly
before him, when he was woken from his reverie by a mellow ringing
laugh, which made him look up--and there before him stood the most
beautiful, fairy-like princess that ever was seen on this earth!
"Well done, kind prince! you have nobly kept your word. And see what
I have gained thereby--instead of that grey fur, I now have a form
which will perhaps make me meet to fulfil the condition your father
has imposed on you for obtaining your throne!"
Her voice, and the glance of her soft eyes, seemed quite familiar
to him--it was the voice which had first inspired him with hope and
enterprise, and the mild light which had beamed on him when he said he
could be happy to be always near her in her bower. How much more now,
when she appeared in such matchless guise!
He remained kneeling at her feet, and asked her if it was indeed true
that she could love him and be with him always as his wife.
"Nay," she replied, raising him up; "it is I who ought to be
astonished. I have nothing to refuse, for I owe you all; and as,
but for you, I should still be nothing but a poor grey Cat, I belong
to you, and am absolutely yours. It is I who have to be astonished,
and to ask you if it is possible you who have known me as a Cat can
really love me and regard me as worthy to be indeed your wife."
"You are mocking me again, I see," he replied; "but you do not
really think me so insensible as not to appreciate your beauty,
and the prudence and generosity of which you have given me such
abundant proof? No; if you will come with me, I have no fear but that
I shall win the trial this time beyond all possibility of demanding
another." He spoke warmly, and his face beamed with joy. The princess
was leaning on his arm, and looked up in his face as he spoke.
"Ah, now you do look!--No, I suppose I mustn't say it now I have
no longer my cat-disguise to hide my blushes," she said, archly;
and they passed on into the reception-hall.
The attendants were no longer invisible. Together with their mistress
they had received their forms and original life; and the corridors
and apartments were filled with her people bustling to serve her. A
banquet was prepared in the dining-hall; and when they had partaken
of it, and had regaled themselves in the bower with happy talk, the
princess reminded the prince--now no longer grave--that it was time
for them to be going back to his father. A great train of carriages
and horses were brought round, with mounted guards and running-footmen,
and all the retinue which became a noble princess.
The princess was carried in a litter by six men in embroidered
liveries, and her ladies with her; and the prince rode on horseback,
close by her side.
This time, though it was near the close of the last day, his brothers
had not appeared when he reached the council-hall. The king and the
queen received the Beneficent Princess with smiles and admiration,
and all the people praised her beauty; and the queen said,--
"There is no fear, my son, that your brothers can demand another
trial this time."
Before she had done speaking, a messenger was hastily ushered into
the hall, covered with dust and stains of travel. He came from the
two younger princes, and had a sorrowful tale to tell.
They had striven to obtain the hands of the princesses of the
neighbouring kingdom; but the king was a prudent sovereign, and
discerned their envious, selfish character. When they found he repulsed
their advances, they had endeavoured to carry off the princesses by
force; but the king had surprised them in the midst of their design,
and had had them shut up as midnight robbers.
The old king was in great distress when he heard the news, for
his sons had manifestly been taken in the midst of wrong-doing,
and he could not defend their acts nor avenge their shame. But the
eldest son took on himself the mission of pacifying the neighbouring
sovereign and delivering his brothers. Having accomplished which, they
were fain to acknowledge that he was not only victor in the trials,
but their deliverer also; and they swore to maintain peace with him,
and obey him as his faithful subjects.
So the old king proclaimed the Grave Prince for his successor, and
married him to the Beneficent Princess, amid great rejoicing of all
the people; and the queen had the happiness of seeing her eldest son
acknowledged as the most prudent prince, and the ruler of the people,
and gifted with a beautiful and devoted wife.
KLEIN-ELSE.
The Passeier-Thal, which at the beginning of the present century
sent Hofer and his famous band of peasant heroes to the defence of
the fatherland, was in ancient times often involved in the wrangles
between its rulers and those of Bavaria. The men of the Passeier-Thal
were no less heroes then than now, but there were heroes in Bavaria
too, so that the success was as often on one side as the other.
Klein-Else [63] was the daughter of a bold baron whose castle was,
so to speak, one of the outposts of the valley; and as he had thus
more often than others to bear the brunt of the feud, his strength
became gradually diminished, and it was only by leaguing himself with
his neighbours that he was enabled to repel the frequent inroads of a
turbulent knight who had established himself on the other side of the
old frontier, but who cultivated a strong passion for annexation. The
Passeier-Thal baron did his best to strengthen his defences and
keep up a watchful look-out; and the moment his scouts perceived
the enemy advancing, their orders were not only to bring word of the
danger to their master, but to hasten at once to the other castles
of the surrounding heights, and summon their owners to his support;
and then the whole valley immediately bristled with valiant defenders
of their country.
But inasmuch as his adversary was reckless and determined, and much
better provided with men and means, he succeeded in laying his plans
so well at last, that he eluded all the vigilance of the baron's
scattered handful of look-out men, and, bursting in upon his domain
by surprise, carried all his defences, laid waste every thing before
him, and marched upon the castle itself.
The bold baron swore he would not remain to be killed like a reptile
in its hole, but sallied out with the few retainers who remained to
him, to sell his life and his possessions as dearly as he might. With
desperate courage he dealt the deadliest blows around which had
been paid out that day. But it was all in vain. Overcome by superior
numbers, he was brought back but a few hours later in piteous plight,
mortally wounded.
Klein-Else bent over her father with despairing cries; and her tears
fell as fast as the blood from the deep wounds she tried in vain
to staunch.
"Leave the bandage, Klein-Else, it boots not," said the baron,
in tones so slow and faint that she could only catch his words by
putting her ear to his lips; and, as she did so, his cold breath
filled her with horror.
"It boots not to staunch the blood, Klein-Else; my life is spent. But
as you have ever obeyed me, listen now to my word. The enemy is at
the door; you have but time to escape falling into his hands. Take
this key--it opens a gate of which no one knows the secret. Count
the tenth buttress in the wall, and where the ivy grows thickest,
there, behind it, feel for the lock and open it. Then creep beneath;
and, once on the other side, replace the branches, that no one may
see they have been disturbed. You will see before you three paths:
one leads down into the smiling plain, where you might think to
find refuge in the houses of our people; but another destiny is for
you. The second leads upwards to the thick pine forest, where you
might think to lie concealed till our friends have time to come and
rout out this vile usurper; but another destiny is for you. Take the
path straight before you, that winds round the mountain; though it
is open and exposed to view, fear not, for it leads to--to----"
And here his voice failed, so that she could no more make out what
he said; and though he continued to exert himself to complete his
directions, it was vain that she attempted to distinguish them. His
power of articulation was gone.
Klein-Else threw herself on his cold body, and clung to it with all
her might. But he who had been her guide and guardian, her will,
till now, was powerless and stark; and for all her beseeching he
could not answer.
The chaplain came and raised her up, and they carried the body to
the sanctuary; but Klein-Else, paralyzed with sadness and despair,
stood and gazed after it as though she knew not where she was.
Suddenly wild shouts broke on her ear, and the sound of many feet,
and the tumult of the servants and men-at-arms bidding her fly,
for the enemy had come.
"Fly, for the enemy is here!" The words recalled her father's counsel,
and mechanically she clasped the key, his last legacy. Scarcely taking
time to change her embroidered garments for a peasant's attire, she
crept along under the wall, counting ten buttresses, with a beating
heart. After the tenth, she put her hand through the thick ivy, and
felt, as her father had foretold, the iron bosses of the lock. It
required all her strength to turn the key; but this accomplished,
there was safety and rest behind the ivy's faithful veil.
It was but just in time; the rough soldiers were close behind.
"Ha! who went there?" she heard a hoarse voice say, as she noiselessly
closed the door. "Saw you not the ivy move? Press through and see
who passed."
"It was but a frightened hare--I saw it run," said another, with a
less terrible voice.
"Nothing taller ever passed that branch," said another; and the
speakers passed out of hearing.
There lay the three paths: the one straight on before--but so open,
so exposed, any one who happened to be passing for miles round might
have seen and pursued her, while either of the others offered instant
cover and security. Klein-Else was sorely tempted to try one of them.
"If I had heard all his instructions," she reasoned, "it would have
been different: I would then have done all he told me, whithersoever
it might have led; but now I know not what he meant. I may go a little
way along this path--and then what shall I do? Maybe, I shall fall
into a greater danger than that from which he would have saved me!"
And she turned to seek the shelter of the friendly cottages in the
valley beneath. But the words seemed to live in the air around her,--
"Another destiny is for you!"
Trembling and confused, she would have plunged into the hiding-place
of the pine-forest above; but the wind that moaned through their
lofty branches seemed charged with the words,--
"Another destiny is for you!"
She was thus impelled forward into the open path; and, creeping close
to the mountain-side, she now pursued her way along it. It was with
no small relief that she noticed the sun was nearly sinking behind
the opposite heights, so that soon she might hope to be safe from
the gaze of men.
And yet, as darkness fell around, it became but the source of other
fears. And the sense of her loneliness and abandonment took away her
courage to proceed any farther.
She leant against the rock for support, and her tears fell fast and
warm upon its stony side--piteously enough, you might have thought,
to move and melt it.
And so it was! for see! the hard rock yielded and made way before the
noble form of a knight in armour, who said, with compassionate voice,--
"Maiden, wherefore these tears?"
"Because my father is dead, and his enemies have taken his castle,
and I have no shelter and nothing to eat!" sobbed Klein-Else.
"If that is all," answered the noble knight, "it is easily made
straight." And with that he turned to the rock, and said,--
"Open, hoary rock!"
And the hoary rock opened, and disclosed a treasure of every imaginable
kind of riches stored around--jewels and coin, and shining armour,
and dazzling dresses.
"All this is yours, Klein-Else," said the knight; "you have but to take
what you will, when you will. It will never grow less. You have only
to say, 'Open, hoary rock!' and these treasures will always appear
at your bidding. Dispose of them as you like; only make a good use
of them, for on that depends all your future happiness. I will come
and see you again in seven years, and I shall see what use you have
made of my gift; but you must remember my name, or woe will be to
you." So he whispered his name in her ear, and disappeared.
Klein-Else was so dazzled and startled that she hardly knew what to
think, or whether what had happened was a dream or reality. To make
sure, she said to the rock, "Open, hoary rock!" and the rock opened at
her bidding as quickly as at the knight's, and disclosed its glittering
treasure. But it was still hard to decide all at once what to take of
it; and knowing that it was in a secure store-house, and that it was
dangerous to burden herself with much riches when travelling alone
in the dark night, she only took a few pieces of money--enough to
pay for food and lodging--and passed on with a lightened heart. The
rock closed up as she went farther--but she took a note of the spot,
so that she might be sure to know it again; and then made for the
lights which appeared with friendly radiance at no great distance
through the trees which now fringed the road, repeating the name of the
knight to herself, as she went along, that she might never forget it.
Klein-Else hasted on, but was rather dismayed to find that the lights
were the lights of a great castle where her money would be of no
use. She could not ask for a lodging and supper for money there, and
there was no other habitation near. So she put by her money again,
and, with the humility befitting her wayworn aspect and lowly attire,
begged the great man's servants to give her some poor employment by
which she might earn a place among them.
"What can a little, dirty, ragged girl like you do?" said the cook,
who was just occupied in fixing the spit through a young chamois that
looked so succulent and tender, one as hungry as Klein-Else might
have eaten it as it was.
"I can do whatever you please to tell me," answered Klein-Else,
timidly.
"A proper answer," replied the cook. "Let's see if you can watch
the poultry-house, then. You must be up by daybreak and go late
to bed, and lie in the straw over the poultry-loft, and keep half
awake all night to scare away the foxes, if any come; and if one
smallest chicken is lost, woe betide you! you will be whipped and
sent away. Here is a piece of dry bread for your supper. Now go,
and don't stand idling about."
Klein-Else was so hungry that she gladly took the piece of dry black
bread, and went to try to sleep on the straw in the poultry-loft. She
had to get up at daybreak, when the cock crew; and she had to keep
her eye on the brood all day; and late at night she had a piece of
dry black bread for supper, and was sent to sleep in the straw of
the poultry-loft. Her only pastime was to recall the memory of her
treasure in the rock, and repeat over and over again the knight's name,
that she might be sure never to forget it.
"But of what use is all my fine treasure," she mused, "if I am never
to be any thing but a wretched Hennenpfoesl [64]? And what can I
do? if I come out with handfuls of gold and fine clothes, they will
take me for a thief or a witch, and I shall be worse off than now;
and if I show them the treasure, who knows but they will take it from
me? The knight said my happiness depended on the use I made of it,
yet I can make no use of it!"
So she sat and counted the hens and chickens, and repeated the
knight's name, and ate her dry black bread, and slept in the straw
in the poultry-loft.
At last Sunday came, and the glad church bells rang merrily, flinging
their joyous notes all abroad; and the servants of the castle put
on their best clothes to go to church. But how could Klein-Else be
seen among them, all in their snow-white linen and bright-
ribbons--Klein-Else, the Hennenpfoesl, with her poor rags?
"Now, at last, I can use my treasury," she said to herself; "I can
at least get some of the pretty clothes that hang there, and go to
church." So she washed herself in the mountain-torrent, and braided
her dishevelled hair in massive golden braids, and crept round to
the rock, and bid it open, saying,--
"Open, hoary rock!"
Of all the treasures it instantly disclosed, she saw none but one
beautiful garment all woven out of sunbeams and glittering with jewels
of morning dew. Having put this on, and once more looking like a
baron's daughter, she made haste to reach the church.
The holy office had already begun, and the church was crowded right
out into the porch. But when the people saw such a dazzling sight,
they all made way for the lady in the shining apparel, none dreaming of
Klein-Else. Now the only part of the church where there was any room
was at the baron's bench. For he was a young lord, and had neither
mother, sister, nor wife; and all the places reserved for his family
were vacant. Klein-Else, moving on till she could find where to kneel,
had thus to come and kneel by him.
The young baron was as much dazzled at the sight as Klein-Else
herself had been at the treasures in the rock, and at every pause in
the service he could do nothing but fix his gaze on her. As soon as
it was over, however, Klein-Else glided out softly, and hasting back
to the rock, hung the sunbeam-dress up again; and once more assuming
her rags, hid herself in the poultry-loft, almost frightened at what
she had done.
All the next week she had new subjects of thought. She felt sure
the young baron had looked at her and admired her; and wasn't it
more meet that she, a baron's daughter, should be kneeling by the
side of the young baron than sleeping in the poultry-loft, a mere
Hennenpfoesl? Ah, if that came true--if the young baron married her;
then she would have some one to tell her good fortune to--some one
to defend her treasure. Then she could make the good use of it the
knight had manifestly intended. She could wipe away the tears of all
those who went without shelter, as she had once; every desolate orphan
who had none to defend her; every poor Hennenpfoesl, the drudge of the
menials. "How strange," she said to herself, "there should be people
blessed with friends, and riches, and enjoyments, who live full of
their own happiness, and who have no thought for the forsaken and the
outcast! She would never be like them, not she! her happiness should
be in making others happy."
But, in the meantime, was she sure the baron had looked at her
otherwise than out of curiosity? Was he really interested in her? and
if he was, would he continue to care for her when he found she was
only a Hennenpfoesl? She must put him to the test; and she sat and
thought how to arrange this. This was subject enough for thought;
and this week was at an end only too soon.
The next Sunday came; and when the church bells rang, Klein-Else ran
to her rock, took out of her store this time a garment woven out of
moonbeams, and having arranged her luxuriant hair in massive tresses,
once more proceeded to the church. But with all the haste she had
made, she could not arrive before the holy office had begun, and the
church was once more full. The people fell back again, in awe of her
shining garments, and made way for her to kneel beside the baron,
who could scarcely suppress a gesture of delight at beholding her
once again. Nor did his joy escape Klein-Else's observation; and many
a blushing glance they exchanged.
"What a noble cavalier!" thought Klein-Else; "and just such a one as
my father always told me my husband should be."
"What a lovely maiden!" mused the young baron; "where can she have
sprung from? Is she of earth or heaven?"
All that last week, while Klein-Else was thinking of him, he had
been thinking still more of her; and had ordered his waiting-men to
surround her as she came out of church, and beg her to come to him
at the castle. But Klein-Else had no idea of suffering herself to
be so easy a prize; so she fled so fast the baron's men could hardly
approach her. And when at last she found they were gaining upon her,
and that her fleet step availed her not, she threw down the pieces of
money which she took the first night from the rock; and while they
stopped to pick them up, pursued her way unperceived, and let the
rock close on her till they had lost the trace. Then, assuming her
poor rags once more, she returned silently to her poultry-loft.
Her thoughts had food enough now; but it was less with the poor
orphans she was to console, than with the young baron, and how to
test his love, that they were occupied.
Next Sunday she chose a garment blue like the sky, and all sparkling,
as with living stars. She presented herself at the church, and
found herself again placed beside the young baron. At the end of the
service she went out quickly, as before, only this time he contrived,
as she rose to leave, to seize her hand, and slip a gold ring on
her finger. Nevertheless, Klein-Else slipped out through the midst
of the congregation, and though the serving-men had had stringent
orders to follow her, she had prudently provided herself with gold
pieces enough to disperse the whole lot of them while she escaped.
The young baron sat alone in his castle, as he had sat this fortnight
past, taking no notice of any one, but as if his whole soul was
wrapt up in the fair apparition, and he was in despair, since her
hiding-place could not be traced. He sat nursing his grief, and could
neither be distracted from it, nor comforted. His friends sent for the
most famous physicians of the country to attend him, but none of them
could do any thing for his case; and daily he grew paler and gloomier,
and none could help him. At last the Graefin Jaufenstein, his aunt,
came and insisted that some amusement must be found to divert him;
but the young baron refused every proposal, till at last she begged
him to give a great banquet, to which every one from far and near
should be invited, every kind of game and every kind of costly
diet should be afforded, and nothing spared to make it the most
magnificent banquet ever given. To the great surprise and delight
of all, he consented to this; but it was because it occurred to him
that inviting the whole country, the chances were that the beautiful
maiden of his choice, who yet hid herself so persistently from him,
might once more mysteriously appear before him too: so he gave his
aunt the Countess Jaufenstein free leave to give what orders she
liked, and go to what expense she liked, only providing that she
should have the invitations publicly published, so that there might
be every chance of their reaching the ears of the mysterious maiden.
At last the day of the banquet came, and there was a running hither
and thither in the baron's castle, with the preparations, such as can
be better imagined than described. The guests swarmed in the halls,
and the servants in the kitchen; and Klein-Else, creeping up from her
poultry-loft, could hardly make her way up to the fire where the cook
was preparing all manner of deliciously scented dishes.
"I don't know what ails the things!" cried the cook; "these pancakes
are the only thing the baron will eat, and, as fate will have
it, I cannot turn one of them to-night! Three and thirty years
I have made pancakes in this castle, and never did I fail before
to-night--to-night, when it is most important of all!" and she poured
another into the pan. But as she did so, with a hand trembling with
anxiety, the oil ran over the side of the pan, and the great heat
of the stove set it on fire, so that a great flame curled over the
pancake--and there was nothing left of it but a black, misshapen mass.
"Oh, dear! oh, dear! what shall I do?" cried the cook; "there is not
one of the whole lot fit to send up, and if this dish is not the best,
I had just as lief I had prepared no dish at all!"
"May I have a try, friend cook?" said Klein-Else, coaxingly.
"You, indeed!" screamed the cook, indignation and envy added to her
former despair; "and a little, dirty, ragged, misbegotten starveling--a
vagabond--a Hennenpfoesl like you, who never saw a kitchen, or a stove,
or a frying-pan, or any thing else! to suppose that you can turn a
pancake, when I, who have turned pancakes in this castle for three and
thirty years, have failed! A likely matter indeed! What is the world
coming to? Begone, with your impudence, and mind your hens! Ah! now
I think of it, I believe it is you that have bewitched the eggs, and
that's why the pancakes won't turn! Begone, I say, out of my kitchen,
and out of the poultry-house too--I'll have no more of your tricks
with my eggs!" and she turned, with a menacing gesture at Klein-Else,
to try her luck once more.
But at the sight of the black mass in the frying-pan, she grew fairly
discouraged, and throwing herself down in a chair, wrapt her face in
her apron, and wept like a child.
Meantime Klein-Else advanced with light step to the stove, took up
the frying-pan, and cleaned it out in a trice, then poured fresh oil
into it, and held it over the stove till it boiled; then, while it
spluttered cheerily, she deftly poured in the batter, gliding into it
the ring which the baron had stealthily put on her hand at church,
and along with it, one with a magnificent diamond, which she had
taken from her treasury in the rock.
The boiling oil danced and chirped merrily round the cake, the batter
rose as batter never rose before; and when Klein-Else shifted it
lightly on to the dish, it wore a bright, golden hue, matched only
by her own radiant hair.
The cook, waking from her stupor, was in a transport of delight at
beholding the effect of her skill, and sent the dish at once to the
baron's table, while Klein-Else took her place in an out-of-the-way
corner to hear what should befall.
Nor had she long to wait. The dish had not been gone ten minutes,
when the baron's body-servant came solemnly into the kitchen, with
the announcement that the baron demanded the immediate attendance
of the cook. "It's because I kept him waiting for the pancakes, and
because the one of that little hussey's making is not so good as those
I have made for him all his life, and his father before him;" and,
all trembling and afraid, she rose to follow his messenger. Espying
Klein-Else watching anxiously behind a pillar as she passed along,
she could not forbear calling out to her, "Ah, wretched child, it is
you have got me into this scrape! But you shall pay for it! Why did I
let you touch the frying-pan! Why did I let you enter the castle! You
had better not come under my sight any more, or I'll soon show you
where the builder made the hole in the wall [65]!" and she dragged
herself along slowly, in great fear of the apprehended displeasure
of the baron, but comforting herself with the determination to let
him know the whole fault lay with the Hennenpfoesl.
Great was her surprise, however, to find that it was with no intention
of chiding that the baron had summoned her. On the contrary, the
gloomy cloud his brow had lately worn had disappeared; he not only
looked gay and joyous as of old, but a special radiance of pleasurable
expectation lit up his countenance.
"Why, cook," he said, "you have made me good pancakes all my life,
but never one like this! Now tell me honestly who made this one?"
"Nay, but if it is so, I may as well have the credit of it," thought
the cook; "and, after all, I did make the batter, and that's the
chief part of the work."
"Oh, I made it myself, baron, upon my soul! no one but myself makes
any thing for the high table."
The baron's countenance fell. He began to look gloomy and disappointed
once more--was the clue to escape him after all? He roused himself
again, as with one flash of hope.
"Did no one help you to make it?"
("If I tell that she had any part in it, it is obvious, from the tone
he takes, he will give the whole merit to her. No, I'll not mention
her; and besides, she didn't help me to make it.")
"Oh, baron, it don't want two people to make a pancake! I've always
made pancakes for this castle these three and thirty years without
help;" and she tried to talk as if she felt hurt, and thus bring the
conversation to an end.
The baron passed his hand roughly across his forehead, and stamped
his foot in despair.
Once more a hopeful thought flashed across his mind.
"These rings! tell me, how did they get into the pancake, if you made
it?" he exclaimed, in peremptory accents.
"Those rings? I never saw those rings before," stammered the cook,
beginning to get a little confused.
"And what did you mutter as you passed the Hennenpfoesl coming
along, about it's being all her fault, and making her suffer for
it?" interposed the body-servant.
"Ha! said she so?" cried the baron. "Speak, woman, what meant you by
those words? Beware, and speak the truth this time, for it is matter
of terrible consequence!"
"Who ever would have thought such a fuss would come of turning a
pancake!" thought the cook to herself; but she said out aloud, "Well,
it is true, the Hennenpfoesl did hold the frying-pan while it was on
the stove; I didn't know it was worth while to mention that. But what
could she have to do with the beautiful rings?"
"True," replied the servant, "that can have nothing to do with it,
as you say."
"Nay," replied the baron, "I'm not so clear of that. Let the
Hennenpfoesl, as you call her, be brought here, and let's see what
account she has to give of it."
"But it's impossible; she isn't even a servant of the house. She is
a little whining beggar brat, that I took in scarce three weeks ago
and put in the poultry-loft, to keep her from starving."
"Three weeks!" exclaimed the baron; "said you three weeks? Let her
be brought to me instantly."
"But she isn't fit to come into your presence; she's grimed with dirt,
and covered in rags."
"Reason not, but send her hither," said the baron, his energy returning
as his hopes kindled.
"If she is the maiden to whom I gave the ring, she is of no low birth:
there is some mystery which I must penetrate. If she were nothing
but a 'Hennenpfoesl,' whence could she have had this brilliant ring,
which puts mine to shame?" he mused within himself, as he waited
impatiently for the maiden of his dreams to appear.
Klein-Else, meantime, had made no doubt that since the baron had
sent for the cook, his wisdom would enable him to discover that she
must be sent for next, and had accordingly repaired to her treasury
in the rock, and had taken thence a resplendent attire. It was no
longer now the simple gifts of nature which furnished her wardrobe;
she was decked as became a baron's daughter, with all the resources of
the milliner and the jeweller's art. Cavaliers and ladies-in-waiting
walked beside her, and twenty pages dressed in pink and white satin,
with plumed bonnets, carried her train behind, while men in rich
liveries, bearing torches, ran by the side of the procession.
Graefin Jaufenstein was at the head of the hall welcoming the guests,
and doing the honours of the castle, to supply what the moody humour
of its lord left lacking in courtesy. But while she courtesied to
noble lords and ladies with queenly grace, and, with imperceptible
asides, at the same time gave directions that every one should have
his due place, and that every thing should proceed with the due order
of etiquette, it never for a moment escaped her practised eye that
something unusual was going on in the neighbourhood of the young
baron. That he should summon the cook to his presence, probably to
chide her justly for some breach of the rules of her art, if such
had befallen, was indeed no unreasonable distraction for the baron's
melancholy, and she hailed it as a token of returning interest in the
ordinary affairs of life, which had occupied him so little of late;
but when she heard him order the Hennenpfoesl to be brought there in
the midst of his guests, she thought it time to interfere--it became
a matter of eccentricity passing all bounds. Dexterously excusing her
momentary absence from her guests, she accordingly made her way up to
her nephew, preparing to wrap up her remonstrance in her most honeyed
language, so as better to convince without provoking him. Before she
could reach his chair, there was a movement of astonishment in the
vast assembly, and a cry of admiration, while the heralds proclaimed,--
"Place for the most noble baron's daughter!" And then, surrounded
by her shining crowd of attendants, and glittering in her jewelled
robes, Klein-Else made her way with modest, but at the same time
noble carriage towards the young baron.
The young baron recognized her the moment the tapestry was raised
for her to pass, and instantly went forth to meet her with courteous
gestures, and led her up to the seat next his own at the banquet.
The stately countess looked on a little perplexed, for the first time
in her life, but with admirable serenity and self-possession inquired
the name of the fair guest who did their poor banquet the honour of
attending it in so great state.
"I am the poor Hennenpfoesl, madame, whom your noble nephew has done
the honour to summon to his presence; and I hope you will not think
I disgrace his command," replied Klein-Else, with a reverence at once
lowly and full of accomplished dignity.
"The Hennenpfoesl!" repeated the countess, returning the salute
mechanically. "But surely there is some mistake--some----"
"Yes, dearest countess, some mystery there is," interposed her nephew;
"but we will not seek to penetrate it till it shall please the lady
herself to reveal it. Why she should have chosen to pass some time as
the Hennenpfoesl, I know not; but this is not the first time we have
met, and I am sufficiently satisfied of her grace and discretion to
know that for whatever reason she chose it, she chose aright. I have
further determined this very night to lay myself and my fortune at
her feet!"
Klein-Else started, with a little cry of satisfied expectation,
then modestly and looked down.
"But the lady will at least favour us with her name?" urged the
countess, but half satisfied. Klein-Else turned to her chamberlain
with dignity, and whispered an order; and then the chamberlain stood
forward and proclaimed aloud the names and titles of the deceased
baron of the Passeier-Thal, her father.
"Oh!" said the lady, in a tone of disparagement, "methinks his was
a fortune which could scarcely be united with that of my nephew!"
"Countess!" exclaimed the young baron, furious at the suggestion;
but before he could proceed the chamberlain once more intervened.
"There need be no difficulty on that score," he said; and he made a
sign to the attendants who were behind. They came up in brave order,
two and two, each pair bearing a casket in which was a thousand
crowns. "A thousand such caskets contain the dowry of the baron's
daughter; and she has priceless jewels without number."
"A million crowns!" echoed the whole assembly, in chorus; "was there
ever such a fortune known?"
The countess was absolutely speechless, and turned to participate in
the astonishment of her guests.
The young baron and Klein-Else, thus left to each other's conversation,
were not slow in confessing their mutual love.
"And now all our friends are gathered round us," he exclaimed, at last,
"what better time to proclaim our happiness? My friends! I present
you the fair lady who has consented to become my bride!"
There was a general sound of jubilation and praise. All gathered
round to felicitate the baron, and the minstrels sang the charms of
the bride.
The baron begged them all to stay with him ten days, to celebrate the
nuptials. And for ten days there was revelry and rapture, singing
and merry-making; and when at last the guests returned home, every
one carried back to his own neighbourhood the tale of the surpassing
beauty, riches, and grace of Klein-Else. Every body had been won by
her, there was no dissentient opinion; and even Graefin Jaufenstein
acknowledged that her nephew could not have made a nobler or better
choice.
When they were left alone, the days seemed hardly long enough to
tell their love. Never was there happiness equal to theirs. Before
the guests left, the baron had invited them all to come back every
year on the anniversary; and every year, as they gathered round,
they found them more and more wrapt in each other's love.
On the second anniversary they found that their happiness had been
increased by the birth of an heir; and the next year there was a little
daughter too, the delight of her parents. Year by year the children
grew in beauty, and grace, and intelligence, and others were added
to their numbers. And every one envied the unequalled happiness of
the baron and baroness.
Meantime the years were passing away, though Klein-Else had taken
no account of them. To her it was one continual round of enjoyment,
uncrossed by any care; each season had its own joys, and she revelled
in the fresh variety of each, but counted them not as they passed.
One day they sat together under a shady grove: the baron was weaving
a chaplet of roses, Klein-Else was fondling her latest-born upon her
knee; round them sported their little ones, bringing fresh baskets
of roses for the chaplet the baron was weaving for Klein-Else; while
Otto the heir, a noble boy who promised to reproduce his father's
stately figure and handsome lineaments, rejoiced them by his prowess
with his bow and arrow.
"How the time has sped, Klein-Else!" whispered the baron; "it seems
but yesterday that you first came and knelt beside me in your sunbeam
garment. Then, just as now, it was happiness to feel you beside me. I
knew not who was there, but as I heard the flutter of your drapery a
glow of joy seemed to come from its shining folds, and I, who had never
loved any one else, loved you from that moment as I love you now!"
"How well you say it, love!" responded Klein-Else. "Yes; where is
the difference between to-day and yesterday, and last year and the
year before that? Ever since that first day it has been one long love,
nothing else! Yes; well I remember that day. I was poor, and despised,
and had no one to talk to, and never thought any one would ever look
at me again--except to scold me. And then I went into the church and
knelt by you; and I felt as the new ivy twig must feel when it has
crept and tossed about in vain, and then at last finds, close under
its grasp, the strong, immovable oak, and clasps it--clasps it never
to loose its hold again, never! but grows up clasping it ever closer
and closer, till it grows quite one with it, and no one can separate
them any more for ever!"
"Yes," replied the baron; "nothing can separate them any more--nothing
can separate us now! We have grown together for years, and have only
grown the closer. It is now--let me see--five, six, seven years,
and we have only grown the closer to each other! To think it is
seven years! no, it wants a few weeks; but it will soon be seven
years. Seven--" he turned to look at her, for he perceived that as
he spoke she had loosened her hold of him, and now he saw she was
pale and trembling.
"But what ails you, Elschen [66]? Elschen dear! speak to me,
Elschen!" he added, with anxiety, for she sank back almost unconscious
against the bank.
"I shall be better presently," stammered the baroness. "I think the
scent of the flowers is too powerful. I don't feel quite well--take
me down by the side of the water; I shall be better presently." An
attendant took the babe from her arms--and the baron remembered
afterwards, that as she parted from it she embraced it with a
passionate flood of tears; then he led her to the side of the stream,
and bathed her burning forehead in the cooling flood.
Suddenly voices in angry altercation were heard through the trees,
and the servants summoned the baron with excited gesticulations,
saying there was a strange knight, all in armour, who claimed to see
the baroness.
Klein-Else was near fainting again when she heard them say that.
"Claims to see the baroness, say you?" replied their lord, with
menacing gesture. "Where is he? Let him say that to me!" and he darted
off to meet him, without listening to the faint words Klein-Else
strove to utter.
Now she was left alone by the side of the stream where, as the
Hennenpfoesl, she had first washed away the stains of servitude and
dressed herself to meet him who was to teach her to love. It was
beside that stream she had sat, and her tears had mingled with it,
as she had vowed that if ever such joy was hers as now she owned,
her treasure should be for those who were outcast and suffering as
she had been, and her happiness should be in making others happy!
How had she fulfilled her vow? From that time to this it had passed
out of her mind. Filled with her own gratification, she had left
the orphan in her bereavement, the suffering in their misery, nor
stretched out a helping hand.
The seven years were spent, and there was no doubt the knight was come
to seek an account of the treasure he had entrusted to her. She had
not only to meet him with shame for its misuse, but even his name she
had forgotten! And he had said, "Woe be to you, if you have forgotten
that name!"
But she had forgotten it. She pressed her hands against her throbbing
temples as if to force it from her brain, and swept away the mantling
hair--if but the cool breeze might waft it back to her! But the
forgotten name came not.
Suddenly the knight stood before her, and terrible he was to look at
in his shining armour! As she saw him she screamed and swooned away.
But he touched her, and bade her rise, then beckoned her to follow him;
and she could not choose but obey. He led her over the stream and along
the path in the mountain-side where the trees fringed the way; and when
they reached the rock she knew so well, with its treasury whence all
her means of happiness had been derived, he said in solemn accents,--
"Open, hoary rock!"
But to her he said,--
"Look!"
Then she could not choose but look. But oh, horror! in place of the
coin and jewels, armour and apparel, it was filled with wasted forms
bowed with misery and distress! the tear-worn orphan, the neglected
sick. Here she saw lying a youth, wan and emaciated, struck down in
all the promise of boyhood, and his mother tore her hair in agony by
his side. And there stood a father, gaunt and grey, vainly grappling
with Hunger, who was stealing away his children one by one from before
his face. Here----
But she could bear no more. She sank upon the ground, and hid her
face for very shame.
"The ransom of these, it is, you have spent upon yourself!" thundered
the pitiless knight; and every word was a death-knell....
The baron and his servants continued their search for the unknown
knight, but for long they found him not; one said he had seen him go
this way, and another that. Till at last an artless peasant maiden
told them she had seen him take the path of the mountain, across
the stream, and the baroness following behind with weak and unsteady
steps. The baron hasted his steps to pursue the way she pointed.
But he only found the lifeless body of Klein-Else kneeling against
the hoary rock!
PRINCE RADPOT [67].
Radpot succeeded early to the throne of his fathers. When on his
deathbed, his sire had called him to his side, and said to him,
"In leaving you my kingdom, I leave you a counsel with it which is
worth the kingdom itself. In all things be guided by the advice of
my wise counsellor Rathgeb, and you shall do well."
But Radpot had a stepmother who hated him, and was determined to
destroy him if possible; so she devised a plot against him, the first
step of which was to get him out of the kingdom. She therefore advised
that he should take a year's journey to perfect himself in knowledge
of the world, before assuming the reins of government.
Radpot was not devoid of shrewdness, and readily suspecting some evil
intention in his stepmother's advice, at first resisted following it;
but afterwards, submitting the matter to Rathgeb, according to his
father's desire, he received from him a different counsel from that
which he had expected.
"Though your stepmother may have evil intentions," he said, "you
need not therefore be afraid; we shall be able to baffle them. In
the meantime, it is well that you should travel to see the world,
and learn experience. We will so establish a council of regency that
the queen shall not be able to do great mischief during your absence."
Radpot was nothing loath to follow this advice, as he was of an
adventurous disposition; so, all things being ordered for the due
conduct of the affairs of the kingdom, he set out with Rathgeb for
his year's journey.
During all the days of preparation for the journey, the queen, who
had always heretofore shown herself harsh and hostile to Radpot,
behaved with the utmost tenderness and devotion, which the young
prince ascribed to her satisfaction at his having followed her advice,
and returned her advances with an ingenuous cordiality. She, in turn,
received his deference with an increase of solicitude, and nothing
could be more affectionate than their leave-taking. Every thing the
thoughtfulness of a fond mother could have suggested was provided by
her for his safety and comfort on the journey; and as he had his foot
in the stirrup she still had one more token of her care of him.
"Take this vial," she said, "it is a precious cordial; and when
you are worn and wearied with heat and travel, a few drops of
its precious contents will suffice to restore you to strength and
vigour. Farewell! and when you taste of it, think of me."
Radpot, with all the openness of his generous nature, assured her
that he should never forget her kindness for him, and stowing away
the vial in his belt, waved his hand to her as he rode away.
Two days and two nights Radpot and his trusty counsellor journeyed
through the cool forest; and then for another day along its border,
exposed to the heat of the sun upon the mountain-sides, till they
came to a vast plain where there was no shelter of hill or tree,
no hospitality of human dwelling. With unbroken courage, however,
the young prince commenced crossing it. It was only when, after three
days' more hard riding, they still seemed as far as ever from a place
of rest, that, wearied and dispirited, he took out his stepmother's
vial, to try the effect of her cordial.
It was the moment Rathgeb had feared. He had observed how completely
the queen had lulled Radpot's suspicions, and that every attempt
at suggesting there was any hypocrisy in her conduct had appeared
to vex him. To have now spoken of any danger in trying her cordial
would probably have provoked his resentment--Rathgeb took another
way of saving his charge.
"Think you not our mounts deserve more than we to taste this precious
restorative? whatever labour we have endured, theirs has been tenfold."
"True," said the good-natured prince; and, dismounting, he opened
the mouth of his steed, and poured some drops of the liquid on his
tongue. He had scarcely done so, however, when the poor beast stretched
out his long neck with an air of agony, then fell over on its side,
and expired!
Rathgeb left the incident to produce its own effect on the mind of
his pupil, who stood gazing as one bewildered.
"What can it be that killed my good horse?" he exclaimed, at length;
"it could not be the cordial! no, never! the queen could not have
been so base! It was, that he has been so long unused to exercise,
this terrible journey has overcome him, and the cordial was too late
to save him."
"Try it on mine," answered Rathgeb; "he is a battle-charger, used to
endurance, and delighting in labour. See," he said, jumping to the
ground, and patting his neck, "he is as fresh now as when we started,
not a hair turned!"
"Be it so," replied Radpot, not without some asperity. "I would not
suffer the trial, could I suspect it possible the queen could be
capable of so horrible a plot as you evidently suppose; but I cannot
believe it--so give the cordial to your horse."
Rathgeb took the vial, and poured not more than three drops on the
tongue of his thirsty beast. Both watched the effect with a tension
akin to awe.
In the first few moments no change was apparent, but Radpot was too
generous to give utterance to the triumph he began to experience.
Suddenly the faithful beast started as if it had been transfixed with
the sharpest arrow, directed one piteous look towards the master it had
served so well, and fell down lifeless by the side of its companion.
"There is no doubt it is as you say," Radpot now confessed, at
once. "Forgive me for the haste with which I spoke."
"Nay, prince, there is no need of excuse. Though it behoved me to
stand on guard and see no harm befell you, it became you to trust
her whose duty it was to befriend, not to harm you."
"And now try this cordial of mine, maybe it will fulfil what the
other promised."
The prince gladly accepted the proffered gift; and both, wonderfully
restored by its effects, continued their journey on foot.
They had not gone far when three ravens passed them on the wing. The
prince and his companion turned back to watch their flight, and saw
them alight on the carrion of their dead horses, immediately after
tasting of which they all three fell to the earth dead, by the side
of the dead horses.
"There may be some profit to be gained from these," said Rathgeb;
and going back to the spot, he picked up the dead ravens and took
them with him.
Their journey was without further incident, till they at last espied
a welcome hut completely sheltered in the border of a vast stretch
of forest land.
The sight gave them courage for renewed exertion; and in a few minutes
more they stood before the door. An old woman came out to ask them
in, but observing the youth and noble mien of the prince, she seemed
to be moved with compassion, and cried out, with great earnestness,
entreating them not to come in, for the place was the resort of a
band of twelve robbers, and that no one could deliver them out of
their hands. They would not be home till the next day for dinner;
in the meantime, by a path she indicated, the travellers could easily
make good their escape.
The prince would have rewarded her for her advice, and have set out
again to find a safer shelter; but Rathgeb remarked to him, that a
prince should rather find means to overcome a danger than fly from it,
and promised to carry him through this one, if he would be guided
by him. Radpot, mindful of his father's desire, promised to do all
he proposed.
"Then, when the robbers come in, do exactly as I do," said Rathgeb;
"in the meantime keep up your courage." And so they supped on what the
old woman set before them, and went to bed and slept peacefully. The
next day, an hour before dinner-time, Rathgeb went into the kitchen
and handed the three ravens to the old woman to cook, giving her
very particular directions as to the sauces that were to accompany
it, as if it were a dish for which they had a particular liking,
and wished dressed entirely for themselves. He was still watching
the confection of the dish when the twelve robbers came back. They
gave the two strangers a friendly reception, and invited them to
dinner. This, the old servant had told Rathgeb, was their custom,
and that after dinner they fell upon their guests and slew them at
the moment when they least expected any onslaught.
Rathgeb accepted the invitation, with the proviso that he and his
companion should be allowed to eat their own food, being some game
they had brought down by the way. The robbers made no objection, and
they sat down to table. While waiting for the repast to be brought,
the robbers entertained their guests with lively conversation, in
which Rathgeb joined with great show of cordiality, the young prince
sustaining his part admirably.
When the dishes were brought, the old woman was careful to set the
three ravens before Rathgeb, as he had bidden her; but the chief
robber interposed, and said they must really allow him to offer him
and his young friend of their hospitality. Rathgeb made a little
courteous difficulty; and the robber-chief, whose object was not
to thwart his guests in any thing, but make a show of the greatest
civility, said he must really consent to exchange dishes, and not
deprive him of the pleasure of providing in one way or another for
his guests. After holding out for some pressing, Rathgeb consented,
and each set to work to help himself to what was before him, the
dish which had been before the robber-chief having been exchanged
with that which was before Rathgeb.
Rathgeb helped the prince without appearing to take any notice of
what befell the robbers; and Radpot, understanding how important
it was to engender no suspicion, fell with a hunter's appetite upon
the viands without taking his eyes from his plate, after the manner
of a famished man. Before he had devoured many mouthfuls, however,
the poison of the ravens had done its work; one after another, or
rather all together, the robbers fell under the table as suddenly as
the horses and the ravens themselves--all but one. For one of the
robbers had felt suspicious of the unusual circumstance of guests
bringing their own food with them; when he pointed this out to his
companions, they said it was clear there could be no guile in the
matter, seeing the guests had so manifestly prepared it for their
own eating. But he had abstained from incurring the risk himself,
and now alone stood erect amid the dead bodies of his companions.
"Draw, prince," said Rathgeb, "and rid us of this scum of the
earth! cross not your sword with the defiled one, but smite him down
as a reptile."
Radpot did not wait to be twice told; before Rathgeb had done speaking
he had hewn down the swaggering bully, who had thought to make easy
work of an unpractised foe.
Radpot and his counsellor lost no time in continuing their journey. The
store of the robbers they left to the old woman who had befriended
them, and took only a provision of wine and bread with them for their
necessities by the way.
Skirting along the borders of the forest, they shortly came to a fine
city, where they established themselves at the first inn. They sat
down in the Herrenstuebchen [68] while other guests came in. "What
news is there?" asked Rathgeb of the new comers.
"News?" asked the person addressed; "it's always the old story
here--but that's as strange for those who don't know it as any news."
"What may it be, then?" pursued Rathgeb.
"Why, the princess to whom all this country belongs is going on with
her old mad pranks. She is perpetually propounding some new stupid
riddle, promising her hand and kingdom to whomsoever divines it. But
no one can divine the meaning of her nonsense; and the penalty is,
that whoever attempts and fails is dressed like a fool or jester,
with long ears and bells, and is made to ride backwards all through
the city, with all the people hooting and jeering him."
Rathgeb then informed himself as to the appearance and character of
the princess.
"Oh, as to that, she is charming in appearance--radiant as the sun,
dazzling all beholders; that is how so many heads are turned by
her. And as for her mind, there is no fault there either, except
that just because she is more gifted than all other women, she is
thus proud and haughty and unbearable."
Rathgeb had heard enough, and went out with the prince to make
acquaintance with the city and people, and to talk to him of the
plans which had suggested themselves to him for courting and taming
the princess. Radpot, who had been much pleased with the account
of the princess they heard repeated all around, entered fully into
his projects.
As the princess was much interested in conversing with foreigners,
it was not difficult for Rathgeb and the prince to obtain admission
to her. Rathgeb then proposed the prince to her as suitor; but on
the condition that, instead of the princess proposing the riddle
for him to guess, he should propose one for the princess to guess,
and that if she failed, she must marry his prince.
Two motives urged the princess to accede to this arrangement: first,
she felt her credit was staked on not refusing the trial; and then,
she was so struck by the appearance of the handsome young suitor,
that she was very glad to be saved the chance of losing him by his
not guessing her riddle.
Rathgeb's riddle was: "What is that of which one killed two, two
killed three, and three killed eleven?"
The princess asked for three days to consider her answer; during
which time she consulted all her clever books and all the wise men
of the kingdom, but she was unable to arrive at any answer which had
a chance of proving to be the right one.
The third day came, and with it Rathgeb and the prince. The
princess was obliged to acknowledge herself vanquished, but found an
over-payment for her vexation in having to marry the handsome prince,
who, however, had, by Rathgeb's advice, not told her that he was a
prince, and they passed for two travelling pedlars.
At first all went well enough. In the happiness of living with a
husband she loved, the Princess forgot many of her haughty ways, and
as the prince governed the kingdom wisely, under Rathgeb's advice,
every body was content.
This happy state of things was not destined to last. By degrees the
princess's old habits of self-sufficiency, haughtiness, and bad temper
came back, and Radpot found he had hard lines to keep peace with
her. From day to day this grew worse; and at last he found it hardly
possible to endure her continual reproaches and causeless vituperation.
In the meantime Rathgeb received the intelligence from the council of
regency that the queen, Radpot's stepmother, was dead, and that all
the people were impatient that he should return and place himself at
the head of the nation.
In communicating this news to the prince, the old counsellor propounded
a scheme for reducing his wife to a better frame of mind which pleased
him well. In accordance with it, Radpot absented himself from the
palace for several days. At the end of that time he returned; but
instead of waiting to listen to the fierce invectives with which the
princess met him on his return, he interrupted her at the beginning of
the discourse by informing her that he had been engaged on important
affairs which did not concern her. Before she had time to recover
from her surprise at his audacity in treating her thus, he went on
to say that this absence was only the prelude to a much longer one,
as he was now called home by his mother, who was a very old woman, and
who entreated him to remain with her during the rest of her days; that
he was about to set out, therefore, to go to her, and he could not say
if he should be able ever to come back again--so he must bid her adieu.
The princess could not for a long time be induced to believe that he
was serious; but when she really found him making preparations for his
departure, without any allusion to the idea of her accompanying him,
she was so softened and distressed that, but for a sign from Rathgeb,
he would have forgiven her at once, and told her all.
By Rathgeb's advice, he determined to put her change of demeanour to
the test before giving in, and he told her it was impossible to take
her with him.
At this, her pleading to accompany him became still more earnest.
"But if you came with me, you would not have a palace-full of servants
to wait upon you; you would have to live in a poor hut with a cross
old woman, to whom I could not bear that you should answer a cross
word, however peevish she might be; you would have to live on the
poorest fare, and to earn something towards the support of your life."
The princess, who really loved Radpot devotedly, and was of a good
and noble nature, having erred chiefly through thoughtlessness and
want of self-control, accepted all these hard terms cheerfully,
rather than be separated from her husband.
So they set out the next day. And when they got near Radpot's capital,
Rathgeb went on before to a poor cottage in the outskirts, where there
lived a lone old woman whom he could trust to carry on his plan by
acting as Radpot's mother, without her ever knowing who he really was.
When Radpot and the princess arrived before this cottage, by Rathgeb's
instruction, the old woman came out and welcomed him as her son,
and Radpot introduced the princess to her as his wife.
"Not much like the wife of an honest workman either!" grumbled the
old woman, according to Rathgeb's instructions. "That's the kind of
wife a man picks up when he goes to foreign parts--a pert, stuck-up
minx! But she'll have to learn to dress like a sober woman, and make
some use of her fingers, now she's come here, I can tell her!"
The princess could have thrust these words back down the old woman's
throat in an instant, but Radpot imposed silence by a severe look; and
then he reminded her that she had promised submission and obedience to
his mother, adding that, if she desired it, and shrank from sharing
his poverty and hard fare, his friend Rathgeb would even now take
her back to her own country.
But the very idea of parting from him produced immediate submission;
and the old lady happening to be leaning against the table, as if
tired of the exertion of welcoming her son, she even fetched and
placed a chair for her, and helped her gently into it.
But still, bad habits cannot be changed all at once, and before the
day was out Radpot had had to put on his severe look to arrest her
angry answer many a time.
The next morning, as soon as they had breakfasted, he said he was
going out for his work as a mason's labourer, and she must choose
what work she would do, as no one must be idle in his house. The
princess timidly replied that she knew many kinds of fine embroidery,
which she thought would sell for a great price, and as she had some
such with her, she would set to work to finish a piece of it, so that
Rathgeb might take it into the town and get it sold.
Radpot came back in the evening, and flung down a few pence on the
table, which he said was the amount of his daily wage, and told
her to go out and get the supper with it. It was little she knew of
how to buy a workman's supper; and what she brought Radpot was so
dissatisfied with that (having dined himself in the palace) he threw
it out of window, so that she had to go supperless to bed. Before
she went up-stairs, however, Radpot told her to show him her day's
work; and when she brought it, expecting him to admire its delicacy
and finish, he at once threw it on one side, saying he was sure such
coarse stuff would not sell there.
The princess spent the night more in weeping than in sleeping. In the
morning she had to get up and prepare the breakfast, in doing which
she not only burnt her hands, but, by her general awkwardness at the
unusual work, incurred a storm of vituperation from the old mother
such as she had often been wont to bestow, while no rough word had
ever been spoken to her before in her whole life.
All through the day she had to attend to all the old woman's whims;
and in the evening when Radpot came home it was nearly the same thing
again with the supper, and he would scarcely suffer her to snatch
more than a few mouthfuls, so angry he showed himself at her mistakes
in the manner of preparing it. He told her, too, that so long as she
did not know how to earn her food, she must not expect to have much
of it. This made her the more desirous that Rathgeb should take her
work to the town. When he had done so, however, he brought it back,
saying no one would buy such coarse, common work there. Then she
tried other kinds, each finer and more delicate than the last; but
all were brought back to her with the same answer. At last they gave
her a basket of common pottery, and told her to go and sell it to the
poor people in the market-place. This answered rather better than
the work. There were plenty of people who wanted to buy crockery,
and the most of them came to her basket in preference to others',
because of her beautiful face all bathed in tears. But just as she
was reckoning up what a nice sum we should now have to take home, and
that it would be acknowledged she had done something right at last,
a smartly caparisoned cavalier came riding past, and, without heed
to her cries, upset the whole of her stock upon the road, smashing
every thing to atoms, and scattering her heap of halfpence into the
gutter. She was so bewildered she hardly thought to look at him,
and yet, from the single glance, he appeared so like Radpot that she
almost called to him by name; he dashed away, however, so quickly,
that he was out of hearing in a minute.
In the evening, when she came to detail her mishap to him, he appeared
to be very angry at her ill success in every thing she attempted,
adding, "I'm afraid you'll never be any use to a poor man--we must
get Rathgeb to take you back home again."
But at this she threw herself at his feet in despair, begged him
so piteously to do any thing but that, promised so earnestly to
apply herself to any mode of life he prescribed for her, provided
only he would keep her by him, that he could hold out no longer,
and determined to put an end to her trial.
"I will give you one chance more," he said, trying to assume a tone
of indifference: "you shall bring me my dinner while I am at work
to-morrow; that will save me the time of going to get it, and will be
worth something--only mind, now, don't make any mistake this time! I
am working at the palace; bring the dinner there, and ask for the
mason's labourer."
The next day she took good care to have the dinner ready in time;
and though she was filled with confusion at having to go through the
public streets carrying the humble provision of the labourer's dinner,
and every one gazing after her beautiful, tearful face, she yet went
her way bravely, and came at last to the palace.
The moment she asked at the door for the mason's labourer, a page was
sent with her, who conducted her through suites of apartments vaster
and more magnificent than those of her own palace; and, while she was
lost in bewilderment, the page suddenly stopped short and pointed to
a drapery hanging, saying, "Pull that aside, and you will find within,
him you seek," and then darted away.
She scarcely dared do as she was bid; but then the clocks began to
strike the midday hour, and, fearful of keeping her husband waiting,
she lifted the drapery with a trembling hand.
On a royal throne, and habited in royal state, there sat Radpot
himself, Rathgeb standing respectfully by his side. The princess
thought to have fainted at the sight, for she could in no way
understand how they came to be there.
"Come in, princess!" said Radpot, encouragingly; and Rathgeb went to
the door, and conducted her up to him. He bid her welcome, and kissed
her tenderly, told her frankly what had been his plans with her, then
led her into an adjoining room, where there were ladies-in-waiting
ready to attire her in a robe of cloth-of-gold and coronet of diamonds,
which they held in readiness, with many choicest ornaments of gold
and precious stones. When she was ready, pages in court suits went
before her, and heralds proclaimed her aloud.
As soon as the prince saw her arrive, he ordered his high council
to be called in, and presented his consort to them, declaring her as
virtuous as she was fair.
After this they lived together many years in great happiness, for
the princess had had a life-long lesson, and never relapsed into her
foolish ways.
THE THREE BLACK DOGS.
The wind roared through the tall fir-trees, and swept the
snow-flakes in masses against the window-panes; the rafters rattled
and the casements clattered; but dismally, above the roaring and
the clattering, sounded the howling of three black dogs at the
cottage-door; for their good master lay on the pallet within, near
his end, and never more should he urge them on to the joyous hunt.
The old man was stark and grey; one bony hand held fast the bed-clothes
with convulsive clutch, and one rested in benediction on the dark
locks of his only son kneeling by his side. Long he lay as if at the
last gasp. Then suddenly raising his weary head from the pillow, he
exclaimed, "Joessl, my son, forget not to pray for your father when
he is no more." And Joessl sobbed in reply.
"Joessl," continued the old man, with painful effort, "you know fortune
has never favoured me in this world: you are my noble boy, and I would
have left you rich enough to be a great man, as your looks would have
you--but it was not to be! Joessl, it was not to be!" and the old man
sank back upon the bed, and hid his face and wept.
"Father, you have taught me to labour, to be honest, to face danger,
and to fear God!" said the brave youth, throwing himself upon him
and caressing his hollow cheeks; "that was the best inheritance you
could leave me."
"Well said! my noble son," replied the father. "But you are young to
rough the world by yourself; and I have nothing to leave you but the
Three Black Dogs--my faithful dogs--they are howling my death-knell
without. Let them in, Joessl--they are all you have now in the world!"
Joessl went to let them in; and as he did so the old man's eyes glazed
over and his spirit fled, and Joessl returned to find only a corpse.
The Three Black Dogs ceased their howling when they saw his grief,
and came and fawned upon him and licked his hands. For three days
they remained mourning together; and then the men came and buried the
father. Other people came to live in the cottage, and Joessl went out
to wander over the wide world, the Three Black Dogs following behind.
When there was a day's work to be done they fared well enough. Though
he had so fair a face and so noble a bearing, Joessl was always ready
to apply his stalwart limbs to labour, and what he earned he shared
with the Three Black Dogs, who whined and fawned and seemed to say,--
"We are eating your bread in idleness now; but never mind, the day
will come when we will earn you yours."
But when there was no work to be had, when the storm beat and
the winter wind raged, Joessl was fain to share a peasant's meal
where he could find pity by the way, and many there were who said,
"God be gracious unto thee, my son," when they saw his comely face;
but the Black Dogs slunk away, as if ashamed that their master's son
should have to beg, not only for himself, but for them also.
Better times came with the spring; and then there was the hay-cutting,
and the harvesting, and the vintage, and Joessl found plenty of
work. But still he journeyed on, and the Three Black Dogs behind.
At last he saw in the distance the towers of a great city, and he
hasted on, for all his life he had lived in the mountains, and had
never seen a town.
But when he reached it, he found that though it was a vast city, it
was empty and desolate. Broad well-paved roads crossed it, but they
were more deserted than the mountain-tracks. There were workshops,
and smithies, and foundries, and ovens, but all silent and empty, and
no sound was heard! Then he looked up, and saw that every house was
draped with black, and black banners hung from the towers and palaces.
Still not a human being appeared, either in the public squares or
at the house-windows; so he still wandered on, and the Three Black
Dogs behind.
At last he espied in the distance a waggoner with his team coming
through the principal road which traversed the city, and lost no
time in making his way up to him and asking what this unearthly
stillness meant.
The waggoner cracked his whip and went on, as if he were frightened
and in a hurry; but Joessl kept up with him. So he told him, as they
went along, that for many years past a great Dragon had devastated the
country, eating up all the inhabitants he found in the way, so that
every one shunned the streets; nor should he be going through now,
but that need obliged him to pass that way, and he got through the
place as quickly as he could. But, he added, there was less danger
for him now, because lately they had found that if every morning some
one was put in his way to devour, that served him for the day, and
he left off teasing and worrying others as he had been used to do;
so that now a lot was cast every day, and upon whomsoever of the
inhabitants the lot fell, he had to go out upon the highway early
the next morning that the dragon might devour him and spare the rest.
Just then a crier came into the street, and proclaimed that the lot
that day had fallen on the king's daughter, and that to-morrow morning
she must be exposed to the dragon.
The people, who had come to the windows to hear what the crier had
to say, now no longer kept within doors. Every one was so shocked to
think that the lot had fallen on their beautiful young princess, that
they all came running out into the streets to bewail her fate aloud;
and the old king himself came into their midst, tearing his clothes
and plucking out his white hair, while the tears ran fast down his
venerable beard.
When Joessl saw that, it reminded him of his own father, and he could
not bear to see his tears.
Then the king sent the crier out again to proclaim that if any one
would fight the dragon, and deliver his daughter, he should have her
hand, together with all his kingdom. But the fear of the dragon was
so great on all the people of the city that there was not one would
venture to encounter it, even for the sake of such a prize.
Every hour through the day the crier went out and renewed the
proclamation. But every one was too much afraid of the dragon to make
the venture, and Joessl, though he felt he would have courage to meet
the dragon; could not find heart to come forward before all the people
of the king's court, and profess to do what no one else could do. So
the hours went by all through the day and all through the night,
and no one had appeared to deliver the princess.
Then daybreak came, and with it the mournful procession which was to
conduct the victim to the outskirts of the city; and all the people
came out to see it, weeping. The old king came down the steps of the
palace to deliver up his daughter; and it was all the people could
do to hold him back from giving himself up in her place.
But when the moment of parting from her came, the thought was so
dreadful that he could not bring himself to make the sacrifice; and
when he should have given her up he only clasped her the tighter in
his arms. Then the people began to murmur. They said, "The hour is
advancing, and the dragon will be upon us, and make havoc among us
all. When the lot fell upon one of us, we gave up our wives, and our
fathers, and our children; and now the same misfortune has visited
you, you must do no less;" and as the time wore on they grew more
and more angry and discontented.
This increased the distress and terror of the king, and he raved
with despair.
When Joessl found matters as bad as this, he forgot his bashfulness,
and coming forward through the midst of the crowd, he asked permission
to go out to meet the dragon; "and if I fail," he added, "at least I
shall have prolonged the most precious life by one day;" and he bent
down and kissed the hem of the princess's garment.
When the princess heard his generous words she took heart, and
looked up, and was right glad to see one of such noble bearing for
her deliverer. But the old king, without stopping to look at him,
threw himself on his neck and kissed him with delight, and called
him his son, and promised him there was nothing of all the crier had
proclaimed that should not be fulfilled.
The discontent of the people was changed into admiration; and they
accompanied Joessl to the city gates with shouts of encouragement as
he went forth to meet the dragon, and the Three Black Dogs behind.
If the king's daughter had been pleased with the appearance of her
deliverer, Joessl had every reason to be no less delighted with that
of the lady to whom he was about to devote his life.
Full of hope and enthusiasm, he passed on through the midst of the
people--regardless of their shouts, for he was thinking only of
her--and the Three Black Dogs behind.
It was past the time when the dragon usually received his victim, and
he was advancing rapidly towards the city walls, roaring horribly,
and "swinging the scaly horrors of his folded tail." The fury of
the monster might have made a more practised arm tremble, but Joessl
thought of his father's desire that he should be a great man, and
do brave deeds, and his courage only seemed to grow as the danger
approached. He walked so straight towards the dragon, with a step so
firm and so unlike the trembling gait of his usual victims, that it
almost disconcerted him. When they had approached each other within
a hundred paces, Joessl called to his dog Lightning, "At him, good
dog!" At the first sound of his voice Lightning sprang to the attack,
and with such celerity that the dragon had no time to decide how to
meet his antagonist.
"Fetch him down, Springer!" cried Joessl next; and the second dog,
following close on Lightning's track, sprang upon the dragon's neck,
and held him to the ground.
"Finish him, Gulper!" shouted Joessl; and the third dog, panting for
the order, was even with the others in a trice, and fixing his great
fangs in the dragon's flesh, snapped his spine like glass, and bounded
back with delight to his master's feet.
Joessl, only stopping to caress his dogs, drew his knife, and cut out
the dragon's tongue; and then returned to the city with his trophy,
and the Three Black Dogs behind.
If the people had uttered jubilant shouts when he started, how much
more now at his victorious return! The king and his daughter heard
the shout in their palace, and came down to meet the conqueror.
"Behold my daughter!" said the old king: "take her; she is yours,
and my kingdom with her! I owe all to you, and in return I give you
all I have."
"Nay, sire," interposed Joessl; "that you give me permission to
approach the princess is all I ask, and that she will deign to let
me think that I may be one day found not unworthy of her hand. But
as regards your kingdom, that is not for me. I am but a poor lad,
and have never had any thing to command but my Three Black Dogs:
how should I, then, order the affairs of a kingdom?"
The king and all the people, and the princess above all, were pleased
with his modesty and grace; and they sounded his praises, and those of
his Three Black Dogs too, and conducted them with him to the palace,
where Joessl received a suit of embroidered clothes and the title of
duke, and was seated next the princess.
The king, finding that he was resolute in refusing to accept the crown,
determined to adopt him for his son; and had him instructed in every
thing becoming a prince, so that he might be fit to succeed him at
his death. To the Three Black Dogs were assigned three kennels and
three collars of gold, with three pages to wait on them; and whenever
Joessl went on a hunting-party, his Three Black Dogs had precedence
of all the king's dogs.
As time wore on Joessl had other opportunities of distinguishing
himself; and by little and little he came to be acknowledged as the
most accomplished courtier and the most valiant soldier in the kingdom.
The princess had admired his good looks and his self-devotion from
the first, but when she found him so admired and courted by all the
world too, her esteem and her love for him grew every day, till at
last she consented to fulfil the king's wish, and they were married
with great pomp and rejoicing. Never was there a handsomer pair; and
never was there a braver procession of lords and ladies and attendants,
than that which followed them that day, with music and with bells,
and the Three Black Dogs behind.
* * * There are countless spots in Tirol in which tales are
traditional of brave peasants, hunters, and woodmen delivering the
place out of some need or danger, symbolized as "a dragon," similar
in the main to the above, but with varieties of local colouring. I
gave the preference to the above for the sake of the Three Black Dogs.
OTTILIA AND THE DEATH'S HEAD; OR, "PUT YOUR TRUST IN PROVIDENCE."
In the little town of Schwatz, on the Inn, the chief river of Tirol,
there lived once a poor little peasant-girl named Ottilia. Ottilia
had been very fond of her dear mother, and cried bitterly when she
had the great misfortune to lose her. She tried hard to do all she
had seen her mother do: she swept the house and milked the cow,
and baked the bread, and stitched at her father's clothes; but she
could not, with all her diligence, get through it all as her mother
did. The place began to get into disorder, and the pigs and the fowls
fought, and she could not keep them apart, and she could not manage
the spinning; and what was worst of all, she could not carry in the
loads of hay, by which her mother had earned the few pence that eked
out her father's scanty wages.
To keep the house straight, the good man found himself obliged to
take another wife; and one day he brought home the tall Sennal,
and told Ottilia she was to be her mother.
When poor little Ottilia heard the tall, hard, bony woman called
"her mother," she burst out into passionate tears, and declared she
should never be her mother, and she would never pay her obedience!
Now the tall Sennal was not a bad woman, but she was angry when the
child set herself against her; and so there was continual anger between
the two. When she told Ottilia to do any thing, Ottilia refused to
do it, lest she should be thought to be thereby regarding her as
her mother, which seemed to her a kind of sacrilege; and when she
tried to do any of the work of the house, her childish inexperience
made her do it in a way that did not suit the tall Sennal's thrift,
so there was nothing but strife in the house. Yet the good father
contrived, when he came home of an evening, to set things straight,
and make peace; and though Ottilia had little pleasure, like other
children of her years, yet she had a good woollen frock to keep out
the cold, and bread and cheese and milk enough to drive away hunger,
and, what she valued most, a father's knee to sit on of an evening in
the well-warmed room, while he kissed her and told her weird stories
of the days long gone by.
But a day came--a day darkened by a terrible storm--on whose evening
no father came home. The long Sennal went out with the neighbours
with lanterns and horns, but the fierce winds extinguished their
lights and drowned the sound of their horns; and Ottilia knelt by
the side of her father's chair, praying and crying.
She prayed and wept, and only slept a little now and then, all through
the night; and in the morning some carters came in, and brought
her father's dead body, which they had found on their mountain way,
under the snow, where it lay buried.
But Ottilia still knelt by her father's chair, and felt like one
in a dream, while they put him in his coffin and carried him to the
churchyard ground, and the sad bells mourned.
"Go, child, and feed the pig!" exclaimed the harsh voice of the tall
Sennal--and it sounded harsher than ever now, for there was none left
to apply the curb. "Crying's all very well for a bit; but you're not
going on like that all your life, I suppose?"
Ottilia felt her helplessness, and therefore resented the
admonition. Without stopping to consider its reasonableness, she
retorted, fiercely,--
"'Child!' I am no child of yours! I've told you so before, a thousand
times; and it's not because my father's dead that you're going to
come over me. You think you'll make me forget him by forbidding me
to cry for him; but never, never will I forget him! nor shall you
forget how he made you behave properly to me!"
The tall Sennal had more patience with her than might have been
expected, and said no more for that time; but Ottilia was not won by
her forbearance, and only reckoned it as a victory.
It was strife again the next day, and the next, and there was no good
father to make peace. And at last the tall Sennal's patience fairly
gave way, and one day, in her provocation, she drove the child from
the door, and bid her never come under her eyes again!
Her anger cooled, she could have recalled the words, but Ottilia was
already far away up the mountain-path, and out of sight, gone she
knew not whither.
Ottilia had no experience of want, and knew not what it was to be
alone upon the mountains; all her full heart felt at the moment was,
that it would be a boon to get away from the reproaches her conscience
told her were not undeserved, and be alone with her parent's memory.
Thus she wandered on, with no more consciousness of her way than just
to follow it to the spot where her father died, and which had been
marked by pious custom with a wayside cross, on which was painted in
vivid strokes the manner of his end.
Ottilia gazed at the cruel scene till fresh tears started to her eyes,
and she threw herself on the ground beside it, and cried till she knew
no more where she was. Then it seemed to her as if the ground were
again covered with snow, and that from under it she heard her father's
voice; and he talked to her as he used to talk of an evening by the
fireside, when she was on his knee after work and he made her peace
with the tall Sennal. And now he brought home to her all her naughty,
senseless ways, not scolding without reason, but making all allowance
for the filial love which had been at the bottom of the strife. Ottilia
seemed to herself to be listening to him with great attention, but her
heart misgave her. She was ready to own now that she had been very
wrong, very unreasonable, and she felt really sorry for it all--so
sorry that, had her home still been his, she felt that she could have
brought herself to obey Sennal, so that she might not grieve him; but
now--now that he was not there--suppose he should require of her that
she should go back now and live with the tall Sennal, all alone! But
he did not require it of her; or, at all events, in her excitement
she woke with his last words sounding in her ear, which were nothing
more severe than, "Put your trust in God, and all will yet be well."
The sun had already sunk behind the mountains, the chill night air
began to penetrate Ottilia's clothing, and hunger stared her in the
face. She felt very humble now, but she had no mind to go back. She
rose and walked on, for numbness, as of death, was creeping over her,
and she knew the mountain-folk said that to yield to that lethargy
of cold was death.
On she walked, and on, and the darkness gathered thicker and
thicker round her; but she thought of her guardian angel, and she
was not afraid. Still the way was weary, and the air was keen, and
her strength began to fail. Then suddenly, on a neighbouring peak,
she descried the broken outline of a castellated building standing
out against the now moonlit sky.
Gathering fresh force from hope, she picked her way over steep and
stone, guiding her steps by the friendly light which beamed from a
turret window. She had had time to realize her whole desolation. If
heaven vouchsafed her another chance of finding a home, she mentally
resolved, she would behave so as to win its blessing, with all
her might.
When at last she reached the castle-gate her courage once more began
to fail--what would the great people at the castle say to a poor
little half-starved peasant-girl, who came without friend or warrant
to disturb their rest? "Where is your trust in Providence?" said
a voice within, which sounded like a memory of her father's, and
rekindled her courage.
A horn hung beside the broad portal; and when, after many timorous
efforts, Ottilia had succeeded in making a note resound, she stood
anxiously wondering what stern warder or fierce man-at-arms would
answer the summons.
None such appeared, however. But after some moments of anxious waiting,
the window whence the friendly light beamed was opened, with noise
enough to make her look up, and then--what do you think she saw?
Nothing but a Death's Head looking out of the window! Almost before
she had time to be frightened, it asked her, in a very kindly voice,
what was her pleasure.
"A night's lodging and a bit of bread, for the love of Christ!" said
Ottilia, faintly; and then she looked up again at the Death's Head,
and she could not resist a sense of horror and faintness that crept
over her. "Put your trust in God," whispered her father's voice,
and she made an effort to stay her teeth from chattering together.
Meantime, the Death's Head had answered cheerily enough, "Will you
promise to carry me up here again faithfully, if I come down and draw
the bolt for you, and let you in?" and scarcely knowing what she said,
Ottilia gave an assent.
"But think what you are saying, and swear that I can rely on you,"
persisted the Death's Head, "for, you see, it is a serious matter
for me. I can easily roll down the steps, but there are a good many
of them, and I can't get up again by myself."
"Of course you may rely on me," now answered Ottilia, for she saw
it was but her bounden duty to perform this return of kindness--and
conscience seemed to have a reproach for her courageous alacrity,
saying, "The tall Sennal never required of you any thing so hard as
this." "I know she didn't," answered Ottilia, humbly; "and this is
my punishment."
All this time the Death's Head was coming rumbling down the stone
stairs--and a hard, dismal sound it was. Clop, clop, clop, first round
the turret spiral; then r-r-r-r-r-roll along the long echoing corridor;
and then, clop, clop, clop once more all down the broad main staircase;
then another r-r-r-r-r-roll; and finally, klump! bump! it came against
the massive door.
Ottilia felt her heart go clop, clop, clop, clop, too, but she
struggled hard; and the cold, and the faintness of hunger made her
yet feel rejoiced to hear the Death's Head take the bolt between its
grinning teeth and draw it sharply back. The great door flew open,
and Ottilia trod timidly within the welcome shelter. The memory of
her father's fate was fresh upon her, and the Death's Head was less
terrible than the pitiless snow.
Not without some difficult mental struggles Ottilia faithfully
fulfilled her promise. A temptation, indeed, came to let the skull
lie. It could not pursue her--it could not possibly climb up all those
stairs, though it could roll down them; besides, it had declared
its incapacity for the task. She could let it lie and enter into
possession of the castle--it was clear there was no one else there,
or the skull would not have put itself in danger by coming to the
door. But honest little Ottilia repelled the thought with indignation,
and, bending down, she picked up the skull, and carried it carefully
up the stairs folded in her apron.
"Lay me on the table," said the Death's Head, when they got into
the turret-chamber where the light was; "and then go down into the
kitchen and make a pancake. It won't be for want of eggs and flour
and butter if it is not good, for they are there in plenty."
"What! go all the way down to the kitchen alone, in this great
strange place?" said poor little trembling Ottilia to herself. "This
is worse than any thing the tall Sennal ever gave me to do indeed;"
but she felt it was a punishment and a trial of her resolution,
and she started to obey with brave determination.
It was a harder task even than she had imagined, for if the Death's
Head was safe up-stairs in the turret tower, the "cross-bones" were
at large in the kitchen, and would get in her way whatever she turned
to do.
True, her impulse for a moment was to turn and scream, and run away,
but there came her father's voice, bidding her trust in God, "And
besides," she said to herself, "what is there so very dreadful about
the sight of dead bones, after all? and what harm can they do me?" So
she took no notice of what was going on around her, but beat her eggs
and mixed her batter, and put it on to fry, till the appetizing odour
and the warmth of the fire brought back life and renewed her courage.
When Ottilia brought the pancake up into the turret-room, and laid the
dish with it on the table, she observed that the side of the pancake
which was turned towards the skull became black, while that nearest
herself retained its own golden colour; so that her curiosity was
piqued, and she was much inclined to ask about it, but she managed
to keep quiet and eat her share in silence. When she had finished
she took the dish and washed it up, and put all away carefully; and
she was just feeling very tired when the Death's Head said to her,
"If you go up that staircase on the left, you will come to a little
bedroom where you may sleep. About midnight a skeleton will come to
your bedside, and try to pull you out of bed; all you have to do is
not to be afraid of it, and then it can do you no harm."
So Ottilia thanked the skull, and went up to bed. She had not been in
bed more than three hours when she heard a great noise and rattling
in the room, much like the noise the cross-bones had made in the
kitchen while she was cooking the pancake. Then she heard the skull
call up to her, "It is just midnight--remember you have only to be
brave!" And as it spoke she saw a great skeleton come and stand in
the bright moonbeam by her bedside! It stretched one of its long bare
arms out towards her, and pulled off the bed-clothes with one bony
hand and seized her by the hair with the other. But Ottilia listened
for her father's voice bidding her put her trust in Providence; and
she remained quite quiet in her bed, giving no sign of fear. When the
skeleton found that she was so brave, it could do nothing against her,
but, after two or three ineffectual tugs, turned and went away; and she
saw nothing more of it, but slept out the rest of the night in peace.
When she woke the next morning the bright sun was pouring cheerfully
into the room, and by the bedside, where the skeleton had stood the
night before, was a beautiful form of a woman, all clothed in white
and surrounded by golden rays, to whom Ottilia said, "What do you
want me to do, bright lady?"
And the vision answered, "I was the mistress of this castle, who, for
my pride and vanity, was condemned to dwell in my bare bones on the
same spot where I had sinned by my extravagance in dress, and other
wanton habits, until one should come, for the sake of whose thrifty,
humble ways, and steadfast trust in God, I should be set free.
"This you have accomplished, and now I can go to my rest; while,
in gratitude, I endow you with this castle and all its lands and
revenues."
With that the bright form disappeared; and a moment afterwards Ottilia
saw, through the window, a milk-white dove winging its upward flight
towards heaven.
So Ottilia became a rich countess, and mistress of the lordly castle
which she had entered as a suppliant. But no sooner was she installed
than she sent for the long Sennal; and, having besought her pardon
for all the trouble she had given her, begged her to come up to the
castle and be with her. So they lived very happily together for the
rest of their lives.
THE TWO CASKETS.
It was a summer holiday; the sun shone with burning rays on the
newly-mown banks; the roads and paths seemed knee-deep with dust;
the flowers by the wayside hung their heads, as if praying for the
refreshing shower; the very waters of the streamlet were heated as
they passed along, and Franzl, lying indolently on its bank, plunged
his hands beneath its bright surface, but found no cooling. With a
peevish exclamation, he rose and sauntered away, and wished there
were no holidays.
"Nay, don't wish that!" said a gentle fair-haired maiden by his side;
"and just on this one, too, which I have been longing for, to fill
the basket I made for mother with fresh strawberries from the wood."
"Not a bad idea of yours, Walburga; they all call you the 'wise'
Walburga," replied Franzl. "There's shade in the wood, and the
strawberries will be cooler and more refreshing than this nasty
stream."
And with that he strolled away towards the wood.
The cottage of Franzl and Walburga was nestled into the side of a steep
hill, the summit of which was mantled with a forest of lofty pines;
and up the precipitous path, which wound past the very chimneys of
the cottage, Franzl now strolled alone, without troubling himself
to offer his hand to the patient little maiden who toiled painfully
behind him, with many a slip upon the loose stones and sunburnt moss.
This was Franzl's character. He was always thus: his own amusement,
his own enjoyment, and his own ease, were his sole care. Nor had the
example of Walburga's loving thoughtfulness for others any effect
upon him. If he took any notice of her at all, it was only to laugh
and rail at her for it, till her silence shamed his reproaches.
At the pinnacle of the path there was a venerable stone cross, shaded
from the weather by a little pent-house covered with ivy. Walburga
knelt before it as she passed, and prayed for help to be always a
good, obedient child, and a blessing to her dear parents. Franzl
raised his hand to his cap mechanically, because it was the custom,
but no holy thought crossed his mind.
"At last there is some coolness after all this horrid heat! and now
we are close to those nice refreshing strawberries." These were his
only ideas.
To Walburga, as she knelt, there came sweet lessons she had been
taught to associate with the cross--of abnegation of self, obedience
to higher powers, and loving devotion to others.
Franzl looked with all his eager eyes to discern the bright red
berries where the shade lay diapered with the light darting between
the thick clothing of the pine-trees, without so much as casting a
glance at the sacred token.
"Oh, what a splendid haul!" he cried, and plunged through the thick
leafage to where the ripe, rich berries clustered closest, and,
without troubling himself to learn whether Walburga was as well
supplied, began helping himself to his heart's content.
Walburga lined her basket with fresh green leaves, and laid the
strawberries in tasteful order upon them, only now and then taking
the smallest and most worthless for herself.
Though possessed with different objects, both were equally eager in
the pursuit, and they pushed deeper and deeper into the thick pine
forest, Walburga always keeping near Franzl, by reason of her tender,
confiding spirit, which loved to be near those dear to her, though he,
intent on his own gratification, had no cheerful word to enliven her.
At last they came to where the dark pines closed thick overhead--so
thick that no golden rays pierced through; all was shade and
silence. But here the strawberries were no longer ripe and red, for
there was no sun to bring them to maturity, so Franzl peevishly turned
to go, and Walburga followed gently behind. Suddenly their progress
was arrested by a bright light--brighter than the burning summer
sun shining beneath the gloom of the dark pines--and in the centre
of that light stood a beautiful queen, and the light seemed to come
from the diadem on her forehead and the garments that encompassed her!
"What are you doing here?" she said, in soft sweet accents, addressing
herself to Walburga.
And Walburga, dropping her eyelids with maiden modesty, replied, hardly
able to force her voice above a whisper, "Gathering strawberries for
mother dear."
The beautiful Lady smiled a smile of approval; and the bright light
seemed brighter when she smiled, and a sweet and balmy breeze stirred
the air when she spoke again.
"Here, my child," she said, "take this casket;" and she handed her
a casket made just like the strawberry-basket she had woven for her
mother, only it was all of pure gold filigree, and, in place of the
piled-up strawberries, it had a lid of sparkling carbuncles. "Take
this, my child; and when you open it think of me."
"And what are you doing?" she said, with something less of mildness,
to Franzl, who, having his hat full of strawberries, was so busy
devouring them that he had not even noticed the beautiful present
his sister had received.
Nor did he stop now even to reply to her; but between throwing away
one chuck and picking out another fruit, he muttered, rudely,--
"I should think you might see that, without asking!"
The beautiful Lady looked at him sadly, and tears like pearls fell
fast down her fair cheeks, as she gave him a dark iron casket, with
the same words she had used to Walburga.
The light disappeared, and the fair Lady was seen no more.
"Who can that bright Lady be? and what can these caskets be that
she has given us?" said Walburga, timidly. "Let us come home quick,
and show them to mother;" and she ran onwards gaily, calling out,
"Mother, mother dear, see what I have got!"
"Stuff!" replied Franzl; "I'm not going to wait for that: I want to
see what's in them now." But Walburga had passed on out of hearing.
He pulled the lid off his dark iron casket; and immediately there
wriggled out two great black ugly snakes, which grew bigger and longer,
dancing round him; nor could he escape from their meshes. Then,
finally, they closed their coils tightly round him, and carried him
away through the thick, sunless forest, and no one ever saw him again!
Meantime Walburga was making her way home with all the speed she
could down the dangerous mountain track, her strawberry-basket in one
hand and the golden casket in the other. Her mother sat spinning in
the luxuriant shade of the climbing plants over-shadowing the broad
cottage-eaves.
"Mother, dear mother!" cried the child; "see what I have got. Here
is a basket of fresh cool strawberries I have gathered for you in the
wood, and here is a golden casket which a beautiful Lady brought me,
with a great shining light! But stop till Franzl comes home, for he
is coming behind, and she gave him a dark iron casket too, and we
will open them both together; so eat the strawberries, mother dear,
till Franzl comes."
The mother kissed her child fondly, and stroked her fair, soft, curling
hair, but turned her head and wept, for she knew what had befallen.
But Franzl came not; and when Walburga had sought him every where,
she said, "He must be gone round by the woodman's track to meet father,
so let us open the casket, mother dear."
So she put the casket in her mother's lap, and lifted the beautiful
carbuncle lid. And see! there flew thereout two tiny beings,
all radiant with rainbow light, and they grew bigger and bigger,
fluttering round her till they appeared two holy angels, who folded
the child softly in their arms, then spread their wings and flew away
with her, singing enchanting melodies, above the clouds!
THE PRUDENT COUNSELLOR.
Alois Zoschg was a peasant of the Sarnthal; his holding was
inconsiderable, but it sufficed for all his needs; his cottage was
small, but his family consisted of only himself and his daughter,
and they found room for all their requirements.
Katharina was bright enough, however, to make any home happy. Though
she shared the cottage with her father alone, she never seemed to
feel the want of younger companions; thoughtful and prudent beyond her
years, and thrifty and notable with all the work of the place, she was
at the same time always ready with her joke and her song. It was no
wonder that her father doated on her, and looked forward all through
the day's toil to the evening spent in cheerful conversation with her.
There were thus the elements of a pleasant existence in Alois' lot,
but there were two disturbing causes also. One was his own temper,
which was violent and ungovernable at times, when he was seriously
provoked. The other was the jealousy and animosity of a rich peasant
neighbour, Andrae Margesin, the owner of a considerable Hof [69]
situated at no great distance from Zoschg's cottage, auf der Putzen.
Circumstances had constantly brought the two neighbours into collision;
the fault generally lay, in the first instance, on the side of the
rich Andrae Margesin, who was grasping and overbearing, but Alois Zoschg
once roused, would never let a quarrel rest, and his irritability and
revengeful spirit were oftentimes enough to disturb the peace of the
whole neighbourhood. No one could say where such quarrels might have
ended, what crimes might perhaps have been the result, but for the
wise interposition of Katharina, who knew how to soothe her father's
ruffled spirit without ever exceeding the limits of filial respect,
as well as how to conciliate the rich neighbour, without condescending
to the use of any servile arts.
By her extraordinary good sense and good temper alone, she would,
time after time, bring both the men back to sober reason from the
highest reach of fury.
Once, however, they had a dispute which was beyond her competence to
decide for them, for it involved a question of law. Andrae Margesin
accused Alois Zoschg of an encroachment, while Alois Zoschg maintained
he was justified in what he had done, by prescriptive right. The
dispute raged high, but all Katharina could do in this case to restore
peace, was to exact a promise from both parties that they would cease
from all mutual recrimination, and carry the matter to be decided
for them by the judge in Botzen.
When the day of hearing came on, the two disputants went up to Botzen
to plead their cause; but each was so determined not to give way,
and had so much to say in defence of his own position, and to the
disparagement of his antagonist, that they carried their pleadings on
for six days, and yet there seemed no chance of arriving at a decision
which should be thoroughly justified by the evidence, so contradictory
was it. At last, the judge, getting tired of the prolonged controversy,
and finding it impossible to moderate the virulence of the combatants,
told them that he could have no more wrangling, they had so confused
the case with their statements and counter-statements, that it was
impossible to say which of them was right, or, rather, which of them
was least in the wrong; but he gave them one chance of obtaining a
decision of the matter, and that was by accepting a test, which he
would propound, of their ability and judgment, and whichever succeeded
in that, he should pronounce was the one who was in the right in the
original pleading.
The rivals looked somewhat disconcerted at this mode of procedure,
but, as they found they could not get the affair decided on any other
terms, they at last agreed to accept the proposal.
"You must tell me, then," said the judge, "by to-morrow morning at
this hour, what is that which is the Strongest, the Richest, and
the most Beautiful;" with these words he left the judgment-seat,
and the two peasants were left standing opposite each other, looking
very foolish, for they both thought that it would be impossible ever
to answer such a question.
After a few moments' consideration, however, Andrae Margesin, who
was a very vain man, bethought himself of an answer which, to his
mind, seemed indisputably the right one. "To be sure! Of course! I
wonder I didn't see it at once! There can be no doubt about it!" he
exclaimed, aloud; and clapping his hands, and making other triumphant
gesticulations, he stalked off homewards, telling all his friends
that he had no doubt of the result.
But poor Alois Zoschg, the more he thought, the more puzzled he got,
and the boasts of Andrae Margesin only made him more furious. There
he stood, crying out against the judge, and against his ill-luck,
against his poverty and the opulence of Margesin, till it became
necessary to close the court, and his friends prevailed on him to
go home. But all the way his passion grew more and more outrageous,
and by the time he reached his cottage he was raging like a maniac;
the other men could do nothing with him, and slunk away one by one,
some in disgust, some in despair.
It was now Katharina's turn; and Katharina came out to meet him with
her brightest smile and her filial greeting, just as if he had been
in the best humour in the world.
But, for the first time, the sight of Katharina seemed rather to
increase than allay his anger; for he found her dressed in all her
festal attire--a proceeding which was quite out of character with
his present disposition.
There was he, worn out with the long dispute, the weariness of the
delayed decision, the provocation of his enemy's insulting mien,
and still more, perhaps, by his own ill-humour; and there she stood,
all smiles and bright colours, as for a joyful occasion--the white
Stotzhaube [70] coquettishly set on her braided hair, the scarlet
bodice tightly embracing her comely shape, with "follow-my-lads
[71]" streamers from her shoulder-knot, the bright red stockings
showing under her short black skirt, and the blue apron over it,
in place of the white apron of working days! Could any thing be more
incongruous? was it not enough to increase his madness?
Nevertheless, Katharina's judgment so uniformly approved itself to
his better reason, that, the first impulse passed, he gulped down
the rising exclamation of annoyance until he had heard what Katharina
had to say.
"Well, father, so you're all right! and I'm the first to congratulate
you," she cried, and flung her arms round him with an embrace,
of which, even in his present state of excitement, he could hardly
resist the tenderness and effusion, and as if she did not perceive
the traces of his ill-humour.
"'Right,' wench! what mean you? all wrong you should say."
"No, no, I mean it is all right; and it only remains for you to hear
it pronounced by the judge to-morrow--and haven't I put on my gala
suit to celebrate your success?"
"Success! speak! what mean you?" cried Alois, eagerly, his stormy
vexation melting away before the sunbeam of her encouragement.
"Why, what has the judge told you to do, to decide the case?" asked
Katharina, who had heard it all from a neighbour who came home hours
before, while Alois was still standing perplexed in the court.
"That I should tell him by to-morrow morning," replied Alois, softened
already by her consoling manner, "what it is which is the strongest,
the richest, and the most beautiful--and how am I ever to guess all
that? And what's more," he continued, relapsing into his former state
of vexation, "that fellow Andrae Margesin has guessed it--guessed it
already! and is gone off proclaiming his triumph!"
"No, father!" exclaimed Katharina, with a mocking laugh, all of fun,
however, not of scorn; "you don't mean to say you believe that great
bully Andrae Margesin could have guessed the right answer?"
"But he said so! he went off telling every one so," rejoined Alois,
positively.
"Oh, you dear, good, simple father! do you really believe it is so
because he boasts of it? Do rest easy; he's not got it."
"Well, but if he hasn't, I haven't either. How am I to guess such
captious absurdities? Why couldn't the man judge the thing on its
merits, instead of tormenting one to this extent?" and Alois was
getting cross again.
"Why, it is the best chance in the world, you couldn't have been
more favoured! As to Andrae, he'll never guess it. Now just think what
answer you'll give."
"Oh, I should never guess any, if I thought till doomsday! But
you"--and he started with the clever thought--"you, of course, who
always find a way out of every thing--what do you say?"
"Why," answered Katharina, readily, "what is Stronger than the
earth on which we stand, which bears up our houses and buildings,
our rocks and mighty mountains, which all our united efforts could
not suffice to move one inch from its place, and on which we all rest
secure, confident that none is strong enough to displace it? What more
Beauteous than spring, with its fresh, soft tints on sky and mountain,
on alp [72] and mead, on blossom and flower--spring, with its promise
and its hope? And what Richer than autumn, with its gifts which make
us glad for all the year--its bursting ears of grain, its clustered
grapes, its abundant olives and luscious fruits?"
"Katharina, girl, I believe you've found it!" said her father, with
enthusiasm. "My bonny girl has saved me this time also!" and he clasped
her in his arms. Though misgivings would come back when he recalled
Andrae's assurance, he yet went to bed happy in the consciousness of
at least having a good chance of not being beaten.
In the morning he was up betimes, and, having taken great pains to
learn what he had to say from Katharina, who walked a good stretch
of the way through the valley with him, he arrived at the court in
tolerably good humour.
Andrae was there before him, and in high good humour too; taking for
granted that, as the richer and more important man, and, moreover, as
the victor (so he felt assured), he had the right to speak first. As
soon as the judge had taken his seat, and even before he had called
on him for his answer, he began,--
"Sir judge, I have the answer to your enigma; and as soon as I have
told it, you will please give judgment in my favour. It was indeed easy
enough to find, so I claim no merit in the discovery," he added, with
the pride that apes humility. "The most Beautiful thing on earth is
my wife, of course; the Strongest, are my oxen; and the Richest, am I."
The judge listened without moving a muscle of his countenance, as
became a judge, and for those who were too obtuse to perceive the fine
irony of the smile with which he bowed to the speaker at the conclusion
of his harangue--and among these was certainly Andrae himself--it
seemed as if he was quite satisfied with the answer. Nevertheless,
he turned to Alois, and said,--
"Well, my man, and what is your answer?"
"But the judgment, good sir judge! would your honour be pleased to
pronounce the sentence in my favour, seeing I have given your worship
the answer?" interposed Andrae Margesin, fussily.
"Gently and fairly!" replied the judge; "wait only a little: we must
hear what friend Alois has to say. He might have an answer, you know;
and, anyhow, we must give him the opportunity."
Andrae chafed, but could not resist; and, at an encouraging word from
the judge, Alois stood forward and repeated word for word the answer
Katharina had taught him.
Though the judge had preserved his imperturbability through the
expression of Andrae's silly bombast, this answer of Alois was
too much for his composure. He had only proposed the enigma as
the means of getting rid of a perplexing case. He had no idea but
that both peasants would bring an answer of which he could easily
expose the folly; and thus, neither having fulfilled the prescribed
terms, the case would fall through of itself, and he be saved from
further trouble. But he saw nothing to reply to Alois' solution
of his question, nor any means of escaping from giving judgment in
his favour. Every body acquiesced in the justice of the decision;
and even Andrae himself had nothing to say, but, crestfallen, and in
very different style from his confidence of the day before, he made
his exit while people were yet engaged with the discussion of Alois'
success, so as to avoid alike scorn and condolence.
The session over, the judge called Alois aside, and inquired how he
had come to find so accurate an answer; upon which Alois, who burnt
to proclaim the merit of his child, at once referred the honour
to Katharina.
"That is it, is it?" replied the judge. "I have often seen the girl at
church, and am not surprised that so comely a form is inhabited by so
clever a mind. Now, go home, and tell your daughter that if she finds
out the way to come to me without any clothes on, and yet not naked;
not by day, and yet not by night; and by a way which shall be neither
a high-road nor yet a by-path, I shall take the opportunity of her
so coming to ask her to be my wife."
Alois lost no time in returning home to tell the good news to his
daughter. "I suppose you'll find one of your clever ways of doing it,
though, for myself, I confess I don't understand a word of it."
"But do you really mean that that good, noble, handsome judge really
means to make his wife of a poor peasant girl like me?"
"He might do worse," answered her father, with archness and pride. "But
there is no doubt he was in earnest. You should have seen the fire
in his eye when he spoke!"
"In that case, you may depend I will find the way to fulfil his
directions: trust me for that!"
Nor was she long in finding a way which satisfied the judge
completely. She took off all her clothes, and then covered herself
with fishing-nets; this for the first condition. Then, for the second,
she timed her journey in the dusk of evening, which is neither called
day nor night; and, for the third, she had previously had the road
covered with boards, and upon these she walked, so that she neither
trod the high-road nor yet a by-path.
Delighted at acquiring such a prize, and having so clever a maiden for
his future companion through life, the judge married Katharina before
the end of the month. There were great rejoicings at the wedding, to
which all the country-side was invited; and then the poor peasant girl
was installed in the judge's house. The judge, however, had exacted of
her one condition, which was that she should never interfere with any
of her clever suggestions in any case brought before him for decision,
but let justice take its free and uninterrupted course.
Years passed by happily enough. The judge rejoiced more and more
every day over the wisdom of his choice, and Katharina sedulously
observed the condition imposed upon her, and never interfered with
her husband's dealings in the court.
Nevertheless, it happened one day that a peasant whom she had known
from her infancy had a case before the judge which was nearly as
perplexed as that of her father had been, and, despairing of making
his right apparent, the peasant came to Katharina, and begged her,
by their lifelong friendship, to give him one of those good counsels
for which she had been so famous at home in the days gone by.
Katharina urged her promise to her husband, and for a long time refused
to break it; but the wily peasant contrived to work on her vanity so
effectually, that at last, in an evil moment, she consented this once
to give her advice, exacting first a promise he would never tell any
one she had done so.
The case was this. Her friend's Senner [73] had been visited in the
night by a Saligen Fraulein [74], who had promised to milk his cow for
him, and every one knew that when a Saligen Fraulein milked a cow,
it gave three times as much milk as the wont. But being a poor man,
and having only one cow, he eked out his living by taking in cows to
graze on his allotment; and he also only had one milking-pail. The
Saligen Fraulein, therefore, when she had milked his pail full, had
been obliged to take a pail belonging to the man to whom the other
cows belonged, who was a rich man, and had a store of all sorts of
utensils. But the milk being in one of his pails, his Senner swore
that it had been milked from one of his cows, and refused to give
it up, though he had no right to it whatever; and he had declined
payment for the use of the pail.
Though the case had been argued since the first thing that morning,
they were no nearer arriving at a decision. Now the disputants had been
ordered to stand back while another case was called, but it would come
on again immediately; and in the meantime the poor peasant entreated
Katharina's counsel as his only chance of rescuing his milk before
it turned sour.
"I see one means, I think, of bringing him to his senses,"
said Katharina, after she had yielded to her poor friend's
importunity. "When your case is called on again, show as much
indifference about the result as you have hitherto shown anxiety;
then tell your adversary that during this interval, which you spent
in the shade of the woods, a Saligen Fraulein had appeared to you
and advised you not to use any of the milk the one who appeared to
the Senner had milked for you, because she was a mischievous one,
and the milk she milked was bewitched, so that all who drank of it,
or of any milk mixed with it--were it only one drop of it--would be
turned into asses. Then add, 'But of course, if your pailful is really
the milk of your own cow, you have nothing to fear; so there's an
end of the dispute.' Then he will probably be so frightened by the
threat of this calamity that he will probably have nothing more to
do with the pail; and that will suffice to prove that it is not the
milk of his cow, and expose his deceit."
The peasant was so delighted with the wise counsel that he hardly
knew how to thank his benefactress, and readily gave her the promise
she required of not letting any one know he had even seen her.
He had scarcely got back to the court when the case was called on
again. The peasant carried out the advice he had received with great
shrewdness, and found it answer completely. Every body applauded
the craft by which he had confounded his would-be oppressor, and
the judge himself was very much pleased to see the end of such a
troublesome case.
A few minutes' thought, however, suggested to him that there was
more than a peasant's shrewdness in the matter, and he was not slow
to discern the guiding of his wife in it; so he called the peasant
apart, and had little difficulty in wringing from the simple clown
a confession of who had been his prompter.
The discovery made the judge set off homeward in great anger. His wife
had broken her promise--the fundamental condition of their union;
and he would have nothing more to say to her! Out of his house she
must go, whithersoever she would, but far away out of his sight.
Katharina, who had so often calmed her father's anger by her prudent
reasoning, exerted herself to the utmost to bring her husband back to a
better mind; but in vain. And all the concessions he would yield were,
to consent that they should eat their last dinner together, and that
she should take away with her one thing out of the house, whatever
she had most fancy for. It was not much to obtain when required to
part for ever from her home, and her hopes, and all to which she had
grown united and attached--but it was all she could obtain.
Dinner-time came, and the judge, who was devotedly fond of his wife,
seemed lost in sorrow at the calamity about to befall him; still he
would not yield. Though she caressed him and entreated him to forgive
her, he still said he could not depart from his word, and he would
not allow her to speak of it. They sat down to their silent meal;
and as the time of separation drew nearer he grew more sombre and
sad, and at last determined to console himself with the red wine that
sparkled by his side. Katharina encouraged him to drink, and as his
bottle got exhausted deftly replaced it by a full one, so that he
was quite unconscious of the depth of his potations.
Presently the steward came into the room ready to drive Katharina
to whatever destination she should select, and, as he had heard it
stipulated that she was to take with her whatever she liked best,
proffered his services to assist in the removal--for she had won the
respect and affection of all her dependants, and they delighted to
be occupied for her.
Katharina rose to depart, thanked the man for his attention, and,
in answer to his question as to the object she would take with her,
pointed to her husband, who now lay helpless across his settle,
his head drooping over the table.
The steward could scarcely believe his eyes, but Katharina had a way
of giving orders which did not admit of being questioned. The first
surprise over, too, it struck him as a capital device, and he entered
heartily into the spirit of the scheme. With the help of a couple
of serving-men the judge was deposited safely in the lumbering old
carriage, and Katharina having taken her place beside him, they drove
away by her direction over one of the worst and most uneven roads in
the neighbourhood. The shaking of the vehicle presently awakened the
sleeper, who was, of course, quite at a loss to conceive where he
was, but, perceiving that he cut a rather silly figure, was ashamed
to ask his wife, who sat by his side as if there was nothing amiss,
and said nothing.
At last his curiosity got the better of his self-respect, and he
begged her to tell him what all this trundling and shaking meant.
Katharina in a few words recalled to him his cruel decree, at the
same time reminding him of his promise that she might take with her
what she liked best, and, throwing her arms round him, asked him if
there could be any doubt as to what that could be.
The judge perceived that his wife had once more shown her sense
and judgment, and was not sorry to find she had contrived this
opportunity of making up their difference. On renewing her petition
for forgiveness, he frankly gave her his pardon; and they drove back
home to live together in love and union to the end of their days.
THE GEESEHERDS.
There was once a peasant who had three sons, Karl, Stefan, and Josef;
but, as he was very poor, they often had scarcely enough to eat, and
were always complaining. So one day he told them that they should go
out, one at a time, into the world, and see whether they could do
better for themselves than he could do for them; and, having drawn
lots which should go first, it fell upon the youngest.
Josef was not altogether sorry to see a little of what the world was
made of, and started with break of day next morning on his travels. He
went begging about the country, but for a long time could find nothing
to make a living by. At last he came to a splendid mansion on the
borders of a large forest. When he asked his usual question, whether
there was any place vacant for him, the servants took him into the
big house; and, after conducting him through a number of apartments,
each more beautiful than the other, he was ushered into a vast hall,
all panelled round with carved wood, with windows of painted glass,
and filled with handsome furniture. Reclining in an easy-chair, sat an
aged nobleman, the owner of the mansion, who, when he heard Josef's
request, took compassion on him, and told him he would take him into
his service, beginning with giving him a very easy employment, and if
he proved himself faithful in that, he would promote him to something
higher. At first, then, he would only have to keep his geese; but
there was one condition he would bid him observe.
Josef was so delighted with the prospect that he hastily interposed
a promise of obeying it, before it was even uttered.
And that condition was, that if at any time he should hear any music
or singing in the forest, he should never listen to it, however much
he might be inclined, for that if he did, he would inevitably lose
his place.
Josef repeated his promise, and swore that he would never listen to
the music. He was then led down to the place where the other servants
were gathered for supper, and as there was a whole crowd of them,
and plenty of good food and drink, Josef began to think that he had
fallen on to his feet indeed!
After supper, Josef was shown into a tidy little room as big as his
father's whole cottage, where was a nice little white bed, and a suit
of clothes ready to put on when he got up. Though Josef liked good
food and a good bed, he was by no means an idle boy, but rose very
early in the morning for his new employment; and, having received
from the cook his breakfast, and his wallet of provisions for the
midday meal, turned out the geese, and drove them before him to the
meadow skirting the forest.
Josef had never seen so many geese together before, and all the
morning long he was never tired of looking at them, and counting them,
observing their ways, fancying he discerned various peculiarities
in each, by which to know one from another henceforth; and he began
to give them all different names. When one showed an inclination to
stray, what fun it was to drive her back, and see her flap her great,
soft, white, awkward wings, and stretch out her great yellow bill,
as with awkward gait she shambled back to the flock! So the morning
went by; and it was long past the hour of dinner before Josef found
any need of it, but when he did, he was astonished at the abundant
supply which had been provided for him. "Truly, I did well to come
out into the world," he thought, as he lounged upon the greensward,
eating the good food. "What a contrast between having this splendid
mansion to live in, and my father's poor cabin; between the dry
crusts we had to eat there, and the princely food allowed to us here;
between the toil and slavery there, and this easy kind of work, which
might more properly be called a pastime! My father thought to punish
me for grumbling, he would be astonished if he could see what a fine
exchange I have made!" and he laughed aloud, though all alone. But
presently the effects of the full meal, the heat of the afternoon, and
the excitement of his new position brought on sensations of lassitude
and somnolence--and soon you might have seen him stretched upon the
grass at full length, and snoring to his heart's content.
It is uncertain how long he had slept, but erewhile his slumber was
disturbed by the sound of the most enchanting strains of music. Josef
raised himself on his elbow, and listened; he had never imagined any
thing so beautiful! and when he had listened a little while, he grew
so rapt that he could not forbear going a little way into the forest
to hear it better, and then a little farther, and farther, till, by
the time it ceased, he was a long way from his charge. Then, as he
perceived this, for the first time he remembered the condition his
master had laid upon him, and his own positive promise to observe
it! In shame and confusion he hasted back; but in place of his
splendid flock of geese, there were but half a dozen, and those the
worst favoured, to be seen! It was vain he called after them, and
tore his hair, and ran hither and thither--no geese appeared! and as
it began to get dark, he found his best plan was to hurry home with
the few that remained.
When he arrived a servant was waiting to conduct him to the master. He
no longer wore the benevolent smile with which he had first instructed
Josef in the terms of his service. He looked so black and angry that
the boy was frightened to approach him--too frightened to find a word
in his defence.
"I had pity on you," said the master, "because you entreated me to
try you: you have broken your word, and I can trust you no more. I
told you the penalty; now you have chosen to incur it, you must go."
Josef could do nothing at first but cry, as he contemplated this sudden
extinction of his dreams of ease and plenty, but he took courage to
throw himself on his knees, and entreat one more trial. The master
was inexorable--only, as he was rich and generous, he would not let
him go away empty-handed, and he took out of a casket before him a
gold pin, as a memorial of his good intention, and dismissed the boy
with a gesture which admitted of no further parleying.
Josef was allowed to sleep in the mansion that night, but the next
morning, instead of carrying on his agreeable occupation of geeseherd,
he had to leave the place ignominiously, his rags being returned to him
in place of the smart livery of the castle. Uncertain whither next to
bend his steps, he determined to go home in the first instance and show
his gold pin, and then take a fresh start in search of another chance.
As he toiled up a steep Joch [75], feeling so thirsty that his eyes
went searching every where for a cottage where he might beg a sup of
milk, a hay-maker turned off the Hoch Alp [76] on to the road just
in front of him, with a cartload of hay he was hastening to take
home before rain fell. But, for all his urging, the oxen could not
turn the cart, and there it stuck in the edge of the road. Seeing
our stout youth coming along, the man called to him to help him
lift the wheel, promising him a bowl of milk in return. Josef was a
good-natured lad, and, as we have said, by no means indisposed for
exertion, so he set to work with a will, and the team was very soon
put in motion. He travelled on by the side of the cart, and when they
reached the Hof for which it was destined, Josef received a bowl of
milk, which refreshed him for the rest of the journey.
As he got near his father's cottage he went to take out the gold
pin with glee, to have it ready to display. Great was his vexation,
therefore, at discovering he had it no longer--nor could his searching
bring it back any more than the geese! Josef burst into tears,
and joined the family meal at home, which was just prepared as he
arrived, with his head low bowed, as if he sought to hide himself
for very shame.
When his father saw him in such melancholy plight, his compassion
warmed to him, and he asked him kindly what had befallen. Josef told
all his adventures, crying afresh as he came to the narration of how
he had lost on the way the gold pin, to display which he had come
home before starting in search of another chance of employment.
"Such chances don't grow as thick as black-berries," said Stefan, the
second son: "instead of your going in search of another, I'll go to
the same grand house; and I won't lose such a fine situation for the
sake of 'tweedle-dum,' I can tell you! And whatever I get for wages,
you may depend, I won't stick it in my belt where it is sure to be
brushed away, but on the brim of my hat, to be sure!"
Josef, who had had enough of trying to provide for himself, and was
not sorry to be at home again, even with its scanty means, made no
objection, and their father, thinking it well Stefan should have his
experience of life too, approved the plan.
Stefan set out next morning, therefore; and by following Josef's
directions soon discovered the stately palace for which he was
bound. The noble owner received him as kindly as Josef, and sent
him out to the same employment, first binding him to observe the
same condition. Stefan readily promised to keep it, and was formally
installed into his office of geeseherd.
All went well enough at first, as with Josef; but it was at an even
earlier period of the day than with him that his curiosity was roused
by the fairy-like music. Then he, too, followed it through the forest;
and when it ceased at sound of the church bells ringing the Ave,
he found not more than three or four geese left of all his flock!
On his return the master was full of anger at his breach of trust,
and inexorably resolved to turn him away; but not to let him go
empty-handed, he gave him a little lamb to take home.
Stefan was pleased enough with his prize, but was somewhat embarrassed
as to the manner of carrying it safely home. He had declared that
whatever he got he would bring home on his hat, and though he had
never thought of so embarrassing a present falling to him, at the
time he spoke, he resolved to keep his word, and so used his best
endeavours to fix the little creature round the brim.
He carried it thus great part of the way in safety, but having to cross
a somewhat rapid stream, a projecting bough of a tree lifted his hat
from his head--and both hat and lamb fell in, and were carried fast
away by the torrent!
Stefan came back even more crestfallen than Josef; and, having told his
story, Karl, the eldest, with great indignation at the carelessness of
his brothers, declared that he would make the trial next. He would not
stick his prize in his belt or his hat, not he! he would carry it by
a string, and then it couldn't get loose; and as for the music, he had
no fear of being led away by that. Josef, indeed, had had some excuse,
as the strains took him by surprise, but to be so foolish as Stefan,
after the warning example of another, was perfectly contemptible. He
couldn't be so silly as that, not he!
He started on his way betimes, and toiled along not without some
misgivings lest he should find so good a post already occupied by
another. But it was not so: the owner of the mansion gave him the same
reception, the same charge, and the same warning as the other two;
and, full of confidence in his superiority, he went forth to his work.
The weather was cool, and he had no need to seek the shade of the
forest trees; and for more than a week he brought the full tale of
geese home day by day. "What idiots those were to throw away their
place for the sake of a little music!" he thought to himself one day
later. "I told them I should not be so foolish--not I! I told them
I shouldn't be led away by it, and I haven't been."
But it was hotter that day, and in the afternoon, when the sun's
power was greatest--forgetting the warning of his brothers' example,
or rather setting it at defiance, with the assurance that though he
sought the shade he need not listen to the music--he crept within
the border of the cool forest, and lay down.
He had hardly done so when his senses were rapt by the delicious but
deceitful strains. "The woods must be full of fairies!" he cried;
"this can be no earthly music--I must follow it up and see what manner
of instruments they are, for never on earth was heard the like!" But
as he went on, the music always seemed farther off, and farther again,
till at last the church bells rang the Ave, and the music ceased.
Then Karl woke to a sense of his weakness and folly; and though he ran
every step of the way back to his geese, only two were there! Though
he had now found the same fate befall himself as his brothers, in all
particulars, yet he could not forbear searching for the lost geese;
but of course it was in vain, and he had to return to the castle
with but two. Nothing could look more miserable, or more ludicrous,
than this diminished procession--Karl at the head of his two geese,
who had gone out in the morning with such a goodly flock.
He would have gladly slunk away without exchanging a word with any
one, but he could not escape being taken before the master, who
scolded him in the same words in which he had chided his brothers,
but gave him a fine rich cake to take home.
The cake was round, and it was very inconvenient to attempt to secure
it by means of a string, but Karl had declared he would bring home
his reward that way, and so it was a point of honour with him to
do it. But passing by a Hof, on his way home, where was a large
and powerful watch-dog on guard, he set off running to escape its
grip. This was the very way to attract the beast's notice, however;
and off it set in pursuit, much faster than Karl's legs could carry
him away--and then, having jumped upon him and knocked him down,
seized his cake, and devoured it before his eyes!
Karl had now to go home as empty-handed as his brothers, and as full
of tears; but his father comforted him, and checked the rising gibe
of his youngers by reminding them that all had failed equally; so
they all joined in a good-humoured laugh in which there was nothing
of bitterness.
The father then asked them if any of them wished to go out into
the world and seek fortune again; but they all agreed that there was
nothing to be gained by the move, and that though there were positions
which at first sight seemed more brilliant and more delectable than
their own, yet that each had its compensatory trials, and that they
were best where God had placed them.
Henceforth, however, they were ashamed of renewing their grumblings,
but, each making the best of his lot, they became noted as the most
contented and, therefore, happiest family of the whole valley.
ST. PETER'S THREE LOAVES [77].
In the days when our Lord and Saviour walked this earth with His
apostles, it happened one day that He was passing, with St. Peter
for His companion, through a secluded valley, and that discoursing,
as was His wont, of the things of the Kingdom of God, and raising the
mind of His disciple from the earthly to the heavenly, they noticed
not how the hours went by. Nevertheless, they had been walking since
daybreak over rough mountain tracks and across swollen torrents many
a weary mile, and had eaten nothing all day, for their way had led
them far from the haunts of men; but as noon came down upon them they
approached the precincts of a scattered hamlet. The bells of all the
large farm-houses were ringing to call in the labourers from the field
to their midday meal, and announced a community of sensations in the
world around akin to those with which St. Peter had for a long time
past been tormented. The heat increased, and the way grew more weary,
and St. Peter found it more and more difficult to keep his attention
alive to his Master's teaching.
The merciful Saviour was not slow to perceive what ailed His disciple,
and kept on the look-out for any opportunity of satisfying him as
anxiously as if the need had been His own; and thus, while St. Peter
was still wondering how long he would have to go on fasting, He
remarked to him the smell of fresh-baked loaves proceeding from a
cottage at the bottom of the valley.
St. Peter could as yet perceive neither the scent nor the
cottage. Nevertheless, used as he was to trust his Lord's word
implicitly, he started at His bidding, following the direction pointed
out just as if both had been patent to himself.
The way was so steep and rough that St. Peter, in his eagerness, had
many falls, but at last, without much damage, reached nearly the foot
of the mountain range along the side of which they had been journeying;
and then suddenly the smell of a wood fire, mingled with the welcome
odour of fresh-baked bread, greeted him. The roof of the cottage
was just beneath his feet, and the smoke was curling up through the
chimney, telling of a well-provided stove, burning to good purpose,
close at hand. One or two more winds of the road, and only one more
slip over the loose stones, brought him to the door.
A comely peasant wife opened it at his knock with a cheerful greeting:
"Gelobt sei Jesus Christus [78]!"
The apostle, having given the customary response, "In Ewigkeit! Amen,"
the peasant wife asked him to come in and rest--an offer which
St. Peter gladly accepted.
The peasant woman wiped a chair, and presented it to him, and, with
some pleasant words about his journey, returned to her occupation at
the fire. The moment had just arrived when she should take her loaves
from the oven, and nothing could smell more tempting to a man whose
appetite was seasoned by a long walk in the fresh mountain air.
"Good woman, I come from far, and the whole of this blessed morning,"
he exclaimed, speaking as one of the people, "I have tasted
nothing! ... nor my companion," he added, with some embarrassment
lest he should seem encroaching, yet full of anxiety to provide for
his Master's needs as well as his own.
"Tasted nothing all this morning!" exclaimed the compassionate peasant
wife, scarcely leaving him time to speak; "poor soul! Why didn't you
say so at first? Here, take one of these loaves; they are the best
I have, and, if humble fare, are at all events quite fresh. And your
companion too, did you say? Take one for him also;" and then, as if
she found so much pleasure in the exercise of hospitality that she
could not refrain from indulging it further, she added, "and take this
one too, if you will; maybe you may want it before the journey is out."
St. Peter thanked her heartily for her generosity, and hasted to take
the loaves to the Master, that He might bless and break them. But
as they were hot, being just out of the oven, he had to wrap them in
the folds of his coarse grey mantle, to be able to hold them without
burning his hands.
As he toiled up the steep, the thought came to him, "It will most
likely be long before we have a chance of meeting with provisions
again, and I always seem to want food sooner than the Master; I might
very well keep this third loaf under my cloak, and then in the night,
while He is lost in heavenly contemplation, and I am perishing with
hunger, I shall have something to satisfy it. I do Him no wrong, for
He never feels these privations as I do--at all events," he added,
with some misgivings, "He never seems to."
With that he reached the place where he had left the Saviour. He was
still kneeling beneath the shade of a knoll of pines. As St. Peter
approached, however, though He was not turned so as to see him coming,
He rose, as if He knew of his presence, and, coming to meet him,
asked him cheerfully what success he had in his catering.
"Excellent success, Lord," replied St. Peter. "I arrived just at the
right moment. The woman was taking the loaves out of the oven, and,
being a good-hearted soul, she gave me one; and when I told her I
had a companion with me, she gave me another, without requiring any
proof of the assertion; so come, and let us break our fast, for it
is time." But he said no word about the third loaf, which he kept
tight in a fold of his mantle under his arm.
They sat down on a rock by the side of a sparkling rivulet, hasting
along its way to swell the far-off river, and its cool crystal waters
supplied the nectar of their meal.
St. Peter, who had now long studied in the school of mortification
of his Master, was quite satisfied with this frugal repast, and,
no longer tortured by the cravings of nature, listened with all
his wonted delight and enthusiasm to every word which fell from the
Lord's lips, treasuring them up that not one might be lost. It was
true that he could not suppress some little embarrassment when the
thought of the third loaf occurred to him; "But," he said, to himself,
"there could be no possible harm in it; the woman had clearly given it
to him; his Lord didn't want it, and he was only keeping it for his
needs. True, if He were to suspect it, He would not quite like that;
but then, why should He? He never suspects any one."
Never had the Saviour been more familiar, more confiding. St. Peter
felt the full charm of His presence and forgot all his misgivings,
and the cause of them, too, in the joy of listening to Him. Then came
a friendly bird, and hopped round Him, feeding on the crumbs that had
fallen. The Saviour, as He watched its eagerness, fed it with pieces
from His own loaf. Another bird was attracted at the sight--another,
and another, and another, till there was a whole flock gathered
round. The Saviour fed them all, and yet He seemed to take His own
meal too.
"It is just as I thought," St. Peter reasoned with himself; "His
needs are not as our needs. Decidedly I do Him no wrong in keeping
the loaf for my own." And he felt quite at ease.
The simple repast was at an end; the birds chirped their thanks and
flew away; and the disciple and the Master rose from their rocky seat.
St. Peter, leaning on his staff, set out to resume the journey,
but the Lord called him back.
"Our Father in heaven has fed us well, shall we not thank Him as is
our wont?"
St. Peter laid aside his staff, and cheerfully knelt down.
"But as He has dealt with particular loving-kindness in the abundance
with which He has provided us this day, let us address Him with
arms outstretched, in token of the earnestness of our gratitude,"
continued the Saviour; and as He spoke He flung His arms wide abroad,
as if embracing the whole universe and its Creator, with an expression
of ineffable love.
He knelt opposite St. Peter, who was not wont to be slow in following
such an exhortation.
"He only suggested it; He didn't command." reasoned St. Peter to
himself. "I need not do it."
But a furtive glance he could not repress, met the Master's eye fixed
upon him with its whole wonted affection--there was no resisting the
appeal. With the spontaneity of habitual compliance, he raised his
arms after the pattern of his Lord; but the loaf, set free by the
motion, fell heavily to the ground beneath the Master's eye.
The Master continued praying, as though He had perceived nothing,
but St. Peter's cheeks were suffused with a glow of shame; and before
they proceeded farther he had told Him all.
THE TWO COUSINS OF ST. PETER [79].
St. Peter had two young cousins whom he sought to bring up in the
way of righteousness according to Christian doctrine. As they were
very docile, and listened gladly to his word, he strove to lead them
in the way of all perfection; and to this end counselled them to give
themselves up entirely to serve God in a community of His handmaidens,
where they should live for the Divine spouse of their souls, and for
Him alone.
The work of the Church called St. Peter away from the East, and he
was already gone to establish the faith in Rome before the maidens
had decided as to their vocation. It was not till many years after
that St. Peter heard, to his surprise, on occasion of St. Timothy
coming to visit St. Paul in Rome, that while the youngest indeed had
fulfilled his expectations, and had given herself up to the religious
life, the elder had married and established herself in the world,
and become the mother of a large family.
During his long confinement in the dark dungeon of the Mamertine
prison, St. Peter's thoughts would often revert from the immense
cares of his sublime office to the quiet hours he had passed in the
lowly dwelling by the Lake of Tiberias, where his pious cousins had
so often sat at his feet listening to his instructions. And he found
a peaceful pleasure in recalling the way in which they had responded
to them; the spontaneity with which they had apprehended the maxims
of the new religion; their fervour in applying them to their own rule
of life; their readiness to go beyond what was bidden them, that so
they might testify their love for their Divine Master; their delight
in all that reminded them of God and His law.
"And to think that one of them should have gone back from
all this! should have been content to give up these exalted
aspirations! How sadly her ardour must have cooled! What could have
worked this change?" the apostle would muse, in his distress, and pray
silently for her forgiveness and guidance; but his thoughts would
revert with greater affection and satisfaction to the more favoured
state of the soul of the younger sister.
It was not long before the terrible decree of Nero consigning St. Peter
to the death of the cross was pronounced, and from the height of the
Janiculum he was received into the celestial mansion to keep the gate
of the Kingdom of Heaven.
He had not exercised this office many years when our Lord called him
to Him one day, and bid him open the gate of heaven to its widest
stretch and deck its approaches as for a high festival, for that one
of the holiest of earth and the dearest to Himself was to be received
into the abode of the Blessed.
"That must be my youngest cousin," said St. Peter, "there is no doubt;
she who generously gave up a world in which she was so well adapted
to shine, to live a life of perfection with God above only for its
object;" and he strained his eyes to see far along the approach to
Paradise, that he might catch the first glimpse of her glorified soul
and greet it with the earliest welcome.
How great was his surprise then, when roused by the melodious
strains of the angelic host escorting her, to hear in the refrain
of their chant the name of the Sorellona [80], not of the younger
of the sisters! Meantime the celestial cortege was wafted by, and
the beautiful spirit was welcomed by the Divine Master Himself,
and placed on one of the highest seats in His kingdom.
Not many days after our Lord called St. Peter to Him again, and
told him to open the gate a little, very little way, and to make no
preparations for rejoicing, for He had promised admission to a soul
who, though of his family, yet had only escaped being excluded by a
hair's breadth.
St. Peter went away perplexed, for he knew there was no one of his
family who could be coming to heaven just at that time except the
younger of the two cousins, and how could the Lord's words apply
to her?
He durst do no more than open the gate a very little way, but stationed
himself opposite that small cleft to obtain the earliest information
as to who the new comer really was.
Presently a solitary angel came soaring--the only escort of a
trembling soul--and, as he approached, without chorus or melody,
he begged admission for one whom, by the name, St. Peter discerned
was actually the Sorellotta [81] he had deemed so meritorious! With
great difficulty, and by the help of the angel who conducted her,
and of St. Peter himself, she succeeded in passing the sacred portal;
and after she had been led to the footstool of the Heavenly Throne in
silence, He who sat on it pointed to a very little, low, distant seat,
as the one assigned to her.
When St. Peter afterwards came to discourse with the Lord about His
dealings with the two souls, he learnt that she who performed her duty
with great exactness and perfection in the world was more pleasing
in His sight than she who, while straining after the fulfilment of
a higher rule, yet fell short of correspondence with so great a grace.
LUXEHALE'S WIVES.
The Devil goes wandering over the earth in many disguises, and that
not only to hunt souls; sometimes it is to choose for himself a wife,
but when he goes on these expeditions he calls himself "Luxehale."
There was once a very beautiful princess, very proud of her beauty, who
had vowed she would never marry any but the handsomest prince. Numbers
of princes, who heard the fame of her beauty, came to ask her hand,
but directly she saw them she declared they were not handsome enough
for her, and drove them out of the city. Her parents were in despair,
for there was scarcely any young prince left in the world whom she
had not thus rejected.
One day the trumpeters sounded the call by which they were wont to
announce the arrival of a visitor.
The princess sat with her mother in an arbour.
"Ah!" said the queen, "there is another come to ask your hand. How
I wish he may be the really handsome one you desire, this time!"
"It is all useless, mother; I don't mean to see any more of them--they
are all uglier, one than the other."
The queen was about to answer by instancing several noted paragons of
manly beauty whom she had rejected like the rest, but the chamberlain
came in with great importance just at that moment, to say that the
prince who had just arrived appeared to be a very great prince indeed,
and that he was in a great hurry, and demanded to see the princess
instantly.
The princess was very indignant at this abrupt proposal, and refused
absolutely to see him; but at last the queen got her to consent to
place herself in a hollow pillar in the great reception-hall, and
through a little peephole, contrived in the decorations, take a view
of him without his knowing that she did so.
When the princess thus saw the stranger, she was dazzled with
the perfection of his form and the surpassing beauty of his
countenance, and she could hardly restrain herself from darting
from her hiding-place and offering him her hand at once; in order
to preserve herself from committing such a mistake, she immediately
let herself down through a little trap-door into the room below,
where it had been agreed that her mother should meet her.
"Well, what did you think of him?" said the queen, who did not keep
her long waiting.
"Oh! I think he might do," said the princess, with an assumed air
of indifference, for she was too proud to acknowledge how much she
admired him.
The queen was overjoyed that at length she consented to marry, and so
put an end to the anxiety she was in to see her established before she
died. That she might not take it into her head to go back from what she
had said, her parents hastened on the wedding preparations, and the
prince seemed very anxious, too, that no delay should occur. As soon
as the festivities were over, he handed his bride into a magnificent
gold coach, and drove off with her, followed by a retinue which showed
he was a very great prince indeed.
Away they rode many days' journey, till at last they reached a palace
of greater magnificence than any thing the princess had ever conceived,
filled with crowds of servants, who fulfilled her least wish almost
before it was uttered, and where every pleasure and every gratification
was provided for her in abundance.
The prince took great pleasure in conducting her frequently over every
part of the palace, and it was so vast that, after she had been over
it many times, there was still much which seemed strange to her;
but what was strangest of all was, there was one high door, all of
adamant, which the prince never opened, and the only cross word he
had spoken to her was once when she had asked him whither it led.
After some time it happened that the prince had to go on a considerable
journey, and before he left he confided to his wife the keys of all
the apartments in the palace, but she observed the key of the adamant
door was not among them, and ventured to ask why it was not.
"Because no one passes through that door but myself; and I advise
you not to think any thing more about that door, or you may be sure
you will repent it," and he spoke very sternly and positively.
This only whetted her curiosity still more; and she was no sooner
sure he was at a safe distance, than she determined to go down and see
if some of the keys would not open this door. The first she tried in
it showed there was no need of any, for it was unlocked, and pushed
open at her touch. It gave entrance to a long underground passage,
which received a strange lurid light from the opening at the far end.
The princess pursued the ominous corridor with beating heart; and,
when she reached the other end, made the frightful discovery that it
was--the entrance to hell!
Without losing a moment, she rushed up-stairs, regained her own
apartment, and sat down to contrive her escape, for she now perceived
that it was the Devil, disguised as a beautiful prince, that she
had married!
As she sat, pursued by a thousand agonizing thoughts, the gentle
cooing of two pigeons in a cage soothed her, and reminded her of home.
Her father's fondness had suggested that she should take the birds
with her that she might have the means of communicating to him how
it fared with her in her married home. Quickly she now wrote a note
to tell him of the discovery she had made, and begging him to deliver
her. She tied the note to one of the pigeons, and let them fly.
The Devil came back in the same disguise, and was profuse in his
caresses; and he never thought of her having opened the door. But all
the princess's affection and admiration for him were gone, and it was
with the greatest difficulty she contrived to keep up an appearance
of the fondness she had formerly so warmly and so sincerely lavished.
Meantime the pigeons went on their way, and brought the note home. The
king and queen were having dinner on the terrace, and with them
sat a young stranger, named Berthold, conversing with them, but too
sad to taste the food before him. He was one of those the princess
had rejected without seeing, but as he had seen her, he was deeply
distressed at the present separation. The pigeons flew tamely in
narrowing circles round the king's head, and, at last, the one which
carried the note came fluttering on to the table before him. He would
have driven them away, the rather, that they were all distressed and
bleeding, and with scarcely a feather left, but the young stranger's
eye discovered the note, which was quickly opened and read.
"Oh, help me! What can I do?" exclaimed the king; "give me some
counsel. How can I ever reach the Devil's palace--and how could I
fight him, if even I did get there?"
"May I be permitted to undertake the deliverance?" asked the stranger.
"Oh, in heaven's name, yes!" cried the king.
"And shall I have your permission to pay my addresses to her when I
bring her back?"
"Why, she will be yours--yours of right, if you succeed in rescuing
her; altogether yours!"
"That must depend on herself. Nevertheless, if I have your consent
to ask her in marriage, that is all I desire."
"Go, and succeed!" devoutly exclaimed the king. "And whatever you
stand in need of, be it men or money, or arms, you have but to command,
and every thing shall be given you that you require."
But the prince, who knew not what sort of enemy he had to encounter,
or which way he had to go, knew not what assistance to ask for,
but set out, trusting in God and his own good sense to guide him.
As he passed out of the castle enclosure his eyes were rejoiced to see
lying on the ground some of the white feathers of the carrier-pigeons,
and then he perceived that, not having been duly matched, they had
fought all the way, and that the whole track was marked with their
feathers. But as they, of course, had come by the directest course,
it led him over steep precipices and wild, unfrequented places;
still Berthold pursued his way through all difficulties without losing
courage, and ever as he went pondering in his own mind with what arts
he should meet the Devil.
He was passing through a desolate stony place, which seemed far from
any habitations of men, when he saw a man crouching by the wayside,
with his ear close against the rock.
"What are you doing there?" said Berthold.
"I am listening to what is going on in the Devil's house," answered the
man, "for my sense of hearing is so fine, it carries as far as that."
"Then come with me," said Berthold; "I will find work for you which
shall be well repaid."
So the man left off listening, and walked on behind him.
A little farther on, he observed a man sitting on a ledge of the
precipice, with his back to the road, and with all the world before
him; and he gazed out into the far distance.
"What are you staring at?" said Berthold.
"I am gazing into the Devil's house," said the man, "for my sight is
so sharp, it carries as far as that."
"Then come along with me; I will give your eyes work that shall be
well paid." said Berthold.
So the man left off gazing, and turned and walked behind him.
"But stop!" said the prince; "let me have some little proof that you
are as clever as you say. If you can see and hear into the Devil's
house, let me know what the Devil's wife is doing."
Then the first man crouched down with his ear against the rock;
and the second man sat himself astride on a jutting projection of
the precipice, and gazed abroad over the open space--Berthold taking
care that they should be far enough apart not to communicate with
each other.
"What do you see?" he said, when the second man had poised himself
to his own satisfaction.
"I see a vast apartment, all of shining crystal, and the Devil lying
fast asleep on a ledge of the flaming spar, while the Devil's wife
sits with averted face, and weeps."
"And what do you hear?" he said, returning to the first man.
"I hear the Devil snore like the roaring of a wild beast, and I hear
great sighs of a soft woman's voice; and every now and then she says,
'Why was I so foolish and haughty, as to send away all those noble
princes whom I might have learnt to love? and above all, Berthold, whom
I would not see, and who my mother said was better than them all; and I
would not see him! If I could but see him now, how I would love him!'"
When Berthold heard that, he could not rest a minute longer, but told
them he was satisfied; and hurried on so fast that they could scarce
keep up with him.
On they went thus; and presently they saw a man amusing himself with
lifting great boulders of rock, which he did so deftly that no one
could hear him move them.
"You have a rare talent," said Berthold; "come along with me, and I
will pay your service well."
So the man put down a great mass of rock he had in his arms, and
walked on behind the prince.
Presently there were no more pigeons' feathers to be seen, and Berthold
wrung his hands in despair at losing the track.
"See!" said the man with the sharp sight, "there they lie, all down
this steep, and along yonder valley, and over that high mountain! it
will take three months to traverse that valley."
"But it is impossible to follow along there at all!" cried all the
men. But Berthold said they must find their way somehow.
While they were looking about to find a path to descend by, they
saw a great eagle soaring round and round, flapping her wings, and
uttering plaintive cries.
"I'll tell you what's the matter," said the man with the sharp hearing:
"one of her eggs has fallen down this ledge, and it is too narrow
for her to get it out; I can hear the heart of the eaglet beating
through the shell."
"Eagle," said the prince, "if I take out your egg, and give it to you,
will you do something for me?"
"Oh, yes, any thing!" said the eagle.
"Well, that is a hot, sunny ledge," said the prince; "your egg won't
hurt there till we come back--I have seen in my travels some birds
which hatch their eggs entirely in the hot sand. Now you take us
all on your back, and fly with us along the track wherever you see
the pigeons' feathers, and wait a few minutes while we complete our
business there, and then bring us back; and then I'll take your egg
out of the fissure for you."
"That's not much to do!" said the eagle; "jump up, all of you."
So they all got on the eagle's back, the prince taking care so to
arrange his men that the great neck and outstretched wings of the eagle
should hide them from the Devil's sight, should he have happened to
be outside his house.
It took the eagle only two or three hours to reach the journey's end,
and by this time it was night.
"And now it is dark," said Berthold, to the sharp-visioned man, as
they alighted from the eagle's back, "you cannot help us any more
with your sight."
"Oh, yes; the crystals of the Devil's apartment always glitter with
the same red glare by night or day. I see the Devil rolled up in bed
fast asleep, and his wife sits on a chair by his side, and weeps."
"And what do you hear?" he said, addressing the first attendant.
"I hear snoring and weeping, as before," said the man addressed.
"Now you, who are so clever at lifting weights without being heard,"
said the prince, "lift the great door off its hinges."
"That's done," replied the man, a minute later, for he had done it
so quietly Berthold was not aware he had moved from the spot.
"Since you have done this so well, I'm sure you'll do the next
job. You have now to go up into the Devil's room, and bring the lady
down without the least noise; if you show her this token, she will
recognize it for her father's device, and will come with you."
The sharp-visioned man told him how he would have to go, for he could
see all the inside of the house, lighted up as it was with the glaring
crystals. But just as he was about to start,--
"Stop!" cried the man with the sharp ears; "I hear the Devil turn
in his bed; our talking must have disturbed him." So they all stood
stock still in great fear.
"He seems to be getting up," whispered the man with the sharp
sight. "No; now he has turned round and rolled himself up once more."
"And now he is snoring again," continued the other.
"Then we may proceed," replied the prince; and the third attendant
went his way so softly that no one heard him go.
"Get up on the eagle's back," said Berthold to the other two, "that
we may be ready to start immediately." So the men took their places.
They had hardly done so when the man came back bearing the princess,
and at a sign from Berthold sprang with her on to the eagle's neck. The
prince got up behind, and away flew the eagle--so swiftly that had
he been less collected he might have lost his balance before he had
secured his seat.
By daybreak they had reached the spot where the eagle's egg had
fallen. Berthold willingly exerted himself to restore her treasure
to her, and she was so grateful that she proposed to fly with them
home the remainder of the journey--an offer which they gladly accepted.
The Devil was still sleeping and snoring, they were assured by the
clever attendants; and away they sped, reaching home just as the king
and queen were sitting down to breakfast.
Great was the rejoicing in all the palace. The princess gladly
acknowledged Berthold's service by giving him her hand; and to all
three attendants high offices were given at court. To the eagle
was offered a gold cage and two attendants to wait on her, but she
preferred liberty on her own high mountain, and flew away, accepting
no reward but a lamb to carry home to her young ones.
When Luxehale woke next morning great was his fury to find that the
princess was gone.
"Order out a troop of horse, and send and demolish her palace, and
kill all belonging to her, and bring her home again," was the advice
of his chamberlain.
"No," replied Luxehale; "I hate violence: I have other ways at command
which I find answer better. There are people enough in the world
glad enough to follow me willingly. It is not worth while to give
myself much trouble with those who resist." And he dressed himself,
and walked out.
This time his steps were not directed towards a grand palace. He didn't
care particularly about birth or cultivation. There was a cottage
situated just above one of the alleys of his pleasure-grounds where
lived three beautiful peasant girls with an old father. Luxehale had
often listened to their merry laugh and thought how he should like
to have one of them for his wife; but he never could find any means
of getting at them, as they were very quiet and modest, and never
would enter into conversation with any stranger.
As he now walked along he heard their voices in earnest talk.
"It's great nonsense of father selling all the celery, and not letting
us have a taste of it!" said one, in a discontented voice.
"Yes, it is; I don't mean to submit to it either," said another.
"Oh, but you wouldn't disobey father!" said the first.
"Well, it's not such a great matter," replied the other; "only a foot
of celery [82]!"
Luxehale was very glad when he heard that, for he had never been able
to catch them in an act of disobedience before. He placed himself
under the celery-bed and watched all the roots. The moment one began
to shake, showing that they were pulling it up, Luxehale took hold of
the root, and held it hard, so that, instead of their pulling it up,
he contrived to drag down the girl who was trying to gather it.
It was the peasant's eldest daughter Lucia; and much surprised was she,
after passing through the hole Luxehale had made in the earth, to find
herself in the arms of a handsome cavalier, who lavished the greatest
care on her! Lucia had never been spoken to by such a good-looking
gallant before, and felt much pleased with his attention. She begged
him, however, to let her go; but he told her that was impossible. She
was his captive, and he never meant to let her go again; but if she
would only be quiet and reasonable she would be happier than any queen;
that he would take her to a magnificent palace where she would have
every thing she desired, and be as happy as the day was long, for
he would make her his wife. In fact, he succeeded in dazzling her so
with his promises that she began to feel a pleasure in going with him.
Nor did he break these promises. She was installed into all the
enjoyments of which we have seen the former wife in possession;
and as the Devil admired her beauty, and flattered and fondled her,
she did not altogether regret her captivity. But when the time came
that he had to go upon earth about his business, he brought her all
the keys of the place, with the express recommendation that she was
never to attempt to open the adamant door; then he plucked a red rose,
and placed it in her bosom, as a memorial of him, which he promised
should not fade till his return, and departed.
Lucia amused herself very well at first with various occupations and
amusements the palace afforded, and which were new to her; but as the
days fled by she began to grow weary, and at last, from being tired
and out of spirits with her loneliness, she became possessed with
so intense a curiosity to see what lay hid behind the adamant door,
that she could not resist it.
Accordingly she went down at last, with the bunch of keys in her
hand, and with trembling steps made her way up to it. But, without
even trying one of the keys, she found her touch pushed it open,
and made the terrible discovery, that it was the gate of hell! She
turned to escape, and rushed back to her apartment, to weep bitterly
over her forlorn condition.
Two or three days later a train of waggons came laden with beautiful
presents Luxehale had bought and sent home to amuse her, and she
became so interested in turning them all over, that when he returned
she was as bright and smiling as if nothing had happened.
Luxehale ran to embrace her, but suddenly observed that the rose
had withered on her bosom! When he saw that, he pushed her from
him. He had given it to her as a test to ascertain whether she had
gone through the adamant door, for the heat of the fire was sure to
tarnish it--and now he knew she was in possession of his secret.
"You have opened the adamant door!" he exclaimed, fiercely; and she,
seeing him so fierce, thought it better to deny it.
"It is useless to deny it," he replied; "for nothing else would have
tarnished that rose." And saying that, he dragged her down to it and
thrust her within its enclosure, saying, "You wanted to know what
there was behind the adamant door; now you will know all about it."
Luxehale now had to look out for another wife. He at once bethought
him of Lucia's sisters, and went pacing up and down under their garden,
as before. The two sisters were talking with some warmth.
"I don't see why father should have forbidden us to look through the
trellis!" said the voice which had spoken first on the former occasion.
"Nor I," said the other. "And I don't mean to be kept in in that
style either," said the other.
Quick as thought the Devil transformed himself into a serpent and
worked his way up through the earth to the other side of the trellis,
where he waited till the maiden put her head through, as she had
threatened. She had no sooner done so than he caught her in his coils
and carried her down under the earth. Before she had time to recover
from her surprise, he had transformed himself back into the handsome
cavalier who had charmed Lucia.
It was the second sister, Orsola; and her opposition to his advances
was as easily overcome as Lucia's. She lived in the palace as Lucia
had done, and learnt to feel great delight in its pleasures. At last
the day came when the Devil had to go upon earth about his business,
and he left her with the same charge about the adamant door, and
placed a red rose on her breast, which he promised should not fade
till his return. After a time her weariness induced Orsola to peep
through the fatal door; and the hot blast which escaped as she opened
it would have been sufficient to drive her away, but that it came
charged with the sound of a familiar voice!
"Lucia!" she screamed, in a voice thrilled with horror.
"Orsola!" returned her unhappy sister, in a tone of agony.
Orsola knew enough. She did not dare venture farther; and as she made
her way back to her apartment she saw in the court below the retinue
which had escorted her husband back. Assuming as composed a mien as
possible, she went out to meet him, and he ran towards her with every
appearance of affection--but his eye caught the withered rose.
"You have opened the adamant door," he said, sternly. "There is no
help for you; those who once pass it cannot live up above here any
more. You must go back, and live there for ever!" And, regardless of
her entreaties and cries, he dragged her down, and thrust her into
the burning pit.
Luxehale now had to search for another wife, and he determined it
should be no other than the third of the sisters. "But," he reflected,
as he walked towards her cottage, "now she has no one left to talk to,
how shall I manage? Ah, well, I generally find a way to do most things
I take in hand--and if I don't catch her I needn't break my heart;
there are plenty of girls in the world whom I have arts to enthrall."
But he did hear her voice. As he got near she was singing, very
sadly and sweetly, a verse which told her regrets for her sisters,
and called on them to return.
"That's all right!" said Luxehale, "she is sure to come to the spot
where she last saw her sister. I'll be there!"
So, transforming himself once more into a serpent, he wriggled
through the earth and took up his place of observation beside the
trellis. He had not been there long, when she actually came up to it,
singing the same melancholy strains; and then she stopped to call,
"Lucia! Orsola! Lucia! Orsola!" till the woods rang again. Then she
seemed to get weary with calling, and she leant against the trellis.
"Ha! she'll soon put her head through now," chuckled Luxehale. And so
she did, sure enough; and no sooner did her head appear on the other
side than he twisted his coils round her and dragged her down under
the earth.
Before she recovered herself he once more appeared as a handsome
cavalier.
It was Regina, the youngest and best-conducted of the sisters.
"Let me go! let me go!" she cried, refusing to look at him.
"I thought I heard you calling for your sisters," he replied,
soothingly; "don't you want to see them?"
"Oh, yes! tell me where they are."
"I can't tell you where they are," he answered; "and if I did, it
would be of no use, because you would not know the way to where they
are. But if you come with me, it is possible we may be able to hear
something about them some day. One thing is certain, no one else is
so likely to be able to hear of them as I."
Regina was terribly perplexed, something within her said she ought
not to speak to the stranger gallant. "And yet, on the other hand,
if, by going with him, I can do any thing to recover my dear sisters,"
she thought, "I ought to risk something for that."
When he saw her hesitate, he knew his affair was won; and, indeed,
it required little persuasion to decide her now. As they went along
he said so many soft and flattering things as to make her forget
insensibly about her sisters. But when they got to the palace there
were such a number of beautiful things to occupy her attention, so
much to astonish her--a poor peasant maid who had never seen any of
these fine things before--that she soon got habituated to her new life,
and the fact of her having come for her sisters' sake went quite out
of her remembrance.
Luxehale was delighted to have brought things so far; and in proportion
to the difficulty he had had in winning her, was the satisfaction he
felt in being with her; thus he spent a longer time with her than
he had with either of the other sisters. But the time came at last
when he had to go upon earth about his business; and then he gave her
the same charge as the others about the keys and the adamant door,
and the rose which was not to fade till his return.
It was not many days either before the desire to see what was
hid behind it took possession of her; but as she approached it she
already perceived that the air that came from it was dry and heated,
and as she really regarded the rose as a token of affection, she was
concerned to keep it fair and fresh, so she went back and placed it
in a glass of water, and then pursued her investigation of the secret
of the adamant door.
She had learnt enough when she had but half opened it, and smelt
the stifling fumes of sulphur which issued from the pit it guarded,
and would have turned to go, but then her sisters' voices, wailing
in piteous accents, met her ear.
"Lucia! Orsola!" she cried.
"Regina!" they replied; and then, courageously advancing farther by
the light of the lurid flames, which burnt fitfully through the smoke,
now red with a horrid glare, now ashy grey and ghastly, she descried
the beloved forms of her sisters writhing and wailing, and calling
on her to help them.
She promised to use all her best endeavours to release them, and,
in the meantime, bid them keep up their courage as best they might,
and be on the look-out to take advantage of the first chance of
escape she could throw in their way. With that she returned to her
apartment, replaced the rose in her bosom, and looked out for the
return of Luxehale. Nor did he keep her long waiting; and when he
saw the rose blooming as freshly as at the first he was delighted,
and embraced her with enthusiasm. In fact, he was so smiling and well
inclined that she thought she could not do better than take advantage
of his good humour to carry out the plan she had already conceived.
"Do you know," she said, "I don't like the way in which your people
wash my things; they dry them in a hot room. Now I've always been
accustomed to dry them on the grass, where the thyme grows, and then
they not only get beautifully aired, but they retain a sweet scent
of the wild thyme which I have always loved since the days when I was
a little, little girl, and my mother used to kiss me when she put on
my clean things."
"It shall be done as you like," said Luxehale. "I will order a field
of thyme to be got ready immediately, and your things shall always
be dried upon it. Is there nothing else, nothing more difficult,
I can do for you?"
"Well, do you know," she replied--for this would not have answered
her purpose at all--"do you know, I don't fancy that would be quite
the same thing either; there is something peculiar about the scent
of our grass and our thyme at home which is very dear to me. Wouldn't
it be possible to send the things home?"
Luxehale looked undecided.
"It's the only thing wanted to make this beautiful place perfectly
delightful," she continued.
He couldn't resist this, and promised she should do as she liked.
Regina then ordered a large box to be made, and packed a quantity of
her things into it. But in the night when all slept she went down to
the adamant door, and called Lucia.
Both sisters came running out. "One at a time!" she said. "Lucia has
been in longest; it will be your turn next." So she took Lucia up with
her, and hid her in the box under the clothes, and told her what she
had to do. She was to send all the linen back clean at the end of the
week, and well scent it with thyme, and to fill up the vacant space
with more linen, so that it might not seem to return with less in it
than when it went. She told her also, if the porter who carried the
box should take into his head to peep in, "all you have to say is,
'I see you!' and you will find that will cure him." Then she went to
bed, and slept quietly till morning.
Early next day Luxehale called a porter to carry the box, to whom she
overheard him giving secret instructions that, as soon as he had got
to a good distance, he should search the box, and let him know what
was in it before he sent him up to her for final orders.
Regina told him all about the situation of her father's cottage. "But,"
she added, "I've had my eye on you a long time--you're not a bad sort
of fellow, but you're too curious."
"Why, I've never been where your worship could see me!" answered the
porter; "I've always worked in the stables."
"I can see every where!" replied Regina, solemnly. "I can see you in
the stables as well as I can see you here, and as well as I shall be
able to see you all the way you are journeying; and if an impertinent
curiosity should take you to look at my clothes, I shall see you,
you may be sure, and shall have you properly punished, so beware!"
The porter planted the chest on his strong shoulders and walked
away. He was a devil-may-care sort of fellow, and didn't altogether
believe in Regina's power of seeing "every where," and, as his
master's injunction to look into the box accorded much better with
his own humour than Regina's order to abstain from opening it, before
he had got halfway he set it down on the ground, and opened it.
"I see you!" said Lucia, from within; and her voice was so like her
sister's that the fellow made no doubt it was Regina herself who
really saw him as she had threatened; and, clapping the box to again
in a great fright, lifted it on to his shoulders with all expedition.
"I've brought your daughter's linen to be washed!" cried the porter,
when he arrived at the cottage, to the father of the Devil's wives, who
was in his field "breaking" Indian corn. "I've got a message to carry
about a hundred miles farther and shall be back by the end of the week,
so please have it all ready for me to take back when I call for it."
The good peasant gave him a glass of his best Kuechelberger [83],
and sent him on his way rejoicing.
He had no sooner departed than Lucia started up out of the box of
linen, and hastily told her father all the story. The peasant's hair
stood on end as he listened, but they felt there was no time to be
lost. All the linen Regina had sent, and all that remained in the
cottage, was washed and well scented with thyme, and packed smoothly
into the box for the porter to take back with him. They had hardly
got it all ready when he came to the gate to ask for it.
"Here you are!" said the peasant; and the porter lifted the box on
to his strong shoulders, and made the best of his way home.
"What did you find when you looked into the box?" asked the Devil,
the first time he could catch the porter alone.
"Oh! nothing whatever but dirty linen," replied he, too much of a
braggadocio to confess that he had been scared by a woman's voice.
After receiving this testimony the Devil made no sort of obstacle any
more to his wife sending a box home whenever she would, and as soon
as she collected sufficient to justify the use of the large chest
she ordered the porter to be ready over night, and then went down
and called Orsola.
Orsola came quickly enough, and was packed into the linen chest as
her sister had been, and with the same instructions. "Only, as I don't
mean to stay here much longer behind, there is no reason why we should
lose all our best linen, so don't send a great deal back this time,
but fill up the box with celery, of which Luxehale is very fond."
The porter, feeling somewhat ashamed of his pusillanimity on the last
occasion, determined this time to have a good look into the box,
for the effect of his fright had worn off, and he said to himself,
"It was only a foolish fancy--I couldn't really have heard it."
So he had hardly got half way when he set the box down, and lifted
the lid.
"I see you!" exclaimed Orsola, in a voice so like Regina's that the
lid slipped out of his hand, and fell upon the box with a crash which
startled Orsola herself. He loaded the box on his shoulders once more,
nor stopped again till he reached his destination.
Hearty was the greeting of the two sisters and their father as soon
as he was gone; and then they set to work to get the washing done.
"The weather has been so bad," said the father, when the porter
returned, "that we could not dry all the linen, please to say to your
mistress, but we hope to have it ready to go back with next week's;
beg her acceptance, however, of the celery which I have packed into
the box in its place."
"Did you look into the box this time?" said Luxehale, as soon as he
got the porter alone.
The porter did not like to acknowledge that he had been scared by
a woman, and so declared again that there was nothing in the box
but linen.
It was more difficult to arrange for her own escape, but Regina had
a plan for all. The box had now gone backwards and forwards often
enough for the porter to need no fresh directions, so she told him
over-night where he would find it in the morning; and he, finding it
seem all as usual, loaded it on his shoulders, and walked off with
it by the usual path.
He had not performed half the journey when he determined to
have a serious look into the box this time, and be scared by no
one. Accordingly he lifted the lid, but this time the words,--
"I see you!" came out of the box so unmistakably in Regina's voice,
that there was no room for doubt of her power of seeing him, and with
more haste than ever he closed it up again, and made the best of his
way to the peasant's cottage.
Both sisters and their father greeted Regina as their good angel and
deliverer when she stepped out of the box; and they went on talking
over all their adventures with no need to make haste, for Regina had
brought away with her money and jewels enough to make them rich for the
rest of their lives, so that they had no need to work any more at all.
When the porter returned to ask for the linen-chest, the peasant came
out with a humorous smile, and bid him tell his master that they had
not time to do the washing that week.
"But what shall I tell my mistress?" asked the man.
As he said so, Regina and her sisters came into the room, striking
him dumb with astonishment.
"No, you had better not go back to him," she said, compassionating
him for the treatment that would have awaited him, had he returned
without her; "Luxehale would doubtless vent his fury on you for my
absence. Better to stay here and serve us; and you need not fear his
power as long as you keep out of his territory."
After this, Luxehale determined to give up young and pretty wives,
since they proved sharp enough to outwit him, as he had before given
up rich and titled ones, who were like to have knights and princes
to deliver them.
This time he said he would look out for a bustling woman of good
common sense, who had been knocked about in the world long enough to
know the value of what he had to offer her.
So he went out into the town of Trient, and fixed upon a buxom woman of
the middle class, who was just in her first mourning for her husband,
and mourned him not because she cared for him, for he had been a bad
man, and constantly quarrelled with her, but because, now he was dead,
she had no one to provide for her, and after a life of comparative
comfort, she saw penury and starvation staring her in the face.
He met her walking in the olive-yard upon the hill whence her husband's
chief means had been derived. "And to think that all these fine trees,
our fruitful arativo, and our bright green prativo [84], are to be
sold to pay those rascally creditors of my brute of a husband!" she
mused as she sat upon the rising ground, and cried. "If he had nothing
to leave me, why did he go off in that cowardly way, and leave me
here? what is the use of living, if one has nothing to live upon?"
The Devil overheard her, and perceived she was just in the mood for
his purpose, but took care to appear to have heard nothing.
"And are you still charitably mourning because the Devil has taken
your tyrant of a husband?"
"Not because he has taken him, but because he didn't take me too,
at the same time!" answered the woman, pettishly.
"What! did you love the old churl as much as all that?" asked Luxehale.
"Love him! what put that into your head? But I didn't want to be left
here to starve, I suppose."
"Come along with me then, and you shan't starve. You shall have a
jollier time of it than with the old fool who is dead--plenty to eat
and drink, and no lack, and no work!"
"That's not a bad proposition, certainly; but, pray, who are you?"
"I am he who you regretted just now had not taken you. I will take you,
if you wish, and make you my wife."
"You the Devil!" exclaimed the woman, eyeing the handsome person he
had assumed from head to foot; "impossible, you can't be the Devil!"
"You see the Devil's not so black as he's painted," replied
Luxehale. "Believe me that is all stuff, invented by designing knaves
to deceive silly people. You can see for yourself if I don't look,
by a long way, handsomer and taller than your departed spouse, at
all events."
"There's no saying nay to that," responded the widow.
"Nor to my other proposition either," urged Luxehale; and, as he found
she ceased to make any resistance, he took her up in his arms, and,
spreading his great bat's wings, carried her down to his palace, where
he installed her as lady and mistress, much to her own satisfaction.
As she was fond of luxury and ease, and had met with little of it
before, the life in the Devil's palace suited her uncommonly well,
and yet, though she had every thing her own way, her bad temper
frequently found subject for quarrel and complaint.
It was on one occasion when her temper had thus been ruffled, and she
had had an angry dispute with Luxehale, who to avoid her wrangling
had gone off in a sullen mood to bed, that some one knocked at the
door. All the servants were gone to bed, so she got up, and asked
who was there.
"I, Pangrazio Clamer of Trient," said a somewhat tremulous voice.
"Pangrazio Clamer of Trient!" returned the widow; "come in, and
welcome. But how did you get here?"
"It's a longish story; but, first, how did you get here, and installed
here too, it seems? Ah, Giuseppa, you had better have married me!"
"I've forbidden you to talk of that," answered Giuseppa. "Besides,
I had not better have married you, for I have married a great prince,
who is able to keep me in every kind of luxury, and give me every
thing I can wish. You couldn't have done that."
"No, indeed," he sighed.
"Well, don't let's talk any more about that. Tell me how every one
is going on in Trient."
"By-and-by, if there is time. But, first, let me tell you about myself,
and what brought me here. That's strange enough."
"Well, what was it, then?"
"You know that you refused to have me, because I was poor----"
"I have already forbidden you to allude to that subject."
"You must know, then, that though I worked so hard to try and make
myself rich enough to please you, I only got poorer and poorer; while
at the same time, there was Eligio Righi, who, though his father left
him a good fortune to begin with, kept on getting richer and richer,
till he had bought up all the mines and all the olive-grounds, and
all the vineyards and mulberry-trees that were to be sold for miles
round--yours among the rest."
"That too?"
"Yes; and I often felt tempted to envy him, but I never did. One day
he came to me while I was hard at work, and said, 'You know, Pangrazio
Clamer, that I am very rich;' and I thought he didn't need to have come
and said that to me, who had all the labour in life to keep off envying
him, as it was. 'Pangrazio,' says he, 'I am not only rich, but I have
every thing I can wish, but one thing; and if I meet any one who will
do that one thing, I will take him to share my riches while I live,
and make him my heir at my death. I come first to ask you.' 'Tell me
what it is,' says I; 'I can't work harder, or fare worse, than I do
now, whatever it may be--so I'm your man.' 'Well, then, it's this,'
he continued. 'My one great unfulfilled wish through life has been
to give the Devil three good kicks, as some punishment for all the
mischief he does in the world; but I have never had the courage to
make the attempt, and now I have got old, and past the strength for
adventures, so if you will do this in my stead, I will put you in
my shoes as far as my money is concerned.' Of course, I answered I
would set out directly; and, as he had made the road by which men
get hither his study, for this very purpose, all through his life,
he could give me very exact directions for finding the Devil's abode.
"But, to get here, I had to traverse the lands of three different
sovereigns; and, as I had to go to them to get my passport properly
in order, they learned my destination, and each gave me a commission
on his own account, which I accepted, because if I should fail with
Eligio Righi's affair, I should have a chance of the rewards they
promised me to fall back on."
"And what were these three commissions?"
"The first king wants to know why the fountain which supplied all
his country with such beautiful bright water has suddenly ceased
to flow. The second king wants a remedy for the malady of his only
son, who lies at the point of death, and no physician knows what
ails him. And the third king wants to know why all the trees in his
dominions bear such splendid foliage, but bring forth no fruit."
"And you expect me to help you in all this?" said the Devil's wife.
"Well, for our old acquaintance' sake, and the bond of our common
home," said Clamer, "you might do that; and for the sake of the nearer
bond that might have united us."
"I would have refused you all you ask, to punish you for going back to
that story," said Giuseppa, "but I really desire to see old Luxehale
get a good drubbing, just now, for he has been very tiresome to-day. I
daren't give it him myself, but I'll help you to do it, if you have
a mind."
"Never mind the motive, provided you give me the help," replied
Clamer. "And will you help me to trick him out of the answers for
the three kings, as well as to give him a good drubbing?"
"That will I; for it will be good fun to counter-act some of his
mischief."
"How shall we set about it then?"
"I am just going to bed; he is asleep already. You must conceal
yourself in the curtains, and bring a big stick with you; and when
I make a sign, you must, without a moment's notice, set to and give
it him. Will that do for you?"
"Admirably! Only, remember, I have to do it three times, or I shan't
get my guerdon."
"And do you think you are certain of getting all Eligio Righi's
fortune?" said Giuseppa, earnestly.
"Oh, as sure as fate!" replied Clamer; "he's a man who never goes back
from his word. But I must fulfil all he says with equal exactness."
"And when I've helped you with half your labour, I don't see why I
shouldn't have half your guerdon."
"Nor I! You'll always find me faithful and true; and what I offered
you when I was poor, I offer you with equal heartiness when I have
the prospect of being the richest man in Trient."
"When you have done all you have to do, then, will you take me back
with you?"
"Nothing would make me happier than your consent to come with me. And
when I'm rich enough to be well fed and clothed, you'll find I'm not
such a bad-looking fellow, after all."
"Ah, you'll never be so handsome as Luxehale! But then I don't half
trust him. One never knows what trick he may take into his head to
play one. I think I should have more confidence of being able to
manage you."
"Then it's agreed; you come back with me?"
"Yes; I believe it's the best thing, after all. And now we must make
haste and set about our business."
She crept up-stairs with soft steps, and Clamer still more softly after
her. The Devil was sleeping soundly, and snoring like the roar of a
wild beast. Giuseppa stowed Clamer away in the curtains, and went to
bed too. When she heard what she reckoned one of the soundest snores,
she lifted the bed-curtains, and whispered, "Now's your time!"
Clamer did not wait to be told twice, but raised his stick, and,
as Giuseppa lifted the bed-clothes, applied it in the right place,
with a hearty good will.
Luxehale woke with a roar of pain, and Clamer disappeared behind
the curtains.
"Forgive me, dear lord!" said Giuseppa; "I had such a strange dream,
that it woke me all of a start, and I suppose made me knock you."
"What did you dream about?" said Luxehale, thinking to catch her at
fault; but Giuseppa had her answer ready.
"I thought I was travelling through a country where all the people
were panting for want of water, and as I passed along, they all
gathered round me, and desired me to tell them, what had stopped
their water from flowing, saying, 'You are the Devil's wife, so you
must know!' and when I couldn't tell them, they threw stones at me,
so that I seemed to have a hard matter to escape from them."
The Devil burst out into a loud laugh, which absorbed all his
ill-humour, as he heard this story, and Giuseppa made a sign to Clamer
to pay attention to what was to follow.
"You see, you never tell me any thing," she continued, pretending
to cry; "I never know any thing about your business, and, you see,
all those people expected I knew every thing my husband knew, as
other wives do."
"I didn't suppose you'd care to know any thing about it," replied
Luxehale, trying to soothe her; "and really there was nothing to
tell! It's an every-day matter. There was a pilgrimage chapel near
the city, to which the people used to go in procession every year;
and as long as they did that, I never could get past to get at the
fountains. But now they have left off the procession, and so I got by,
and had the fun of stopping the water."
Clamer winked to Giuseppa, to show he understood what the remedy was,
and Giuseppa said no more, so that the Devil very soon fell off to
sleep again.
When he began to snore again very soundly, she lifted the bed-clothes,
and made the agreed sign to Clamer. Clamer came forward, and applied
his stick with a hearty will in the right place, and the Devil woke
with a shout of fury.
"Oh, my dear husband!" cried Giuseppa, deprecating his wrath by her
tone of alarm; "I have had another dreadful dream!"
"What was it?" growled the Devil.
"I thought I was going through a great city where all the people were
in sorrow, and sat with ashes on their heads. And when they saw me
pass, they said they sat so because the king's son was at the point
of death, and no one could tell what ailed him, and all the doctors
were of no use; but that as I was the Devil's wife, I must know all
about it. When I couldn't tell them, they began pelting me; as they
kept putting fresh ashes on their heads each had a pan of fire by his
side, in which they were making, and they actually took the red-hot
cinders out of the pan of fire to pelt me with, and my clothes were
all on fire; so you may believe if I tried to run away fast--and it
is no wonder if I knocked you a little."
The Devil's fancy was more tickled than before with this story,
and he laughed fit to split his sides, as she proceeded, so that he
forgot all about the beating.
"It is all very well for you to lie there and laugh, but you wouldn't
have laughed if you had been treated as I was, I can tell you!" sobbed
Giuseppa. "And it's all because you never tell me any thing, as other
husbands do."
"Bosh!" answered the Devil; "I should have enough to do, if I told
you all the stories like that! Why, it's the commonest thing in the
world. That king's son was a good young man, obedient to all the
advice of his elders. But after a time he got with bad companions,
who introduced him to some of my people. After they had played him
a number of tricks, one day one of them took into his head to give
him a stunning good illness, to punish him for some luck he had had
against them at cards. And that's the history of that--there's nothing
commoner in life."
Giuseppa made a sign to know if Clamer had heard all he wanted to know,
and, finding he was satisfied, let the Devil go to sleep again.
As soon as he began to snore very soundly, Giuseppa lifted the
bed-clothes, and Clamer once more applied his stick. Whether by getting
used to the work and therefore less nervous, he really hit him harder,
or whether the previous blows had made the Devil more sensitive,
he certainly woke this time in a more furious passion than ever, and
with so rapid a start that it was all Clamer could do to get out of
his sight in time.
"What have you been dreaming now?" he exclaimed, in his most fearful
voice. "I declare, I can scarcely keep my hands off you!"
"Don't be angry," answered Giuseppa; "it is I who have had the worst
of it. I dreamt I was passing through a country where the trees had
given up bearing fruit; and when the people saw me go by, they all
came round me, and said, as I was the Devil's wife, I must know what
ailed their trees; and when I couldn't tell them, they cut down great
branches, and ran after me, poking the sharp, rough points into my
sides! You may believe if I tried to run away fast."
The Devil had never had such a laugh since he had been a devil,
as at this story, and the whole palace echoed with his merriment.
When Giuseppa found him once more in such good humour, she went on,--
"And why do you do such mischievous things, and make people so
savage? It isn't fair that they don't dare to touch you and all their
ill-will falls on me."
"As it happens, it's not my doing at all this time; at least, I didn't
go out of my way to do it for any sort of fun. It all came about in
the regular way of business."
"What do you mean?" pursued Giuseppa, who knew it was necessary to
probe the matter to the bottom.
"Why, the king of that country is a regular miser. He is so afraid
that any body should get any thing out of their gardens without paying
the due tribute to him first, that he has built such high walls round
all the orchards, and vine-gardens, and olive-yards, that no sun can
get at them. And he is so stingy, he won't pay people to dig round
them and manure, and prune, and attend to the property; so how can
the fruit grow? As long as he defrauds the poor people of their work,
he can have no fruit. It's not my fault at all!
"But, really, I've had enough of this. You'd better go and sleep
somewhere else for the rest of the night, for I can't stand being woke
up any more. If you do it again, I am sure I shall strangle you--and
that would be a pity! Go along, and dream somewhere else--and I hope
you may get properly punished before you wake next time!"
Giuseppa desired no command so much; but pretending to cry and
be much offended, she got up and went to lie down in another bed
till the Devil began to snore soundly again. Then she rose up, and,
taking all her fine clothes and jewels, went out softly, and beckoned
to Clamer to follow her.
"Suppose the Devil wakes before we get far away?" said Clamer,
beginning to get frightened as the time of trial approached.
"Never fear!" answered Giuseppa; "when he gets disturbed like that,
he sleeps for a week after it."
Then she clapped her hands, and a number of great birds came flapping
round. She helped Clamer on to the back of one, and, loading her
jewels on to another, sprang on to a third, and away they flew,
while she beckoned to three more to follow behind.
When they came to the first kingdom, Clamer left the strange cortege
behind a mountain, and went alone up to the court, to tell the
king he was a miser, and that if he gave up his sordid ways and set
the people, who were starving for want of work, to pull down half
the height of his walls, and to dig, manure, and prune his trees,
he would have as good a crop of fruit as any in the world. Then the
king acknowledged his fault, and praised Clamer for pointing it out,
and gave him a great bag of gold as his reward.
Clamer packed the sack of gold on to the back of one of the birds
which were following them, and away they sped again. When they arrived
at the second kingdom, Clamer hid his cortege in a pine forest,
and went alone to the court, to tell the king that if his son would
give up his bad companions, and live according to the advice of his
elders, he would be all well again as before. The prince was very
much astonished to find that Clamer knew about his bad behaviour,
for he had concealed it from his parents and all about him, but this
convinced him that he must be right in what he said, so he promised to
alter his life and behave according to the wise counsel of his elders
in future. From that moment he began to get better; and the king,
in joy at his restoration, gave Clamer a great sack of gold, which
he laded on to the back of the second bird; and away they flew again.
When they arrived at the third kingdom, Clamer hid his retinue in
the bed of a dried lake and went alone to the court, to tell the
king that if he would order the procession to the pilgrimage chapel
to be resumed, the Devil would not be able to get in to stop the
fountains. The king at once ordered the grandest procession that had
ever been known in the memory of the oldest inhabitant, and all the
people went out devoutly praying. Immediately the springs and fountains
began to flow again; and the king was so pleased that he gave Clamer a
great sack of gold, which he packed on to the back of the third bird;
and away they flew again, till they reached the gloomy shades of the
Val d'Ombretta, under the cold, steep precipices of the Marmolata [85].
"Here will be a good place to hide all this treasure," said Giuseppa;
"it will never do to take it into Trient all at once. We will bury
it here where foot of man seldom falls, and my birds will keep good
watch over it and defend it, and yet by their services we shall be
able to fetch down any portion of it as we want it."
Clamer saw there was some good in the proposal, but he hardly liked
giving up the possession of the treasure to Giuseppa's birds, neither
did he like to show any want of confidence.
"Don't you think it an excellent plan?" asked Giuseppa, as she saw
him hesitate.
"I think I could stow it away as safely in an old well at home," said
Clamer. "This is an uncanny place of evil renown, and I had just as
lief have nothing to do with it."
"What's the matter with the place?" asked Giuseppa.
"Oh, you know, the Marmolata was as fertile as any pasture of Tirol
once," answered Clamer; "and because the people had such fine returns
for their labour from it, they grew careless and impious, and were
not satisfied with all the week for working in it, but must needs be
at it on Sundays and holidays as well. One Sunday an ancient man came
by and chid them for their profanity. 'Go along with your old wives'
stories!' said a rich proprietor who was directing the labourers;
'Sunday and working-day is all alike to us. We have sun and rain
and a fine soil, what do we want with going to church to pray?' And
they sang,--
'Nos ongh el fengh en te tabla,
E i autri sul pra [86]!'
"The old man lifted up his finger in warning, and passed on his way;
but as he went it came on to snow. And it snowed on till it had covered
all the ground; covered all the hay up to the top; covered over the
heads of the labourers and their masters; snowed so deep that the sun
has never been able to melt it away again! A curse is on the place,
and I had rather have nothing to do with it."
"Oh, I've lived long enough where curses abound to care very little
about them," answered Giuseppa, "or I could tell you the real story
about that, for you've only got the wrong end of it. But it doesn't
do to think of those things. The only way is to laugh at all that
sort of thing, and make yourself jolly while you can."
"My story's the right one," replied Clamer, "and you won't laugh me
out of believing it."
"Oh, dear no; the right story is much more serious than that! But
I lose my patience with people who trouble themselves about those
things."
"I don't believe there's any more of the story," continued Clamer,
who was dying to hear it, and knew that the best way to get at it was
by provoking her. Had he merely begged her to tell it, she would have
found a perverse pleasure in disappointing him.
Giuseppa was very easily provoked. "The right story proves itself,"
she cried, pettishly; and Clamer chuckled aside to see his plan
succeed. "Your way of telling it only accounts for the snow; how do
you account for the ice?"
"Oh, there's no way of accounting for that," replied Clamer, with a
malicious laugh.
"Yes, there is," rejoined Giuseppa, fairly caught. "It wasn't an old
man at all who came to give the warning. It was a very young man,
for it was no one else but St. John."
"St. John!" cried Clamer; "how could that be?"
"Don't you know any thing, then?" retorted Giuseppa. "Don't you know
that there was a time when our Lord and His Apostles went walking
over the earth, preaching the Gospel?"
"Yes, of course I know that," replied Clamer, much offended.
"Well, then, in process of travelling they came here just the same as
every where else--why shouldn't they? The Apostles had been sent on to
prepare a lodging for the night, and St. John, being the youngest and
best walker, outstripped the rest, and came by first. But he was so
soft and gentle in his warning that the labourers laughed at him, and
he went on his way sighing, for he saw that their hearts were hardened.
"Then St. Peter and St. Paul came by----"
"But St. Paul--" interposed Clamer.
"Don't interrupt, but listen," said Giuseppa. "St. Peter and St. Paul,
though not younger than the others like St. John, were always in the
front in all matters, because of their eagerness and zeal, and the
important post which was assigned them in the Church. They came next,
therefore; but they, seeing the men working on Sunday, were filled
with indignation, and chid them so fiercely that they only made them
angry, and they took up stones to throw at them, and drove them out
of the ground. One by one the other Apostles all came by and warned
them, but none of them seemed to have the right way of getting at
their hearts. And they went on working, with a worse sin on them for
having been warned.
"Last of all, the Lord Himself came by, and His heart was moved
with compassion by the perversity of the people. He saw that all the
preaching of all His Apostles had been in vain, and He resolved to
save them in another way, and prove them, to see if there was any
charity or any good in them at all.
"Instead of threatening and warning, He came leaning on His staff,
weary and way-sore.
"'You have a fine Berg-Segen [87], my friends,' He said, sweetly,
as He sat on a great heap of fresh hay placed ready to load the
returning wain.
"'Oh, yes! first-rate crops,' replied the rich proprietor, with a
look of contempt at the travel-stained garments of the wayfarer;
'but they're not meant to serve as beds for idle fellows who go
prowling about the country and live by begging instead of by work,
so you just get up and take yourself off!'
"Our Lord looked at him with a piteous glance, but his heart was not
softened. 'Move off quicker than that, or you'll taste my stick!' he
cried, assuming a threatening attitude.
"Our Lord passed on, without uttering a word of complaint, till He
reached the holding of the next proprietor.
"'Where there are such fine pastures there must be fine cattle and
a fine store of produce,' He said.
"'Oh, yes, I've plenty of stores!' said the man addressed; 'and
that's just why I don't like to have loafing vagabonds about my place;
so please to move on quicker than you came.'
"'But I'm weary, my good man, and have come a long journey this day,
and have nothing to eat: give me, now, but one sup of milk from your
bountiful provision there.'
"'Give!' answered the man; 'I've nothing to give away. I work hard
for all I gain, and I don't encourage those who don't work.'
"'But you won't miss the little I ask--and I have travelled very far
and am very weary,' replied our Lord, condescending to speak very
piteously, to see if He could not by any means move the man's heart.
"'Hola! you there! Domenico, Virgilio, Giacomo, Rocco, Pero! come
along here, and throw this fellow out!' shouted the proprietor.
"The men turned with their pitchforks, and drove the wayfarer rudely
away, without pity, notwithstanding that His legs trembled with
weariness and the way was so steep.
"Our Lord uttered not a word, and hasted on, that He might not increase
their condemnation by resistance.
"But the heavens grew black with anger at the sight; the storm-clouds
gathered in vengeance; grey and leaden, mass above mass, they thickened
over the devoted peak of the Marmolata; the sun ceased to smile,
and a horrible darkness fell around.
"Closer and closer lowered the clouds, till they fell, enveloping
the mountain-top with white fields of snow.
"'Nay!' cried the Saviour, compassionately; 'Father, stay Thine
hand!' And for a moment the convulsion of the angry element was
stilled. 'They knew not what they did,' He pleaded; and He passed
down the path to the next holding.
"'See,' He said to the proprietor, who was watching the strange storm
with some alarm, 'see how terrible are the judgments of God! Give Him
praise for the blessing He has poured out on you, and save yourself
from His anger.'
"'What have I to do with the misfortunes of others? Every thing goes
right with me.'
"'But it may not always. Be wise betimes, and render praise to God.'
"'What do I know about God?' answered the man; 'I've enough to do with
taking care of the earth; I don't want to puzzle my head about heaven!'
"'All good gifts are from heaven.' replied the Lord, faintly; and He
sank upon the ground exhausted.
"'See!' cried a woman who had come out with her husband's dinner,
'see, He has fallen; will you do nothing to restore Him?' And she
ran to raise Him up.
"'Let Him lie.' said her master, pushing her roughly away; 'it were
better the earth were rid of such idle fellows.'
"He had filled up the measure of his iniquity. 'Hard and icy as his
heart has been, so shall his pasture be!' proclaimed the Angel of
Judgment. And as he spread his arms abroad, the clouds fell over the
sides of the mountain; the cold blast turned them into ice, and it
became a barren glacier for evermore.
"But the angels carried the Lord to the place the Apostles had prepared
for Him. And the woman who had pitied Him alone escaped and recorded
the story."
A shudder had fallen over Clamer, and he seemed hardly inclined
to break the silence which reigned around. There was not a bird to
chirp a note, nor a leaf to flutter, nor a blade of grass to gladden
the eye. Meantime they had reached the Fassathal, which, though so
fruitful farther along, is scarcely more smiling at its east end.
"Were it not well, Pangrazio," urged Giuseppa, "to bury our treasure
here, before we get nearer the habitations of men? Ah!" she added,
"I see what it is, it is not of the weird neighbourhood that you are
shy, it is that you trust not me! you think if my birds guard the
treasure you will have less control over it than I!"
"Oh, no!" answered Clamer, ashamed to have been found out; "it is not
that; but there are as many weird warnings rife here as concerning the
Marmolata. Does not the Feuriger Verraether [88] haunt this place? and
does not the Purgametsch conceal a village which was buried for its
sins? Is it not just here that lurk the Angane and the Bergostanoe
[89]?"
"Really, I can undertake to defend you against all these chimerical
fancies," replied Giuseppa, scornfully; "but if you don't feel any
confidence in me, it is absurd our attempting to live together."
"It is not that--I have told you it is not that!" cried Clamer.
"Then shall we do it?" urged she. Thus driven, Clamer could not
choose but give in; and Giuseppa sent her monster birds to conceal the
treasure they bore, in the hole she pointed out high up in the rocks,
and remain in guard over it.
This done they sped over the pleasant Fleimserthal and Cembrathal
to Trient.
Eligio Righi received his returning envoy with a hearty welcome, and
listened without wearying to his frequent repetition of the tale of
his adventures. The part where he described the manner in which he
had administered the chastisement on the Devil was what delighted
him most, and the account of the roaring of the Devil with the pain.
Moreover, he kept his word, and opened his house and his purse to
Clamer, who shared every thing as if it had been his own, and even
obtained his sanction to bring home his wife, though he durst not
tell him how he obtained her.
Giuseppa had now not only a fine house and broad lands, and plenty of
servants and clothes, and every thing she wished for, but she had only
to send one of her birds to the treasury in the Fassathal to supply
all her caprices as well as wants--yet she was always complaining
and quarrelling. Pangrazio often found her quite unbearable; but he
remembered she was his wife, and he forgave her, though the more he
gave in, the more unreasonable she got.
In the meantime, it must not be supposed that Luxehale had never
awaked. True, he slept on for a good week, as Giuseppa had predicted,
but that over, he woke up in a pretty passion at finding she had
escaped.
With all her evil temper, Giuseppa had suited him very well; he rather
enjoyed an occasional broil, it was much more to his taste than peace
and amity--and besides, he was sure always to get the best of it. So
he determined that this time, instead of going in search of a new wife,
he would get the old one back.
"Those who come to me in the way she did," he reflected, "don't
escape so easily. The others I more or less deceived. They came
with me thinking I was one of their own sort; but she followed me
with her eyes open--she knew all about me before she came. Besides,
they hated the place the moment they found out where they were, but
she knew what it was, and yet liked it all along. No, I don't think
she's of the sort that go back in thorough earnest."
So he dressed himself up in his best, put a plume in his hat and a
flower in his button-hole, and went off to Trient. He had not watched
the house where Giuseppa lived many days before he heard her voice
raised to that angry key he knew so well.
"That'll do for me," he said, rubbing his hands. "It's all going
on right."
"What do you want more?" he heard Clamer plead. "If there is any
thing I can do to please you, I will do it!"
"You are a fool! and there's nothing in you can please me," screamed
Giuseppa, too angry to be pacified; "you're not like Luxehale. Why
did you ever take me away from him? He was something to look at!"
"It's going on all right!" said Luxehale, chuckling.
"Why did you come away?" said Pangrazio, quietly.
"I didn't know what I was about! Would that I had never done it!" she
added.
"Oh, don't say that!" replied Pangrazio, imploringly. But instead
of being won by his kindness she only grew the more noisy, till at
last Pangrazio could stand it no longer, and he went out to avoid
growing angry.
"Now is my time!" said the Devil; and he slipped round to the
window. Giuseppa was still fretting and fuming, and invoking Luxehale
at the top of her voice.
"Here I am!" said Luxehale. "Will you come back with me, and leave
this stupid loafer?"
"What you there!" cried Giuseppa, rushing to the window, and kissing
him. "Of course I'll go with you. Take me away!"
"All right; jump down!" said Luxehale, helping her over the
window-sill. Giuseppa threw herself into his arms, and away they
walked. Arrived outside the town, Luxehale lifted her up, spread his
black bat's wings, and carried her off.
"Go through the Fleimserthal and the Fassathal," said Giuseppa;
"I've got something to show you there."
"Any thing to please you!" answered Luxehale.
"Oh, it's not to please me!" cried Giuseppa, taking offence.
"Now don't begin again; it won't do with me!" replied Luxehale,
with a sternness he had never before exercised. "Mind, I don't mean
to allow any more of it."
"Oh, if that's to be it," said Giuseppa, "I'll go back again to
Pangrazio."
"No, you won't!" replied Luxehale; "you don't go back any more,
I'll take good care of that! And now, what did you want to come by
the Fassathal for?"
"Only because it's the way I passed with Pangrazio, and it renewed
a sweet memory of him."
"That won't do for me! What was the real reason?"
"What will you give me if I tell you?"
"Nothing. But if you don't tell me, I shall know how to make you."
Giuseppa's courage failed her when she heard him talk like this. She
knew she had given herself to him of her own will, and so she belonged
to him, and she could not help herself; and now, the best course she
could think of was to tell him of the treasure, and trust to the good
humour it would put him in, for he was very avaricious, to get her
forgiveness out of him.
Clamer came back from a walk outside the town--where he had gone to get
cool after his wife's scolding--just in time to see Luxehale spread
his wings and fly away with Giuseppa in his arms. He called to her,
but she did not hear him; and all he could do was to stand watching
them till they were out of sight.
He came back so gloomy and dejected that his friend Eligio Righi was
quite distressed to see him. He was so sympathizing, indeed, that
Pangrazio could not forbear telling him the whole story. "Then, if
that is so, you need not regret being quit of her," moralized his sage
friend: "she was no wife for an honest man. And as for the treasure,
you have enough without that. It was but ill-gotten gain which came
to you for knowledge obtained from such a source."
ZOVANIN SENZA PAURA [90]; OR,
THE BOY WHO WENT OUT TO DISCOVER WHAT FEAR MEANT.
Zovanin was a bold boy, and never seemed to be afraid of any
thing. When other children were afraid lest Orco [91] should play
them some of his malicious tricks, when people cried out to him,
"Take care, and don't walk in those footprints, they may be those of
Orco!" he would only laugh, and say, "Let Orco come; I should like
to see him!" When he was sent out upon the mountains with the herds,
and had to be alone with them through the dark nights, and his mother
bid him not be afraid, he used to stare at her with his great round
eyes as if he wondered what she meant. If a lamb or a goat strayed
over a difficult precipice, and the neighbours cried out to him,
"Let be; it is not safe to go after it down that steep place," he
would seem to think they were making game of him, and would swing
himself over the steep as firmly and as steadily as if he were merely
bestriding a hedge. He saw people shun passing through the churchyards
by dark, and so he used to make it his habit to sleep every night on
the graves; and as they said they were afraid of being struck blind
if they slept in the moonlight, he would always choose to lie where
the moonbeams fell. Nor thunder, nor avalanche, nor fire, nor flood,
nor storm seemed to have any terror for him; so that at last people
set him to do every kind of thing they were afraid to do themselves,
and he got so much wondered at, that he said, "I will go abroad over
the world, and see if I can find any where this same Fear that I hear
people talk of."
So he went out, and walked along by the most desolate paths and through
frightful stony wildernesses, till he came to a village where there
was a fair going on. Zovanin was too tired to care much for the dance,
so instead of joining it he asked for a bed.
"A bed!" said the host; "that's what I can give you least of all. My
beds are for regular customers, and not for strollers who drop down
from the skies;" for, being full of business at the moment, he was
uppish and haughty, as if his day's prosperity was to last for ever.
While Zovanin was urging that his money was as good as another's, and
the host growing more and more insolent while repeating that he could
not receive him, a terrific shouting of men, and screeching of women
made itself heard, and pell-mell the whole tribe of peasants, pedlars,
and showmen came rushing towards the inn, flying helter-skelter before
a furious and gigantic maniac brandishing a formidable club. Every one
ran for dear life, seeking what shelter they could find. The inn was
filled to overflowing in a trice, and those who could not find entrance
there hid themselves in the stables and pig-styes and cellars. But
no one was in so great a hurry to hide himself as mine host, who had
been so loud with his blustering to a defenceless stranger anon. Only,
when he saw the baffled madman breaking in his doors and windows with
his massive oaken staff, he put his head dolefully out of the topmost
window, and piteously entreated some one to put a stop to the havoc.
Zovanin was not quick-witted: all this noisy scene had been transacted
and it had not yet occurred to him to move from the spot where he
originally stood; in fact, he had hardly apprehended what it was that
was taking place, only at last the host's vehement gesticulations
suggested to him that he wanted the madman arrested.
"Oh, that's it, is it?" said Zovanin. "All right, I'm your man!" and
walking up coolly to the cause of all this disturbance, he said,
in the tone of one who meant to be obeyed, "Give me your club."
The poor imbecile was usually harmless enough; he lived in
an out-of-the-way hut with his family, where he seldom saw a
stranger. They had incautiously brought him up to the fete, where
he had first become excited by the sight of the unwonted number of
people; then some thoughtless youths had further provoked him by
mocking and laughing at him; and when the people ran away in fear of
his retaliation, he had only yielded to a natural impulse in running
after them. But when Zovanin stood before him, fearless and collected,
and said, in his blunt, quiet way, "Give me your club," his habitual
obedience prevailed over the momentary ebullition, and he yielded
himself up peaceably to the guidance of the young giant. Zovanin first
secured the club, and then desired the madman to bestow himself in
an empty shed, of which he closed and made fast the door. When the
landlord and people saw the coast clear they all came out again,
the latter losing no time in going back to their games, the former
to resume his preparations for the entertainment of his guests.
"Well," said Zovanin, "I suppose now you'll make no difficulty in
providing me a bed? I think that's the least you can do for me,
after my befriending you as I have. I have earned it, if any one has."
"What! you think that such a great feat, do you?" said the landlord,
who, deeming the madman well secured, felt no compunction in disowning
Johnny's service. "Do you suppose any other couldn't have said,
'Give me your club,' just as well as you?"
"Perhaps you would like to try," replied our hero; and he went to
unbar the shed-door.
"For heaven's sake, no!" screamed the cowardly landlord, preparing
to run away. "Don't let him loose on any account; I'll do any thing
for you sooner than that!"
"Well, you know what I want; it's not much, and reasonable enough,"
replied Fearless Johnny, relaxing his hold of the door.
"But that's just the one thing I can't do," lamented the host. "My
beds are bespoken to customers who come every year to the fair,
and if I disappoint any of them I'm a ruined man."
"Very well then, here goes!" and Zovanin once more prepared to open
the shed-door.
"Oh, no; stop!" roared the landlord. "Perhaps there is a way,
after all."
"Ah!" ejaculated Johnny; "I thought as much."
"There is a room, in fact a whole suite of rooms, and a magnificent
suite of rooms, I daren't give to any one else, but I think they will
do for you, as you are such a stout-hearted chap."
"Where are they?" said Johnny.
"Do you see that castle on the tip of the high rock yonder, that
looks like an eagle perched for a moment and ready to take flight?"
"I should rather think I did, seeing it's one of the most remarkable
sights I have met with in all my travels."
"Well, that castle was built by a bad giant who lived here in former
times; and he balanced it like that on the tip of the rock, and only
he had the secret of walking into it. If any one else steps into it,
they are pretty sure of stepping on the wrong place, and down will
go the whole castle overbalanced into the abyss. When he was once
inside it, he had an iron chain by which he made it fast to the rock;
and when he went out he used to set it swinging as you see, so that
no one might dare to venture in and take back possession of the
booty which he seized right and left from all the country round. If
you don't mind trying your luck at taking possession of the castle,
you can lodge there like a prince, for there are twelve ghosts, who
come there every night, who will supply you with every thing you can
ask for. So there is all you desire to have, and more, provided only
the idea does not strike you with fear."
"Fear, say you?" said Zovanin, opening his great round eyes; "do you
say I shall find 'Fear' in yonder castle?"
"Most assuredly. Every body finds it in merely listening to the story."
"Then that's what I came out to seek; so show me the way, and there
I will lodge."
The host stared at his crack-brained guest, but, glad to be rid of
his importunity for a night's lodging in the inn, made no delay in
pointing out the path which led to the giant's castle.
Zovanin trudged along it without hesitation, nor was he long in
reaching the precariously balanced edifice. Once before the entrance,
he had little difficulty in seeing what was required in order to take
possession. Just in the centre of the building a large stone stood
up prominently, and though at a great distance from the threshold,
was probably not more than a stride for the giant of old--as a further
token, it was worn away at the edge, evidently where he had stepped
on to it. Zovanin saw it could be reached by a bold spring, and,
having no fear of making a false step, he was able to calculate his
distance without disturbance from nervousness. Having balanced himself
successfully on the stone, he next set himself to fix the chain which
attached his airy castle to the rock, and then made his way through
its various apartments. Every thing was very clean and in good order,
for the twelve ghosts came every night and put all to rights. Zovanin
had hardly finished making his round when in they came, all dressed
in white.
"Bring me a bottle of wine, and some bread and meat, candles and
cards," said Fearless Johnny, just as if he had been giving an order
to the waiter of an inn; for he remembered that the landlord had
said they would supply him, and he felt no fear which should make
him shrink from them.
"I wonder where this same Fear can be?" he said, as the ghosts were
preparing his supper; "I have been pretty well all over the castle
already, and can see nothing of him. Oh, yes! I will just go down
and explore the cellars, perhaps I shall find him down there."
"Yes; go down and choose your wine to your own taste, and you will
find him there, sure enough," said the twelve ghosts.
"Shall I, though?" said John, delighted; and down he went.
The bottles were all in order, labelled with the names of various
choice vintages in such tempting variety that he was puzzled which to
choose. At last, however, he stretched his hand out to reach down a
bottle from a high shelf, when lo and behold a grinning skull showed
itself in the place where the bottle had stood, and asked him how he
dared meddle with the wine! Without being in the least disconcerted
at its horrid appearance, Fearless Johnny passed the bottle into his
left hand, and with his right taking up the skull, flung it over his
shoulder to the farthermost corner of the cellar. He had no sooner
done so, however, than a long bony arm was stretched out from the same
place, and made a grab at the bottle. But Fearless John caught the
arm and flung it after the skull. Immediately another arm appeared,
and was treated in the same way; then came a long, lanky leg, and
tried to kick him on the nose, but Johnny dealt with it as with the
others; then came another leg, which he sent flying into the corner
too; and then the ribs and spine, till all the bones of a skeleton
had severally appeared before him, and had all been cast by him on
to the same shapeless heap.
Now he turned to go, but as he did so a great rattling was heard in
the corner where he had thrown the bones. It was all the bones joining
themselves together and forming themselves into a perfect skeleton,
which came clatter-patter after him up the stairs.
Zovanin neither turned to look at it nor hurried his pace, but walked
straight back, bottle in hand, into the room where the supper was laid
ready, and the pack of cards by the side, as he had ordered. All the
while that he was supping, the skeleton kept up a wild dance round him,
trying to excite him by menacing gestures, but Fearless Johnny munched
his bread and meat and drank his wine, and took no more notice than
of the insects buzzing round the sconces.
When he had done he called to the ghosts in the coolest way imaginable
to clear away the things, and then dealt out the cards, with one hand
for a "dummy" and one for himself. He had no sooner done this than the
skeleton sat down, with a horrid grimace of triumph, and took up the
"dummy's" hand!
"You needn't grin like that," said Johnny; "you may depend on it I
shouldn't have let you take the cards if it hadn't pleased me. If
you know how to play, play on--it is much better fun than playing
both hands oneself. Only, if you don't know how to play, you leave
them alone--and you had better not give me reason to turn you out."
The skeleton, however, understood the game very well, and with
alternate fortune they played and passed away the hours till it was
time to go to bed. Johnny then rose and called the twelve ghosts to
light him up to bed, which they did in gravest order. He had no sooner
laid himself to sleep than, with a great clatter, the skeleton came
in and pulled the bedclothes off him. In a great passion Fearless
Johnny jumped up, and brandishing a chair over his head, threatened
to break every one of his bones if he didn't immediately lay the
clothes straight again. The skeleton had no defence for his bones,
and so could not choose but obey; and Johnny went quietly to bed again.
"It was a pity I didn't ask the poor fellow what ailed him, though,"
said Johnny, when he was once more alone. "Perhaps he too is tormented
by this 'Fear' that every one thinks so much of, and wanted me to
help him. Ah, well, if he comes again I will ask him;" and with that
he rolled himself up in the quilt, and went to sleep again. An hour
had hardly passed before the skeleton came in again, and this time
he shook the bedpost so violently that he woke Johnny with a start.
"Ah! there he is again!" cried Johnny; "now I'll ask him what he
wants;" so he jumped out of bed once more, and addressed the skeleton
solemnly in these words:--
"Anima terrena,
Stammi lontana tre passi,
E raccontami la tua pena [92]!"
Then the skeleton made a sign to him to follow it, and led him down to
the foundations of the castle, where there was a big block of porphyry.
"Heave up that block," said the skeleton.
"Not I!" replied Johnny; "I didn't set it there, and so I'm not going
to take it up."
So the skeleton took up the block itself, and under it lay shining
two immense jars full of gold.
"Take them, and count them out," said the skeleton.
"Not I!" said Johnny; "I didn't heap them up, and so I'm not going
to count them out."
So the skeleton counted them out itself, and they contained ten
thousand gold pieces each.
When it had done, it said, "I am the giant who built this castle. I
have waited here these hundreds of years till one came fearless enough
to do what you have done to-night, and now I am free, because to you
I may give over the castle; so take it, for it is yours, and with it
one of these jars of gold, which is enough to make you rich, but take
the other jar of gold and build a church, and let them pray for me,
and learn to be better men than I."
With that he disappeared, and Fearless Johnny slept quietly for the
rest of the night.
In the morning, when the sun was up, and the birds began to sing
cheerily on the branches, the landlord began to feel some compunction
for having abandoned such a fine young Bursch to a night by himself
among the unquiet spirits; so he summoned all his courage, and all
his servants, and all his neighbours, and, thus prepared, he led the
way up to the haunted castle. Finding that it was firmly fixed by the
chain, they all entered in a body, for none durst be the first; and
the entrance, having been made for the giant, was big enough for all.
Zovanin having had such a disturbed night was still fast asleep, but
their footsteps and anxious whisperings woke him. In answer to all
their questionings he gave an account of what had happened to him, but
still complained that, after all, he had not been able to find Fear!
Zovanin was now a rich man, and had a mighty castle to live in where
he might have ended his days in peace, but he was always possessed
by the desire of finding out what Fear was, and this desire was too
strong to let him rest.
The neighbours, however, told him he might find Fear out hunting;
and many were the hunting-parties he established, and wherever the
wild game was shyest, there he sought it out. Once, as he sprang
over a chasm his horse made a false start, and was plunged into the
abyss, but Fearless Johnny caught at the bough of a birch-tree that
waved over the mountain-side. The branch cracked, and it seemed as
if nothing could save him, but Fearless Johnny only swung himself on
to another on the ledge below, and climbed back by its means to the
path. Another time, as he was pursuing a chamois up a precipitous
track, a great mass of loose rock, detached from the height above,
came thundering down upon him. An ordinary hunter, scared at the sight,
would have given himself up for lost, but Fearless Johnny stood quite
still and let it bound over his head, and he came to no harm.
So he still was unable to find Fear. After some years, therefore,
he once more went abroad to seek it. This time, however, he provided
himself with a fine suit of armour and a prancing charger, and a
noble figure he cut as he ambled forth.
After a long journey, with many adventures, he came one hot day,
as he was very thirsty, to a fountain of water in the outskirts of a
town, and as he dismounted to drink he observed that the whole place
looked sad and deserted; the road was grass-grown, and the houses
seemed neglected and empty. As he went up to the fountain to drink,
a faint voice called to him from the wayside, "Beware, and do it
not! Think you that we all should be lying here dying of thirst if
you could drink at that fountain?"
Then he looked round, and saw that, as far as eye could reach, the
banks of the wayside were covered with dying people heaped up one on
the other, and all gazing towards the fountain!
"Know you not," continued the weary voice, "that a terrible dragon
has taken possession of all the fountains; and that the moment one
goes to drink of them he appears, as though he would eat you up,
so that you are bound to run away for very fear?"
"'Fear!'" cried Zovanin; "is Fear here at last?" and he joyfully ran
to the side of the well.
All the weary, dying people raised themselves as well as they could, to
see what should befall him who was not afraid of the terrible dragon.
But Fearless Johnny went up to the fountain's brim to dip his hand
into the cooling flood. Before he could do so, however, the terrible
dragon put his head up through the midst, with a frightful howl, and
spueing fire out of his nostrils. Zovanin, instead of drawing back,
instantly took out his sword and, with one blow, severed the monster's
head from the trunk! Then all the people rushed to the fountain,
hailing him as their deliverer. But ere they had slaked their thirst,
the dragon, which had sunk back into the depth of the water, reappeared
with a new head, already full grown, and more terrible than the last,
for it not only spued out fire from its nostrils, but darted living
sparks from its eyes.
When the people saw this they all ran away screaming, and Zovanin was
left alone; but, as usual, he did not lose heart, and with another
well-aimed blow sent the second head of the monster rolling by the
side of the first!
The people came back, and began to drink again when they saw the huge
trunk disappear beneath the surface; but it was not many minutes
before another head cropped up, more terrible than either of the
preceding, for it not only spued fire from its nostrils and darted
living sparks from its eyes, but it had hair and mane of flames,
which waved and rolled abroad, threatening all within reach. All the
people fled at the sight, and Zovanin was once more left alone with
the monster. Once more he severed the terrible head; and after this
the dragon was seen no more.
"That must be very wonderful blood out of which three heads can
spring," thought Fearless Johnny; and he filled a vial with the
dragon's blood, and journeyed farther.
After a time he came to the outskirts of another town. It was
not deserted like the last. The streets were full of people making
merry--in fact, every one was so very merry that they seemed a whole
community of madmen. Another might have been afraid to encounter
them at all; but not so Fearless Johnny, he spurred his horse and
rode right through their midst. But for all his seeming so fearless
and self-possessed, the people got round him, and seized his horse's
bridle, and dragged him from the saddle.
"What do you want with me, good people?" cried Zovanin; "let me hear,
before you pull me to pieces."
When they found him so cool, spite of the wild way in which they had
handled him, they began to respect him, and loosed their hold.
"If you want to know," answered one, "it is soon told. We are all
in this town wholly given up to amusement. We have done with work
and toil, and do nothing but dance, and drink, and sing, and divert
ourselves from morning to night. But after enjoying all this a long
time, we begin to find it rather wearisome, and we are almost as
tired of our pastime as we used to be of our labour. So the king has
decreed that every stranger who comes by this way shall be caught,
and required to find us a quite new diversion, and if he cannot do
that, we will make him dance on red-hot stones, and flog him round
the town, and get some fun out of him that way, at all events; as
you don't look very likely to find us a new pastime, we may as well
begin with putting you on your death-dance."
"Don't make too sure of that!" said Fearless Johnny, not at all
disconcerted; "take me to your king, and I'll show you a diversion
you never heard of before."
When he came to the king, the king laughed, and would hardly listen
to him, because he looked so broad and heavy, and not at all like
one who could invent a merry game.
But Johnny protested that if they would let him cut off any one's
head, he would stick it on just as before, and the man should be
never the worse.
The king was greatly delighted with the idea, and most anxious to see
the performance, promising that he would not only let him go free if
he succeeded, but would load him with honours and presents into the
bargain. Zovanin professed himself quite ready to prove his skill,
but no one could be found who was willing to let the experiment be
tried on him.
This angered the king greatly; and at last he called forward his
jester, and ordered Zovanin to make the trial on him.
The jester, however, objected as much as any one else, only, as he
belonged entirely to the king, he could not disobey him. "But think,
your majesty," said the poor hunchback, "what will your majesty do
without his jester, if this quack does not succeed in his promises?"
"But I shall succeed!" thundered Fearless Johnny; and he spoke with
such assurance, that the king and all the people were more desirous
than ever to see the feat, and cried to him to commence. When the
jester found that all hope of wriggling out of the cruel decree was
vain, he threw himself on his knees, and begged so earnestly that
the king would grant him two favours, that he could not resist. The
two favours were, that he should have the satisfaction of repeating
the trick on Johnny, if he allowed him to try his skill on him, and
also that he should first give proof of what he could do on the ape,
with whose pranks he was wont to amuse the king.
The king and Zovanin both agreed to the two requests, and the poor
ape was brought forward, and delivered over to make the first essay.
Zovanin did not keep the breathless multitude long in suspense;
with one blow he severed its head, threw it up high in the air,
that all might see it was well cut off, and then placed it on again,
smearing in some drops of the dragon's blood by way of cement. The
head and trunk were scarcely placed together again, with the dragon's
blood between, than the ape bounded up as well as before, and just
as if nothing had been done to him; but, on the contrary, finding
himself the object of great attention, and excited by the shouts of
the people, he sprang and gambolled about from side to side with even
greater alacrity than his wont.
"Now, Sir Hunchback!" cried Zovanin, "it is your turn. You see it's
not very bad; so come along, and no more excuses."
"Go it, hunchback!" said the king; and all the people shouted, "The
hunchback's head! the hunchback's head!" with such vehemence that it
was evident there was no means of getting out of the trial. It was
true, Zovanin had proved he could put a head on again; but the jester
shrank from the cold steel nevertheless, and it was only with a look
which concentrated all his venom that he yielded himself up. Fearless
Johnny struck off his head in a trice, then threw it up high in the
air, as he had done the ape's, and then cemented it on again with
the dragon's blood as well as ever.
"Now for you!" screamed the hunchback, when he found his head back
in its right place once more.
Zovanin had no fear, but sat down on the ground instantly, so that
the hunchback might reach him more conveniently. "This is all you
have to do," he said: "take my sword in your two hands, and swing it
round across my throat. Then pour the contents of this vial over the
stump of the throat, and clap the head down on it again."
"Yes, yes! I think I ought to know how it's done, as well as you,"
answered the dwarf, hastily; and he swung the sword round with a will,
sending Johnny's head rolling at the king's feet. The people caught
it up and handed it round; and it might soon have got lost in the
crowd, but that the king shouted to them to bring it back, because he
wanted to see it stuck on again. So they gave it back to the jester,
and he smeared the rest of the dragon's blood over the stump of the
throat--but in putting the head on, took care to turn it the wrong
way, which, as he managed to bend over Johnny's recumbent body, no
one observed till he rose to his feet. Then all the people screeched,
and yelled, and shouted, so that John could not make out what was
the matter, but, getting angry, demanded his horse, that he might
ride away from them all.
The king ordered his horse to be brought, and Johnny sprang into the
saddle, and the cries of the people made the beast start away faster
even than Johnny himself wished; only Johnny could not make out why
he seemed to him, for all his urging, always to go backwards.
At last, he got quite away from the shouts of the people, into a
calm, quiet place, where there was a lake shut in by high hills,
which, with the mulberry-trees, and vines, and grassy <DW72>s, were
all pictured in the lake's smooth face.
Zovanin was hot with his ride, and so was his mount; so he walked
him into the shallow water, while he himself dismounted, and bent
down to drink.
At the sight that met his gaze in the water, a shout burst from his
lips more terrible than the shouts of all the people. He gazed again,
and couldn't think what had befallen him; but, so horrified was he
at the sight of his own back where he was wont to see his breast,
that he fell down and died of fear on the spot! And thus Fear visited
him at last--in a way which would certainly never have occurred, if
the jester had put his head on again in the way nature designed for it.
THE DOVE-MAIDEN.
In the days when heathenism still disputed the advance of Christianity
in Tirol, there lived a nobleman in a castle, of which no trace now
remains, overlooking the egg-shaped Lago di Molveno. The nobleman
and his family had embraced the teaching of St. Vigilius, and were
among his most pious and obedient disciples. Eligio, his eldest son,
however, had two faults which led him into great trouble, as our
story will show; but as he was of a good disposition, and was always
desirous to make amends for his wrong-doing, he found help and favour,
which kept him right in the main. His two faults were--an excess of
fondness for card-playing and an inclination to think he knew better
than his elders, which led him to go counter to good advice.
It so happened that whenever he played at cards he always won; and this
made it such a pleasure that he could not be persuaded to leave it off,
though he knew he was wasting all the time he ought to have devoted to
more manly pursuits. Nor was there for a long time any lack of people
to play with him, for every one said his luck must turn at last, and
each thought he should be the fortunate person in whose favour this
would happen. But when at last they found he still won, and won on,
they got shy of the risk, and refused to incur it any more.
When Eligio found this to be the case, he determined to travel abroad,
and play against strangers. His parents tried to make use of the
opportunity to lead him to break with his bad habit, but it was of
no avail, and, as experience is a good school, they agreed to let
him go forth and see what the world was made of.
It was a brave sight as he descended the terrace of the castle
accoutred in the noble array befitting his rank, and with a retinue
of followers handsomely attired too. But his lady mother watched him
depart with a boding heart, and then went into the chapel to pray
that he might be preserved amid all dangers.
Nothing particular occurred to mar the pleasure of travel for several
days, till he came to a large and fertile plain, studded with many
towns, whose white stone-built houses sparkled in the sun. "Ha! now
we come to life and human kind again!" cried Eligio; and putting spurs
to his steed he rode joyously to the first of these smiling towns. It
had no lofty towers, no heaven-pointing spires--nowhere was seen the
sign of the saving cross, which from boyhood he had been taught to
reverence and to see planted every where before him in consecration of
every affair of life. But there were sounds of mirth and revelry, as
of a perpetual feast, and all around the place was gay with dancers
and mummers, musicians, dice-throwers, and card-players. Eligio
wandered about till he saw a number of these making up a fresh party,
and courteously asked to be allowed to join them. They accepted his
company willingly, and fortune favoured him as usual. Again and again
he tried, and it was always the same. It was as much as his train
of followers, numerous as they were, could do to gather in and take
charge of all his gains. The stranger's unvarying luck became the
talk of the place, and all the people collected to see him play.
Towards evening there came amid the crowd a tall man of serious mien,
who, having watched his play with much attention, said to him, as
he saw him complete a game which gave him once more the benefit of
a considerable haul,--
"Truly, you are an expert player, young man; I had thought myself
hitherto the best of our countryside, but I doubt me if I should be
right to measure my skill with yours. However, you must be tired with
your long travel and with the excitement of the day's play, and if
you will honour my poor board with your presence at dinner I will ask
you afterwards to let me try my power against yours with the cards."
Eligio thanked him for his courteous speech, and assured him he should
have the greatest pleasure in doing as he wished.
The stranger then led him to his abode, which was appointed with a
sumptuousness such as had never entered into Eligio's dreams in his
mountain home. Marble courts and fountains, surrounded by bowers of
exquisite flowers, formed the approach, and then they passed beneath
endless-seeming arcades of polished marble into a vast alcove encrusted
with alabaster of many colours, the dim light only reaching through
its clear golden veins, no sound disturbing its still repose but
the cool murmur of a fountain which fed a marble lake. Here noiseless
attendants advanced, and, having helped Eligio and his host to undress,
afforded him a delicious bath, complete with ministrations of unguents
and scents--very different from the plunge into the icy waters of
the Lago di Molveno, which was his greatest luxury at home.
They now arrayed him in an entirely new suit of superb attire; and
then, to the sound of hushed music, led him and his host through
the arched corridors to a banqueting-hall, where every thing of the
choicest was ready laid.
Nothing could have been more delightful than the charming and
accomplished conversation of his hospitable entertainer, who, when the
long succession of various viands was at length exhausted, proposed
that they should repair to an upper room and commence their game.
Delighted as Eligio had been with his extraordinary entertainment,
he was yet burning to try his luck with his obliging host, and
accordingly followed him with alacrity to a divan spread on the roof,
having for its only covering a leafy pergola [93], and lighted by
lamps contrived with such art that they seemed to be the very bunches
of grapes themselves which gave the rays.
The cards were brought, and the friends set to work. The first game was
a long one; the host seemed to be in great fear of not succeeding, and
pondered every throw. Eligio played in his own rough-and-ready style,
expecting luck to come as it always had--he never troubled himself how.
But this time luck did not come to him, and his entertainer was the
winner! The stakes were large, but his hospitable friend had been so
urbane throughout, that he could not show any ill-will. His attendants
were called in, and paid the debt.
The winner put up the cards as though he did not wish to play again.
"Come, you must give me my revenge," said Eligio.
"Oh, certainly, if you wish it," he replied; and they played
again. This time Eligio paid more attention to his style, and
calculated every card he played; but it was of no use, he was beaten
again. Caring more for the disappointment than the loss, he saw the
money counted out without a sigh; but the unusual sense of having been
overcome rankled in his mind. He had offered to play high because it
seemed required by the princely character of the house where he had
been so sumptuously received; and of all the treasure he had brought
with him, and of all he had won through a day's undeviating luck,
there only remained enough to repeat the stakes. Nevertheless he
pledged the same sum once more, and they played again.
This time fortune seemed to have come back to him. All went right up
to the end; Eligio's heart felt lightened. So luck was coming back,
was it? He played with an interest which he had almost ceased to
find--but his adversary threw down his last card which reversed every
thing, and once more he was the winner!
Eligio called in his followers, and ordered them to pay out the
last farthing of his treasure; but even this distressed him less
than having nothing more to stake, whereby to have a chance of
retrieving his luck. "Let be," said his new friend, soothingly;
"perhaps to-morrow your luck will turn. Come down with me to supper,
and have a quiet night's rest, and think no more about the play."
"I can't rest, and I can't eat!" said Eligio; "I can do nothing till
my luck turns. I must stake something. Ah! there's my horse--but
that's not enough. Put along with it all my retainers. If I lose,
they shall be yours, and serve you."
"Since you insist, I have no objection," said his host. "My men
know their service well, and will not shame you if you win and I
have to render you an equal number of them; and for your horse,
I can match him, how good soever he may be, with the swiftest Arab
in the whole world."
Eligio sat down, hardly heeding his words, intent only on
re-establishing his success. But his pains were vain; the game went
against him like the last, and, scarcely mastering his vexation,
he called in his retainers and told them they had passed into the
service of the new master.
But this only left him in the same position as before. Still he wanted
to retrieve his fortune, and again he had no stake.
"Leave it for to-night," recommended his host; "better times will
come with the morrow." But Eligio would not hear of it; the passion
and excitement were too strong within him; he could not turn to
other thoughts.
"Myself! my life! that is all I have left to play. Will you accept
the wager of my life?"
"If you insist," replied his host, "I have no objection, but it is
an odd sort of play. I really never heard of such a thing before;
but any thing to oblige you--though I really advise you to leave it
till the morning, when you are cooler."
And all the time he was a magician of the heathen, who had invited
Eligio for the express purpose of bringing him to this strait; but,
as he saw how impetuous and excited he was, he knew that he would
but fall into his snare the more surely for whetting his ardour with
a little opposition.
Eligio would, indeed, listen to no mention of delay, and they sat
down and played--with the same result as before! His life was now
at the magician's disposal, and he stood in a desponding attitude,
waiting to hear what the magician decided to do with him.
As he stood there, however, a great cry rose in the room beyond--a
cry of a young maiden's voice in distress--and from under the usciale
[94] came running, in terror for its life, a sleek white rat, and
behind it, in close pursuit, a bouncing cat. "Save my rat! oh, save
my white rat!" cried the maiden's voice; and her steps approached
as if she would have run into the room after her pet. "Keep back,
child! keep back! Enter not, for your life!" cried the magician,
sternly; and nothing more was heard but the gentle maiden's sobs.
Quick as thought, however, Eligio had started from his despondent
attitude at the sound of her distressful voice, and with one blow had
stamped the life out of the treacherous cat. The little white rat,
freed from fear of its tormentor, returned softly to its mistress,
and an exclamation of joy was Eligio's reward.
"Who have you got there, father? Mayn't I come in and thank him?" said
the maiden, prettily pleading.
"On no account. Don't think of it!" was the magician's angry reply.
"Then you must do something for him instead. Ask him what he wants,
and do it for him, whatever it is."
"Very well, that'll do; go back to your own apartment," replied the
magician, impatiently.
"No, it won't do, like that. You don't say it as if you meant
it. Promise me you will give him something nice, and I will go. It's
only fair, for he has done me a great pleasure, and you mustn't
be ungrateful."
"It is enough reward, fair maiden, to hear from your sweet voice that
you are satisfied with me," Eligio ventured to say; but this made
the magician more angry, and, to ensure his daughter's departure,
he promised he would do as she wished, but forbade either of them to
speak a single word more to the other.
"I have promised my daughter to give you a good gift," he said,
when he had satisfied himself that she was gone to a distance;
"and under present circumstances I do not see that I can give you a
better boon than to grant you a year of the life which you have lost
to me. Go home and bid adieu to your friends, and be sure that you
are back here by this day year, or woe be to your whole house!"
Eligio now began to suspect that he had fallen into the power of
one of those against whom he had been often warned. No ordinary
mortal could have cared to win his life; no ordinary mortal could
have threatened woe on his whole house. But the more convinced he
felt of this, the more terrible he felt was the spell that bound him.
Sad and crestfallen he looked as he toiled his way back to the castle
on the Lago di Molveno, and very different from the brave order with
which he had started.
When his parents saw him all alone, and looking so forlorn, they knew
that his bad habit had got him into trouble, but he looked so sad
that they said nothing; but by little and little he told them all. It
was a year of mourning that succeeded that day; a year so sad that it
seemed no boon the maiden had procured him, but a prolonged torment,
yet when that thought came he spurned it from him, as ungrateful to
her who had meant him well. In fact his only solace was to recall that
clear, ringing voice so full of sympathy, and to picture to himself
the slender throat and rosy lips through which it must have passed,
the softly-blushing cheeks between which those lips must have been
set, and the bright, laughing, trusting eyes that must have beamed
over them, till he seemed quite to know and love her.
But then, again, of what use? was not his year nearly run out? Was
not her father determined they should not meet? Was it not a greater
torture to die knowing there was one left behind he might have loved,
than to have died that night alone, as he had been then?
Meantime the year was drawing to a close, and, not to give an
appearance of shrinking from his plighted word, Eligio started
betimes to render his life up to him who had won it of him. It was
a sad parting with his parents, but he held up through it bravely;
and when they advised him to take a large sum of money with him to
buy himself off, though he felt it would be of no use, he would not
say them nay, as he had so often done before.
With a heavy heart he set out; and first he stopped at the chapel
of St. Anthony, at the foot of the hill, where dwelt an old hermit,
to make his peace with heaven before he was called to lay down his
life. Then he rose and pursued his way.
As he journeyed farther he met a hermit coming towards him who he
thought was the same he had spoken with in the chapel. "Tell me,
father," he said, "how comes it that you, whom I left behind me in
the chapel, are now coming towards me on the road?"
"I am not the hermit whom you left behind you in the chapel," replied
the advancing figure, gravely. "But I have heard all you confided to
him, for I am St. Anthony; and because I am satisfied with the good
disposition I have observed in you, I am come to give you help."
Eligio fell on his knees full of thankfulness, for never had he felt
more in need of help than now.
"Something I know, my son, of the ways of these men who hunt the lambs
of our flock to destroy them, and I am minded to save you from the
one into whose power you have fallen, and with you the fair maiden
whose voice charmed you in his house."
Eligio started with joyful surprise, and clasped the saint's feet in
token of gratitude.
"She is not his daughter, as you have supposed," continued the saint,
"but a child of our people, whom he stole from us. And now you must
attend to my bidding, and do it exactly, or you will fail, and lose
her life as well as your own."
Eligio felt the reproach, for he knew how often he had preferred his
own way to the advice of his elders, but he was humble now in his
distress, and listened very attentively to the directions prescribed
to him.
"Continue this public road towards the city," then said St. Anthony,
"till you get to the last milestone; then count the tenth tree
that you pass on the right hand and the eleventh on the left hand,
and you will see a scarcely perceptible track through the brake to
the right. Follow that track till you come to a knoll of ilex-trees,
there lie down and rest; but to-morrow morning awake at daybreak and
lie in wait, and you shall see a flock of white doves come before
you. They will lay aside their feathers and hide them, but you must
watch them very closely, for they are the magician's daughters;
but among them will be she whom I commission you to deliver. You
must observe where she puts her feathers, for the maidens will all
then go away for the rest of the day in their own natural form. As
soon as they are gone, take her feathers from their hiding-place and
possess yourself of them. In the evening they will all come back and
resume their dove form and fly away, but your maiden will continue
seeking hers; then come forward and tell her that you want her help
to overcome the sorceries of the magician. Remember this well, my son,
and for the rest do as she bids you." So saying, the saint raised his
hands in blessing, and passed on his way to the chapel, where he had
to instruct the hermit in the conduct he had to pursue in the manifold
dangers with which he was surrounded from the malice of the heathen.
Eligio walked briskly along, once more filled with the hope and
energy incident to his youth and character. "Why should I count the
trees?" he said to himself; "surely, it will do if I look out for the
track when I come to the brake!" But the terrible warning he had had
was too recent that he should forget its lessons already. "Perhaps
it's better to keep to the letter. The saint laid great stress on
my doing exactly as he bid me; it is better to be on the safe side,
for another worthier life than mine is concerned with me, this time."
So he walked on steadily till he came to the last mile, and then
counted the trees conscientiously, till he found the path through
the brake, and made his way to the ilex grove, where he laid him down
and slept peacefully. But long before daybreak he was awake with the
anxiety not to be behindhand, and closely he watched for the arrival
of the enchanted doves.
With the first streaks of daylight they came flying, as the saint had
predicted, and, having flung off their covering of white feathers,
each sought out a snug place under the heather where to deposit
them. It required close watching, indeed, to make out which was
his maiden; but, as they all chatted together, after the manner of
maidens, Eligio knew he could trust himself to recognize her voice,
and, guided by that, he kept his eyes hard fixed on her whose tones
he recognized, that he might be sure to distinguish where she laid by
her disguise. It was not light enough to satisfy himself whether her
features corresponded with the idea he had built up in his own mind;
but the grace of her form, as she passed by in her simple white,
loosely-flowing dress, with a chaplet of roses for her only ornament;
only made him the more anxious to behold her face.
The maidens walked away, and Eligio took possession of the feathery
covering, which he laid up in his bosom as a precious token, and took
it out again and gazed at it, and kissed it, and laid it by again
a thousand times, for it was his only solace through the long day
of waiting.
At last evening came, and he resumed his post of observation. The
maidens returned; each sought out and resumed her dove's feathers and
flew away; only the one was left, seeking hers in vain. Then Eligio
came forward, and said, respectfully, "Fair lady, I know what it is
you seek, and I will help you to find it; but first promise to do me
a great favour."
The maiden started, for she too recognized his voice. Their eyes met,
and both owned, in the depth of their own hearts, that the other bore
the very image which for a year past their fancy had conjured up.
"That will I, willingly, good sir!" she replied, in her sweetest
tones; "for, an' I mistake not, I owe you a debt of gratitude before
to-day. The treacherous cat that you killed so opportunely was no cat,
but a cruel Angana [95]; and the white rat concerned me so nearly,
because it was no rat, but my dear nurse, whom the magician turned
into a rat when he stole me from my father's house. So believe if
I was not anxious to save her, and if I ought not to be grateful to
him who preserved her to me! so tell me, what can I do to help you,
and, whatever it may be, I will do it to the utmost of my power."
"St. Anthony appeared to me as I came along this way," rejoined Eligio,
"and he told me that you had been stolen from Christian parents and
brought up by this heathen mage, and that you would help me to get
out of his power; but he also seemed to say that I should have the
happiness of helping you to leave this dreadful abode, and restoring
you to Christendom."
"Said he so?" answered the maiden, with intense earnestness; "then
my heart did not deceive me when I first heard your voice: you are
indeed he with the thread of whose life mine is woven, and without
whom I could not be set free."
When Eligio heard that, he was full of gladness, and he said, "Let
us escape, then! What should prevent us from leaving this country
together? When I saw the magician before, the laws of hospitality
made him sacred to my sword; but now--now that I have learnt I have
a right to defend your life--I defy him, and all his arts!"
"You are brave, I see; and it is well," she replied; "but it is not
so you can discard his power. By your own error you gave him power
over you, and now you are his; you can only be free by his will."
"By his will!" cried Eligio, in despair; "then shall I never be free!"
"Art must be met by art," she continued. "His art is all round
you, though you see not its meshes; and by art we must bring him to
renounce his claim on you. Trust me, and I will show you how it is to
be done. He would force me to learn his arts when I begged him not,
and now I know many things which will serve us. I can see the threads
of his toils woven all around you; you cannot escape from them till
he speaks you free."
"Tell me, then, what I must do," said Eligio; and he mentally resolved
as he spoke, that he would this time implicitly obey what she told him.
She remained thinking for a time, as if reckoning out a problem. Then
she said, "For this first time I must act. On the fatal day you
must present yourself according to your oath. I will take care to
be with him when they tell him you are come; and when I hear your
name, I will plead, as I did before, that he should not sacrifice
you at once, but give you some hard trial in which, if you succeed,
he shall speak you free. To silence my importunity, he will agree
to this, intending to give you so hard a trial that you should not
succeed. But you come to me in my bower, cooing three times like a
dove, for a signal, at this same evening hour, and tell me what it
is, and I will find the means in my books to carry you through the
trial. So that, whatever he proposes to you, be not disconcerted,
but accept and undertake it with a good heart. And now, give me my
dove's feathers quickly, for already they will be questioning why I
am so long behind." And without waiting to let him take so much as
another gaze at her, she assumed her dove shape, and flew away.
The next day Eligio went, with a lighter heart than he had borne for
a long time past, to give himself up to the magician. The magician,
won over by the maiden's importunity, offered him his liberty on
condition of his performing successfully the difficult feat that he
should impose on him.
"Any thing you please to impose on me, I am ready to perform,"
replied Eligio.
The magician smiled, with a ghastly, sardonic smile, while he paused,
and tried to think of the most terrible trial he could impose.
"Since you were here last," he said, at length, "I have grown a little
deaf, and I am told that the only cure there is for me is the singing
of the phoenix-bird. The first thing you have to do is to find me
the phoenix-bird, that its singing may heal me."
"I will do my best; and hope I may be the means of curing your malady,"
said Eligio, courteously; but the magician, seeing him of such good
courage, began to fear he really might succeed, and added, hastily,
"But, mind, I only allow you three days for your search!"
"Three days are but little to find the phoenix-bird," replied Eligio;
"nevertheless, I will do my best;" and without waiting to listen to
any further restrictions, he started on his way, saying, "If I have
only three days, I have no time to lose."
At the approach of the evening hour Eligio found his way to his
maiden's bower, and having attracted her attention by cooing three
times like a dove, told her what was the trial the magician had
imposed.
"The phoenix-bird!" she said, and she looked rather blank; "he has
chosen a difficult task indeed. But wait a bit; I think I can find it
out;" and she went back and took down scroll after scroll, and turned
them over so long, that Eligio began to fear that she would not be able
to help him after all. At last she came back to him, looking grave.
"It is more difficult even than I thought," she said; "and three
days is but short time to do it in. You must start this night,
without losing a minute. Set out by the stony path outside the town,
and ride ahead till you come to a forest, where a bear will come out
upon you. The moment you see him, spring from your horse, and cut
its throat with your hunting-knife; but if you hesitate a moment
he will fall upon you, and devour you. If, however, you kill your
horse dexterously, as you will, the bear will be satisfied with its
flesh. You must wait standing by till he has eaten his fill, and watch
for the moment when he is about to turn away again, then spring on
his back, and he will take you to the castle where the phoenix-bird
is kept; but if you lose that particular moment, he will return to his
cave, and you will never have a chance of reaching the phoenix-bird!"
"Rely on me; your directions shall be punctually obeyed," said Eligio,
and he stooped to kiss her hand. But she would not allow this, and
told him he had not an instant to spare.
Eligio mounted his horse, and rode away over the stony path outside
the city, and pursued it all night, till at daybreak he reached the
thick forest, when a bear came out upon him; Eligio sprang deftly from
his horse, and plunging his hunting-knife into his throat, flung the
carcase across the path. The bear fell upon the dead horse, and Eligio
watched for the moment when he should have finished his repast; but,
as he was long about it, he thought to himself, "Why not jump upon him
at once? and then I shall be ready to start with him when he has done,
without so much anxiety about catching the right instant." So said,
so done; but the bear was not at all the docile animal he had expected.
"Don't disturb me when I'm feeding!" he growled, and shook our hero
off into a bed of nettles.
Eligio owned to himself he would have done better to follow the
directions of those wiser than he, and waited, with as much patience
as the stinging of the nettles would allow him, till the brute was
ready to start, and then made a bold leap on to his back, which made
him turn round.
"Well sprung, this time!" growled the bear; "and as you have managed
that part of the business so well I have no objection to do what
you require. But you must attend to what I have to tell you. Keep
your seat steadily, for I have to go swiftly; but speak not a word,
and when I bring you to the palace where the phoenix-bird is kept,
look not to the right hand or the left, but walk straight before
you, through terrace, and galleries, and corridors, till you come to
a dismal, deserted-looking aviary, where the phoenix-bird evermore
sits on his perch. Put this hood over him, and bring him away with
you; but listen not to the songs of the other birds all around, and,
above all, touch not the golden owl which sits in the shade above!"
Eligio promised to attend to all the bear told him, and took a firm
seat on his back. The bear bounded away with an awkward gait, but
Eligio was an accomplished cavalier, and was nothing daunted. After
many hours' rough riding, they came to a vast palace, which he
understood by the bear's halting was the abode of the phoenix-bird; so
he dismounted, and walked straight along the terraces, and galleries,
and corridors, till he came to a sorry aviary where a thousand birds
of gay plumage fluttered and chirped around. Faithful to his promise,
Eligio stopped to look at none of them; but walked straight up to
the perch of the phoenix-bird. When, however, he saw him, he began
to reason in place of obeying. "What can be the use of taking a
shabby old bird like that? he looks like a fowl plucked ready for
cooking! surely, some of these gay-plumaged birds are better worth
taking!" and then his eye caught the golden owl snugly ensconced
in the shady bower above. "Ah! that's a bird worth having, that is
now! that's worth coming a perilous journey for; something to be proud
of when you've got it! That's the bird for me!" and, springing upon
a ledge of rock, he threw the hood the bear had given him over the
head of the golden owl, and brought it down. He had scarcely touched
the golden owl, however, when the whole assemblage of other birds,
which had taken no notice of him before, suddenly began screeching
forth their highest notes. Their cries brought a crowd of servants,
who surrounded him and held him fast, while the lord of the palace
came down, and severely asked an account of his conduct.
Eligio told his story with a frankness which, in some measure,
conciliated the old lord; but the offence was too great to be passed
over. "The phoenix-bird," he said, "might have been taken by him who
had courage to take it after the prescribed manner; but the other
birds it was sacrilege to meddle with, and the golden owl he had been
expressly forbidden, of all others, to touch; and though he granted
him his life, he condemned him to perpetual durance." The servants
dragged him off to a deep dungeon, where he had nothing to do but to
bewail his folly.
Night fell around, and nothing could be more hopeless than his
position. His cell was hewn out of the earth; the iron door through
which he had been thrust had been made fast with bolts and chains,
and the only window which admitted the free air was strongly fitted
with iron bars.
Eligio was generous enough, in his utter desolation, to grieve more
over his unfulfilled mission and wasted opportunities, than over his
personal hardships. "Oh, my beautiful Dove-Maiden!" he exclaimed,
"shall I, then, never see you again? Must you be left for aye to
the power of the horrid pagan enchanter, because I, by my insensate
folly, have failed in restoring you to the brightness of the Christian
faith?" and when he thought of her fate, he wept again.
"St Anthony! St. Anthony!" he cried, a little after, "you befriended
me once; give me one chance again! This once but send me forth again
with the mission of liberating her, and then let me come back and
pass my life in penance; but let not her suffer through my fault!"
By a mechanical instinct he had placed himself near the window,
as the type of freedom to him, and now he thought he heard a low
grunt on the other side of it, close to his ear. The sound was not
melodious, but yet he fancied there was something friendly in its
tone. He raised himself up, and saw two white boar's tusks between
the bars. His solitude was so utter that even the visit of a wild boar
was a solace of companionship; but much greater was his pleasure when
he found that his uncouth visitor was grubbing up the earth round the
iron bars and the stones which held them, and had already loosened one.
"How now, good boar!" cried Eligio; "are you really come to release
me?"
"Yes," said the boar, as he paused for a moment to take breath;
"St. Anthony has heard you, and has sent me to give the fresh chance
you ask for; and if you this time but keep your promise, and do as you
are bid, he will not exact the performance of the lifelong penance
you offer to perform; but after you have released the Dove-Maiden,
you shall live with her the rest of your life in holy union and
companionship."
In a transport of delight Eligio set to work to co-operate with the
boar in unearthing the massive stanchions; and when they had loosened
three he was able to force himself through the narrow opening.
"Now return to the aviary," said the boar; "look neither to right nor
left, but bring away the phoenix-bird; and speak not a word, but mount
on my back, and I will carry you back to the city. But make all haste,
or the three days will have expired, and then all will be lost!"
This time Eligio followed his instructions implicitly, and got
back to the town just in time to present the magician with the
phoenix-bird before the expiring of his three days' grace. The
magician was surprised indeed to find he had been successful, but
could not recall his word, so he was forced to pronounce him free;
and Eligio immediately repaired to the Dove-Maiden to thank her for
her succour, and to ask what was next to be done to set her free too,
that they might go away together to Christian lands, and live for
each other in holy union.
"As for me," replied the maiden, blushing, "I shall be free by virtue
of your freedom when you have performed one trial well, and without
altering according to your own ideas the directions prescribed for
you. And now the first thing is, to obtain the release of my dear
nurse from the horrid form in which the magician has disguised her. To
keep her in that shape, she is forced to eat a live mouse every week;
and as nothing else is given her that she can eat, and as she is very
ravenous by the time the week comes round, she is forced to eat the
mouse. But if the mouse be killed by a sword consecrated to Christian
chivalry, and it is dead before she eats it, the spell will be broken,
and she will resume her natural form."
Eligio said this was an easy matter. She had only to tell him on what
day the feeding took place, and where.
"It has its difficulties, too," replied the Dove-Maiden; "for if
any blood of the mouse be spilt, the magician will know that I have
instructed you, and he will play us some bad turn. To prevent this,
you must cut the mouse in two by drawing your sword towards you;
then all the blood will be caught on the sword, and you must make the
rat lick it off afterwards." Then she showed him where the mouse was
brought, and told him to be on the watch at sunset that very night.
Sunset accordingly found Eligio in close watch, his sword ready
in his hand. But he thought, "As for how to use a sword, my pretty
Dove-Maiden knows nothing about that. Who ever heard of drawing a
sword towards one? Why, if any one saw me they would laugh, and say,
'Take care of your legs!' I know how to cut a mouse in two so quickly
that no blood shall be spilt; and that's all that matters." So, you
see, he would do it his own way; and the consequence was that three
drops of blood were spilt on the ground However, the white rat got
a dead mouse to eat instead of a live one, and immediately appeared
in her proper woman's form.
When Eligio went to visit the Dove-Maiden after this, she spoke no word
of reproach, but she told him she knew some trouble would befall them
in consequence of those three drops of blood. She could not tell what
it would be: they must do their best to provide against it when the
time came. The next thing he had to do was, to go by midnight to the
magician's stables under the rock, and take out thence the swiftest
horse in the whole world, and he was to know it by the token that it
was the thinnest horse he ever saw; its eyeballs and its ribs were
all that could be seen of it; and its tail was only one hair! This
he was to saddle and bring under her window; and then all three would
ride away on it together.
Eligio went down into the magician's stable under the rock by midnight,
and there he saw the lean horse, with his protruding ribs and eyeballs,
and whose tail was only one hair. But he said to himself, "My pretty
Dove-Maiden hasn't much experience in horseflesh; that can't be the
swiftest horse in the world. Why, it would sink to the ground with our
weight alone, let alone trying to move under us! That high-couraged
chestnut there, with the powerful shoulders--that is the horse to
hold out against fatigue, and put miles of distance behind you! I
think I know a good horse to go when I see one!" So he saddled the
high-couraged chestnut, and led it under the Dove-Maiden's window.
When she saw the stout chestnut instead of the lean horse, she could
not suppress a cry of disappointment.
"What have you done?" she said. "You have left the swiftest horse in
the world behind; and now the magician can overtake us, nor can we
escape him!"
Eligio hung his head, and stammered out a proposal to go and change
the horse. But she told him it was too late; the stable-door was only
open at midnight. He could not now get in till the next night; and
if they left their escape till then, the magician would find out the
disenchantment of the white rat, and from that suspect their scheme;
and would then surround them with such a maze of difficulties, that
it would take her years to learn how to solve them; whereas she had
promised St. Anthony to have nothing more to do with the books of
magic, but to burn them all, and go and live with a Christian husband,
far from all these things. There was nothing to be done, therefore,
but to start at once with their best speed, only keeping on the watch
for the pursuer, who would inevitably come.
Away went the high-couraged chestnut, with the speed of the wind,
and as if his threefold burden had been light as air. But how swiftly
soever he went, the lean horse was swifter; and before the end of
the second day's journey they saw, at no distance, his fire-darting
eyeballs and smoking ribs, and his tail of one hair stretched out
far behind.
When the Dove-Maiden saw the magician coming after them on this weird
mount, she called to her companions to jump down; and she turned
the horse into a wayside chapel of St. Anthony, and herself into a
peasant girl weaving chaplets on the grass outside.
"Have you seen a chestnut steed pass this way, with a young man and
maiden, pretty child?" said the magician, bending low over his horse's
neck to pat the peasant girl's cheek, but without recognizing her. The
Dove-Maiden started aside from his touch; but she answered,--
"Yes, good sir; they are gone into the chapel; and if you will go in,
there you will find them."
"Oh! I've got into the land of the Christians, have I?" said the
magician to himself. "I think I had better make the best of my way
home, and not trust myself there." So he mounted his fiery steed,
and rode away.
Then the Dove-Maiden restored herself and her companions to their
former shapes, and they soon reached home, where Eligio was received
with joyful acclamations by all. But to his intense surprise and
disappointment, his mother did not welcome his beautiful Dove-Maiden
with any thing like satisfaction.
"That is because of the three drops of the mouse's blood incautiously
spilt," she whispered, when he deplored it to her; "but I have a spell
against that also. Let me into your mother's room when she is asleep,
to-night, and I will anoint her eyes with an ointment with shall make
her look on me for ever after with a loving glance. It was done as
she said, and next morning Eligio's mother received her lovingly to
her arms as a daughter.
After this, the Dove-Maiden burnt her magic books, and her nuptials
with Eligio were celebrated with great rejoicings throughout the
valley. They lived together for the rest of their days, in holy union,
and the poor Christians of the whole countryside blessed their charity.
KRISELDA.
Long, long ago, in the days when the light of Christian teaching yet
struggled with the gloom of heathendom, there lived in the Edelsitz of
Ruggburg, by Bregenz, a most beautiful maiden--Kriselda by name. So
fair she was that, from far and near, knights and nobles came to ask
her hand; but though she was not proud or haughty, she would have
none of them, because there was not one of them all that came up to
her expectations. It was not that she said they were not good enough
for her, but high or noble, or rich or renowned, as they might be,
they all failed to satisfy her longings; and with gentle words and
courteous demeanour, she dismissed them all. And yet she looked out
with hope, too, that the next should supply the bright ideal of her
heart; but when that other came he always still fell short of what
she had imagined.
One evening she went out to walk amid the dark pines, where the golden
light of the setting sun gleamed between their bare stems. At the
foot of one of them lay a poor wayworn beggar woman, fainting with
hunger and fatigue.
Kriselda was full of compassion for her sad state, and sent her maidens
to fetch restoratives, and ministered them to her with her own hands.
But the beggar woman, instead of cringing with gratitude and surprise
at the interest the noble lady had shown in her, was no sooner able
to speak than she reproached her bitterly.
"It is well for you," she said, "who live daintily, and have your
will every day, now and then to show a little charity for those who
suffer! but what is it, think you, to suffer every day, and to have
your own will never?"
"It must be very sad!" said Kriselda, compassionately; "that is not
your case, I hope?"
"How can you know it is sad? How can you hope any thing about
it?" retorted the beggar woman, sternly; "you who know not what it
is to suffer. Believe me, it is not fine clothes and a grand palace,
a beautiful face, or deeds of fame which make one great. Those to whom
all these things appertain are, for the most part, little worth. To do
well is so easy to them, that what merit have they to boast? The truly
great is one who suffers, and yet does well; who goes through toil
and travail, sorrow and grief, and bears it in silence, and in secret,
and has no fame and no praise of men to sound sweetly in his ears."
Kriselda listened to her words full of excitement, for it seemed as
if a chord in her heart had been touched which none had ever reached
before. And the picture the old beggar woman had drawn was nearer
her mind's bright ideal than any image she had approached heretofore.
"What, then, is this same travail and grief?" she asked, with
simplicity.
"If you really desire to know with good desire," answered the beggar
woman, "take this end of a hank of yarn, and follow its leading,
winding it up as you go along, till you come to the bobbin, where it
is made fast; and when you arrive there you will know what travail
and grief are. But you must go forth alone."
Kriselda dismissed all her maidens, and taking the yarn, cheerfully
followed the steep path through which it led. On it led her, and on
and on. Her light garments were rent by the thorns and briars, and
her hands and delicate cheeks too; her feet were cut by the stones of
the way, and her knees began to tremble with fatigue. Darkness fell
around, and loneliness crept over her, with fear, for she had never
been in the forest by night alone before; but still the yarn led on,
and on, and it was thick night before she reached the bobbin, where
it was made fast.
When she reached the place a dim light gleamed around, and in the
midst of the dim light a Kreuzstoecklein [96]: and on the cross,
One fairer than the sons of men, but wan and wayworn, even as the
fainting beggar woman; His brow rent by thorns, even as her own;
His knees bent with weariness; His body wasted by want.
But in His face the majesty and sweetness she had sought so long;
the perfect ideal of her heart, which none who had approached her
had ever presented before.
"This, then, is He for whom my soul longed!" she cried, and clasped
her hands. "I have found Him, and will not leave Him more! But who
is He? what does He here? and is it He who knows travail and grief?"
"In truth, have I known travail and grief!" He sighed, and the silvery
tones of His plaintive voice filled her with unutterable joy; "and,
in truth, must all those who would abide with Me know travail and
grief too!"
She strained her ears that she might hear those sweet notes again, but
she listened in vain; only its echoes seemed to live on in her heart,
as though they would never die there. But without, there was no sound,
save of the terrible Foehn [97] moaning through the tall black pines,
and drifting round her masses of heaped-up snow, which had long lain
by the wayside. Even the Kreuzstoecklein she saw no more, nor the dim
light, nor knew how to find the way home. All alone, with terror only
for her companion, she stood and wondered what that cross could mean,
and who He could be who hung thereon. Soon she ceased to wonder,
for numbness crept over her, and unconsciousness which was not sleep.
When she opened her eyes again the grey light of morning had fallen
around, and there was a sound as of men in deadly combat. A terrible
sound, yet less terrible than the deathly stillness of the night.
It was a hermit and a giant who strove, as men who give no quarter,
and yet neither prevailed against the other. The giant was accoutred in
burnished steel; and his polished weapons flashed with angry fire. The
hermit bore no arms--or rather, those he bore were invisible, for when
he wielded them you saw the giant shrink, though you saw not the blow;
and, in like manner, many a stroke of the giant's sword was harmlessly
warded off, though no shield was seen.
"Wherefore fight you so furiously?" said Kriselda, at length. "Put
up your arms, and be at peace."
"We fight for you, fair maiden!" said both, speaking together.
"For me!" cried Kriselda.
"Yes, even for you," said the giant; "anon you were lying here asleep,
and I would have carried you to rule over my castle, when up started
this puny man in brown, and dared me to lay finger on you; and till
you have pronounced which of us you approve, neither can prevail. Say
only one word, and I will hurl him down the cliff, like this pebble,
with one spurning of my foot; and you shall come and reign with me
in my castle, where I will fulfil your every desire."
A brave enthusiasm kindled his eye as he spoke; his well-knit frame,
terrible in its strength, was bowed to hear her word; and his arms,
anon so furiously raised, were now folded before her, seemingly
awaiting his life to be rekindled at her lips.
Kriselda looked at him, and met his rapt gaze, and asked herself
was there not here the strength, the majesty, the nobility, her soul
had desired. Almost she had spoken the word he craved. But first she
addressed the hermit.
"And you--why measured you your strength with him for my sake?"
"Because," said the hermit, meekly, "I am the servant of Him who knows
travail and grief; because you have lifted up your eyes to Him, and to
all such He sends help, that they may be strengthened to follow Him."
Then the dim light seemed once more present to Kriselda's mind, and
she recalled the Kreuzstoecklein, and the majesty and beauty of Him who
hung thereon; and the musical tones of His plaintive voice which said,
"Truly I have known travail and grief; and all they who would abide
with Me must know grief and travail too!"
The giant's nobility paled before the thought; she looked at him
again, and his strength and his power had lost their charm, for the
image of One stronger than he was present to her mind. Then she turned
and followed the hermit, and said, "Where is He whom I seek? Take me
to Him."
The hermit raised his hand and beckoned her to follow still higher
up the steep path.
But the giant was forced to sheathe his sword and to depart alone;
Kriselda had spoken, and he knew he could not prevail against the
hermit contrary to her will. He turned away sorrowful, casting in his
mind who it could be whose attractions were more powerful on Kriselda
than his own; and as he walked he determined he would not sleep
or eat till he had found out Him who hung upon the Kreuzstoecklein,
and knew travail and grief.
Kriselda, meantime, followed the hermit to where the crystal brook
flowed, and there he signed her with the token of Him who knew travail
and grief. Then he took her to where other maidens dwelt who loved
that same ideal; and there she lived many years, waiting for the time
when the hermit promised her she should be united with Him for ever.
That day came at last; and she called her sisters round her, and told
them the joy of her soul. Already she saw a dim light, as on that
first night under the black pines, and she knew it was the dawn of
the bright unending day, and the soft voice that had spoken to her
there was calling to her to come to Him.
But when they carried her earthly form out to burial, they found one
already lying in the grave. It was the giant, who had journeyed thus
far, and had there laid him down and died in the place where Kriselda
should be laid; and he held, clasped to his breast, the Kreuzstoecklein
of the black pine-forest.
THE GOLDEN PEARS.
There was once a poor peasant of Buers who had nothing in the world
but three sons, and a pear-tree that grew before his cottage.
But as his pears were very fine, and the Kaiser was very fond of them,
he said to his sons one day, that he would send the Kaiser a basket
of them for a present.
So he plaited a nice Krattle [98] and lined it with fresh leaves,
and laid the pears on them, and sent his eldest son with it to make
a present to the Kaiser, giving him strict charge to take care and
not let any one rob him of them by the way.
"Leave me alone, father!" replied the boy; "I know how to take care
of my own. It isn't much any one will get out of me by asking; I can
find as good an answer as any one."
So he closed up the mouth of the basket with fresh leaves and went
out to take the pears to the Kaiser.
It was autumn, and the sun struck hot all through the midday hours, and
at last coming to a wayside fountain, he sat down to drink and rest.
A little doubled-up old woman was washing some rags at the same
fountain, and singing a ditty all out of tune. "A witch, I'll be
bound!" said the boy to himself. "She'll be trying to get my pears,
by hook or by crook, but I'll be even with her!"
"A fair day, my lad!" said the little old wife; "but a heavy burden
you have to carry. What may it be with which you are so heavily laden?"
"A load of sweepings off the road, to see if I can turn a penny by
it," replied the boy, in a moody tone, intended to arrest further
questioning.
"Road-sweepings?" repeated the hag, incredulously. "Belike you don't
mean it?"
"But I do mean it," retorted the boy.
"Oh, well, if you mean it, no doubt it is so. You will see when you
get to your journey's end!" and she went on washing and singing her
ditty that was all out of tune.
"There's mischief in her tone," said the boy to himself, "that's
clear. But at all events I'm all right: I haven't even let her look
at the fruit with her evil eye, so there's no harm done." But he felt
perplexed and uneasy, so it was no good taking rest, and he went on
to the end of his journey.
Though he was only a country lad, the Kaiser was so fond of pears that
he had only to say he had brought some to obtain immediate admittance
to his presence.
"You have brought me some pears, have you, my boy?" said the Kaiser,
in a tone of satisfaction; and he licked his lips with pleasurable
expectation.
"Yes, your majesty; and some of the finest golden pears in your
majesty's whole empire."
The Kaiser was so delighted to hear this that he removed the covering
of leaves himself. But proportionately great was his fury when he
found that under the leaves was nothing but offensive sweepings off
the road! The attendants who stood by were all equally indignant,
and waited not for an order from the Kaiser to carry the boy off to
close prison, in punishment for so great an insult as he appeared to
have offered.
"It is all that old hag by the fountain," he said to himself, the first
day and the second; but when the penitential discipline of the prison
led him to think more closely over his own conduct, he acknowledged
that he had himself been in the wrong in telling a falsehood.
Meantime, his father, finding he did not return, said to his other
sons, "You see what it is to be as wide-awake as your elder brother;
he has obviously taken care of his basket of golden pears, and so
pleased the Kaiser that he has given him some great office near his
person, and made him a rich man."
"I am just as sharp as he," said the second brother: "give me a Krattle
of the pears, and let me take them to the Kaiser, and become a rich
man too; only I won't keep it all for myself. I will send for you,
and make you a rich man too."
"Well said, my son," replied the father; "for I have worked hard for
you all my life, and it is meet that in my old age you should share
your ease, which I helped you to attain, with me."
And as the season for pears had just come round again, he plaited
another Krattle, like the first, and lined it with fresh leaves,
and laid in it a goodly show of the golden pears.
The second son took the basket, and went his way even in better spirits
than his elder brother, for he had the conviction of his success to
encourage him. But the sun was as hot as it had been the previous year,
and when, in the middle of the third day, he came to the fountain by
the wayside he was glad to sit down to rest and refresh himself.
The doubled-up old woman stood washing her rags at the fountain and
singing her ditty all out of tune. She stopped her croaking, however,
to ask him the same question as she had asked his brother; and,
as he and his brother had agreed together on what they considered
a clever answer, he now gave her the same, which she received by
repeating the menace she had ejaculated the first time. And when he
brought his basket to the Kaiser it also was found to be filled with
street-sweepings instead of pears! With even more of indignation they
hurried him off to prison, putting him in the next cell to his brother.
Meantime the year was wearing away, and the promised tidings of good
fortune not reaching the father, he got very uneasy. The third son
had no pretension to the sharpness his brothers boasted. He was a very
dull boy, and often had to endure being laughed at by the others for
his slow parts.
"What a pity it is you are so heavy and stupid!" his father now would
often say. "If I only dared trust you, how glad I should be to send
you to see what has befallen your brothers!"
The lad was used to hear himself pronounced good-for-nothing,
and so he did not take much notice of these observations at first,
but seeing his father really in distress, his affectionate heart was
moved, and he one day summoned courage to say he would go and see if
he could not find his brothers.
"Do you really think you can keep yourself out of harm's
way?" exclaimed his father, glad to find him propose to undertake
the adventure.
"I will do whatever you tell me," replied the lad.
"Well, you shan't go empty-handed, at all events," said the
father. And, as the pears were just ripe again, he laid the choicest
of the year's stock in another Krattle, and sent him on his way.
The boy walked along, looking neither to right nor left, but with his
heart beating, lest he should come across the "harm" out of the way
of which he had promised to keep himself. All went smoothly, however,
except that he got terribly scorched by the sun, and when he reached
the fountain, he was glad to sit down to rest and refresh himself.
The old wife was washing her rags in the water, and singing, as she
patted the linen, a ditty all out of tune. "Here comes a third of those
surly dogs, I declare!" she said to herself, as she saw him arrive
with another lot of the magnificent pears. "I suppose he'll be making
game of me too--as if I didn't know the scent of ripe golden pears from
road-sweepings! a likely matter! But if they enjoy making game of me, I
have a splendid revenge to enjoy upon them, so I oughtn't to complain."
"Good-morrow, little mother!" said the boy, in his blunt way, ere
he sat down, at the same time not omitting to doff his cap, as he
had been taught, because she happened to be old and ugly--matters of
which he had no very nice appreciation.
"He's better mannered than the other louts, for all he doesn't look so
bright-faced," said the hag to herself; and she stopped her discordant
song to return his greeting.
"May I sit down here a bit, please, good mother? asked the boy,
thinking in his simplicity the fountain must belong to her.
"That you may, and take a draught of the cool water too," replied
the dame, wondrously propitiated by his civility.
"And what may it be with which you are so laden, my pretty boy?" she
continued. "It ought to be a precious burden to be worth carrying so
far as you seem to have come. What have you in your Krattle?"
"Precious are the contents, I believe you," replied the simple boy; "at
least, so one would think from the store my father sets by them. They
are true golden pears, and he says there are no finer grown in the
whole kingdom; and I am taking them to the Kaiser because he is very
fond of them."
"Only ripe pears, and yet so heavy?" returned the old wife; "one
would say it was something heavier than pears. But you'll see when
you come to your journey's end."
The boy assured her they were nothing but pears, and as one of his
father's injunctions had been not to lose time by the way, he paid
the old dame a courteous greeting and continued his journey.
When the servants saw another peasant boy from Buers come to the
palace with the story that he had pears for the king, they said,
"No, no! we have had enough of that! you may just turn round and go
back." But the poor simple boy was so disappointed at the idea of
going back to be laughed at for not fulfilling his message, that he
sank down on the door-step and sobbed bitterly, and there he remained
sobbing till the Kaiser came out.
The Kaiser had his daughter with him, and when she saw the boy sobbing,
she inquired what ailed him; and learnt it was another boy from Buers
come to insult the Kaiser with a basket of road-sweepings, and asked
if they should take him off to prison too.
"But I have got pears!" sobbed the boy; "and my father says there
are no finer in the empire."
"Yes, yes; we know that by heart. That's what the others said!" replied
the servants, jeering; and they would have dragged him away.
"But won't you look at my pears first, fair lady? the pears that I
have brought all this long way for the Kaiser? My father will be so
sorry!" for he was too ignorant to feel abashed at the presence of
the princess, and he spoke to her with as much confidence as if she
had been a village maiden.
The princess was struck by the earnestness with which he spoke, and
decided to see the contents of his basket. The moment he heard her
consent, he walked straight up with his Krattle, quite regardless of
the whole troop of lacqueys, strong in the justice of his cause.
The princess removed the covering of leaves, and discovered that what
he had brought were golden pears indeed, for each pear, large as it
was, was of solid shining metal!
"These are pears indeed worthy to set before the Kaiser!" she said,
and presented them to her father.
The Kaiser was pleased to see his favourite fruit so splendidly
immortalized, and ordered the pears to be laid up in his cabinet of
curiosities; but to the boy, for his reward, he ordered that whatever
he asked should be given.
"All I want is to find my two brothers, who hold some great office
at court," said the boy.
"Your brothers hold office in prison, if they are those I suspect,"
said the Kaiser, and commanded that they should be brought. The boys
immediately ran to embrace each other; and the Kaiser made them each
recount all their adventures.
"You see how dangerous it is to depart from the truth!" he said,
when they had done. "And never forget that, with all your cleverness,
you might have remained in prison to the end of your days but for
the straightforward simplicity of him you thought so inferior to
yourselves."
Then he ordered that the tree which brought forth such excellent
pears should be transplanted to his palace; and to the father and
his three sons he gave places among his gardeners, where they lived
in plenty and were well content.
HOW THE POOREST BECAME THE RICHEST [99].
There was once a poor peasant, named Taland, who lived in a poor
cottage in the Walserthal, a valley of Vorarlberg. He was as poor
in wits as in fortune, so that he was continually making himself the
laughing-stock of his neighbours; yet, as he possessed a certain sort
of cunning, which fortune was pleased to favour, he got on better in
the long run than many a wiser man.
Plodding along steadily, and living frugally, Taland, in process of
time, laid by enough money to buy a cow; and a cow he bought without
even stopping to consider that he had no means of pasturing it.
The cow, however, provided for that by her own instinct; there were
plenty of good pastures in the neighbourhood, and the cow was not
slow to discover them. Wherever the grass was freshest and sweetest,
thither she wandered, and by this token Taland had no difficulty in
finding her out at milking time; and in the whole country round there
was no sleeker or better-favoured animal.
But the neighbours at whose expense she fed so well in course of
time grew angry; and finding remonstrance vain, they met together and
determined to kill the cow; and, that none might have to bear the blame
of killing her more than another, every one of them stuck his knife
into her. By this means, not only was poor Taland's cow destroyed,
but even the hide was riddled with holes, and so rendered useless.
Nevertheless, Taland skinned his cow, and plodded away with the hide
to the nearest tanner, as if he had not the sense to be conscious
that it was spoilt. The tanner was not at home, but his wife was able
to decide without him, that there was no business to be done with
such goods, and she sent him away with a mocking laugh, bidding him
remember she dealt in hides, and not in sieves.
Taland, however, had come a long way, and having no money to buy food,
he begged so piteously for a morsel of refreshment, that the good wife
could not refuse, and having spread a table before him with good cheer,
went on about her business.
Taland, delighted with the spread, determined to do justice to it;
and as he sat and ate he saw the tanner's son, an urchin full of
tricks, hide himself, while his mother's back was turned, in an old
corn-bin which stood before the door. He went on eating and drinking,
and watching the corn-bin, and still the boy never came out, till
at last, he rightly judged, he had fallen asleep. Meantime, having
finished his meal, he turned to take leave of the tanner's wife; and
then, as he went away, he said, quite cursorily, "If you have no use
for that old corn-bin yonder--it's just the thing I want--you may as
well give it to me, and you won't have sent me away empty-handed."
"What! you want that lumbering, rotten old corn-bin?" cried the
tanner's wife; and she laughed more heartily than even at the riddled
cow-hide. "And you would carry it all the way home on your shoulders?"
The peasant nodded a stupid assent, without speaking.
"Then take it, pray, and be welcome; for I just wanted to get rid of
the unsightly old rubbish!"
Taland thanked her, and loaded the chest on his shoulder, but
carefully, lest he should wake the child too soon. And carefully
he continued to walk along with it till the tan-yard was left far,
far out of sight. Then he stopped short, and, setting the corn-bin
down with a jerk calculated to wake its inmate, he holloaed out,--
"I be going to fling the old corn-bin down the precipice!"
"Stop, stop! I'm inside!" cried the child, but with a tone of
conviction that he had only to ask, to be let out. This was not
Taland's game, who wanted to give him a thorough frightening; so he
shouted again, taking no heed of the child's voice,--
"I be going to fling the old corn-bin down the precipice!"
"Stop! stop! I tell you; I'm inside it!" repeated the boy, in a louder
tone, thinking he had not made himself heard before.
"Who be you? and what be you to me?" replied Taland, in a stupid
tone of indifference. "I be going to chuck the old corn-bin down
the precipice."
"Oh, stop! for heaven's sake, stop!" screamed the now really affrighted
child; "stop, and spare me! Only let me out, and mother will give
you ever such a heap of gold!"
"It's a long way back to 'mother,'" replied the peasant,
churlishly. "I'd much rather chuck the old thing over, and have done
with it. You're not worth enough to repay the trouble."
"Oh, but I am though!" answered the boy, in a positive tone. "There's
nothing mother wouldn't give to save my life, I know!"
"What would she give, d'you think? Would she give five hundred
thalers, now?"
"Ay, that she would!"
"Well, it's a longsome way; but if you promise I shall have five
hundred thalers, I don't mind if I oblige you."
"You shall have them, safe enough, never fear!"
On this promise, Taland took the boy home, and made up a story of his
surprise at finding him at the bottom of the old chest, and how hardly
he had saved his life. The mother, overjoyed at the idea of her son
being restored to her under such circumstances, readily counted out
the five hundred thalers, and sent Taland home a richer man than when
his fortune consisted of a cow.
Elated with his good fortune, our hero determined to have a bit of
fun with his spiteful neighbours, and accordingly sat himself down
in an arbour, where there was a large round table, in front of the
Wirthshaus, and spreading his heap of gold before him, amused himself
with counting it out. Of course the sight attracted all the peasants
of the place, who were just gathering for a gossip on their way home
from work.
"And where did you get such a heap of gold from?" asked a dozen
excited voices at once.
"From the sale of the cow-hide, to be sure," replied Taland, in an
inanimate voice.
"What! the cow-hide all riddled with holes?" vociferated his
interlocutors, in a chorus of ridicule.
"To be sure; that's just what made it so valuable," persisted Taland,
confidently.
"What! the tanner gives more for a hide all full of holes than for
a sound one?"
"What's the use of asking so many silly questions?" returned the
imperturbable peasant "Do you see the money? and should I have
got such a sum for an ordinary cow-hide? If you can answer these
two questions of mine, you can answer your own for yourselves;"
and gathering up his gold, he walked away with a stolid look which
defied further interrogations.
The village wiseacres were all struck with the same idea. If riddled
cowhides fetched five hundred thalers apiece, the best way to make a
fortune was to kill all the cows in the commune, pierce their skins all
over with holes, and carry them to the tanner. Every one went home to
calculate what he would make by the venture; and the morning was all
too long coming, so eager were they to put their plan into execution.
Taland, having now plenty of money, had nothing to do next day but to
dress himself in his feast-day clothes and play at dominoes in the
Bier-garten; but though this was a favourite enjoyment, far sweeter
was that of observing the running hither and thither of his spiteful,
mocking neighbours, slaughtering their sleek kine--the provision of
their future lives--skinning them, and destroying the very skins out
of which some small compensation might have been earned.
Taland hardly knew how to contain his inclination to laugh, as he saw
them caught in his trap so coarsely baited; and the good landlord,
as he saw the irrepressible giggle again and again convulse his stupid
features, thought that the gain of the five hundred thalers had fairly
turned his weak head.
The peasants had gone off to the tan-yard with their riddled cow-hides,
merrily shouting and boasting; and Taland sat at home, drinking
and laughing. But it was a different story by-and-by. There was a
sound like the roar of a wild beast, which stopped even Taland's
inclination to laugh, and made him shrink in his chair. It was the
lament of the long file of peasants returning from the tan-yard from
their bootless errand, filling the air as they went along with yells
of fury at their ruin, and imprecations and threats of vengeance on
him who had led them into the snare.
Taland had meant to have had his laugh over their discomfiture,
but finding them in this mood, he thought his wisest course was to
keep out of their sight, lest they should take summary vengeance on
him. So he found a corner to hide himself in; and he thus overheard
their debate on the means of punishing their deceiver.
"He's such shifts for getting out of every thing, that one doesn't
know where to have him," said the noisiest speaker; and the rest
re-echoed the sentiment.
"Ay; it'll never do to let him get scent of what we're up to!"
"But how to avoid it?"
"Take him asleep."
"Ay; take him when he's asleep; that's the way!"
"Go up the stairs and rattle at his window, and when he comes out,
knock him on the head!"
"And every one have a go at him, as we did at his cow."
"That's the plan!"
"And the sooner the better."
"This very night, before we go to bed!"
"To be sure; we won't sleep tamely upon such an affront."
"No; we'll make an end of it, that we will!"
"And it's time we did."
"Another day would be unbearable!"
"Another hour is bad enough; but we must keep quiet till he's well
asleep."
"Yes; there's nothing to be done till midnight."
"We'll meet again at midnight, then."
"All right; we shall all be there!"
"Good-bye, then, till midnight!"
"Good-bye, till midnight; good-bye!"
They all spoke at once, and the whole dark plan was concocted in a
few minutes; then they dispersed to their homes with resolute steps.
Taland listened to the sound with beating heart, and as soon as
silence once more prevailed, he stole stealthily homewards.
His wife was sitting over her spinning-wheel.
"I've caught a cold wearing these holiday clothes out of their turn,"
said Taland; "will you do me the favour to sleep in the window-sill,
and keep that flapping shutter close, good wife?"
"With all my heart," responded the compliant spouse; and thus disposed,
they went to rest.
At midnight the villagers came, faithful to their appointment, in a
strong body, and mounted the stairs [100] as quietly as might be. The
foremost pushed open the shutter, and exclaimed, "Why, here's the
old idiot lying ready for us, across the window-sill!"
"Then we're spared the trouble of hunting for him," exclaimed the next.
"So here goes!" cried all together; and they showered their blows on
the devoted body of the old wife, while Taland, comfortably enveloped
in his coverlet, once more laughed at the success of his deceit,
and the discomfiture of his foes.
Towards morning he rose, and taking up the dead body, placed it in
a chair, and bore it along, together with the old spinning-wheel,
a good distance down the high road; and there he left it, while he
sat behind a bush to see what would happen.
Presently a fine lord came along the road driving a noble chariot.
"Holloa, good woman! get out of the way!" shouted the lord, while yet
at a considerable distance; for he thought the old woman was silly,
spinning in the roadway. But the corpse moved not for his shouting.
"Holloa! holloa, I say! you'll be killed! move, can't you?" he cried,
thinking she was deaf, and hadn't heard his first appeal. But still
the corpse moved not.
"Get out of the way! get out of the road! can't you?" at last fairly
screamed the lord; for, never dreaming but that the woman would move
in time, he had not reined in his fiery steeds--and now it was all
too late! On one side went the old lady in the chair, and on the
other side the fragments of the spinning-wheel, while the chariot
dashed wildly on between them.
"What have I done?" said the lord, alighting from his chariot as soon
as he could stop, and looking round him in wild despair.
"Why, you've run over and killed my old mother! that's what you've
done!" said Taland, emerging from his hiding-place. "And now you must
come with me before the judge."
"Really, I meant no harm," pleaded the good lord; "I called to her to
get out of the way, and I couldn't help it if she was deaf. But I'll
make you what compensation you like. What do you say to accepting my
chariot full of gold, and the horses and all, to drive home with?"
"Why, if you say you couldn't help it, I suppose you couldn't," replied
Taland. "I don't want to hurt you; and since you offer fair terms,
I'm willing to accept your chariot full of gold, and the horses to
drive it home. I'll square the account to your satisfaction."
So the lord took him home to his castle and filled up the chariot
with gold, and put the reins in his hands, and sent him home richer
and merrier than if the neighbours had never attempted his life.
When these same envious neighbours, however, saw him coming home in the
chariot full of gold, driving the prancing horses quite gravitetisch
[101], they knew not what to make of it. And that, too, just as they
were congratulating themselves that they had made an end of him!
"It must be his ghost!" they cried. There was no other way of
accounting for the reappearance. But as he drove nearer, there was
no denying that it was his very self in flesh and blood!
"Where do you come from? where did you get all that heap of money
from? and what story are you going to palm off on us this time?" were
questions which were showered down on him like hail.
Taland knew how easily they let themselves be ensnared, and that the
real story would do as well this time as any he could make up, so he
told them exactly what had happened, and then whipped his horses into
a canter which dispersed them right and left, while he drove home as
gravitetisch as before.
Nor was he wrong in expecting his bait to take. With one accord the
peasants all went home and killed their wives, and set them, with their
spinningwheels before them, all along the road. Of course, however,
no lucky chance occurred such as Taland's--no file of noblemen driving
lordly chariots, and silly enough to mistake the dead for the living,
came by; and while Taland was rich enough to marry the best woman in
the place, they had all to bury their wives and live alone in their
desolated homes.
To have been so tricked was indeed enough to raise their ire; and the
only consolation amid their gloom was to meet and concoct some plan
for taking signal and final vengeance. This was at last found. They
were to seize him by night, as before; but this time they were not
to beat him to death in the dark, but keep him bound till daylight,
and make sure of their man, then bind him in a sack and throw him
over the precipice of the Hoch Gerach.
As Taland was not by to overhear and provide against the arrangement,
it was carried out to the letter this time; and all tied in a sack
the struggling victim was borne along in triumph towards the Hoch
Gerach. They had already passed the village of St. Gerold, and the
fatal gorge forced through the wall of living rock by the incessant
world-old wear of rushing torrents was nearly reached. Taland,
paralyzed with fear and exhaustion, had desisted from his contortions
for very weariness.
The Haeusergruppe [102] of Felsenau, standing like a sentinel on guard
of the narrow hollow, had yet to be passed. It was near midday, and
the toil of the ascent had been great. Not one of the party objected
to take a snatch of rest and a sip of brandy to give them courage to
complete the deed in hand.
While they sat drinking in the shade of the cottage which stood
Felsenau in lieu of a Wirthshaus, Taland was left lying on a grassy
bank in the sun. About the same time a goatherd, driving his flock into
Bludenz to be milked, came by that way, and seeing the strangely-shaped
sack with something moving inside, arrested his steps to examine into
the affair. Taland, finding some one meddling with the mouth of the
sack, holloaed out,--
"List'ee! I'll have nothing to do with the princess!"
"What princess?" inquired the goatherd.
"Why, the princess I was to marry. B'aint you the king?"
"What king?" again asked the goatherd, more and more puzzled.
"I can't talk while I'm stifled in here," replied Taland. "Let me out,
and I'll tell you all about it."
The curious goatherd released the captive from the bag, and he told his
tale as follows. "The king has got a beautiful daughter--so beautiful
that such a number of suitors come after her she cannot decide between
them all. At last the king got tired, and said he would decide for her;
and this morning he proclaimed that whoever could bear being carried
about for seven hours in this sack should have her, be he peasant or
prince. So I thought I might try my luck at it as well as another;
and those chaps you hear talking in the little house yonder have been
carrying me about for three hours, but I can't stand more of it,
and away I go;" and he looked up anxiously to see if the bait had
taken; for he wanted the other to propose to get into the sack, as,
if he had walked away and left it empty, he knew the villagers would
pursue and overtake him. Nor was he mistaken in his calculation.
"It doesn't seem so hard to bear," said the goatherd, after some
moments' consideration.
"Would you like to try?" inquired Taland, anxiously; "it won't be so
bad for you, as, if you get in now, the men won't perceive we have
changed places, and you'll get the benefit of three hours for nothing."
"You're really very kind!" responded the goatherd, drawing the sack
over him; "I don't know how to thank you enough. I'm sure I can stand
four hours easily enough, for the sake of being reckoned a king's son
at the end. I shan't want the goats, however, when I'm married to a
princess, so pray take them at a gift--only make fast the cords of the
sack so that the men may not perceive that it has been meddled with."
Taland tied up the sack exactly as it had been before, and drove home
the flock of goats.
He was scarcely out of sight when the men, now well rested, came out,
and having taken up the sack again, carried it up the Hoch Gerach;
and just as the unhappy goatherd within fancied he was reaching the
top of some high terrace leading to the royal palace, bang, bang from
rock to rock he found himself dashed by the relentless villagers!
Confident that the job was now effectually completed, they trooped
home full of rejoicing over their feat.
The first thing that met their eye, however, was Taland seated before
his door, just as if nothing had happened, milking the goats which
browsed around him, making a goodly show.
Too much awed at the sight to rush at and seize him, they once more
asked him to give explanation of his unlooked-for return, and of how
he became possessed of such a fine herd of goats.
"Nothing easier!" replied Taland, gravitetisch. "Where shall I begin?"
"From where you were thrown over the mountain-side."
"All right!" pursued Taland. "Well, then, as you may suppose, I
struggled hard to get out of the sack, but it was too tough, and
I could do nothing with it at first; but, by-and-by, from knocking
against the jutting rocks again and again, it got a rent, and this
rent I was able to tear open wide, so that by the time I got to the
bottom there was a big hole, big enough to get out by. And where do
you think I found myself when I got out? In the enchanted regions of
the underground world, where the sky is tenfold as blue as it is here,
and the meadows tenfold as green! It was so beautiful to look at that I
gladly wandered on a little space. Presently I found a way that led up
home again; but I had no mind to come away from the beautiful country
till I saw, climbing the rocks by the side of the path, numbers of
goats, much finer than any goats we ever see in these parts."
"So they are! so they are!" chimed in the gullible multitude.
"Then I thought it would be fine to bring a flock of such fine goats
home--and, after all, it was easy to go back again when I wanted
to see that deep blue sky and those rich pastures again; so home I
came. Here am I, and here are the goats; and if you don't believe I
got them there, you can go and fetch some thence and compare them."
"But shall we really find such goats if we go?" eagerly inquired the
credulous villagers.
"To be sure you will--and sheep, and oxen, and cows too, without
number."
"Cows too! Oh, let's come and supply ourselves, and make good our
losses! But first show the way you came up by."
"Oh, it's a long, steep, weary way, and would take you two days
to get down! Much the nearest way is to jump down the side of the
Hoch Gerach."
"But are you sure we shan't hurt ourselves? Didn't you get hurt
at all?"
"Not a bit. Feel me; I'm quite sound."
"To be sure, you couldn't hurt falling on to such soft, beautiful
meadows!" they replied; and off they set, only eager which should
reach the Hoch Gerach first, and which should be the first to make
the bold spring, and which should have the first pick and choice of
the fair flocks and herds in the enchanted world underground!
Slap! bang! plump! they all went over the side of the Hoch Gerach,
one after the other, never to return! And Taland thus alone remained
to inherit the houses and goods of the whole village, all for
himself--and, from being the poorest of all, became possessed of the
riches of all.
THE END.
NOTES
[1] It is common in England to speak of Tirol as "the Tyrol;" I have
used the name according to the custom of the country itself.
[2] The name for "the wild huntsman" in North and South Tirol.
[3] The Beatrik of the Italian Tirol is, however, a milder spirit
than the Wilder Jaeger of the northern provinces. He is also called
il cacciatore della pia caccia, because he is supposed only to hunt
evil spirits.
[4] The name in Vorarlberg.
[5] The three helpers against the plague. There are many churches so
called in Tirol.
[6] "Schliess die Kammer fein,
Sonst kommt der Norg herein."
[7] The Meierhof was the homestead of a small proprietor standing
midway between the peasant and the noble.
[8] Mistress of the Meierhof.
[9] Literally, "high lakes;" i. e. lakes on a high mountain
level. There are three such in the valley of Matsch, the inundations
of which often work sad havoc.
[10] "Morgen oder Heut
Kommt die Zahlzeit."
[11] The "home of the wolves;" a nickname given to Matsch, because
still infested by wolves.
[12] On Midsummer-day.
[13] The local names of two favourite kinds of grass.
[14] St. Martin is considered the patron of mountain pastures in Tirol.
[15] That the Norgs should be at one time represented as incapable of
comprehending what death was, and that at another their race should
be spoken of as dying out, is but one of those inconsistencies which
must constantly occur when it is attempted to describe a supernatural
order of things by an imagery taken from the natural order.
[16] From tarnen, to conceal, and Haut, skin; a tight-fitting
garment which was supposed to have the property of rendering the
wearer invisible. It was likewise sometimes supposed to convey great
strength also.
[17] Literally, "crystal palace." Burg means a palace no less than
a citadel or fortress; the imperial palace in Vienna has no other name.
[18] Ignaz von Zingerle, in discussing the sites which various local
traditions claim for the Rosengarten of King Lareyn, or Laurin, says,
"Whoever has once enjoyed the sight of the Dolomite peaks of the
Schlern bathed in the rosy light of the evening glow cannot help
fancying himself at once transported into the world of myths, and
will be irresistibly inclined to place the fragrant Rose-garden on
its strangely jagged heights, studded by nature with violet amethysts,
and even now carpeted with the most exquisite mountain-flora of Tirol."
[19] Cornfield.
[20] Nobleman's residence.
[21] In the mediaeval poems the shade of the Lindenbaum is the favourite
scene of gallant adventures.
[22] The heroes of the old German poetry are frequently called by the
epithet "sword"--ein Degen stark; ein Degen hehr; Wittich der Degen,
&c., &c.
[23] Hildebrand, son of Duke Herbrand and brother of the Monk Ilsau,
one of the persons of the romance of "Kriemhild's Rose-garden," is the
Nestor of German myths. He was the instructor of Dietrich von Bern
(Theodoric of Verona). We find him sought as the wise counsellor in
various undertakings celebrated in the mediaeval epics; he is reputed
to have lived to the age of 200 years.
[24] This was commonly the office of the daughter of the house.
[25] This would appear to have been the usual custom in the middle
ages after a meal.
[26] See note, p. 35.
[27] The German legends are inclined to extol the heroism of Dietrich
von Bern, better known to us as Theodoric, King of the Visigoths, who,
after his conquests in Italy, built a palace at Verona, and made it
his seat of government; but the traditions of Verona ascribe his great
strength and success, both as a hunter and warrior, to a compact with
the Evil One. His connexion with the Arians, his opposition towards
the Popes, and his violent destruction of the churches of Verona,
were sufficient to convince the popular mind at his date that his
strength was not from above. Procopius relates that his remorse for
the death of Symmachus haunted him so, that one day when the head of a
great fish was served at table, it appeared to him as the head of his
murdered relative, and he became so horrified that he was never able to
eat any thing afterwards. The Veronese tradition is, that by his pact
with the devil, evil spirits served him in the form of dogs, horses,
and huntsmen, until the time came that they drove him forth into
their own abode (Mattei Verona Illustrata). In the church of St. Zeno
at Verona this legend may be seen sculptured in bas-relief over the
door. In the mythology of some parts of Germany he is identified,
or confounded, with the Wild Huntsman (Boerner, Sagen aus dem Orlagau,
pp. 213, 216, 236). In the Heldenbuch he is called the son of an evil
spirit. He is there distinguished by a fiery breath, with which he
overcomes dwarfs and giants; but he is said to be ultimately carried
off into the wilds by a demon-horse, upon which he has every day to
fight with two terrible dragons until the Judgment Day. Nork cites
a passage from Luther's works, in which he speaks of him (cursorily)
as the incarnation of evil, showing how he was regarded in Germany at
his day ("It is as if I should undertake to make Christ out of Dietrich
von Bern"--Als wenn ich aus Dietrich von Bern Christum machen wollte).
[28] Wigand, man of valour.
[29] We often find the heroes' trusty swords called by a particular
name: thus Orlando's was called Durindarda, it is so inscribed in
his statue in the porch of the Duomo at Verona; and the name of King
Arthur's will occur to every one's memory.
[30] Him of Verona.
[31] This hardening power of dragons' blood was one of the mediaeval
fables.
[32] Bearing a red banner thus was equivalent to a declaration of
hostile intent.
[33] These it was knightly custom for the vanquished to surrender to
him who had overcome him.
[34] The Styrian.
[35] A Schirmschlag was a scientifically-manoeuvred stroke, by which
he who dealt it concealed himself behind his shield while he aimed
at any part of his adversary's body which presented an undefended
mark. But Theodoric drew the stroke without even having a shield for
his own defence.
[36] The Norgs are not always spoken of as pagans; many stories of
them seem to consider them as amenable to Christian precepts. The
ancient church of the village of St. Peter, near the Castle of Tirol,
is said by popular tradition to have been built by them, and under
peculiar difficulties; for while they were at work, a giant who lived
in Schloss Tirol used to come every night and destroy what they had
done in the day, till at last they agreed to assemble in great force,
and complete the whole church in one day, which they did; and then,
being a complete work offered to the service of God, the giant had
no more power over it.
[37] It was an old German custom that no flagons or vessels of the
drinks should be put on the table; but as soon as a glass was emptied
it was refilled by watchful attendants.
[38] Lautertrank, by the description of its composition, seems to
have been nearly identical with our claret-cup. Moras was composed
of the juice of mulberries mixed with good old wine.
[39] Concerning Theodoric's fiery breath, see note, p. 39. All the
myths about him mention it. The following description of it occurs
in the legends of "Criemhild's Rosengarten:"--
"Wie ein Haus das dampfet, wenn man es zuendet an,
So musste Dietrich rauchen, der zornige Mann.
Man sah eine rothe Flamme geh'n aus seinem Mund."
["As a house smokes when it is set on fire, so was the breath of
Theodoric, the man of great anger; a red flame might be seen darting
from his mouth."]
[40] The power of the Norgs to pass in and out through the rock is one
of the characteristics most prominently fabled of them. Sometimes we
hear of doors which opened spontaneously at their approach, but more
often the marvel of their passing in and out without any apparent
opening is descanted on.
[41] The value and efficacy ascribed in the old myths to a virgin's
blessing is one form in which the regard for maiden honour was
expressed.
[42] The dwarfs who were considered the genii of the mineral
wealth of the country were a sub-class of the genus dwarf.
Their myths are found more abundantly in North Tirol,
where the chief mines were worked.
[43] A deserted mine is called in local dialect taub.
[44] Miners.
[45] i.e. Joseph of Goign, a village near St. Johann. Such modes of
designation are found for every one, among the people in Tirol.
[46] Ann.
[47] Every body wears feathers according to their fancy in their
"Alpine hats" here, but in Tirol every such adornment is a distinction
won by merit, whether in target-shooting, wrestling, or any other
manly sport; and, like the medals of the soldier, can only be worn
by those who have made good their claim.
[48] Hof, in Tirol, denotes the proprietorship of a comfortable
homestead.
[49] To Spaniards the outline of a mountain-ridge suggests the edge
of a saw--sierra; to the Tirolese the more indented sky-line familiar
to them recalls the teeth of a comb.
[50] Garnets and carbuncles are found in Tirol in the Zillerthal,
and the search after them has given rise to some fantastic tales--of
which later.
[51] Carbuncle.
[52] Pray for him.
[53] District.
[54] Wild Georgey.
[55] In some parts of Tirol where the pastures are on steep <DW72>s,
or reached by difficult paths--particularly the Zillerthal, on which
the scene of the present story borders--it is the custom to decide
which of the cattle is fit for the post of leader of the herd by trial
of battle. The victor is afterwards marched through the commune to
the sound of bells and music, and decked with garlands of flowers.
[56] "Just as I wanted you."
[57] Little Frederick.
[58] A local expression for a village fete.
[59] The old race of innkeepers in Tirol were a singularly trustworthy,
honourable set, acting as a sort of elder or umpire each over his
village. This is still the case in a great many valleys out of the
beaten track.
[60] "Have mercy on them!"
[61] The cry which in South Germany is equivalent to our "hurrah!"
[62] Page 76.
[63] Little Elizabeth.
[64] A local word in the Passeier-Thal for a poultry-maid.
[65] "Ich zeige Sie wo der Zimmermann das Loch gemacht hat." A Tirolese
saying for, "I'll soon show you the way to the door."
[66] Another form of Klein-Else: Else, with the diminutive, chen.
[67] This curious name was borne by a Margrave of Istria,
and three other princes at least who ruled over part of Tirol,
and who figure in the authentic history of the country. It
does not appear that the present story, however, is referable
to traditions of the life of any of these.
[68] The best public room in a Tirolese inn is so called.
[69] The homestead of a peasant proprietor.
[70] The local name of the holiday cap of the Sarnthaler women.
[71] Lieblingsbaender.
[72] Alp is used in Tirol for the green mountain pastures.
[73] Alpine herdsman.
[74] See Preface.
[75] Joch is used in Tirol when speaking of a moderately high mountain;
in most other mountain districts of Germany it means only a pass
or col.
[76] A high-lying range of mountain pasture-land.
[77] The stories of our Lord's life on earth, treated with perfect
idealism, sketching His character as He was pleased to
manifest it, or His miraculous acts, pervade the popular mythology
of all Catholic peoples. I have given one from Spain,
by the title of "Where One can Dine, Two can Dine," in
"Patranas," of the same character as this Tirolese one; and
perhaps it is not amiss to repeat the observation I felt called
to make upon it,--that it would be the greatest mistake to
imagine that anything like irreverence was intended in such
stories. They are the simple utterances of peoples who
realized so utterly and so devoutly the facts recorded in the
Gospels that the circumstances of time and place ceased to
occupy them at all, and who were wont to make the study
of our Lord's example their rule of conduct so habitually, that
to imagine Him sharing the accidents of their own daily life
came more natural to them than to think of Him in the far-off
East. These stories were probably either adapted from
the personal traditions which the first evangelists may well
be thought to have brought with them unwritten, or invented
by themselves, in all good faith, as allegories, by means of
which to inculcate by them upon their children the application
of His maxims to their own daily acts. They demand, therefore,
to be read in this spirit for the sake of the pious intention
in which they are conceived, rather than criticised for
their rude simplicity or their anachronisms.
[78] "Praised be Jesus Christ!" This was formerly the universal
greeting all over Tirol in the house or on the road, for friend or
stranger, who answered, "For ever and ever. Amen." It is still in
common use in many parts.
[79] I must beg my readers to apply the apology contained in
the note to the last story, in its measure to this one also.
[80] Sorella, sister; with the augmentative ona, the bigger or
elder sister.
[81] The little, or younger, sister.
[82] We say, "a head of celery;" in Italy they say, "a foot of celery."
[83] A favourite vintage of Tirol.
[84] Arativo and prativo are dialectic in Waelsch Tirol for arable
and pasture land.
[85] "On our right soared the implacable ridges of the Marmolata,"
writes a modern traveller; "the sheer, hard smoothness of whose
scarped rocks filled one with a kind of horror only to look at them."
[86] "We have hay in the stables, and more also in the meadow."
[87] Berg-Segen (literally "mountain-blessing") is the form in which
Tirol in its piety expresses the ordinary word crop.
[88] See Preface.
[89] Two kinds of more or less mischievous strie, or wild fairies.
[90] "Fearless Johnny." John is a favourite name in Waelsch
Tirol, and bears some twenty or thirty variations, as Giovannazzi,
Gianaselli, Gianot, Zanetto, Zanolini, Zuani, Degiampietro
(John Peter), Zangiacomi (John James), &c.
[91] The Latin name of the god of hell remains throughout Italy,
and holds in its nurseries the place of "Old Bogie" with us.
[92] "Earthly soul, stand off three paces, and tell me your grief."
[93] Vine-trellis.
[94] Tapestry hanging before a door.
[95] Witch.
[96] A wayside cross under a little penthouse, such as is to be met
at every turn of the road in Tirol.
[97] The south wind, which does much mischief at certain times of
the year, and is most dreaded in Vorarlberg.
[98] Dialectic for a basket in Vorarlberg.
[99] It has been my aim generally, in making this collection, to
give the preference to those stories which have a moral point
to recommend them; my readers will not, perhaps, take it
amiss, however, if I present them with this specimen of a
class in which this is wanting, and which aims only at
amusement. It is, moreover, interesting from the strong
evidence it bears of extremely remote origin; for the light
way in which putting people to death, deception, and selfishness
are spoken of prove it has a pre-Christian source, while
the unimportant accessories show how details get modified
by transmission.
[100] It must be understood that it is an outside staircase that is
here alluded to, and the shutter of an unglazed window on its landing
serving for a door also.
[101] In a lordly manner.
[102] A cluster of houses too small to be designated a village.
End of the Project Gutenberg EBook of Household stories from the Land of
Hofer, by R. H. Busk
*** | {
"redpajama_set_name": "RedPajamaBook"
} | 3,125 |
//-----------------------------------------------------------------------------
// Filename: TurnServer.cs
//
// Description: Encapsualtes the connection details for a TURN/STUN server.
//
// History:
// 26 Feb 2016 Aaron Clauson Created.
//
// License:
// This software is licensed under the BSD License http://www.opensource.org/licenses/bsd-license.php
//
// Copyright (c) 2016 Aaron Clauson (aaron@sipsorcery.com), SIP Sorcery Pty Ltd, Hobart, Australia (www.sipsorcery.com)
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without modification, are permitted provided that
// the following conditions are met:
//
// Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
// Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following
// disclaimer in the documentation and/or other materials provided with the distribution. Neither the name of SIP Sorcery Pty Ltd
// nor the names of its contributors may be used to endorse or promote products derived from this software without specific
// prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING,
// BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
// IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY,
// OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
// OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
// OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//-----------------------------------------------------------------------------
using System.Net;
namespace SIPSorcery.Net
{
public class TurnServer
{
public IPEndPoint ServerEndPoint;
public string Username;
public string Password;
public string Realm;
public string Nonce;
public int AuthorisationAttempts;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,338 |
Q: In operating system, How MMU searches for virtual page number as key in page table 1)So lets say a single level page table
3)A TLB miss happens
3)The required page table is at main memory
Question : Does MMU always fetch the page table required to a number of registers inside it so that fast hardware search like TLB can be performed? I guess no that would be costly hardware
4)MMU fetch the physical page number (I guess MMU must be saved it with a format like high n-bits as virtual page no. and low m bits as physical page frame no. Please correct and explain if I am wrong)
Question: I guess there has to be a key-value map with Virtual page no as key and physical frame no. as value. How MMU search for the key in the page table. If it is a s/w like linear search than it would be very costly.
5)With hardware it appends offset bits to page frame no.
and finally a read occurs for physical address.
So this question is bugging me a lot, how the MMU performs the search for given key(virtual page entry) in page table?
A:
The use of registers for a page table is satisfactory if the page
table is reasonably small(for example, 256 entries). Most contemporary
computers, however, allow the page table to be very large (for
example, 1 million entries). For these machines, the use of fast
registers to implement the page table is not feasible. Rather, the
page table is kept in main memory, and a page table base register (PTBR) points to the page table.
Changing page tables requires changing only this one register,
substantially reducing context-switch time.
The problem with this
approach is the time required to access a user memory location. If we
want to access location i, we must first index into the page table,
using the value in the PTBR offset by the page number for i. This task
requires a memory access. It provides us with the frame number, which
is combined with the page offset to produce the actual address. We can
then access the desired place in memory. With this scheme, two memory
accesses are needed to access a byte (one for the page-table entry,
one for the byte). Thus, memory access is slowed by a factor of 2.
This delay would be intolerable under most circumstances. We might as
well resort to swapping!
The standard solution to this problem is to
use a special, small, fastlookup hardware cache, called a translation look-aside buffer(TLB) . The
TLB is associative, high-speed memory. Each entry in the TLB consists
of two parts: a key (or tag) and a value. When the associative memory
is presented with an item, the item is compared with all keys
simultaneously. If the item is found, the corresponding value field is
returned. The search is fast; the hardware, however, is expensive.
Typically, the number of entries in a TLB is small, often numbering
between 64 and 1,024.
Source:Operating System Concepts by Silberschatz et al. page 333
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var TodoActions = Reflux.createActions([
'addItem'
]);
var TodoStore = Reflux.createStore({
items: [1, 2],
listenables: [TodoActions],
onAddItem: function (model) {
$.post('/server/add', {data: model}, function (data) {
this.items.unshift(data);
this.trigger(this.items);
});
}
});
var TodoComponent = React.createClass({
mixins: [Reflux.listenTo(TodoStore, 'onStatusChange')],
getInitialState: function () {
return {list: []};
},
onStatusChange: function () {
this.setState({list: TodoStore.items});
},
render: function () {
return (
<div>
{this.state.list.map(function (item) {
return <p>{item}</p>
})}
</div>
)
}
});
React.render(<TodoComponent />, document.getElementById('container')); | {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/www.science.org\/doi\/10.1126\/sciadv.aay1394","text":"OPEN ACCESS\nResearch Article\nAPPLIED SCIENCES AND ENGINEERING\n\n# Extending electron paramagnetic resonance to nanoliter volume protein single crystals using a self-resonant microhelix\n\n4 Oct 2019\nVol 5, Issue 10\n\n## Abstract\n\nElectron paramagnetic resonance (EPR) spectroscopy on protein single crystals is the ultimate method for determining the electronic structure of paramagnetic intermediates at the active site of an enzyme and relating the magnetic tensor to a molecular structure. However, crystals of dimensions typical for protein crystallography (0.05 to 0.3mm) provide insufficient signal intensity. In this work, we present a microwave self-resonant microhelix for nanoliter samples that can be implemented in a commercial X-band (9.5 GHz) EPR spectrometer. The self-resonant microhelix provides a measured signal-to-noise improvement up to a factor of 28 with respect to commercial EPR resonators. This work opens up the possibility to use advanced EPR techniques for studying protein single crystals of dimensions typical for x-ray crystallography. The technique is demonstrated by EPR experiments on single crystal [FeFe]-hydrogenase (Clostridium pasteurianum; CpI) with dimensions of 0.3 mm by 0.1 mm by 0.1 mm, yielding a proposed g-tensor orientation of the Hox state.\n\n## INTRODUCTION\n\nThe catalytic cycle of redox enzymes often contains paramagnetic intermediates, and electron paramagnetic resonance (EPR) spectroscopy is the method of choice used to study these occurrences. Through EPR experiments, information on the electronic and geometrical structure of the active site can be obtained. For typical EPR experiments on proteins, a frozen solution of 0.1 to 1 mM concentration is prepared and placed in a microwave cavity. Standard sample volumes at X-band (nominally 9.5 GHz) are in the 200 \u03bcl range. However, frozen solution EPR experiments only allow the determination of the principal values of magnetic interactions at an active site and, thus, provide only a limited view of the electronic structure.\nTo resolve the full-tensor magnetic interaction parameters, such as g-, zero-field, hyperfine, and quadrupole tensors, single-crystal EPR experiments must be performed. In combination with x-ray crystallography, the magnetic interaction tensors obtained with EPR experiments can be directly related to the protein geometry to help identify and better understand the catalytic mechanism of the enzyme (1, 2). Despite its usefulness, single-crystal EPR is rarely applied to protein systems because of challenges in growing crystals of sufficient quality and volume for these experiments. Many protein crystals used in x-ray crystallography are of dimensions in the 0.05-0.3 mm range and, hence, are too small to be studied using commercial EPR instrumentation. Crystallization methods, such as macroseeding (3), have the potential to increase the volume of the crystals, but these techniques are difficult to implement.\nTo study magnetic interactions of the paramagnetic center with the surrounding nuclear spins, pulse and double-resonance EPR experiments are required. However, these experiments require even higher sensitivity. Unlike nuclear magnetic resonance (NMR), where all nuclei are excited and contribute to the NMR signal, double-resonance experiments, such as electron spin echo envelope modulation (ESEEM), hyperfine sublevel correlation (HYSCORE), and electron nuclear double resonance (ENDOR), probe only the nuclei that are magnetically coupled to the paramagnetic center. Extending these experiments to single crystals provides not only the magnitude of the hyperfine and quadrupole tensors of ligand nuclear spins that interact with the paramagnetic centers but also the associated angles relative to the active site of an enzyme. These interacting nuclei are either naturally abundant, such as 1H and 14N, or the catalytic cofactors can be enriched with nuclei such as 2H, 13C, 15N, and 57Fe, for further analysis of magnetic interaction tensors with respect to the first ligand sphere. Furthermore, the same interaction tensors can be calculated from the molecular structure using quantum chemical calculations (4). These experimentally determined spectroscopic parameters can, therefore, be used to verify the adequacy of the level of theory, which, in turn, gives confidence to the predicted electronic and geometric structure of the involved intermediates and transition states in the whole catalytic cycle. The groundwork for understanding the inner workings of enzymes lies in collecting as much accurate spectroscopic information as possible, including other spectroscopic and structural methods (optical and vibrational spectroscopy, M\u00f6\u00dfbauer, x-ray spectroscopy, and diffraction). Every experiment contributes to the total picture and ultimately leads to a fundamental understanding of the catalytic mechanism of these enzymes.\nCurrently, volume-limited crystals can only be studied using W-band (94 GHz) (or higher) EPR in a single-mode resonator (5) or, for continuous-wave experiments, a Fabry-P\u00e9rot (6) resonator. These high-frequency EPR spectrometers are not widely available, and high-frequency conditions are usually unfavorable for pulse experiments such as ESEEM or HYSCORE, which provide the best response when the nuclear Zeeman and hyperfine\/quadrupole interactions are of the same order of magnitude (7). Instead, at higher frequencies, where the ESEEM effect becomes increasingly difficult to detect, ENDOR- and ELDOR-detected NMR experiments are performed to obtain hyperfine and quadrupole information.\nTo improve the sensitivity for studying single crystals using EPR on readily available spectrometers, typically at X-band, one must abandon the microwave cavity design and move to small-volume resonators based on lumped circuits in the microwave frequency range. This allows the reduction of the sample volume by one order of magnitude, from 200 to 20 \u03bcl using a loop-gap resonator (LGR) (8). Further reductions can be achieved by incorporating materials with a high dielectric constant in a standard resonator to reduce the active volume down to 1 \u03bcl (9). For protein single crystals, one must reduce the volume even further (less than 0.03 \u03bcl), which requires radical new approaches.\nHere, we combine the concept of a self-resonant microhelix, shown in Fig. 1A, with that of a planar microcoupler. The coupling structure on the printed circuit board is a resonant structure, which drives the self-resonant microhelix placed in the center of the coupling loop (10), shown in Fig. 1B. The microhelix geometry offers notable advantages in that the microwave field homogeneity is strongly improved along with volume sensitivity for small samples compared to other microresonator geometries, such as the planar microresonator (PMR) (1113). The microwave characteristics of the self-resonant microhelix are optimal for pulse and continuous-wave experiments requiring very little microwave power, and the microhelix assembly is easily matched and tuned over a variety of samples and temperatures.\n\n### Single-crystal EPR for metalloenzyme research\n\nNature has evolved enzymes with various metallic cofactors (metalloenzymes) to efficiently catalyze a broad variety of chemical reactions (14). These enzymes mostly use first-row transition metals to perform their catalytic functions. One of the main challenges is to fully understand these enzymatic mechanisms and provide a basis for cheap, robust, and highly active molecular catalysts designed for practical applications, e.g., in the field of energy conversion and storage (15). The ultimate goal is to alleviate the requirement of noble metals, such as platinum, that limit the scalability of current technology. This important biophysical and biochemical research seeks metalloenzyme-based and metalloenzyme-inspired systems as an interesting route to advance toward the future of clean energy and efficient energy storage.\nFor hydrogenases, specifically [NiFe]-hydrogenase, the single-crystal EPR strategy has been very successful (2). The active site of this enzyme harbors a [NiFe] binuclear cluster in which the iron carries two cyanide (CN) and one carbon monoxide (CO) ligand. The metals are bridged by two cysteine thiols, and the nickel center is further coordinated by two cysteine thiolate side groups. The paramagnetic states all originate from the nickel center, while the iron center remains Fe(II) during the catalytic cycle (16). The open-coordination site between the two metals can be occupied by an oxygen species leading to the inactive oxidized states or a hydride, which is the key intermediate in the catalytic cycle (2). For all these species, the g-tensor magnitude and orientation were determined and analyzed in terms of ligand-field theory and verified using quantum chemical calculations, providing a fundamental insight into the electronic structure and the dependence on the first ligand sphere (2). The [NiFe]-hydrogenase crystals in these studies were relatively large (2 mm by 0.5 mm by 0.5 mm), enabling measurements in standard X-band probeheads with a measuring time of 2 to 3 hours per angle, stepping over 180\u00b0. However, even with the large crystal volume, ESEEM and HYSCORE experiments on [NiFe]-hydrogenase were only published in frozen solution with volumes of 50 to 200 \u03bcl and experimental times upward of 2 hours for each experiment (2).\nSimilar experiments were anticipated for [FeFe]-hydrogenase, which has a much higher activity and exhibits a different catalytic mechanism compared to [NiFe]-hydrogenase. The [FeFe]-hydrogenase active site carries a [4Fe-4S] cluster linked via a cysteine ligand connecting a [2Fe] site. The [2Fe] cluster contains an iron atom proximal (Fep) and one distal (Fed) to the [4Fe-4S] cluster. Each iron carries a cyanide (CN) and one carbon monoxide (CO) ligand, and the two irons share a bridging carbon monoxide. In addition, the two iron atoms are bridged by an azapropane-dithiolate ligand (ADT ligand). The molecular structure of the [FeFe]-hydrogenase active site, known as the H-cluster, can be found in Fig. 2. The whole active site has a total of six iron atoms at various redox states in the catalytic cycle (17). Unfortunately, the crystals obtained from [FeFe]-hydrogenase are much smaller than those available from [NiFe]-hydrogenase with dimensions less than 0.3 mm. Frozen solution EPR on [FeFe]-hydrogenase has provided a lot of information on the binuclear active site, such as the discovery of a nitrogen in the dithiolate bridge (ADT ligand) (18). However, the full g-tensor and the hyperfine tensor, including angular information, of the active site are still elusive. Therefore, there is continuing interest in developing EPR instrumentation, specifically microresonators, at 9.5 and 35 GHz, X- and Q-band, respectively, optimized for metalloenzyme research. Furthermore, other potentially interesting proteins are rarely studied in single crystals, and doing so would contain a wealth of information on the electronic structure and enzymatic function (14).\nWe report here an EPR crystal rotational study of a [FeFe]-hydrogenase in the active oxidized state (Hox) from Clostridium pasteurianum (CpI) with crystal dimensions of 0.3 mm by 0.1 mm by 0.1 mm using the self-resonant microhelix. The excellent signal-to-noise ratio has also allowed for advanced pulse EPR experiments to be performed. These data demonstrate the utility of the microhelix in studying protein single crystals at volumes relevant for x-ray crystallography.\n\n### Theory and design\n\nWe focus on the development of resonant structures to increase EPR absolute spin sensitivity at X-band without the application of commercial bridge modifications, which require technical expertise not commonly available in EPR laboratories. We start with the well-established EPR signal analysis by Feher (19), which states that the continuous-wave EPR signals for a critically coupled resonator on a reflection bridge is proportional to\n$No alternative text available$\n(1)\nwhere \u03c7 is the magnetic susceptibility, Q is the Q-value of the cavity, P is the incident power, and \u03b7 is the filling factor. The Q-value is defined as the ratio of the stored energy in the resonator to the power loss in the sample and coupled resonator, whereas the filling factor is defined as the ratio of the circularly polarized microwave magnetic field stored energy perpendicular to the static magnetic field that gives rise to EPR transitions and the complete magnetic field stored energy in the geometry\n$No alternative text available$\n(2)\nwhere B1 is the microwave magnetic field in all space (V) and B1r is one component of the clockwise (or counterclockwise) rotational component of the linear B1 field perpendicular to the static magnetic field in the sample volume (Vs) (19, 20).\nFrom Eq. 1, we can derive a metric that encompasses the filling factor \u03b7 and the Q-value, called the resonator efficiency (8). The resonator efficiency (or conversion factor) is defined as\n$No alternative text available$\n(3)\nwhere Pl is the power loss in the system, including sample. However, \u039b assumes that the microwave magnetic field is uniform and does not take into account nonuniformity of the field over the sample. Experimentally, the microwave magnetic field is distributed throughout the sample, and the average of \u039b is measured. We can define the resonator efficiency average as\n$No alternative text available$\n(4)\nThe resonator efficiency average is proportional to the signal and, for a fixed sample volume, is proportional to the absolute spin sensitivity.\nFrom Eqs. 1 and 4, we can outline the criteria needed to increase the sensitivity for fixed sample volume EPR experiments. We choose the target dimensions of 0.3 mm by 0.3 mm by 0.3 mm (27 nl) as a maximum sample size. These dimensions are a reasonable size for protein single crystals used in x-ray crystallography diffraction, and any increase in EPR sensitivity would benefit single crystals of smaller dimensions. To maximize sensitivity, one must increase the \u03b7Q product, as shown in Eq. 1. However, there are practical limitations that must be considered while designing a resonator.\nFor instance, cavity resonators, such as the Bruker Super High-Q probehead, are not sufficiently sensitive for extremely small samples since the very small filling factor \u03b7 is not compensated by the high Q0-value, resulting in an overall poor EPR signal (21). Further increase of the Q0-value, by superconducting materials, is detrimental for pulse experiments where the available bandwidth of the resonator is inversely reduced. The resonator bandwidth filters the excitation pulses as well as the EPR signal, distorting the spectrum. The trade-off required for advanced pulse experiments results in the design parameters of a large filling factor \u03b7 while keeping the bandwidth around 100 MHz (Q0-value of 190 at 9.5 GHz).\nTherefore, one way to maximize \u03b7Q, and consequently \u039bave, is to reduce the size of the resonant structure relative to the sample volume, increasing the filling factor while maintaining a homogeneous magnetic field profile. The challenge arises because of the potential increase of the losses in the system can degrade the Q0-value more than the increase in the filling factor. Two common methods to reduce the size of a resonant geometry are to use either dielectric resonators to reduce the wavelength and, consequently, the size of the resonator or LGRs to reduce the cutoff frequency of a waveguide by introducing protrusions to create regions of inductive loops and capacitive gaps. Dielectric resonator and LGR geometries with commercially available dimensions are illustrated in fig. S1, and the microwave characteristics are tabulated in table S1. However, in order to study sample volumes less than 0.03 \u03bcl, further resonator reduction strategies are needed.\nLimitations to minimizing LGR geometries stem from an increase in ohmic losses due to a reduction in the gap spacing to maintain a constant resonant frequency as the sample-loop radius is reduced. In practice, this has put a limit on the \u039bave obtainable to less than 1 mT\/W1\/2 for X-band. Further EPR signal improvement is possible using dielectric resonators by increasing the dielectric permittivity, and dielectric resonators with permittivity up to 80 have been used for continuous-wave EPR experiments on crystals of porous materials and polymers (9, 22). However, these resonators exhibit Q0-values over 2500 that make pulse experiments problematic.\nHere, we introduce a new type of resonator based on a self-resonant microhelix that is particularly useful for protein single-crystal experiments at X-band and can be used as a drop-in replacement on a standard commercial system. The self-resonant microhelix geometry, illustrated in Fig. 1, solves these challenges by providing good magnetic field homogeneity, a high efficiency parameter, an optimum Q0-value for both pulse and continuous-wave EPR experiments, straightforward impedance matching, and ease of sample placement.\nHelical resonators were first introduced to EPR in the early 1960s as a method to increase the microwave magnetic field at the sample. Resonant helical geometries were affixed to one end of a shorted waveguide creating a slow-wave structure (23, 24). Coupling was achieved by direct connection to a coaxial line with a capacitive matching network or by microwave incident on the helical structure from a waveguide. The sample was placed within the helix and showed reasonable sensitivity increase and larger microwave magnetic field due to higher filling factors compared to typical cavities (25). Broadband slow-wave helical resonators were used for multifrequency experiments, where a nonresonant structure, having a Q-value close to unity, could be matched with a slide-screw tuner over an octave bandwidth (26). However, over time, they were replaced by LGRs, which achieved higher concentration sensitivity for volume-limited samples.\nRecently, microcoils have gained popularity in NMR for nanoliter samples (27, 28) and for microfluidics (29). However, three characteristics differentiate our microhelix configuration from those described in the EPR (23, 24) and NMR literature: (i) The helix is self-resonant, meaning that the self-inductance of n-turns (Ltot) and self-capacitance between the loops (Ctot) resonate at a frequency determined by \u03c92LtotCtot = 1, where \u03c9 is the resonant frequency in radians per second. Since the geometry is self-resonant, no additional capacitors are needed. A self-resonant microhelix has lower ohmic loss, which provides a higher Q0-value than is typically feasible with microcoil geometries in NMR, where, with an NMR microcoil, a typical Q0-value is around 30 (28). With a self-resonant microhelix, the volume-to-surface ratio is maximized and a Q0-value of 300 is achievable. (ii) Unlike the slow-wave structures in (23) and (24), the helix length is much smaller than the wavelength (31.6 mm at 9.5 GHz), which increases the uniformity. In addition, the inner diameter is 0.4 mm, which increases the resonator efficiency. Therefore, the microhelix is not a slow-wave structure but an inductor at self-resonance. The microwave magnetic field profile is shown in Fig. 1D. (iii) Last, the helix is coupled to an inductive coupling loop on a printed circuit board by mutual inductance, which can be designed to minimize noise and further increase the EPR signal-to-noise ratio (10). Mutual inductance does not require a balun or additional capacitive matching networks since impedance matching can be achieved by varying the distance between the microhelix and the inductive coupling loop, simplifying coupling methods.\n\n## RESULTS AND DISCUSSION\n\n### Performance compared to commercially available and state-of-the-art microwave probes\n\nThe self-resonant microhelix geometry wound around a 0.4-mm capillary is shown in Fig. 1A. The final number of the microhelix windings is determined by the pitch of the helix, the quartz capillary sample tube (0.4 mm outer diameter and 0.3 mm inner diameter), and the surrounding Rexolite, which all affect the resonance frequency. The 6.5-turn microhelix had a resonant frequency around 9.8 GHz with sample when coupled to the printed circuit board inductive coupler. The microhelix assembly is attached to a custom insert that is compatible with commercial EPR systems. The complete structure is shown in Fig. 1B with an expanded view of the printed circuit board geometry in Fig. 1C. Comparison of the fabricated microhelix geometry with commercial (Bruker MD5W1 and Bruker MS3) and state-of-the-art microwave probes is provided in Table 1.\nAs a comparison to the state-of-the-art microwave probes, two \u03a9-shaped 0.5 mm inner diameter PMRs (based on Rogers RO6010LM printed circuit board or sapphire substrates) are tested, which were fabricated by printing the microresonator geometry on a substrate using photolithographic techniques (1113). The PMRs are considered state-of-the-art because of the significant improvement in absolute spin sensitivity compared to the best commercial probeheads available (21). However, the self-resonant microhelix introduced in this work exhibits the highest absolute sensitivity with no modification to the commercial bridge.\nAs described in Table 1, if the EPR signal cannot be saturated (Unsat.), a factor of approximately 28 can be achieved compared to commercially available probeheads. EPR signals that cannot be saturated are proportional to the square root of the incident microwave power; therefore, the EPR signal intensity is only limited by the amount of power available. However, most protein samples saturate readily; hence, the maximum signal that can be obtained is determined by the microwave magnetic field at the sample. When the sample is saturable (Sat.), a factor of 5.7 can be achieved. Further experimental details are provided in the Supplementary Materials and fig. S3. The procedure to calculate the relative EPR signal intensities can be found in (20). Also found in Table 1 is an effective filling factor (Eff. \u03b7) calculated by using the ratio of a \u201cpoint\u201d sample (1 nl reference volume) with the effective active volume of the resonator. The effective height of the PMR is assumed to be 1.1 mm, taken from the on-axis profile in fig. S2C. The \u03b7Q0 product can be related to the unsaturable signal as per Eq. 1. In a pulse experiment, the signal enhancement will be proportional to the saturable signal, in this case, approximately a factor of 6.\n\n### Frozen solution EPR of photosystem II tyrosine D radical ($No alternative text available$)\n\nWater oxidation in photosystem II takes place at the tetranuclear manganese cluster, with a redox-active tyrosine radical ($No alternative text available$) as an interface to the light-induced electron transfer process (30). Symmetrically to $No alternative text available$, a long-lived tyrosine radical ($No alternative text available$) exists in the second branch of the photosystem II that contains no manganese cluster. In this work, the $No alternative text available$ radical is used as a standard probe because it is stable for a number of hours under ambient conditions (31) and has been well characterized using a variety of EPR techniques (5, 30). The hyperfine interactions from several protons, both on the phenyl ring and distal CH2 carbon, lead to the distinct splittings of the radical (S = 1\/2). To generate the tyrosine radical ($No alternative text available$) EPR signal, the photosystem II core complex samples are illuminated in ambient light and rapidly frozen.\nThe tyrosine D radical ($No alternative text available$) of photosystem II is measured in two forms: (i) a frozen solution of photosystem II prepared by the method of Berthold, Babcock, and Yocum (BBY particles) (32) placed in a 0.3 mm inner diameter capillary and (ii) a 0.3 mm by 0.18 mm by 0.18 mm single crystal of photosystem II core complexes (33). In both photosystem II samples, the $No alternative text available$ and first ligand sphere are known to be identical. These samples provide a benchmark for future work.\nShown in Fig. 3 is the $No alternative text available$ radical EPR signal in an 85 nl frozen solution of photosystem II BBY particles at a temperature of 80 K using the self-resonant microhelix. A continuous-wave EPR experiment, shown in Fig. 3A, was performed using an Elexsys E580 X-band bridge by sweeping 10 mT in 1 min (4096 points) with a modulation rate of 100 kHz and an amplitude of 0.5 mT. The data were averaged 49 times for a total of 49 min at an incident power of 0.2 \u03bcW. To further improve the signal-to-noise ratio of the continuous-wave experiment, we performed a field-swept nonadiabatic rapid scan (NARS) experiment (data shown in Fig. 3B). The field-swept NARS experiment was performed on the same commercial hardware using the rapid-scan method of M\u00f6ser et al. (34) and processed with the method described by Hyde et al. (35). Here, the scan rate was a sinusoidal 100 kHz field sweep at 1 mT amplitude and a field-step size of 0.05 mT. The collected real and imaginary, pure-absorption and pure-dispersion, spectra were pseudo-modulated with a 0.5 mT moving difference (MDIFF) (35) to compare to the field-modulated continuous-wave experiment. A factor of 2 in signal-to-noise improvement is obtained for the same signal acquisition time.\nA field-swept two-pulse electron spin-echo (ESE) EPR experiment was performed on the same commercial hardware over an 8 mT sweep, shown in Fig. 3C. The field-swept ESE data were pseudo-modulated with a 0.5 mT MDIFF to compare the experiment with the field-modulated continuous-wave experiment in Fig. 3A. Only 9 mW of incident power was needed to obtain a sufficient echo using a 40 ns \u03c0\/2 pulse. The signal-to-noise ratio for all three experiments was calculated and tabulated in Table 2.\nLast, a comparison of $No alternative text available$ radical EPR signal between the MD5W1 dielectric resonator and the self-resonant microhelix was performed at a temperature of 80 K. A 636 nl volume sample was placed in the MD5W1 resonator and a signal-to-noise ratio of approximately 300 was measured (data shown in fig. S5). This represents a volume-normalized absolute spin sensitivity improvement of approximately 5 at similar microwave magnetic field and EPR experimental parameters. These experiments serve to show the versatility of the microhelix to perform EPR experiments on limited sample volumes (less than 85 nl) at X-band.\n\n### Single-crystal continuous-wave EPR of the tyrosine D radical ($No alternative text available$) in the photosystem II core complex\n\nShown in Fig. 4 are continuous-wave EPR data collected at two separate angles of the photosystem II $No alternative text available$ radical in a single crystal at a temperature of 80 K as a sensitivity test for the 0.4-mm\u2013inner diameter self-resonant microhelix. The photosystem II core complex crystal had dimensions of 0.3 mm by 0.18 mm by 0.18 mm. The spectra were collected by sweeping 15 mT in 1 min (4096 points) with a modulation rate of 100 kHz and an amplitude of 0.3 mT. The data are averaged 49 times for a total time of 49 min at an incident power of 0.2 \u03bcW. Simulations using the known g-tensor and hyperfine tensors (5) were performed with an EasySpin (http:\/\/easyspin.org; 36) global fit routine to find the crystal orientation and plotted in red in Fig. 4. At X-band, the g-anisotropy of the $No alternative text available$ radical is very small and is not resolved. Instead, the orientation dependence is primarily determined by the hyperfine interaction pattern of the coupled proton nuclei (5). Using only two angles, a unique fit cannot be found, but a demonstration of the $No alternative text available$ features is shown. A nonspecifically bound Mn2+ signal is also present in the crystal, yielding the signals indicated by an asterisk (*).\nThe use of photosystem II crystals as a benchmark provides a challenging system to measure. The photosystem II core complex has a molecular mass of approximately 350 kDa as a monomer, and each complex contains only one $No alternative text available$ radical. With a crystal size of 0.3 mm by 0.18 mm by 0.18 mm, the size of the unit cell, and the fact that there are eight photosystem II complexes per unit cell, one can calculate approximately 8.9 \u00d7 1012 $No alternative text available$ radicals to be present in the sample. This demonstrates the versatility of the microhelix to study large complexes in small crystal dimensions. A photosystem II core complex crystal can be routinely grown to dimensions of 0.3 mm but requires significant effort to increase in size. Last, the $No alternative text available$ radical is easily saturable with large microwave magnetic fields, which limits the available microwave power and maximum EPR signal at a given temperature. Despite these challenges, a signal-to-noise ratio of approximately 35 could be obtained for the $No alternative text available$ radical.\n\n### Pulse EPR on the H-cluster in single crystals of [FeFe]-hydrogenase\n\nUsing the photosystem II $No alternative text available$ radical, we have established that the self-resonant microhelix is suitable for single-crystal protein samples. However, we now want to demonstrate that (i) a full angular g-tensor determination can be performed and that (ii) advanced pulse EPR experiments, like ESEEM and HYSCORE, are also possible in this setup. The self-resonant microhelix is optimal for these experiments because of the relatively large bandwidth (90 MHz critically coupled); an efficiency of 3.2 mT\/W1\/2, which corresponds to a \u03c0\/2 pulse of as short as 20 ns with an incident power of only 20 mW; and the relatively homogeneous microwave magnetic field incident on the sample.\nFirst, a field-swept two-pulse ESE EPR experiment has been performed every 5\u00b0 on a protein single crystal of the [FeFe]-hydrogenase of C. pasteurianum (CpI) in the oxidized Hox state (16). Under a microscope in an anaerobic chamber, the protein crystal is drawn by capillary action into a 0.3 mm inner diameter capillary with mother liquor and cryoprotectant, centered in the microhelix, and flash-frozen. The microhelix assembly is affixed to a Bruker Flexline\u2013compatible support assembly, placed in a precooled cryostat, and attached to the EPR bridge. The whole assembly is then rotated in 5\u00b0 steps over 180\u00b0 in one plane within the magnet (shown in Fig. 5). A very good signal-to-noise ratio of approximately 290 is calculated for a collection time of 8 min for each spectrum at a temperature of 15 K. The [FeFe]-hydrogenase of C. pasteurianum (CpI) has a molecular mass of 67 kDa. The unit cell has P1 21 1 symmetry with two molecules in the asymmetric unit [Protein Data Bank (PDB) ID: 4XDC], resulting in four distinct signals in the EPR spectrum. The single crystal had dimensions of approximately 0.3 mm by 0.1 mm by 0.1 mm and, on the basis of the unit cell dimensions, approximately 17 \u00d7 1012 enzymes within the crystal are calculated, each containing one active site (H-cluster, shown in Fig. 2). Each enzyme in the unit cell provides one of the four observed signals corresponding to 4.25 \u00d7 1012 spins per peak.\nFrom these spectra, the data can be fitted to simulations that relate the different frames of reference to each other, as defined in the EasySpin simulation package (further details in the Supplementary Materials). The [FeFe]-hydrogenase H-cluster is shown in Fig. 5A with the chosen molecular frame (details in the Supplementary Materials), and a schematic that relates the molecular frame to the laboratory system frame is shown in Fig. 5B. Here, the laboratory system frame is defined and the static magnetic field (B0) is oriented along the L1 axis, while in this work, the microwave magnetic field (B1) is along the L3 axis. The crystal frame with respect to the laboratory frame is unknown and depends on how the crystal lies within the capillary. The molecular frames of each of the four molecules in the unit cell are related to each other by the known crystal symmetry. The two molecules in the asymmetric unit are labeled Molecular-Frame AI and Molecular-Frame BI (Site I). The three axes for the molecular frames (AI and BI) were chosen on the basis of the x-ray crystal structure as (i) the axis from the distal iron (Fed) to the proximal iron (Fep); (ii) the normal of a plane calculated from the proximal iron, distal iron, and the nitrogen of the ADT ligand (Fep-Fed-N1); and (iii) the cross product of (i) into (ii). The crystal symmetry\u2013related molecules, Molecular-Frame AII and Molecular-Frame BII from Site II, are generated by the screw axis along the b axis of the unit cell. The g-tensor frame, which gives rise to the EPR signal, is also unknown and is defined as a rotation with respect to the molecular frame. (Rotational matrices for the relationship of these frames can be found in table S2.) The proposed g-tensor frame was solved using the principal g-values obtained from the frozen solution EPR experiment in (16) (g-values: [1.999, 2.039, 2.097] corresponding to the x, y, and z axes, respectively), and the resonance roadmap is overlaid on the field-swept two-pulse ESE EPR experiment shown in Fig. 5D. Since four molecules with different (known) orientations are present in the P1211 unit cell symmetry, it can be anticipated that a rotational study in one plane might be sufficient to fully define the g-tensor axis with the molecular frame.\nThe preliminary proposed g-tensor orientation is plotted as a stereo view in Fig. 5E. The g-tensor (gx, red; gy, green; and gz, blue) is drawn with an origin at the distal iron (Fed) since density functional theory calculations consistently identify Fed to contain most of the spin density in the Hox state (37). In this work, we find that the g-tensor principal axis does not coincide with the molecular frame, as defined by the two iron atoms and the ADT-amine nitrogen, as suggested by Adamska-Venkatesh et al. (38). Rather, the g-tensor orientation seems to be symmetric with respect to the plane defined by the two ADT sulfurs, the distal iron atom, and the distal CN and CO ligand. The orientation of the largest g-component (gz) dissects the plane between the distal CN and CO ligands, while the gx and gy components tend to be roughly symmetric with respect to the plane spanned by the two iron atoms and the ADT-amine nitrogen. See fig. S6 for a direct comparison. A rotation of the molecular frame by (\u2212142.0, \u221284.1, 137.6) using a ZYZ convention Euler angle produces the proposed g-frame. Rotational matrices relating the g-tensor to the molecular frame within the PDB ID 4XDC crystal structure can be found in table S2.\nIn the study by Adamska-Venkatesh et al. (38), the orientation of the Fed-CN axis was established with respect to the g-frame using the axial components of the 13CN hyperfine and C14N quadrupole interaction, as determined by orientation-selective 13C ENDOR and 14N HYSCORE experiments in frozen solution. The calculated direction cosines of the Fed-CN orientation within the g-frame seem to be consistent with g-frame being coincident with the molecular frame. It turns out, however, that the direction cosines of the Fed-CN axis with respect to the actual g-frame (not coinciding with the molecular frame), as determined in our current single-crystal study, are still similar to those found in the study of Adamska-Venkatesh et al. and only deviate by about 10\u00b0 (see fig. S6). The information obtained from the frozen solution orientation-selective ESEEM and ENDOR experiments was underdetermined and did not allow accurate assignment of the full magnetic interaction tensors. Accurate analysis and refinement is only possible with the collection of hyperfine and quadrupole data originating from single-crystal EPR and relating the whole dataset to quantum chemical calculations.\n\n### Advanced pulse EPR on the H-cluster in single crystals of [FeFe]-hydrogenase\n\nTo illustrate the availability of more advanced experiments for single-crystal studies, a HYSCORE experiment for the 150\u00b0 field-swept ESE EPR dataset was performed on each of the peaks and is plotted in Fig. 6. Each HYSCORE spectrum was collected over approximately 1 hour, using a standard four-pulse HYSCORE pulse sequence (7). To obtain information on the hyperfine and quadrupole tensors, HYSCORE or ESEEM data must be collected on at least one peak and followed through a 180\u00b0 rotation to obtain the axial relationship of the hyperfine interactions. Multiple peaks can be used to overdetermine the system.\nIn a HYSCORE experiment, the two-dimensional (2D) density representation shows correlations between the nuclear-spin transitions (mI) in both projections of the electron spin. Both the 14N nucleus (I = 1) from a distal cyanide ligand ($No alternative text available$) and the secondary-amine group in the ADT ligand can potentially contribute to the HYSCORE spectrum generating three transitions per ligand for each electron-spin transition (ms) manifold for a maximum of 12 modulation frequencies. According to an earlier study on Hox in frozen solution, the features of the distal cyanide ligand spread out up to 6 MHz, while the transitions of the ADT-amine nitrogen are found between 2 and 4 MHz (38).\nIn the single-crystal 2D spectrum, shown in Fig. 6, six main transitions can be identified, which are assigned to the 14N of the distal CN ligand. The modulation frequencies can be grouped into two sets (0.5, 3.7, and 4.2 MHz) and (1.5, 3.5, and 5.0 MHz), each originating from a different ms manifold. The correlation features between these transitions are indicated by the white, red, and green circles and are in agreement with frozen-solution HYSCORE performed in (38). However, the current work only seeks to highlight the feasibility of these advanced EPR experiments. Future ESEEM\/HYSCORE experiments will address the 14N couplings of the CN and ADT ligand in greater detail. Possibly, this will involve selective 15N labeling as has been demonstrated before (38, 39). From these experiments, extracting the magnitude and orientation of the hyperfine and nitrogen quadrupole tensors in the molecular axis frame and relating these to the electronic structure as predicted through quantum chemical calculations are possible.\n\n## CONCLUSIONS AND OUTLOOK\n\nAn application of the self-resonant microhelix geometry and planar-coupling structure that increases the EPR absolute spin sensitivity by a factor of approximately 28 if the signal is unsaturable, and 6 if the EPR signal is able to be saturated, is presented. For saturable EPR signals, such as those found in protein samples, the self-resonant microhelix saves up to a factor of 36 in measuring time. From this gain in sensitivity, the self-resonant microhelix is well suited for EPR studies on protein single crystals with dimensions less than 0.3 mm. Because of the very high efficiency parameter of 3.2 mT\/W1\/2, which corresponds to a \u03c0\/2 pulse of 20 ns with an incident power of 20 mW, the microhelix geometry is advantageous in extending pulse EPR to experiments that usually require costly high-powered microwave amplifiers (e.g., HYSCORE), further expanding the applicability of pulse EPR. The significantly reduced power contributes to reduced dead time and, therefore, potentially expands the use of Fourier transform EPR on systems with short relaxation rates (40). We also show that the microhelix performs well for field-swept NARS techniques because of its small size and \u201copen\u201d structure, which increases the continuous-wave EPR spin sensitivity further by a factor of 2 for the same experimental time. Because of the relatively large bandwidth of the microhelix (90 MHz critically coupled), this geometry is particularly well suited for frequency-swept NARS and rapid scan experiments, which further improve the signal-to-noise ratio for saturable samples (35, 34) and for the use of arbitrary-waveform generators for advanced pulse spectroscopy.\nThis advance in resonator design has allowed the collection of EPR data from a 0.3 mm by 0.1 mm by 0.1 mm single crystal of [FeFe]-hydrogenase in the Hox state from C. pasteurianum (CpI) at a temperature of 15 K. To our knowledge, the HYSCORE spectra collected are the first published results from a protein single crystal with dimensions less than 0.3 mm. Full g-tensor and 14N hyperfine tensor analysis of the active-site cofactor from the collected data is to follow. In addition, further studies of the hyperfine and quadrupole tensor of the [FeFe]-hydrogenase are now feasible in single-crystal experiments. These studies will provide further insight for protein engineering and artificial enzyme research for creating bio-inspired and bio-mimicking hydrogenase systems (16).\nAs this technology matures, further improvements to enhance the sensitivity based on new fabrication techniques and choice of other materials will be explored. Not only does the increase in sensitivity save time in EPR data measurements, but it also reduces the need for the availability of, or necessity to grow, larger crystals. Other microhelix structures, for example those made from superconducting materials have very high Q0-values and, consequently, a limited bandwidth, which makes them ideal for continuous-wave experiments. Because of the extremely high resonator efficiency, only nonsaturable samples will fully benefit from these designs. Therefore, the current microhelix geometry and fabrication provides an optimal compromise between maximum sensitivity and bandwidth for a broad range of temperatures. Specifically, the self-resonant microhelix provides the possibility to study catalytically active proteins at crystal dimensions relevant to x-ray crystallography and, hence, is a significant advancement in the field of enzyme research.\n\n## MATERIALS AND METHODS\n\n### Simulations and experiments\n\nAll microhelix and planar-coupling designs were modeled in the commercially available finite-element modeling program Ansys Electronics Desktop with HFSS (High Frequency Structure Simulator; v. 19.1) using driven mode. In driven mode, Ansys HFSS requires a coupling structure and mimics the output of a network analyzer. All designs were matched to 50 ohms with an S11 < \u2212 35 dB. Frequency and Q-values (\u22123 dB) were read directly from a simulated S11 plot, and Q0-values were calculated by the known equation Q = Q0\/(1 + \u03b2), where \u03b2 is the reflection coefficient at the frequency of resonance (\u03b2 = 1 for critically coupled). EPR signal intensity and resonator efficiency values (mT\/W1\/2) were calculated using Ansys HFSS (20) and tabulated. EPR experimental comparisons were performed on an Elexsys E580 X-band bridge by Bruker Biospin. Four resonators were used for this comparison. The Bruker Biospin (i) dielectric ER4118X-MD-5 W1 (MD5W1) and (ii) LGR ER4118X-MS-3 W1 (MS3; split-ring) resonators were used as comparisons with known commercial resonator geometries. Two \u03a9-shaped 0.5-mm\u2013inner diameter PMR resonators were also tested. The first used (iii) Rogers 6010LM (RO6010LM printed circuit board; Rogers Corporation, Chandler, AZ, USA) substrate as per (1113) and one with (iv) sapphire substrate. The sapphire substrate was fabricated by Technical University Ilmenau (Ilmenau, Germany). Both PMR geometries had a 0.5 mm hole through the substrate. Resonator characteristics are found in table S1.\nThe EPR experiments performed in this work were as follows. The continuous-wave EPR experiment measures the sample using a constant microwave power incident on the sample and slowly sweeping a quasi-static magnetic field through the resonance condition, \u03bd = \u03b3B0, where \u03bd is the operating frequency (nominally 9.5 GHz for X-band), \u03b3 is the gyromagnetic ratio of the spin system (\u03b3 = 2.8 MHz\/G for a free electron), and B0 is the quasi-static magnetic field. The magnetic field was modulated, typically at 100 kHz, and collected using a phase-sensitive detector. Continuous wave was the standard EPR experiment. With modern digital signal processing and fast analog-to-digital converters, the continuous-wave experiment has recently been improved upon. The NARS (35, 41) and adiabatic rapid scan (34, 42) methods collect real and imaginary, pure-absorption and pure-dispersion, EPR spectra using fast quasi-static magnetic field or microwave frequency sweeps without the need for a phase-sensitive detector. Both rapid scan data can be pseudo-modulated to the conventional first-derivative EPR spectrum using an MDIFF pseudo-modulation (35). The NARS experiment uses a field sweep fast enough to overcome 1\/f noise but remains in a thermal equilibrium, while adiabatic rapid scan sweeps the field fast enough to cause passage. The advantage of NARS is the signal-to-noise improvement of collecting pure-absorption EPR spectra, while adiabatic rapid scan can further improve the continuous-wave and NARS experiment by changing the effective microwave magnetic field at the sample. This allows for an increase of microwave power and, thus, increase in EPR signal for saturable signals (42). While the NARS method can be implemented on commercial bridges with no hardware changes, it does require some technical expertise (34). However, to perform adiabatic rapid scan experiments on protein samples, custom current drivers are needed to increase the swept field amplitude. For simplicity, in this work, we implemented only NARS. Other experimental parameters are as stated in the manuscript.\nThe field-swept two-pulse ESE experiment used a \u03c0\/2 \u2212 \u03c4 \u2212 \u03c0 pulse sequence (\u03c0 is 80 ns) that resulted in an echo \u03c4 seconds; herein, \u03c4 is 300 ns, after the \u03c0 pulse. The field was stepped, and a whole spectrum was acquired. This is the standard pulse sequence to collect an EPR spectrum (7). The HYSCORE experiment performed in this work has a \u03c0\/2 \u2212 \u03c4 \u2212 \u03c0\/2 \u2212 t1 \u2212 \u03c0 \u2212 t2 \u2212 \u03c0\/2 pulse sequence, which resulted in an echo \u03c4 seconds after the last \u03c0\/2 pulse. The values for t1 and t2 were swept to form a 2D experiment at a fixed magnetic field position. Here, the magnetic field was set to one of the peaks measured in the two-pulse ESE experiment: \u03c4 is 280 ns, t1 and t2 start at 300 ns with 256 48 ns steps, and \u03c0 is 80 ns.\n\n### Fabrication techniques\n\nThe microhelix was fabricated by hand winding 5 to 8 turns of 0.125 mm diameter silver wire with polytetrafluoroethylene (PTFE) coating (0.0255-mm thickness, total 0.18 mm diameter; Science Products GmbH, Hofheim, Germany) around a 0.4 mm drill bit and placed inside a Rexolite cylinder (0.8 mm inner diameter and 1.2 mm outer diameter) with a length of 10 mm. The drill bit was removed as the coil was affixed with super glue by capillary action, waiting for 1 min and blowing out the excess. The assembly was left to dry for several days.\nIn the current setup, 6.5 turns are necessary to obtain a helix that has a resonant frequency around 10 GHz without sample and 9.8 GHz with a 0.4 mm outer diameter and 0.3 mm inner diameter quartz capillary. The microhelix is reproducible within 500 MHz (9.3 to 9.8 GHz). Further reduction of the variability is feasible with the use of an assembly jig. The resonator withstands many freeze-thaw cycles and is quite robust.\nThe coupling loop was designed in Ansys HFSS and prepared for fabrication in AutoDesk Inventor Professional 2019. The printed circuit board designs were emailed to Streamline Circuit (Santa Clara, CA, USA) engineers and manufactured on a PTFE substrate. The printed circuit board was connected to the bridge by a high-frequency SubMiniature version A (SMA) end launcher (AmphenolRF, 901-10510-1). Impedance matching was achieved by moving the microhelix relative to the coupling loop until critically coupled on a network analyzer. Fine-tune matching was obtained with a slide-screw tuner at the bridge output. Bench tests of resonator characteristics\u2014such as the frequency measurements, Q0-value, and sample frequency shifts\u2014were performed on an Agilent 8722ES (now Keysight Technologies, Santa Rosa, CA, USA) vector network analyzer.\n\n### Sample preparation\n\nThe photosystem II complex sample from spinach was prepared by following the BBY method and placed in a 0.4 mm outer diameter capillary with a 0.3 mm inner diameter (32). The tyrosine D ($No alternative text available$) radical was generated by ambient light at room temperature for a few minutes and then rapidly frozen in liquid nitrogen. The $No alternative text available$ radical is well suited as a biological test sample since it has been extensively studied and the g- and hyperfine tensors are known (5). Using ultraviolet-visible (UV-VIS) spectroscopy, the number of chlorophyll molecules in the sample can be determined, and taking into account that there are approximately 250 chlorophyll molecules per photosystem II complex in spinach, the approximate amount of enzyme can be calculated. Each photosystem II complex contains one $No alternative text available$ radical. For the sample prepared in this work, chlorophyll molecules (7.9 \u03bcM\/ml) were measured, resulting in approximately 1.6 \u00d7 1012 $No alternative text available$ radicals in the 85 nl that fill the microhelix.\nIn addition, the photosystem II core complex, extracted and purified from the thermophilic cyanobacterium Thermosynechococcus elongatus, was crystallized to reach a crystal size of 0.3 mm by 0.18 mm by 0.18 mm, using the method of Kern et al. (33). The crystals were gradually transferred to a cryogenic protection buffer [100 mM MES (pH 6.5), 5 mM CaCl2, 30% (w\/w) glycerol, and 16% polyethylene glycol 2000]. The photosystem II core complex forms an asymmetric unit, and the crystal has a unit cell space group symmetry P21 21 21, which generates four sites per unit cell (PDB ID: 1W5C). From this, one can calculate that there are approximately 8.9 \u00d7 1012 $No alternative text available$ radicals in the test crystal. The active center and the $No alternative text available$ of a cyanobacterium were equivalent to that of the photosystem II from spinach.\nLast, [FeFe]-hydrogenase from C. pasteurianum (CpI) was grown and crystallized to dimensions of 0.3 mm by 0.1 mm by 0.1 mm by the method of Esselborn et al. (43) under auto-oxidative conditions, i.e., without reducing agents. This leaves the enzyme in the characteristic active oxidized state (Hox), giving rise to an S = 1\/2 ground state of the H-cluster. The accessory iron-sulfur clusters in the protein were oxidized and remained EPR silent (16). The [FeFe]-hydrogenase crystal has a space group symmetry P1211 with two asymmetric units in two sites per unit cell (PDB ID: 4XDC). With this information, we can calculate that there are approximately 17 \u00d7 1012 single enzymes within the crystal, with each peak corresponding to 4.25 \u00d7 1012 spins.\n\n## Acknowledgments\n\nWe thank M. Chrysina from the Max Planck Institute for Chemical Energy Conversion for guidance on photosystem II spectra and M. Reus for the photosystem II BBY prepared sample and the UV-VIS measurements used in this work. Funding: The research reported in this publication was supported by funding from the European Union Horizon 2020 Marie Sk\u0142odowska-Curie Fellowship (no. 745702; ACT-EPR, https:\/\/act-epr.org), the Max Planck Society, and Sonderforschungsbereich Sfb1078 by Humboldt Universit\u00e4t zu Berlin, Project A5 (R.H. and A.Z.), Cluster of Excellence-2033 RESOLV #390677874 (Deutsche Forschungsgemeinschaft, DFG), DFG Research Training Group GRK 2341 Microbial Substrate Conversion (MiCon), and Volkswagen Stiftung (Design of [FeS] cluster containing Metallo-DNAzymes [Az 93412]) (M.W. and T.H.). Author contributions: J.W.S., E.J.R., and W.L. conceived the project. J.W.S. designed and fabricated the self-resonance microhelix, executed the project, and performed the measurements. R.H. and A.Z. fabricated photosystem II crystals from T. elongatus. J.D., M.W., and T.H. fabricated [FeFe]-hydrogenase crystals from C. pasteurianum (CpI) in the Hox state. D.S., A.S., W.L., and E.J.R. oversaw the project. J.W.S., W.L., and E.J.R. wrote the manuscript. All authors discussed and commented on the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and\/or the Supplementary Materials. MATLAB code and data can be found at https:\/\/act-epr.org\/data. Additional data related to this paper may be requested from the authors.\n\n## Supplementary Material\n\n### Summary\n\nEPR characteristics comparison of multiple resonators\nPower saturation and \u039bave measurements\nSensitivity comparison of the X-band microhelix to a commercial dielectric resonator\nQualitative sensitivity comparison of the X-band microhelix to high-frequency single-mode resonators\nCrystal rotation and simulation\nFig. S1. Dimensions and geometry of the four resonators compared in this paper.\nFig. S2. Ansys HFSS finite-element modeling simulation of the microwave magnetic fields comparing the PMR and microhelix.\nFig. S3. Power saturation curve of LiPC using various resonators.\nFig. S4. Continuous-wave EPR of frozen solution photosystem II BBY particles performed in the Bruker MD5W1 dielectric resonator at a temperature of 80 K.\nFig. S5. Comparison of the g-tensor proposed by Adamska-Venkatesh et al. (38) and the current proposed g-tensor from this work.\nTable S1. Resonator characteristics calculated and measured.\nTable S2. Rotational matrices for the crystal frame with respect to the laboratory frame and the g-tensor with respect to the molecular frame.\nReferences (4456)\n\n### Resources\n\nFile (aay1394_sm.pdf)\n\n## REFERENCES AND NOTES\n\n1\nS. E. J. Bowman, J. Bridwell-Rabb, C. L. Drennan, Metalloprotein crystallography: More than a structure. Acc. Chem. Res. 49, 695\u2013702 (2016).\n2\nW. Lubitz, E. 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Reus for the photosystem II BBY prepared sample and the UV-VIS measurements used in this work. Funding: The research reported in this publication was supported by funding from the European Union Horizon 2020 Marie Sk\u0142odowska-Curie Fellowship (no. 745702; ACT-EPR, https:\/\/act-epr.org), the Max Planck Society, and Sonderforschungsbereich Sfb1078 by Humboldt Universit\u00e4t zu Berlin, Project A5 (R.H. and A.Z.), Cluster of Excellence-2033 RESOLV #390677874 (Deutsche Forschungsgemeinschaft, DFG), DFG Research Training Group GRK 2341 Microbial Substrate Conversion (MiCon), and Volkswagen Stiftung (Design of [FeS] cluster containing Metallo-DNAzymes [Az 93412]) (M.W. and T.H.). Author contributions: J.W.S., E.J.R., and W.L. conceived the project. J.W.S. designed and fabricated the self-resonance microhelix, executed the project, and performed the measurements. R.H. and A.Z. fabricated photosystem II crystals from T. elongatus. J.D., M.W., and T.H. fabricated [FeFe]-hydrogenase crystals from C. pasteurianum (CpI) in the Hox state. D.S., A.S., W.L., and E.J.R. oversaw the project. J.W.S., W.L., and E.J.R. wrote the manuscript. All authors discussed and commented on the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and\/or the Supplementary Materials. MATLAB code and data can be found at https:\/\/act-epr.org\/data. Additional data related to this paper may be requested from the authors.\n\n### Authors\n\n#### Affiliations\n\nMax Planck Institute for Chemical Energy Conversion, Stiftstra\u00dfe 34-36, 45470 M\u00fclheim an der Ruhr, Germany.\nAG Photobiotechnologie, Ruhr-Universit\u00e4t Bochum, Universit\u00e4tsstra\u00dfe 150, 44780 Bochum, Germany.\nAG Photobiotechnologie, Ruhr-Universit\u00e4t Bochum, Universit\u00e4tsstra\u00dfe 150, 44780 Bochum, Germany.\nThomas Happe\nAG Photobiotechnologie, Ruhr-Universit\u00e4t Bochum, Universit\u00e4tsstra\u00dfe 150, 44780 Bochum, Germany.\nInstitut f\u00fcr Biologie, Humboldt-Universit\u00e4t zu Berlin, Philippstra\u00dfe 13, 10115 Berlin, Germany.\nInstitut f\u00fcr Biologie, Humboldt-Universit\u00e4t zu Berlin, Philippstra\u00dfe 13, 10115 Berlin, Germany.\nExperimentelle Physik, Technische Universit\u00e4t Dortmund, Emil-Figge-Stra\u00dfe 50, 44221 Dortmund, Germany.\nAlexander Schnegg\nMax Planck Institute for Chemical Energy Conversion, Stiftstra\u00dfe 34-36, 45470 M\u00fclheim an der Ruhr, Germany.\nMax Planck Institute for Chemical Energy Conversion, Stiftstra\u00dfe 34-36, 45470 M\u00fclheim an der Ruhr, Germany.\nMax Planck Institute for Chemical Energy Conversion, Stiftstra\u00dfe 34-36, 45470 M\u00fclheim an der Ruhr, Germany.\n\n#### Funding Information\n\nHumboldt-Universit\u00e4t zu Berlin: Sonderforschungsbereich Sfb1078\nHorizon 2020: 745702\nMax-Planck-Institut f\u00fcr Chemisch Energiekonversion\n\n#### Notes\n\n*\nCorresponding author. Email: [email\u00a0protected] (J.S.); [email\u00a0protected] (E.J.R.)\n\n## Metrics & Citations\n\n### Citations\n\n#### Export citation\n\nSelect the format you want to export the citation of this publication.\n\n#### Cited by\n\n1. Scalable microresonators for room-temperature detection of electron spin resonance from dilute, sub-nanoliter volume solids, Science Advances, 6, 44, (2020).\/doi\/10.1126\/sciadv.abb0620\nAbstract","date":"2022-08-07 23:27:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 36, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5002089738845825, \"perplexity\": 4835.587090309517}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570730.59\/warc\/CC-MAIN-20220807211157-20220808001157-00779.warc.gz\"}"} | null | null |
Q: IF conditions, data from cookie not validating I need help, I have been at this for too many hours for me to no be embarrassed. Please, why am I not getting this?
function members_only() {
$cookie_name = 'cookie_name';
$error_url = home_url($path = '/403-error/');
global $pagenow;
$array_cookie_value = json_decode( stripslashes($_COOKIE[$cookie_name]), true);
$woo_user_id = $array_cookie_value['user_id'];
$user_meta = get_user_meta($woo_user_id);
$user_order_status = $user_meta['doris_shop_enabled'][0];
if ( is_admin() || is_front_page() || $pagenow == 'wp-login.php' || is_page('403-error') ) {
echo('Do nothing');
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A: The problem was in Chrome data that was saved, outside of the cookie and hard-refresh which I make regularly. I needed to go into the Settings > Cookies and other website data and empty it all from there. After that I got it to work.
Yes I made some minor changes to the code also, but that was the main problem.
Thanks all for trying to help :)
| {
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Hello. A couple of you might remember me by my previous username: GenericUsernameHere. I posted a few questions and answers back in 2014-2016. I haven't used the site in a long time and I figured I wanted to make a more up to date account with a new mehr .. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/docs.mosek.com\/10.0\/pythonfusion\/expression.html","text":"# 14.2.14 Class Expression\u00b6\n\nmosek.fusion.Expression\n\nAbstract base class for all objects which can be used as linear expressions of the form $$Ax+b.$$\n\nThe main use of this class is to store the result of expressions created by the static methods provided by Expr.\n\nMembers\n\nExpression.eval \u2013 Evaluate the expression and push the result onto the work stack.\n\nExpression.getDim \u2013 Return the d\u2019th dimension in the expression.\n\nExpression.getND \u2013 Return the number of dimensions in the expression.\n\nExpression.getShape \u2013 Get the shape of the expression.\n\nExpression.getSize \u2013 Return the total number of elements in the expression (the product of the dimensions).\n\nExpression.index \u2013 Get a single element in the expression.\n\nExpression.pick \u2013 Pick a number of elements from the expression.\n\nExpression.slice \u2013 Get a slice of the expression.\n\nExpression.toString \u2013 Return a string representation of the expression object.\n\nImplemented by\nExpression.eval\neval(WorkStack rs, WorkStack ws, WorkStack xs)\n\n\nEvaluate the expression and push the result onto the rs work stack.\n\nParameters\n\u2022 rs (WorkStack) \u2013 The stack where the result of the evaluation is stored.\n\n\u2022 ws (WorkStack) \u2013 The stack used by evaluation to perform intermediate computations. It will be returned in the same state as when the function is called.\n\n\u2022 xs (WorkStack) \u2013 An auxiliary stack.\n\nExpression.getDim\ngetDim(int d) -> int\n\n\nReturn the d\u2019th dimension in the expression.\n\nParameters\n\nd (int)\n\nReturn\n\n(int)\n\nExpression.getND\ngetND() -> int\n\n\nReturn the number of dimensions in the expression.\n\nReturn\n\n(int)\n\nExpression.getShape\ngetShape() -> int[]\n\n\nGet the shape of the expression.\n\nReturn\n\n(int[])\n\nExpression.getSize\ngetSize() -> int\n\n\nReturn the total number of elements in the expression (the product of the dimensions).\n\nReturn\n\n(int)\n\nExpression.index\nindex(int i) -> Expression\nindex(int[] indexes) -> Expression\n\n\nGet a single element in the expression.\n\nParameters\n\u2022 i (int) \u2013 Index of the element to pick.\n\n\u2022 indexes (int[]) \u2013 Multi-dimensional index of the element to pick.\n\nReturn\nExpression.pick\npick(int[] indexes) -> Expression\npick(int[][] indexrows) -> Expression\n\n\nPicks a number of elements from the expression and returns them as a one-dimensional expression.\n\nParameters\n\u2022 indexes (int[]) \u2013 Indexes of the elements to pick\n\n\u2022 indexrows (int[][]) \u2013 Indexes of the elements to pick. Each row defines a separate multi-dimensional index.\n\nReturn\nExpression.slice\nslice(int first, int last) -> Expression\nslice(int[] firsta, int[] lasta) -> Expression\n\n\nGet a slice of the expression.\n\nParameters\n\u2022 first (int) \u2013 Index of the first element in the slice.\n\n\u2022 last (int) \u2013 Index of the last element in the slice plus one.\n\n\u2022 firsta (int[]) \u2013 Multi-dimensional index of the first element in the slice.\n\n\u2022 lasta (int[]) \u2013 Multi-dimensional index of the element after the end of the slice.\n\nReturn\nExpression.toString\ntoString() -> str\n\n\nReturn a string representation of the expression object.\n\nReturn\n\n(str)","date":"2022-08-12 02:11:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.28242233395576477, \"perplexity\": 2747.4545803124233}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571538.36\/warc\/CC-MAIN-20220812014923-20220812044923-00312.warc.gz\"}"} | null | null |
Катаптерикс кримський (Catapterix crimaea) — вид метеликів родини Acanthopteroctetidae.
Поширення
Ендемік Криму. Відомо два місця знаходження: Карадазька біостанція та біостанція Сімферопольського державного університету у селі Краснолісся.
Опис
Розмах крил 6,0-6,5 мм. Опушення голови світле, помаранчево-жовтого забарвлення. На лобі притиснуті лусочки відсутні. Губні щупики дуже маленькі, приховані під великими вигнутими щелепними. Вусики коричневі, зі слабким жирним блиском. Груди і тегули покриті щільно притиснутими широкими лусочками. На передніх крилах основи всіх радіальних жилок знаходяться на приблизно однаковій відстані одна від одної. Передньоспинка і тегули покриті зеленувато-золотистими, блискучими лусочками. Передні крила одноколірні, зеленувато-золотистого забарвлення, з сильним відблиском. Бахрома крил темна, буро-сіра. Задні крила буро-сірого забарвлення. Бахромка задніх крил трохи світліша. Ноги і черевце однотонні, буро-сірого забарвлення. Гомілки і членики лапок не мають світлих перев'язів.
Примітки
Лускокрилі
Комахи Європи
Метелики України
Тварини, описані 1988 | {
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Big D Gets A Big L to Kick Off 2021 NFL Season
Dane Powell
Dane Powell Published: August 6, 2021
Football is Back!
The 2021 NFL season has officially arrived, and it is certainly a glorious feeling knowing that every Sunday from now until February will have some games to watch.
The season kicked off last night in Canton, Ohio, the home of the Pro Football Hall of Fame. The Cowboys and Steelers met up for the annual Hall of Fame Game to get the 2021 NFL season started.
How'd It Go?
Well, the whole game definitely looked like a preseason game with a whole lot of rusty players and miscommunication from both teams. The Cowboys came out and ended up being the first team to put points on the board, but that was it for their scoring on the night. Steelers put up 16, and the Cowboys start the new season the same way they ended the last one...with a loss.
Pittsburgh Steelers v Dallas Cowboys
The thing about these preseason games is that they are primarily there to take a look at your younger talent and veterans who are on the bubble to make the upcoming roster cuts. With that being said, you can't really put any stock into the outcome of the game. It really only exists as tool for the coaching staff to gauge who makes the final roster.
The New Guy
The Cowboys and Steelers really wanted to take a look at the talent behind their starting quarterbacks. The results are a little concerning for both teams, but Cowboy fans did get a look at their shiny new toy from the first round of the NFL draft. Former Penn State linebacker Micah Parsons got his first game reps. He managed to collect three solo tackles and a fumble recovery before being pulled. Good news for the cowboys, who have been plagued of the past few years at the LB position.
Like I said before, this game's sole purpose is for player evaluation, so it's pretty hard to gauge where the team is at from just one game. There is a lot left to play out in the preseason, as the coaches continue to rotate personnel in hopes of finding the perfect roster for the season.
The Cowboys certainly wish they started on a better note, but hey., at least that Parsons kid looked decent, right?
Texas' Top 15 Pro Athletes from the Past 25 Years in No Particular Order
LOOK: 50 images of winning moments from sports history
Sometimes images are the best way to honor the figures we've lost. When tragedy swiftly reminds us that sports are far from the most consequential thing in life, we can still look back on an athlete's winning moment that felt larger than life, remaining grateful for their sacrifice on the court and bringing joy to millions.
Read on to explore the full collection of 50 images Stacker compiled showcasing various iconic winning moments in sports history. Covering achievements from a multitude of sports, these images represent stunning personal achievements, team championships, and athletic perseverance.
CHECK IT OUT: 100 sports records and the stories behind them
Filed Under: dallas cowboys, football, nfl, This Is NOT The Sports Category! | {
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Student of the Year Winner Gets Noticed
Annie Michalski, one of two winners of the 2016 Collision Repair Education Foundation's Student of the Year Award, says she feels welcome in a world where she never thought she would, and she's planning to stick around. The automotive collision repair industry is lucky to have her; in a time when demands are rising, technology is advancing rapidly, and the vast majority of employees are male and over age 35, it's in desperate need of people like Michalski. She's hoping to make the industry more welcoming to others like her: intelligent millennial women.
Annie Michalski, a 2016 Student of the Year winner, was featured on the cover of the February 2017 issue of Fenderbender magazine.
Currently, Michalski represents two minorities in the industry: women and young people. She hopes that by stepping up and setting an example, she can continue to be a role model for future generations. She said, "I'm a woman, and there aren't a lot of women in the industry. The thing about that is, every single woman I have met in the industry is amazingly strong and pushing us further every day to equality. The thing that I still see needing much work is the age gap. There are so few technicians entering the scene nowadays."
When she was first introduced to the collision repair program at her school, Michalski was a little intimidated. "I won't lie, anytime I was ever approached about the industry… it was older men who were… shown as the standard in the field. I think we need to break that stereotype." Within a few days, she decided it was the opportunity she needed to prove herself as both a young person and a woman. Michalski has studied collision repair since her sophomore year of high school, and this year, was selected as a CREF student of the year.
Michalski said, "I'd like to say I was selected because my teacher (John Warabow) saw the potential in me… I'm drive and enjoy the art and science of collision repair."
On being selected as a CREF student of the year, Michalski said, "The Collision Repair Education Foundation helped me understand how far one could go in this industry; without them I would be a lot more lost and have a foggier future."
The collision repair program at her school has enabled Michalski to complete several certifications. In addition, she is a member of SkillsUSA, she serves on the collision program's Advisory Committee, and she works at a local dealership as part of the cooperative education program. Her accomplishments in collision repair have helped to shape her dream of continuing to a career in the industry, and enabled her to find a path to do it. Michalski said, "I hope to continue forward into the business aspect of the industry. I plan on attending college in the fall for management or marketing."
Michalski was also interviewed by CollisionWeek publisher Russell Thrall. The video of that interview appears below.
Michalski, who attended Western Area Career and Technology in Canonsburg, Pa. was nominated by her instructor, John Warabow, who wrote in his recommendation for the award last year, "Anna entered my program as a sophomore and is now a senior. Her ability to communicate with fellow students, staff and faculty has made her well-known and respected. She has become a true leader in the class and her fellow students are encouraged by her positive attitude and the ability to accomplish any given task."
In the video above, Michalski explains what attracted her to a career in the collision industry, how her family and friends reacted and supported her decision. She also talks about the support she has received from the industry, including from Petra Schroeder, chair of the Women's Industry Network, as well as the Foundation. Michalski graduated high school in 2017 and is attending college, planning to study business. She plans to combine the business skills she learns in college with the technical skills she gained in collision repair to find a career in the industry upon graduation.
Filed Under: News & Events, Slider Tagged With: Student of the Year | {
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\section{Introduction}
Higher Derivative Gravity (HDG) is the theory of gravity based on the metric as the carrier of degrees of freedom,
with an action containing terms of order zero, one and two in the curvature.
It contains both dimensionful couplings (the cosmological and Newton constant) and dimensionless ones
(the coefficients of the HD terms).
When treated perturbatively in the latter, it is renormalizable~\cite{stelle}, but not unitary.
Following some earlier attempts~\cite{julve,ft1},
its one-loop beta functions were correctly derived for the first time in~\cite{avrabar};
for more details and generalizations, see~\cite{deBerredoPeixoto:2003pj,deBerredoPeixoto:2004if}.
Depending on the signs of the couplings, the theory
can be asymptotically free, but it has ghosts and/or tachyons.
There has been recently a revival of interest in this theory,
and proposals to get around its problems in various
ways~\cite{Mannheim:2006rd,Salvio:2014soa,Salvio:2017qkx,Salvio:2018crh,Einhorn:2014gfa,AKKLR,smilga,holdom,donhdg,Anselmi:2018tmf,Donoghue:2019fcb}.
In the asymptotic safety approach to quantum gravity, one tries to construct a continuum limit around an interacting
fixed point (FP)~\cite{Weinberg}.
The main tool to investigate the gravitational renormalization group has been the Functional Renormalization Group Equation
(FRGE), as applied for the first time to gravity by Martin Reuter \cite{Reuter}.
It defines a flow on the theory space consisting of all diffeomorphism invariant functionals of the metric.
One expects that at an interacting gravitational FP, infinitely many gravitational couplings will be nonzero.
In spite of this complication, much evidence for the existence of such a FP has been collected so far \cite{perbook,rsbook}.
In the context of asymptotic safety, when one uses the FRGE,
there is never the need to postulate the form of the bare action
to be used in the path integral.
Instead, one directly calculates the flow of the effective action
as a function of an external ``coarse-graining'' scale,
or IR cutoff, $k$.
In this context, the action
of HDG can be used as an ansatz for the running effective action.
We will call this the ``HDG truncation''.
It tracks the flow of the theory in a five-dimensional
``theory space'' parametrized by the couplings:
${\cal V}$, $Z_N$, $\lambda$, $\xi$ and $\rho$, defined below.
The beta functions of HDG have been studied from this point of view in several papers.
They were obtained in a one-loop approximation to the FRGE in~\cite{Codello:2006in,niedermaier,OP2013,OPP2}.
In these calculations, the beta functions of the HD couplings
are asymptotically free,
in agreement with the old perturbative results,
but the flow of the dimensionful couplings looks very similar to the one of the Einstein-Hilbert truncation,
and exhibits a nontrivial FP for the cosmological and Newton constant.
To go beyond one loop, one has to keep terms involving
the beta functions in the r.h.s. of the flow equation,
and then solve these algebraic equations for the beta functions.
We highlight this process in Section 3.1.
This produces non-linearities that amount to resummations of
infinitely many loop diagrams.
This has been calculated in \cite{lauscher,bms} on a generic Einstein manifold, and a fully interacting FP was found,
but these calculations were limited to one or two,
out of the three HD couplings.
This may seem to be sufficient, since one of the three couplings
is the coefficient of the Euler term, that does not contribute to the local dynamics.
Unfortunately, as we shall see in Sect.2.1,
on an Einstein manifold one computes the beta function of certain
linear combinations of the three couplings, and
it is actually impossible to identify the beta function
of the two dynamically interesting ones: there is an unknown mixing with the beta function of the Euler term.
To compute the beta functions of all the independent couplings
is the main task of this paper.
The main motivation for this is the determination of the dimension of the UV critical surface.
There is evidence from the $f(R)$ truncations that
the scaling exponents at the nontrivial fixed point
are not too different from the classical ones,
so that couplings with positive mass dimension remain relevant
and couplings with negative mass dimension remain irrelevant
FP~\cite{CPR1,CPR2,FLNR2013,FKL2018,Falls:2018ylp}.
The marginal coupling of the $R^2$ term becomes relevant,
so altogether, in this truncation,
the dimension of the critical surface seems to be three.
An attempt to include different tensor structures has been made in~\cite{FKL2018}, where actions of the form
$f_1(R_{\mu\nu}R^{\mu\nu})+R f_2(R_{\mu\nu}R^{\mu\nu})$
are studied, leading to the same conclusion.
A limitation of these calculations is that,
on a spherical background, it is not possible to
properly disentangle independent couplings
with the same number of curvatures.
The case of Ricci tensor squared and scalar curvature squared actions
on an Einstein manifold, has already been cited above \cite{bms}.
While more general than spheres, Einstein manifolds are
still not general enough to distinguish all invariants.
With this limitation, it was found again that the dimension of the critical surface is three.
This suggests that some linear combination of the HD couplings
may be an irrelevant operator.
It seems possible, and even likely, that the dimension of the critical surface in pure gravity is determined entirely
by the fate of the HD couplings, since they are not expected to remain marginal at an interacting FP.\footnote{
So far the only indication that things could be more complicated comes from work in progress by Kluth and Litim
on actions of the form $f_1(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})+R f_2(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})$,
where a term cubic in curvature seems to become relevant~\cite{kluth}.}
We find that of the three dimensionless couplings,
one becomes relevant, one irrelevant and one --
the coefficient of the Euler term -- remains marginal.
The beta function of the Euler term is related to the $a$-function.
The $a$-theorem states that when two fixed points are joined
by an RG trajectory, the value of $a$ at the IR fixed point is lower
than the one at the UV fixed point.
We find some evidence that this may hold also in gravity.
In the present paper we try to shed some light on these issues by computing the beta functions of all the HD couplings
beyond the one-loop approximation, taking the anomalous
dimensions into account.
We shall do this by using the ``Universal RG Machine''~
to compute the r.h.s. of the FRGE on an arbitrary background.
This is a technique based on non-diagonal heat kernel coefficients that can be used to evaluate functional traces
involving covariant derivatives acting on a function of a Laplacian.
The Universal RG Machine has been introduced, and applied to the Einstein-Hilbert truncation, in \cite{BGMS2010}.
Later it was used to calculate the one-loop beta functions
in HDG \cite{SGRZ}.
Technical details are given in \cite{GSZ2011}.
Here we bring that program one step forward by
evaluating the full beta functions of HDG,
including the anomalous dimensions.
The main steps of the calculation are outlined in Sect.2,
and in Sect.3 we describe the results.
We find three fixed points, of which one has vanishing higher derivative couplings, while the others are fully interacting.
In principle, any of these could be a viable UV fixed point.
To have a viable theory, one would also have to prove unitarity.
For the first of these fixed points, one could apply
the arguments developed in perturbation theory
\cite{Mannheim:2006rd,Salvio:2014soa,Salvio:2017qkx,Salvio:2018crh,Einhorn:2014gfa,AKKLR,smilga,holdom,donhdg,Anselmi:2018tmf,Donoghue:2019fcb}.
For the remaining ones, the issue is more involved
and will require a detailed study of the two point function.
\section{Beta functions}
\subsection{Why Einstein backgrounds are not enough}
Let us momentarily concentrate on the HD terms, that we can write
as ${\cal L}_{HD}=\a R^2+\b R_{\mu\nu}^2 + \c R_{\mu\nu\rho\lambda}^2$.
Due to the fact that the Gauss--Bonnet combination
$E = R_{\mu\nu\a\b}^2- 4 R_{\mu\nu}^2+ R^2$
is topological, one of these couplings is uninteresting
as far as local dynamics is concerned.
It is therefore more meaningful to write the Lagrangian as
\begin{equation}
{\cal L}_{HD}=\frac{1}{2\lambda} C^2 + \frac{1}{\xi} R^2 - \frac{1}{\rho} E
\end{equation}
where
\begin{equation}
\frac{1}{\xi}= \frac{3\alpha+\beta+\gamma}{3}\ ,\quad
\frac{1}{2\lambda} = \frac{\beta+4\gamma}{2}\ ,\quad
-\frac{1}{\rho} = -\frac{\beta+2\gamma}{2}\ .
\end{equation}
and
$C^2 = R_{\mu\nu\a\b}^2-2R_{\mu\nu}^2+ \frac13 R^2$
is the square of the Weyl tensor.
We are mainly interested in the beta functions of $\lambda$ and $\xi$.
Calculations are simpler on an Einstein background.
In this case $E=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
and $C^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-R^2/6$, so
\begin{equation}
{\cal L}_{HD}=\left(\frac{1}{\xi}-\frac{1}{12\lambda}\right)R^2
+\left(\frac{1}{2\lambda}-\frac{1}{\rho}\right) E
\ .
\end{equation}
This implies that if we expand the r.h.s. of the functional RG equation
on an Einstein background, and we interpret the coefficients
of $R^2$ and $E=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
as beta functions, we can read off
the beta functions of two combinations of $\lambda$, $\xi$, $\rho$
but we are unable to unambiguously identify $\beta_\lambda$ and $\beta_\xi$.
To do this, we need an additional independent equation,
that in turn requires a more general background.
This is what we do in this paper.
All calculations will be based on the Euclidean action
\begin{eqnarray}
S&=&\int d^4 x \sqrt{g} \Big[{\cal V}-Z_N R+{\cal L}_{HD}
\Big] ,
\label{action}
\end{eqnarray}
where $Z_N=\frac{1}{16\pi G}$, $G$ being Newton's constant,
${\cal V}=2\Lambda Z_N$ and $\Lambda$ is the cosmological constant.
Sometimes we shall use the combinations
\begin{eqnarray}
\omega \equiv -\frac{3\lambda}{\xi},~~~
\theta \equiv \frac{\lambda}{\rho}\ .
\end{eqnarray}
\subsection{Remark on the topological term}
Before embarking in calculations, we can make a general remark
on the Gauss-Bonnet term, that actually holds
{\it independently of the truncation}.
Due to the topological character of the term $E$,
its coefficient $1/\rho$ does not appear in the Hessian
and therefore does not appear in the r.h.s. of the flow equation.
Thus the beta function of $\rho$ must have the form
\begin{equation}
\beta_\rho=-\frac{1}{16\pi^2}a\rho^2\ .
\label{betarho}
\end{equation}
where $a$ is a function of all the other couplings,
but not of $\rho$ itself.
In the search of a fixed point one can solve first the equations
of all the other couplings, which are also independent of $\rho$.
When these fixed point values are inserted in (\ref{betarho}),
$a$ becomes just a number.
The UV behavior of $\rho$ is determined by the value of this number.
If $a=0$, $\rho$ could reach any value in the UV.
If $a>0$ ($a<0$), when all other couplings are very close to
a fixed point, it will run logarithmically to zero
from above (below).
\subsection{Expansion and gauge fixing}
We split the metric $g_{\mu\nu} = \bar g_{\mu\nu} + h_{\mu\nu}$,
where $\bar g_{\mu\nu}$ is an arbitrary background.
For details of the expansion of the action, we refer to \cite{OP2013}.
The gauge-fixing and ghost action can be written
\begin{eqnarray}
{\cal L}_{GF+FP}/\sqrt{\bar g}
= -\frac{1}{2a} \chi_\mu Y^{\mu\nu} \chi_\nu + i Z_{gh} \bar c^\mu \Delta^{(gh)}_{\mu\nu} c^\nu
+ \frac{1}{2} Z_Y b_\mu Y^{\mu\nu} b_\nu + Z_Y \bar{\zeta}_\mu Y^{\mu\nu} \zeta_\nu
\label{gfgh1}
\end{eqnarray}
where $\bar c_\mu$, $c_\mu$ are complex ghosts, $b_\mu$ is a real commuting field, $\bar{\zeta}_\mu$, $\zeta_\mu$ are complex anti-commuting fields and
\begin{eqnarray}
\chi_\mu &\equiv& \bar\nabla^\lambda h_{\lambda\mu} + b \bar\nabla_\mu h\ , \nonumber\\
\Delta^{(gh)}_{\mu\nu} &\equiv & g_{\mu\nu} \bar\nabla^2 +(2b+1)\bar\nabla_\mu \bar\nabla_\nu +\bar R_{\mu\nu}\ , \nonumber\\
Y_{\mu\nu} &\equiv& \bar g_{\mu\nu} \bar\nabla^2+ c \bar\nabla_\mu \bar\nabla_\nu - f \bar\nabla_\nu \bar\nabla_\mu\ .
\end{eqnarray}
where $a$, $b$, $c$ and $f$ are gauge parameters.
There is some freedom in how we choose the wave function renormalisations $Z_{gh}$ and $Z_{Y}$
since they can be rescaled while keeping $Z_{gh}^2 Z_Y=1/a$
fixed without affecting the path integral.
In our calculations we fix
\begin{equation}
Z_{gh}=1\ ,\quad Z_Y=1/a
\label{zgh}
\end{equation}
We make the usual gauge choice
\begin{eqnarray}
a=
\lambda\ ,~~~
b=
-\frac{1+4\omega}{4+4\omega}\ ,~~~
c = \frac23(1+\omega)\ ,~~~
f=1\ ,
\label{gpar}
\end{eqnarray}
leading to a minimal fourth order operator for the fluctuations.
The operators in \p{gfgh1} are then
\begin{eqnarray}
\Delta^{(gh)}_{\mu\nu} &\equiv &
g_{\mu\nu} \bar\nabla^2 -\sigma_{gh}\bar\nabla_\mu \bar\nabla_\nu +\bar R_{\mu\nu}, \nonumber\\
Y_{\mu\nu} &\equiv& \bar g_{\mu\nu} \bar\nabla^2 -\sigma_Y\bar\nabla_\mu \bar\nabla_\nu - R_{\mu\nu},
\end{eqnarray}
with
\begin{eqnarray}
\sigma_{gh}=-1-2b
=-\frac{1-2\omega}{2(1+\omega)}
\ ;\qquad
\sigma_Y=1-2\frac{\gamma-\alpha}{\beta+4\gamma}=\frac{1-2\omega}{3}\ .
\label{sigmas}
\end{eqnarray}
We note that the cancellation between unphysical degrees of freedom
becomes exact in the ``Landau gauge'' limit $a\to 0$,
which happens to be satisfied in the
asymptotically free regime.
Then, the quadratic terms in the action can be written in the form~\cite{OP2013}
\begin{eqnarray}
{\cal L}^{(2)}= h_{\mu\nu}K^{\mu\nu\rho\sigma}
{\cal O}_{\rho\sigma}{}^{\alpha\beta} h_{\a\b},
\end{eqnarray}
where the operator ${\cal O}$ is
\begin{eqnarray}
{\cal O}=\Delta^2 + V_{\rho\lambda} \bar\nabla^\rho \bar\nabla^\lambda +U\ .
\label{hami}
\end{eqnarray}
with $\Delta=-\bar\nabla^2$,
$U=K^{-1}W$ and we write
\begin{eqnarray}
K = \frac{\b+4\c}{4} \Big(\mathbb{I}
+\frac{4\a+\b}{\c-\a}\mathbb{P} \Big)\ ,
\qquad
K^{-1}= \frac{4}{\b+4\c}
\Big(\mathbb{I}
-\frac{4\a+\b}{3\a+\b+\c}\mathbb{P} \Big),
\label{kkinv}
\end{eqnarray}
where
$\mathbb{I}$ is the identity in the space of symmetric tensor
and $\mathbb{P}$ is a projector
\begin{eqnarray}
\mathbb{I}_{\mu\nu,\a\b}\equiv \d_{\mu\nu,\a\b}=\frac{1}{2} (\bar g_{\mu\a} \bar g_{\nu\b}+\bar g_{\mu\b} \bar g_{\nu\a})\ ,\quad
\mathbb{P}^{\mu\nu}{}_{\rho\sigma} \equiv P^{\mu\nu}{}_{\rho\sigma}=\frac{1}{4}\bar g^{\mu\nu}
\bar g_{\rho\sigma}\ .
\end{eqnarray}
The coefficients $V_{\rho\lambda}$ and $U$ are functions
of the curvatures, ${\cal V}$ and $Z_N$, for whose form
we refer again to \cite{OP2013}.
The ``beta functional'' of the theory is the sum of three
contributions coming from gravitons, ghosts and the new ghost $b_\mu$:
\begin{eqnarray}
\dot\Gamma_k &=&T_g+T_{gh}+T_Y\ .
\end{eqnarray}
In order to write these terms more explicitly, we have to choose
a cutoff for each of them.
For a one-loop calculation, where the couplings in the r.h.s. of the equation are treated as fixed,
it was most convenient to think of the cutoff as a function of the whole
operator ${\cal O}$, $\Delta_{gh}$ or $Y$ respectively
(so-called type III cutoff).
In this paper we will not ignore the running of the couplings that may be present in the cutoff,
so it is best to minimize their presence.
This is achieved by choosing the cutoff to be a function of $\Delta$ only (so-called type I cutoff).
The one-loop calculation with this cutoff has been
done before in \cite{SGRZ}.
\subsection{Graviton contribution}
We choose the graviton cutoff to have the form
${\cal R}=K R_k(\Delta^2)$,
where $R_k(\Delta^2)=(k^4-\Delta^2)\theta(k^4-\Delta^2)$
and we define as usual
$P_k(\Delta^2)=\Delta^2+R_k(\Delta^2)=k^4\t(k^4-\Delta^2)$.
Note that it is convenient to view $R_k$ as a function of $\Delta^2$,
although of course one could also view it as a function of $\Delta$.
Then, writing the kinetic operator as $\Delta^2+V+U$,
the graviton contribution to the FRGE is
\begin{equation}
T_g=\frac{1}{2}\mbox{Tr}\frac{\partial_t[K R_k(\Delta^2)]}{K [{\cal O}+R_k(\Delta^2)]}
=\frac{1}{2}\mbox{Tr}\frac{\partial_t R_k(\Delta^2)+\eta_K R_k(\Delta^2)}{P_k(\Delta^2)+V+U}\ ,
\label{frgeg}
\end{equation}
where we defined
\begin{equation}
\eta_K = K^{-1}\frac{dK}{dt} .
\end{equation}
Note that $\eta_K$ is a tensor.
From (\ref{kkinv}) we find
\begin{eqnarray}
\eta_K=\eta_1\mathbb{I}+\eta_P\mathbb{P},
\label{etaK}
\end{eqnarray}
where
\begin{eqnarray}
\eta_1=-\frac{\dot\lambda}{\lambda}\ ,\qquad
\eta_P=
-\frac{\xi\dot\lambda-\lambda\dot\xi}{\lambda(3\lambda-\xi)}\ ,
\label{etap}
\end{eqnarray}
We divide $V$ and $U$ into various terms:
$V=V_0+V_1$ and $U=U_0+U_1+U_2$,
where the subscript counts the power of curvature,
and the remaining dimension is carried either by ${\cal V}$ or $Z_N$:
$$
V_0\sim Z_N\nabla\nabla\ ;\qquad
V_1\sim R\nabla\nabla\ ;\qquad
U_0\sim {\cal V}\ ;\qquad
U_1\sim Z_N R\ ;\qquad
U_2\sim R^2\ .
$$
We now have to decide how to expand the fraction in (\ref{frgeg}).
Since we want to compute the beta functions of all the couplings in
(\ref{action}), we need to expand to second order in curvatures.
It would be natural to assume that $\sqrt{{\cal V}}\sim Z_N\sim R$
(which implies also $\Lambda\sim R$),
but such an expansion would miss important features,
as we shall discuss below.
It is possible without too much effort to keep the {\it full} dependence on ${\cal V}$, and we shall do so.
We will therefore not expand in $U_0$.
It is much harder to keep all dependence on $Z_N$,
therefore we will expand in $V_0$, $V_1$, $U_1$ and $U_2$,
to first order in $Z_N/k^2$, independently of curvatures.\footnote{Note that we wrote ${\cal V}=2Z_N\Lambda$ and treated
$\Lambda$ as an independent coupling, the expansion in $Z_N$
would also entail and expansion in $\Lambda$.
This is not what we do here.}
This corresponds to considering a trans-Planckian regime.
If one considers the Einstein-Hilbert part of the action,
it correspond to a {\it strong gravity} expansion.
See \cite{Niedermaier:2019iqk} for a recent discussion.
Keeping only terms up to linear order in $Z_N$ we thus have
to evaluate:
\begin{eqnarray}
T^{\rm grav}&=&\frac12 \mbox{Tr}\left[
\frac{\partial_t R_k(\Delta)+\eta_K R_k(\Delta)}{P_k(\Delta)+U_0}
\left(1-\frac{1}{P_k(\Delta)+U_0}(V_0+V_1+U_1+U_2)
\right.\right. \nonumber\\
&& \hs{-10} \left.\left.
+\frac{1}{P_k(\Delta)+U_0}V_0 \,
\frac{1}{P_k(\Delta)+U_0}V_1
+\frac{1}{P_k(\Delta)+U_0}V_1 \,
\frac{1}{P_k(\Delta)+U_0}V_0
\right.\right. \nonumber\\
&& \hs{-10} \left.\left.
+\frac{2V_0 U_2}{(P_k(\Delta)+U_0)^2}
+\frac{V_1^2}{(P_k(\Delta)+U_0)^2}
+\frac{2V_1 U_1}{(P_k(\Delta)+U_0)^2}
+\frac{3V_0 V_1^2}{(P_k(\Delta)+U_0)^3}
\right)\right]
\ .
\label{tgrav}
\end{eqnarray}
In the last line we have written the terms only in a schematic way,
without paying attention to their order:
to be precise one has to write out several terms where the
projectors $\mathbb{P}$ appear in different positions.
(For details, we refer the reader to the ancillary file on the arXiv page.)
\subsection{Ghost contribution}
To some extent, it is possible to treat $\Delta_{gh}$ and $Y$
together.
Both operators are non-minimal, and of the form
$\Delta \delta_\mu^\nu+\sigma \bar\nabla_\mu\bar\nabla^\nu+B_\mu^\nu$
(note the overall sign is reversed),
where $\sigma$ is a constant defined in \p{sigmas} and $B_\mu^\nu=s \bar R_\mu^\nu$,
where $s=-1$ for $\Delta_{gh}$ and $s=1$ for $Y$.
In the standard one-loop calculations, one can use the known
heat kernel coefficients for this type of operators.
In contrast to \cite{Codello:2006in,niedermaier,OP2013} and
coherently with the treatment of gravitons,
we use a type I cutoff also for the ghosts.
This type of cutoff for ghosts had been used before in \cite{SGRZ}.
The novelty of our calculation is that we also
take into account the contributions due to the anomalous dimensions
\begin{equation}
\eta_{gh}=0\ ,\qquad
\eta_Y=-\beta_\lambda/\lambda\ .
\label{etaY}
\end{equation}
The type I cutoff has the form\footnote{We observe that
the calculation of the ghost contributions is considerably simpler
with a so-called type-II cutoff
${\cal R}^\mu_{k\,\nu}=Z\delta^\mu_\nu R_k(\Delta+B)$.
The use of this alternative scheme for the ghosts
would lead to only small quantitative differences
in the final results for the fixed points and
we shall not discuss this in detail.}
\begin{equation}
{\cal R}^\mu_{k\,\nu}=Z\delta_\mu^\nu R_k(\Delta),
\end{equation}
where $Z$ is given by (\ref{zgh},\ref{gpar}).
Adding the cutoff, the kinetic operator (aside from the factor $Z$)
becomes
$P_k(\Delta) \delta_\mu^\nu+\sigma \bar\nabla_\mu\bar\nabla^\nu+B_\mu^\nu$.
In the flow equation one needs the inverse of this operator.
We refer to \cite{SGRZ} for some technical details.
The evaluation of the traces to second order in curvatures
is rather laborious.
In the end we arrive at the following
\begin{eqnarray}
&& \hs{-10}T_{gh}= -\frac{1}{(4\pi)^2} \int d^4 x \sqrt{\bar g}
\Bigg\{
\left[ 3-\frac{2}{\sigma_{gh}}-\frac{2}{\sigma_{gh}^2}\log(1-\sigma_{gh})
\right]k^4
\nonumber\\
&& -\frac{1}{12\sigma_{gh}^2}\left[\frac{3\sigma_{gh}(2+\sigma_{gh}(7-5\sigma_{gh}))}{\sigma_{gh}-1}
-2(3-2\sigma_{gh}) \log(1-\sigma_{gh})\right]k^2 \bar R
\nonumber\\
&& -\frac{11}{90}\bar R_{\mu\nu\rho\lambda}^2
+\frac{43-2\sigma_{gh}(13+\sigma_{gh})}{45(1-\sigma_{gh})^2} \bar R_{\mu\nu}^2
+ \left[\frac{5}{18}+\frac{1}{6(1-\sigma_{gh})^2}\right] \bar R^2
\Bigg\} .
\label{tgh}
\end{eqnarray}
Note the appearance of $\log(1-\sigma_{gh})=-\log(2(1+\omega)/3)$,
which forces us to consider only the domain $\omega>-1$.
For $Y$:
\begin{eqnarray}
&& \hs{-10}T_Y=-\frac{1}{2} \frac{1}{(4\pi)^2} \int d^4 x \sqrt{\bar g}
\Bigg\{
\Bigg[ 3-\frac{2}{\sigma_Y}-\frac{2}{\sigma_Y^2}\log(1-\sigma_Y)
\nonumber\\
&&
\qquad\qquad\qquad\qquad\qquad\qquad
+\eta_Y\left( \frac{2-\sigma_Y+\sigma_Y^2}{2\sigma_Y^2}
+\frac{(1-\sigma_Y)}{\sigma_Y^3}\log(1-\sigma_Y)\right)\Bigg]k^4
\nonumber\\
&&\!\!\!\!\!\!\!
+\left[-\frac{2+\sigma_Y}{4\sigma_Y}
-\frac{3+2\sigma_Y}{6\sigma_Y^2}\log(1-\sigma_Y)
+\eta_Y\Big(\frac{6-\sigma_Y}{12\sigma_Y^2}
+\frac{3-2\sigma_Y-\sigma_Y^2}{6\sigma_Y^3} \log(1-\sigma_Y) \Big)\right]k^2 \bar R
\nonumber\\
&&
\!\!\!\!\!\!\!
-\frac{11}{90}\Big(1+\frac{\eta_Y}{2} \Big) \bar R_{\mu\nu\rho\lambda}^2
\!+\!\left[\frac{43}{45}
+\eta_Y \Big(\frac{20\!-\!20\sigma_Y\!-\!39\sigma_Y^2\!+\!29\sigma_Y^3}{120\sigma_Y^2(\sigma_Y-1)}
-\frac{1\!-\!\sigma_Y\!-\!2\sigma_Y^2}{12\sigma_Y^3}\log(1\!-\!\sigma_Y)
\Big)\right] \bar R_{\mu\nu}^2
\nonumber\\
&& \!\!\!\!\!\!\!
- \left[ \frac{2}{9}
+\eta_Y\Big(\frac{4+\sigma_Y^2+\sigma_Y^3-3\sigma_Y^4}{48(-1+\sigma_Y)\sigma_Y^2}
-\frac{2-\sigma_Y-2\sigma_Y^2}{24\sigma_Y^3}\log(1-\sigma)\Big)
\right] \bar R^2
\Bigg\} ,
\label{tY}
\end{eqnarray}
Both agree with \cite{SGRZ} if we put $\eta_Y=0$.
\section{Results}
\subsection{Beta functions}
For the study of the flow, the dimensionful couplings
${\cal V}$ and $Z_N$ have to be replaced by their
dimensionless counterparts $\tilde{\cal V}={\cal V}/k^4$ and
$\tilde Z_N=Z_N/k^2$, or the related quantities
$\tilde G=Gk^2$, $\tilde\Lambda=\Lambda/k^2$.
The beta functions are too complicated to be written here
(they are given in a Mathematica notebook \cite{kevin_falls_2020_4017671}),
but they simplify in two cases.
Expanding for small $\lambda$ we obtain the universal one-loop beta functions
\begin{eqnarray}
\beta_\lambda & =& -\frac{133 \lambda ^2}{160 \pi ^2}+O\left(\lambda ^3\right)
\\
\beta_\omega &=& - \frac{\lambda \left(200 \omega ^2+1098 \omega +25\right)}{960 \pi ^2} +O\left(\lambda^2\right)
\\
\beta_\theta &=& \frac{7(56-171\theta)}{1440\pi ^2}\lambda+O\left(\lambda^2\right)
\end{eqnarray}
while the non-universal beta functions for $\tilde G$ and $\tilde\Lambda$ agree with those found in the one-loop calculation \cite{SGRZ} at $\lambda =0$.
Explicitly they are given by
\begin{eqnarray}
\beta_{\tilde G} &=& 2 \tilde G+\tilde G^2
\left[-\frac{c_1}{72 \pi (1-2 \omega )}+\frac{c_2
\log \left(\frac{2 (1+\omega )}{3}\right)}{12 \pi (1-2 \omega )^2}\right] + O\left(\lambda \right)
\label{oneloopg}
\\
\beta_{\tilde \Lambda} &=& -2 \tilde\Lambda
+\frac{\tilde G}{72 \pi} \left[\frac{c_3+\tilde\Lambda c_4 }{1-2 \omega
}+\frac{6 \left(c_5+\tilde\Lambda c_6 \right) \log
\left(\frac{2 (1+\omega )}{3}\right)}{(1-2 \omega )^2}\right] + O\left(\lambda \right)
\end{eqnarray}
with the coefficients
$c_1= 35-2 \omega (109+176 \omega )$,
$c_2 = 65+4 \omega (7+2 \omega )$,
$c_3 = 162-540\omega$,
$c_4= -35+218\omega+352\omega^2$,
$c_5 = 6-96\omega-48\omega^2$,
$c_6= 65+28\omega+8\omega^2$.
Our calculation differs from one-loop calculations
in that we take into account the anomalous dimensions.
For example, we see $\eta_Y$ appearing explicitly in (\ref{tY}),
which gives contributions to the beta functions of all the couplings.
Equation (\ref{etaY}) tells us that $\eta_Y$ is proportional to
$\beta_\lambda$.
Thus, comparing the terms proportional to $C^2$ on both sides
of the FRGE, we obtain a relation of the form
$\beta_i=B_i+C_{ij}\beta_j$.
At one loop one just keeps $\beta_i=B_i$.
Solving the algebraic equations gives beta functions
that contain contributions with arbitrarily high loop order.
However, from the definitions,
the anomalous dimensions
at a fixed point are known a priori to be
\begin{equation}
\eta_1=0\ ;\qquad
\eta_P=0\ ;\qquad
\eta_Y=0\ .
\label{andimfp}
\end{equation}
So, in the search of fixed points,
one can use simplified beta functions where
these values are used:
the full expressions for the anomalous dimensions
are only needed when one calculates the scaling exponents.
It is easy to see that if we had assumed that
all terms in $V$ and $U$ are of the same order,
namely $\sqrt{\cal V}\sim Z_N\sim R$,
then all the terms containing $V_0$ and $U_1$
would not contribute to the beta functions of
$\lambda$, $\xi$ and $\rho$.
Therefore, these beta functions would not contain $Z_N$
and would be exactly the same
as in
\cite{SGRZ}.
This is why it is important to keep the expansion in $Z_N$
separate from the expansion in $R$.\footnote{It would obviously be even better not to expand
in $Z_N$ at all, but this would be technically much more challenging.}
Even the simplified beta functions with (\ref{andimfp})
are too complicated to be reported in detail.
However, we shall see a posteriori that $\tilde{\cal V}$
is very small at fixed points.
If we put $\tilde{\cal V}=0$, the equations for the remaining variables
become simple enough:
\begin{eqnarray}
\beta_\lambda&=&-\frac{133}{160\pi^2}\lambda^2+\tilde Z_N\lambda^3\frac{251\xi-58\lambda}{120\pi^2\xi}
\label{blam}
\\
\beta_\xi&=&-\frac{5(72\lambda^2-36\lambda\xi+\xi^2)}{576\pi^2}+\tilde Z_N\frac{9720\lambda^3-1980\lambda^2\xi
+489\lambda\xi^2-14\xi^3}{6480\pi^2}
\\
\beta_\rho&=&-\frac{49}{180\pi^2}\rho^2+\tilde Z_N\lambda\rho^2\frac{233\xi-58\lambda}{240\pi^2\xi}
\label{brho}
\\
\beta_{\tilde Z_N}&=&\left(-2+\frac{(30\lambda-\xi)(4\lambda+\xi)}{192\pi^2\xi}\right)\tilde Z_N
+\frac{-3168\lambda^2+654\lambda\xi+35\xi^2}{1152\pi^2\xi(6\lambda+\xi)}\nonumber\\
&&\qquad\qquad\qquad-\frac{72\lambda^2-84\lambda\xi+65\xi^2}{192\pi^2(6\lambda+\xi)^2}
\log\left(\frac23-\frac{2\lambda}{\xi}\right)\ .
\label{bZ}
\end{eqnarray}
\subsection{Fixed points}
Now we recall that already in the one-loop calculation,
the beta functions of $\tilde Z_N$ (and also $\tilde{\cal V}$) have a nontrivial fixed point.
This nonzero value of $Z_N$ enters in the beta functions
of (\ref{blam}-\ref{brho}) in such a way that besides
the asymptotically free fixed point, there are now two
(and only two) new ones. Their coordinates are
given in Table 1.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|}
\hline
& $\lambda_*$ & $\xi_*$ & $\rho_*$ & $\omega_*$ & $\tilde Z_{N*}$ & $\tilde G_*$ \\
\hline
FP$_1$ & 0 & 0 & 0 & $-0.02286$ & 0.00833 & 2.388 \\
\hline
FP$_2$ & 29.26 & $-220.2$ & 0 & 0.4040 & 0.01318 & 1.509 \\
\hline
FP$_3$ & 52.61 & 1672 & 0 & $-0.0944$ & 0.00761 & 2.614 \\
\hline
\end{tabular}
\end{center}
\caption{Fixed points in the approximation $\tilde{\cal V}=0$.}
\label{t2}
\end{table}
The first fixed point is found also in the one-loop approximation,
and it is a non-trivial fact that it persists also when $\tilde Z_N$
is present in the beta functions of $\lambda$ and $\xi$.\footnote{Actually, this fixed point is best studied
using the variable $\omega$ instead of $\xi$. It corresponds to letting $\lambda$
and $\xi$ go to zero with a particular ratio,
and is different from setting e.g. first $\lambda=0$
and then $\xi=0$.}
Note that in the one-loop approximation there is also
another fixed point with $\lambda=\xi=0$, $\omega=-5.467$,
which however is excluded by our condition $\omega>-1$
(otherwise it gives a complex $\tilde Z_N$).
The remaining two fixed points are ``fully interacting''.
It is worth noting that if we treat $\tilde Z_{N}$ as an external parameter in the beta functions
of $\lambda$ and $\xi$, we find that $\lambda_*$ and $\xi_*$
go to infinity for $\tilde Z_{N}\to 0$.
\footnote{and to zero for $\tilde Z_{N}\to\infty$, but this
is outside the domain of our approximation.}
We then come to the solution of the full flow equations,
where we take into account also the running of $\tilde{\cal V}$.
There are now more fixed points, and we report in Table 2 the properties
of the most interesting ones.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|}
\hline
& $\lambda_*$ & $\xi_*$ & $\rho_*$ & $\omega_*$ & $\tilde Z_{N*}$ & $\tilde{\cal V}_*$ & $\tilde G_*$ & $\tilde\Lambda_*$ & $a$ \\
\hline
FP$_1$ & 0 & 0 & 0 & $-0.02286$ & 0.00833 & 0.006487 & 2.388 & 0.3894 & 4.356 \\
\hline
FP$_2$ & 24.91 & $-287.1$ & 0 &0.2603 & 0.01635 & 0.004575 & 1.217 & 0.1399 & $-2.741$ \\
\hline
FP$_3$ & 28.24 & 175.6 & 0 & $-0.4825$ & 0.01499 & 0.006928 & 1.327 & 0.2310 & $-3.566$ \\
\hline
FP$_4$ & 0 & $-312.2$ & 0 & 0 & 0.009222 & 0.006092 & 2.157 & 0.3303 & $4.357$ \\
\hline
\end{tabular}
\end{center}
\caption{Selected fixed points including $\tilde{\cal V}$.}
\label{t2}
\end{table}
We see that in all cases the fixed point value of $\tilde{\cal V}$ is very small,
justifying the earlier approximation ${\cal V}=0$.
In fact, by considering only the beta functions of $\lambda$, $\xi$
and $\tilde Z_N$, and treating $\tilde{\cal V}$ as a parameter,
and letting this parameter vary between zero and $0.004575$,
or $0.006928$, we can see that FP$_2$ and FP$_3$
change continuously from the values of Table 1 to those of Table 2.
We may thus identify the first three fixed points of Table 2
with those of Table 1.
There are several other fixed points with $\lambda=0$,
of which FP$_4$ is a representative example.
We list it here for reasons that will become clear later.
There may also exist other non-trivial fixed points with $\lambda\not=0$,
but this would require a more extensive numerical search that
we have not undertaken.
Besides, these fixed points are probably artifacts of the
truncation, as are known to occur in other similar cases.
We note that also $\tilde Z_{N*}$ is small,
and this justifies {\it a posteriori} the expansion in $\tilde Z_N$
that we used throughout our calculations.
If we change variable from $\tilde Z_N$ to $\tilde G_N$
and set $\lambda=0$, then as seen from (\ref{oneloopg})
there is a fixed point at $\tilde G=0$.
On the other hand, if we first set $\tilde G=0$,
there is no acceptable fixed point for the dimensionless couplings.
In any case, since we have expanded in $\tilde Z_N$,
any result near $\tilde G=0$ is unreliable.
This is unfortunate, because it means that we cannot check whether there exist a RG trajectory joining one of the fixed points listed above
to the standard weak gravity regime in the IR.
\subsection{Scaling exponents}
If we rescale the fluctution field $h_{\mu\nu}$ by a factor
$\sqrt\lambda$, so that the prefactor of its kinetic term is canonical,
the fixed point FP$_1$ is seen to be a Gaussian fixed point,
and indeed we find that the scaling exponents are given by
the canonical dimensions: $4$, $2$, $0$, $0$, $0$.
The scaling exponents of FP$_2$, listed from more to less relevant, are
$$
\theta_{1,2}=2.35191 \pm 1.67715 i\ ,\quad
\theta_3=1.76672\ ,\quad
\theta_4=0\ ,\quad
\theta_5=-3.20030\,,
$$
while those of FP$_3$ are
$$
\theta_{1,2}=2.03270 \pm 1.52155 i\ ,\quad
\theta_3=1.23742\ ,\quad
\theta_4=0\ ,\quad
\theta_5=-5.27685\,.
$$
The marginal coupling is $\rho$, the (inverse of the) coefficient
of the topological term.
At the non-Gaussian fixed points, we find
$\beta_\rho=A\rho^2$ with $A=0.01736$ at FP$_2$
and $A=0.02258$ at FP$_3$.
Thus, at both fixed points, $\rho$ is marginally relevant
when it is negative and marginally irrelevant
when it is positive.
We thus arrive at the conclusion that also in the present
approximation, the dimension of the critical surface
of pure gravity is three, up to the marginal topological term.
\subsection{The $a$-function}
The beta function of $\rho$ is given by (\ref{betarho}).
In an ordinary CFT, the coefficient $a$
appears in the trace anomaly as
\begin{equation}
\langle T^\mu{}_\mu\rangle=
\frac{1}{16\pi^2}(c C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}-a E)\ .
\end{equation}
For example, for a free theory with $N_S$ scalars,
$N_f$ Dirac fields and $N_V$ gauge fields,
\begin{equation}
a=\frac{1}{360}(N_S+11 N_f+62 N_V)\ ,\quad
c=\frac{1}{120}(N_S+6 N_f+12 N_V)\ .
\label{matter}
\end{equation}
According to the $a$-theorem,
if there is a RG trajectory joining two fixed points,
$a$ is higher at the UV fixed point \cite{Cardy:1988cwa,Komargodski:2011vj,Shore:2016xor}.
This accords to the intuition that $a$ is a measure of the number
of degrees of freedom of the theory.
There is no known $a$-theorem for gravity.
However, we can view our calculation as a quantum field theory
in a curved background, and from this point of view the theorem should
be applicable.\footnote{Similar calculations involving gravity have been
reported in \cite{Antoniadis:1992xu,Antoniadis:1996pb}.}
At FP$_1$ we have $a=\frac{196}{45}$. The values of $a$
at the other fixed points can be calculated numerically and
are reported in the last column of Table 2.
Since FP$_2$ and FP$_3$ have a unique irrelevant direction,
there is only one RG trajectory leaving these fixed points,
that can be integrated numerically
in the direction of increasing $t=\log k$ and ends up (in the UV)
at another fixed point.
In this way we have found an RG trajectory that goes from
FP$_1$ to FP$_3$ and one that goes from FP$_4$ to FP$_2$.
The value of $a$ decreases along these trajectories,
in accordance with the theorem.
On the other hand, all the fixed points with $\lambda=0$
have very similar values of $a$
and there is a trajectory that goes from FP$_4$ to another
fixed point with $\lambda=0$ and a slightly larger value of $a$,
in contradiction to the theorem.
Since it is doubtful that these additional fixed points do exist,
the meaning of this result is not very clear,
and will have to be investigated more carefully in the future.
\subsection{Spectrum}
The appearance of several non-trivial fixed points is
not a novelty in this kind of calculations.
Several of these are likely to be spurious,
but we do not see any reasons
why FP$_1$ or FP$_2$ should be rejected {\it a priori},
or to prefer one over the other.
Regarding the spectrum, we recall that in order to avoid tachyons
in the expansion around flat space,
the action for gravity in Lorentzian signature\footnote{we use the Lorentzian signature $-+++$.}
must have
a negative Weyl squared term and a positive $R^2$ term.
A naive Wick rotation of the linearized action around flat space
leads to a Lorentzian action that only differs from the
Euclidean one by an overall sign.
Therefore, FP$_2$ has the correct signs to avoid tachyons.
Although this is not sufficient to guarantee a healthy theory,
it gives us some more room in the search of one.
\\
Note added: After this paper was submitted to the journal the work referred to in footnote 4 has appeared on the arXiv \cite{Kluth:2020bdv}.
\section*{Acknowledgment}
We would like to thank Dario Benedetti, Taichiro Kugo, Frank Saueressig and Omar Zanusso for valuable discussions.
This work was supported in part by the Grant-in-Aid for Scientific Research Fund of the JSPS (C) No. 16K05331.
| {
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} | 4,622 |
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Роммель () — военная драма 2012 года режиссёра Ники Штайна. В главной роли сыграл Ульрих Тукур. В фильме о последних годах жизни фельдмаршала Роммеля начиная с принятия им руководства над Атлантическим валом и до его гибели. Фильм был показан по каналу Das Erste.
Описание сюжета
К фельдмаршалу Роммелю прибывают генералы Вильгельм Бургдорф и Эрнст Майзель. Они обвиняют Роммеля в участии в заговоре против Гитлера. Воспоминания уносят Роммеля в прошлое...
Фельдмаршал Роммель принимает командование над Атлантическим валом. Вопреки мнению генералов он считает, что в случае высадки союзников их надо бить прямо на берегу и скинуть в море. Для этого все силы нужно держать вдоль линии берега. Фельдмаршал Рундштедт напротив считает, что силы нужно стянуть в кулак в глубь территории Франции, дать союзникам высадиться а после разбить их. Роммель возражает, что в этом случае немцам при движении к побережью, придётся пробиваться через засады партизан, плацдармы десантников, подвергаясь ударам ВВС Союзников. Однако фюрер оставляет танковые силы под командованием Рундштедта, взбешённый Роммель покидает заседание.
Время подтверждает правоту Роммеля. Союзники высаживаются и начинают с боями продвигаться к Германии. Роммель начинает понимать, что сопротивление бессмысленно. С ним связываются заговорщики, рассчитывая, что небывалый авторитет Роммеля в войсках позволит им удержать их в подчинении. Однако покушение на фюрера проваливается. Несмотря на то, что никто из заговорщиков не выдает Роммеля, к нему прибывают два генерала и предлагают на выбор пойти под суд или совершить самоубийство. Роммель навсегда прощается с женой и с сыном и спокойно уезжает навстречу смерти. Фильм завершается трансляцией эпизодов помпезных похорон Роммеля.
В ролях
Ульрих Тукур — фельдмаршал Роммель
Тим Бергманн — оберст-лейтенант Цезарь фон Хофакер
Рольф Канис — оберст Эберхард Финк
Патрик Мёллекен — Манфред Роммель
Ханс Цишлер — фельдмаршал Герд фон Рундштедт
Клаус Г. Берендт — генерал-полковник Гейнц Гудериан
Беньямин Задлер — генерал-лейтенант Ханс Шпайдель
Аглая Шишковиц — Люси-Мария Роммель
Роберт Шупп — гауптман Альдингер
Петер Вольф — обергруппенфюрер Эрнст Кальтенбруннер
Хубертус Хартманн — генерал от инфантерии Карл фон Штюльпнагель
Вики Крипс — графиня Ла Рошфуко
Михаэль Кранц — фельдфебель Карл Даниэль (водитель Роммеля)
Йоханнес Зильбершнайдер — Адольф Гитлер
Оливер Нигеле — генерал от инфантерии Гюнтер Блюментритт
Хари Принц — генерал танковых войск Лео Гейр фон Швеппенбург
Ханс Кремер — генерал от инфантерии Маркс
Питер Кремер — генерал от инфантерии Вильгельм Бургдорф
Томас Тиме — фельдмаршал Ханс Гюнтер фон Клюге
Максимилиан фон Пуфендорф — генерал-майор фон Темпельхофф
Джо Бауш — ''фельдмаршал Вильгельм Кейтель
Примечания
Ссылки
Фильмы Германии 2012 года
Фильмы на немецком языке | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,496 |
\section{Introduction}
An elegant mechanism of the generation of (almost) flat spectrum
of scalar perturbations is provided by inflation~\cite{inflation-HZ}.
Yet it is legitimate to ask whether the spectrum of the
Harrison--Zeldovich type can emerge in a different way.
This question is of interest, in particular,
from the viewpoint of alternatives to
inflation --- pre-Big Bang scenario~\cite{pre-BB-i}
(for a review see Ref.~\cite{pre-BB}),
ekpyrotic/cycling
models~\cite{ekpyro-i,cyclic} (for a review see Ref.~\cite{ekpyro}),
starting-the-Universe picture~\cite{starting},
etc. One possibility
is to make use of scalar fields with negative
exponential potentials~\cite{finelli,minus-exp,minus-bis,minus-bis2}
(see also Ref.~\cite{minus-old}).
In this way one indeed obtains almost flat spectrum
of scalar perturbations
in ekpyrotic
models, with overall homogeneity and isotropy of the Universe
preserved
during the contracting phase because of the stiff effective equation
of state~\cite{smooth}. In concrete models of this type
the contracting solutions are
unstable~\cite{koyama,tolley},
so that one has to
fine tune initial data in
our part of
the Universe.
It has been argued,
however, that such
a fine tuning may be automatic, in a sense,
in a cyclic scenario
with long periods of dark energy domination~\cite{antro-fine-tune}.
In this paper we suggest another mechanism for generating
the Harrison--Zeldovich spectrum. The idea is to relate scale invariance
of this spectrum to conformal invariance of
field theory.
We will see in Section~\ref{sec-2}
that this can be done by introducing conformal scalar field
as a source of entropy perturbations, which are converted into
adiabatic perturbations at later epoch.
The three basic ingredients are that (i) the theory possesses
some global symmetry, (ii) the quartic scalar potential is negative
and (iii) there is long enough stage of the cosmological evolution
prior to the conventional
hot Big Bang epoch. At that early stage,
the classical scalar field rolls down its
potential along the late time attractor. In this regime, the
vacuum perturbations of field(s)
orthogonal to the radial direction ---
the direction along which the classical field rolls down ---
get amplified and freeze out having the
spectrum which is
automatically flat.
Hence, getting flat spectrum of the field perturbations
is
reasonably simple.
Implementing this mechanism to obtain adiabatic matter perturbations of
correct amplitude is somewhat less
straightforward. We propose the
corresponding scenario in
Section~\ref{sec-3}. At the end of the rolling stage, both conformal invariance
and global symmetry are assumed to be broken, and the scalar potential
is assumed to have a minimum or a set of minima, see Fig.~\ref{fig1}.
If the scale of global symmetry
breaking is small enough, the situation becomes analogous to the axion
misalignment picture (for a review see Ref.~\cite{axion-misalign}),
the analog of axion being
the orthogonal, pseudo-Goldstone field(s) corrseponding to broken
global symmetry. Unlike in the case of axion, the interactions of the
pseudo-Goldsone field with matter
are not negligible, so this field plays the same role as
curvaton in inflationary models.
It gets frozen at a slope of the potential, later on
it starts to oscillate and eventually these oscillations convert into
conventional particles. In this way the adiabatic perturbations
with flat spectrum are produced.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.5\textwidth,angle=-90]{phi4.eps}
\end{center}
\caption{The scalar potential is negative quartic and respects global
symmetry ($U(1)$ here) at relatively small $\phi$, and has one or more minima
at large $\phi$. Dashed line shows the evolution of the field
(dot with arrow) at the rolling stage. At that stage the perturbations
(shown by double arrow)
of the pseudo-Goldstone field are developed.
At the hot Big Bang epoch, the pseudo-Goldstone field transfers its energy
to hot matter, and its perturbations are converted into adiabatic
perturbations.
\label{fig1}}
\end{figure}
In Section~\ref{sec-4} we discuss a subtle point of our model.
Namely, at the stage when the radial field rolls down, this field
develops its own perturbations. Their growth is fast, and at some point
the linearized theory breaks down. We argue, however, that
the results of the linearized theory concerning the perturbations
of the pseudo-Goldstone field remain valid, and the flat spectrum
of adiabatic perturbations is generated indeed.
We end up in Section~\ref{sec-conclude} with concluding remarks.
\section{Getting flat spectrum}
\label{sec-2}
To begin with, let us consider massless
scalar field $\phi$ conformally coupled to gravity
and evolving in spatially flat
4-dimensional FRW Universe.
Let us assume
for the time being that there is exact global symmetry,
which we take for definiteneness to be $U(1)$
(though the argument is valid for any compact symmetry group).
Hence, the field is comlex, and the global symmetry is
$\phi \to \mbox{e}^{i\alpha} \phi$.
Conformal invariance allows for quartic scalar potential, which
we assume to be negative,
\begin{equation*}
V(\phi) = - h^2 |\phi|^4
\end{equation*}
(modulo an additive constant).
As usual, one defines the field $\chi$ by
\begin{equation*}
\phi = \frac{\chi}{a} \equiv \frac{\chi_1 + i \chi_2}{a} \; .
\end{equation*}
Then the action for $\chi$ in conformal coordinates is
\begin{equation*}
S_\chi = \int~d^3x~d\eta~
\left[
\partial_\mu \chi^* \partial^\mu \chi + h^2 |\chi|^4
\right] \; ,
\end{equation*}
where indices are raised by Minkowski metric.
Let us consider spatially homogeneous background field.
The field equation is
\begin{equation}
- \chi^{\prime \prime} + 2h^2 |\chi|^2 \chi = 0 \; ,
\label{2}
\end{equation}
where prime denotes $d/d\eta$.
In terms of the radius and phase, $\chi=\rho \mbox{e}^{i\theta}$,
one of the equations is the conservation of current,
\begin{equation*}
\frac{d}{d\eta} \left( \rho^2 \theta^\prime \right) = 0 \; .
\end{equation*}
Hence, as the value of $\rho$ increases, the phase $\theta$ freezes
out, and the evolution proceeds along the radial direction.
Without
loss of generality we take $\chi$ to be real
at that time, $\chi=\chi_1$. Then the
first integral of the equation of motion is
$ \chi_1^{\prime \, 2} - h^2 \chi_1^4 = \epsilon = \mbox{const}$.
Hence, the solutions approach the asymptotics
that is independent of $\epsilon$,
\begin{equation}
\chi_1 (\eta) = \frac{1}{h (\eta_* - \eta)} \; ,
\label{16-1}
\end{equation}
where $\eta_*$ is a constant of integration.
Now, let us study field perturbations in the orthogonal direction,
$\delta \chi_2$. They obey the linearized equation, in momentum
representation,
\begin{equation}
(\delta \chi_2)^{\prime \prime}
+ k^2 \delta \chi_2 - 2 h^2 \chi_1^2 \delta \chi_2 = 0 \; .
\label{1}
\end{equation}
At early times, when $k(\eta_* - \eta) \gg 1$, the second term dominates,
and $\delta \chi_2$ oscillates like free scalar field in Minkowski
space-time. At later times the third term dominates instead.
So, the perturbation
``exits the horizon'', but now the ``horizon'' is due to evolving
$\chi_1$ (hereafter we use quotation marks to distinguish this
``horizon'' from the true cosmological horizon).
Outside the ``horizon'', i.e., at $k(\eta_*- \eta) \ll 1$, the perturbation
$\delta \chi_2$ evolves in the same way as the background $\chi_1$.
This is clear from the fact that at $k=0$, eq.~\eqref{1} is the linearized
equation \eqref{2}. Now, alongside with the solution $\chi_1 (\eta)$,
eq.~\eqref{2} has a solution $\mbox{e}^{i\alpha} \chi_1 (\eta)$,
where $\alpha$ is a real constant. For small $\alpha$ the latter solution
is $(\chi_1 + i\alpha \chi_1) $, and
the second part is small perturbation,
which is precisely $\delta \chi_2$. So, if the perturbation $\delta \chi_2$
oscillates with amplitude $c(k)$ at early times, it behaves at late times as
\begin{equation}
\delta \chi_2 = C c(k) \chi_1 (\eta)
= C^\prime \frac{c(k)}{k (\eta_* - \eta)} \; ,
\label{3}
\end{equation}
where the factor $k^{-1}$ is evident on dimensional grounds, $C$ and $C^\prime$
are time-independent, and $C^\prime$
is independent of $k$.
This is precisely the enhancement of perturbations needed for
getting flat spectrum of $\delta \chi_2$.
We stress that this property is entirely due to the symmetries of the
model: the behaviour \eqref{16-1} of the background is a
consequence of conformal invariance, whereas the behaviour
\eqref{3} of perturbations is a consequence of the global
$U(1)$ symmetry.
In more detail, eq.~\eqref{1} reads
\begin{equation*}
(\delta \chi_2)^{\prime \prime}
+ k^2 \delta \chi_2 - \frac{2}{(\eta_* - \eta)^2} \delta \chi_2 = 0 \; .
\end{equation*}
This is formally the same equation as the equation for perturbations of
minimally coupled massless scalar field in de~Sitter space-time.
Hence, the spectrum of $\delta \chi_2$ is the same flat spectrum.
We are interested in the solution that behaves at early times as
\begin{equation}
\chi_2^{(-)} = \frac{1}{(2\pi)^{3/2}\sqrt{2k}} \mbox{e}^{ik (\eta_* - \eta)}
\; .
\label{1b}
\end{equation}
Modulo overall constant
phase, this solution is expressed through the Hankel function,
\begin{equation*}
\chi_2^{(-)} = \frac{1}{4\pi} \sqrt{\frac{\eta_* - \eta}{2}}
H_{3/2}^{(1)} \left[k (\eta_* - \eta) \right] \; .
\end{equation*}
At $k (\eta_* - \eta) \ll 1$ this solution is
\begin{equation}
\chi_2^{(-)} = \frac{i}{2\pi^{3/2}} \frac{1}{k^{3/2} (\eta_* - \eta)}
\; ,
\label{1a}
\end{equation}
in accord with \eqref{3}. Assuming that the quantum field $\delta \chi_2$
is originally in the vacuum state, we obtain its power spectrum
outside the ``horizon'' in the standard way.
The spectrum is flat, and its amplitude is
\begin{equation}
\Delta_{\chi_2} \equiv \sqrt{{\cal P}_{\chi_2}} = \frac{1}{2\pi
(\eta_* - \eta)} \; .
\label{10a}
\end{equation}
As promised, generating flat spectrum of the field perturbations
is fairly straightforward.
We end up this section by noting that
the mechanism requires long time during which
the classical field $\phi$ rolls down its potential. Indeed,
let $\eta_i$ be the time at which the field starts to roll down, and
$\eta_f$ the time at which rolling down terminates.
Then
the existence of the
regimes \eqref{1b} and \eqref{1a} requires that
\begin{eqnarray*}
&& k(\eta_* - \eta_i) \gg 1 \; ,
\\
&& k(\eta_* - \eta_f) \ll 1 \; .
\end{eqnarray*}
These inequalities should hold for all scales of interest, say, for
present momenta (we set the present value of the scale factor equal to 1)
\begin{equation}
k_{min} \sim (1~\mbox{Gpc})^{-1} \lesssim k \lesssim k_{max} \sim
(100~\mbox{kpc})^{-1} \; .
\label{16-2}
\end{equation}
Therefore, the duration of the rolling stage should be large,
\begin{equation}
(\eta_f - \eta_i) \gg (k_{min}^{-1} - k_{max}^{-1}) \approx k_{min}^{-1}
\; .
\label{17-1}
\end{equation}
This excludes the possibility that the mechanism works at the
hot stage of the cosmological evolution: there is simply not enough
time. On the other hand, there may well be enough time in the ekpyrotic
or starting-the-Universe scenario\footnote{Note that the inequality
\eqref{17-1} means that the comoving size of light cone originating from
$\eta=\eta_i$ and measured at $\eta=\eta_f$ exceeds the comoving
size of the visible Universe. Hence, the mechanism can work only in
cosmological models which solve, at least formally, the horizon problem.}.
\section{Implementing the mechanism}
\label{sec-3}
In this Section we suggest a
way to implement the mechanism and generate adiabatic perturbations
from the perturbations $\delta \phi_2$.
Let us assume that as the classical field $\phi_1 = \chi_1/a$ approaches the
region $\phi_1 \sim f$, conformal symmetry gets broken and the
potential along the radial direction
has a minimum at $\phi_1=f$, where $f$ is some high energy
scale.
We assume that $O(2)$ symmetry is broken as well, but that
the scale of
that breaking is small compared to $f$, so we neglect
the effects of $O(2)$ symmetry breaking for the time being. Then the
classical field
$\phi_1$ oscillates near $f$ and eventually settles down to $\phi_1 = f$
(say, by producing conventional particles).
We require that this process occurs
at the time when the energy density of the
field $\phi_1$ is much smaller than the total energy density
in the Universe.
The reason for this requirement
is that the field $\phi_1$ develops its own perturbations.
In terms of perturbations $\delta \chi_1$, the linearized equation for this
mode is
\begin{equation*}
(\delta \chi_1)^{\prime \prime}
+ k^2 \delta \chi_1 - \frac{6}{(\eta_* - \eta)^2} \delta \chi_1 = 0 \; .
\end{equation*}
The solution which is negative frequency at early times is
again expressed through the Hankel function, but with another index,
\begin{equation*}
\chi_1^{(-)} = \frac{1}{4\pi} \sqrt{\frac{\eta_* - \eta}{2}}
H_{5/2}^{(1)} \left[k (\eta_* - \eta) \right] \; .
\end{equation*}
At $k(\eta_* - \eta) \ll 1$ this solution is
\begin{equation}
\chi_1^{(-)} = - \frac{i}{\pi \Gamma (-3/2)} \frac{1}{k^{5/2}
(\eta_* - \eta)^2} \; .
\label{2a}
\end{equation}
This behaviour
again has simple interpretation: if $\chi_1$ is a classical solution
to the field equation \eqref{2}, then
\begin{equation}
\delta \chi_1 \propto \chi_1^\prime
\label{6a}
\end{equation}
is a solution
to the linearized classical field
equation for $k \to 0$. Hence $\delta \chi_1 \propto
(\eta_* - \eta)^2$ on super-''horizon'' scales; the delpendence on $k$
again follows from dimensional argument. In any case, eq.~\eqref{2a}
implies that the spectrum of perturbations of the field
$\chi_1$ is red,
\begin{equation}
\Delta_{\chi_1} \equiv \sqrt{{\cal P}_{\chi_1}} = \frac{3}{2\pi
k(\eta_* - \eta)^2} \; .
\label{5a}
\end{equation}
These perturbations should not leave any trace in the late Universe;
the simplest way to achieve that is to impose the above
requirement.
The perturbations of the field $\phi_1$ may become so strong that
they ruin the linearized analysis.
To be on the safe side, we impose here the
requirement that the field $\phi_1$ is approximately homogeneous
over the whole visible Universe at the time when the scales of interest
exit the ``horizon'' (see, however, Section~\ref{sec-4}).
This is sufficient for
the linearized theory to be valid
in our patch of the Universe at that time;
perturbations $\delta \phi_1$
of longer wavelengths correspond to unobservable
shift of the parameter $\eta_*$ in the whole visible Universe,
see \eqref{6a}.
Our requirement gives
\begin{equation}
\Delta_{\chi_1} (k_{min}, \eta_{max}) \ll \chi_1 (\eta_{max}) \; ,
\label{19-1}
\end{equation}
where $k_{min}^{-1}$ is the comoving size of the visible Universe,
cf. \eqref{16-2}, and $\eta_{max}$ is the time of the
``horizon'' exit of the shortest interesting modes. The value of
$\eta_{max}$ is determined
by
\begin{equation*}
k_{max}(\eta_* - \eta_{max}) \sim 1 \; .
\end{equation*}
Making use of \eqref{16-1} and \eqref{5a} we obtain a constraint
on the coupling constant
\begin{equation}
h \ll \frac{k_{min}}{k_{max}} \sim 10^{-4} \; .
\label{11a}
\end{equation}
We will further discuss the non-linearity issue in Section~\ref{sec-4}.
Let us now turn to the field $\phi_2$. In fact, after the field
$\phi_1$ settles down to $f$, it is more appropriate to
speak about the phase $\theta$
of the original field $\phi$. We keep the notation $\phi_2$,
with understanding that $\theta = \phi_2/f$.
Let us first estimate the amplitude of perturbations
$\delta \phi_2$ at the time $\eta_f$ when the background field
$\phi_1$ reaches $f$. At that time one has
\begin{equation}
\chi_1 (\eta_f) \sim \frac{1}{h (\eta_* - \eta_f)} \sim a_f \cdot f
\; ,
\label{8a}
\end{equation}
where the subscript $f$ refers to the time $\eta_f$. From
\eqref{10a} we obtain
\begin{equation*}
\Delta_{\phi_2} (\eta_f) = \frac{\Delta_{\chi_2}(\eta_f)}{a_f}
\simeq \frac{1}{2\pi} hf \; .
\end{equation*}
We now have to convert these entropy perturbations into adiabatic ones.
Let us recall that
the global $O(2)$ symmetry is broken near $|\phi| = f$.
We have in mind axion-like picture, in which the energy scale of this
breaking is smaller than $f$. Importantly, the phase
field $\phi_2$, as any (pseudo-)Goldstone field, {\it minimally}
couples to gravity~\cite{voloshin}.
From this point on, the discussion is parallel to that in the
curvaton model.
Generically, $\phi_2=0$ is not a minimum of the scalar potential,
so the radial evolution ends up at a slope of the potential
for the pseudo-Goldstone field $\phi_2$.
Let us assume that at the time $\eta_f$,
the Hubble parameter exceeds the mass parameter of this
pseudo-Goldstone field.
In this case the field $\phi_2$
stays close to zero for some time\footnote{This is true both for
expanding Universe and for contracting Universe dominated by matter
with stiff equation of state, $w>1$.},
then rolls down to the nearest minimum
of its potential,
oscillates there and eventually the oscillations decay into usual particles.
We assume that the decay happens at the hot Big Bang
stage of the cosmological
expansion, when the Universe is either radiation-dominated, or
temporarily dominated by the oscillating field $\phi_2$ itself.
Let $M$ denote the distance to the nearest minimum of the
potential for the filed $\phi_2$, so that the number of minima
of the scalar potential is $N \sim f/M$. Then at the beginning
of oscillations and until their decay,
the perturbations in the energy density of the pseudo-Goldstone field are
\begin{equation*}
\frac{\delta \rho_{\phi_2}}{\rho_{\phi_2}} \sim
\frac{\delta \phi_2 (\eta_f)}{M} \; .
\end{equation*}
This leads to adiabatic perturbations with flat spectrum
and the amplitude
\begin{equation}
\Delta \sim r \frac{\Delta_{\chi_2} (\eta_f)}{a_f M}
\sim r \frac{hf}{2\pi M} \; ,
\label{12a}
\end{equation}
where $r$ is the ratio of the energy density of the field $\phi_2$
to the total energy density at the time the oscillations decay.
The pseudo-Goldstone picture with a single minimum of the potential for
the phase would correspond to $M \sim f$. If we insist on
the constraint
\eqref{11a}, the amplitude \eqref{12a}
of the resulting adiabatic perturbations would be too small in that case
(barring fine tuning). On the other hand, with $N \sim f/M \gg 1$ the required
amplitude $\Delta \sim 10^{-5}$ can obtained without fine tuning of
other parameters\footnote{This
discussion assumes that after the rolling stage,
the value of the phase of the field $\phi$
is generic. Alternatively, one could make use of the anthropic argument
for explaining why the field $\phi_2$ ends up close to the minimum of
its potential, i.e., why $M \ll f$, so that the amplitude of the matter
perturbations is large enough,
cf. Refs.~\cite{tegmark,wilczek}.
With this line of reasoning, having large number of minima of the potential
for the pseudo-Goldstone field is certainly
unnecessary.}.
As we discuss in the end of Section~\ref{sec-4}, the constraint
\eqref{19-1}, and hence \eqref{11a} may in fact be unnecessary, so
the mechanism may work for $N=1$ as well.
\section{Generalizing to non-linear radial field perturbations}
\label{sec-4}
Let us come back to the non-linearity issue.
Since the perturbations \eqref{5a} increase in time faster than
the background field \eqref{16-1}, the evolution in the radial direction
may become non-linear at some stage. Naively, one would think that the
analysis of Section~\ref{sec-2} goes through only if
this does not happen, i.e., if the perturbations $\delta \phi_1$ remain
small up to the time $\eta_f$. Such a requirement would push the parameters
of the model to a very contrived range: the coupling contant $h$ would
have to be extremely small, the field value $f$ would have to be
super-Planckian, and the number of minima of the scalar
potential, $N \sim f/M$,
would have to be very large. This is discussed
in Appendix. However, we argue that non-linearity of the
radial field perturbations
does not spoil the result \eqref{12a}.
The argument is as follows. Once the inequality \eqref{19-1} is
satisfied, all interesting scales exit the ``horizon'' when
the linearized theory is still valid. Let $\eta_{lin}$ be some moment
of time when the linearized theory is valid, but all interesting
scales are already
super-''horizon''. At that time, $\delta \phi_2$ may be considered
as classical random field whose spatial gradients are negigible.
According to \eqref{10a} and \eqref{16-1} this field is small,
$\delta \phi_2 / \phi_1 \sim h$, and it evolves in the same way as
$\phi_1$,
\begin{equation}
\delta \phi_2 (\eta) = \frac{\delta \phi_2 (\eta_{lin})}{\phi_1 (\eta_{lin})}
\phi_1 (\eta) \; .
\label{19*}
\end{equation}
The field $\phi_1 ({\bf x}, \eta) = \phi_1 (\eta) + \delta \phi_1
({\bf x}, \eta)$ is also classical random field which is homogeneous
inside the ``horizon''. At late times this field becomes inhomogeneous,
but
on super-horizon scales only.
The point is that the evolution of the
entire system proceeds according to the homogeneous classical field
equation. Therefore, the symmetry
argument presented before eq.~\eqref{3} tells
that $\delta \phi_2$ is still given by \eqref{19*}, but now with
$\phi_1 = \phi_1 ({\bf x}, \eta)$. Note that $\delta \phi_2$ remains small.
The field $\phi_1$ reaches the value $f$
in the end, and at that time
\begin{equation*}
\delta \phi_2 = \frac{\delta \phi_2 (\eta_{lin})}{\phi_1 (\eta_{lin})}
f \; .
\end{equation*}
This coincides with the result of the linearized theory.
Another way to phrase this argument is to write the rolling field $\phi_1$,
with long-ranged perturbations included, as
\begin{equation}
\phi_1 ({\bf x}, \eta) = \frac{1}{h a(\eta) [\eta_{*\; eff} ({\bf x}) - \eta]}
\; .
\label{19-3}
\end{equation}
Comparing this expression with \eqref{5a} at the time when the linearized
theory is still valid, we find that $\delta \eta_*({\bf x})
= \eta_{* \; eff} ({\bf x}, \eta) - \eta_*$ is long ranged
random classical field
whose fluctuation is
\begin{equation*}
\Delta_{\delta \eta_*} = \frac{3h}{2\pi k} \; .
\end{equation*}
At late times, when $\delta \eta_* \gtrsim (\eta_* - \eta)$,
the dependence of $\phi_1 ({\bf x}, \eta)$ on $\delta \eta_*$ becomes
non-linear, but the expression \eqref{19-3} remains valid.
The point is, again, that the expression
\eqref{19*} with
$\phi_1 = \phi_1 ({\bf x}, \eta)$
is a solution to the linearized field equation for $\delta \phi_2$
at all times.
To see this explicitly, let us show
that $|{\boldsymbol \nabla} \delta \phi_2| \ll \delta \phi_2^\prime$,
so that spatial variation of $\eta_{* \; eff}$ can be neglected in
that field equation, and the argument before eq.~\eqref{3}
indeed applies. Using \eqref{19*} we write
\begin{equation}
\frac{|{\boldsymbol \nabla} \delta \phi_2|}{\delta \phi_2^\prime}
= |{\boldsymbol \nabla} \delta \eta_*| \; .
\label{19-2}
\end{equation}
Now, ${\boldsymbol \nabla}\delta \eta_*$ is random field with fluctuation
\begin{equation*}
\Delta_{({\boldsymbol \nabla} \delta \eta_*)} = k \Delta_{\delta \eta_*}
\sim h \; ,
\end{equation*}
so the ratio \eqref{19-2} is indeed small. Since the final state of the
super-''horizon''
perturbation $\delta \phi_2$ is independent of the parameter $\eta_*$,
we see that non-linearity of the evolution of $\phi_1$, encoded in
the non-linear dependence of $\phi_1$ on $\delta \eta_*$,
is irrelevant
for the spectrum of $\delta \phi_2$.
The latter way of presenting the argument suggests that
the condition \eqref{19-1} is in fact unnecessary. The only requirement
that remains is that $h \ll 1$. This ensures
that the perturbations $\delta \phi_1$
become non-linear long after they exit the ``horizon'', and the spatial
variation of $\eta_{* \; eff}$ is irrelevant in the field equation for
$\delta \phi_2$. Hence, the parameters of our model are constrained very
weakly; in particular, there may exist just one minimum of the
potential for the pseudo-Goldstone field, i.e., $M \sim f$.
\section{Conclusions}
\label{sec-conclude}
We conclude this paper with a few remarks. First, the mechanism for
converting the perturbations of $\phi_2$ into adiabatic
perturbations, suggested in Section~\ref{sec-2}, is most probably not
unique. For example, one
can imagine adding other fields which interact with
the pseudo-Goldstone field $\phi_2$ so that the energy density
of these fields,
after $\phi_1$ relaxes to $f$, is linear in $\delta \phi_2$.
In that case the simple formula \eqref{12a} would not be valid.
Second, the resulting spectrum needs not be exactly flat.
One expects that for conformal symmetry slightly broken during
the stage of rolling $\phi_1$, the spectrum becomes tilted.
The tilt may be present also if the global symmetry ($U(1)$ in our
example) is explicitly broken at that stage. How strong is the
variation of the tilt with $k$, and whether there are other
properties of the spectrum in these cases, remains to be understood.
Third, if the coupling constant $h$ is not very small, one would expect
sizeable non-Gaussianity of the perturbations. Also, there may be
an interplay between the perturbations $\delta \phi_1$ and
$\delta \phi_2$ which may result in peculiar properties
of the adiabatic perturbations. It would be interesting to see
whether or not the latter may help discriminate our
mechanism from others (say, from inflationary ones).
Fourth, as in many other cases, the generation of scalar
perturbations in our model is unrelated to the generation of tensor
modes. Thus, there is no reason to expect sizeable gravitational wave
background in our scenario.
Finally, we left aside the issue of the initial conditions for the
field $\phi$. We simply assumed that this field is spatially
homogeneous and starts rolling from near the top of its quartic
potential. Whether these initial conditions may emerge naturally
in the context of some cosmological scenario remains unclear.
We hope to address these and other issues in future.
The author is indebted to D.~Gorbunov, A.~Khmelnitsky, E.~Nugaev,
G.~Rubtsov and I.~Tkachev for helpful discussions. This work
has been supported in part by Russian Foundation for Basic
Research grant 08-02-00473.
\section*{Appendix}
In this Appendix we discuss what kind of constraints
on the parameters one would obtain by
insisting that the entire evoluton of both perturbations
$\delta \phi_1$ and $\delta \phi_2$
occurs in the linear regime, so that the analysis of Section~\ref{sec-2}
is literally valid. The strongest constraint comes from the
requirement that the perturbations of the radial field $\delta \phi_1$
are small by the end of the regime of rolling $\phi_1$,
\begin{equation}
\delta \phi_1 (\eta_f) \sim \Delta_{\chi_{1}} (\eta_f)/a_f \ll f \; .
\label{4a}
\end{equation}
Let us pretend that this inequality should be valid
for length scales up to the present
horizon size $k_{min}^{-1}$;
perturbations of longer wavelengths correspond to unobservable
shift of the parameter $\eta_*$ in the whole visible Universe.
Making use of \eqref{8a} and \eqref{5a} one finds that the condition
\eqref{4a} gives
\begin{equation}
\frac{k_{min}}{a_f} \gg h^2 f \; .
\label{7a}
\end{equation}
Given the very small value of $k_{min}$ of the order of
the present Hubble parameter, the latter inequality
impies that the coupling constant $h$ is extremely small.
Indeed, assuming that the Universe did not expand significantly
before the beginning of the conventional hot Big Bang stage
(no inflation) and ignoring weak dependence on the effective number of
degrees of freedom $g_*$, one finds that $a_f \gtrsim T_0/T_{max}$,
where $T_{max}$ and $T_0$ are the maximum temperature in the
Universe and the present temperature, respectively.
Hence, the constraint \eqref{7a} reads
\begin{equation}
h^2 f \ll \frac{k_{min}}{T_0} T_{max} \sim 10^{-29} T_{max} < 10^{-29} M_{Pl}
\; .
\label{2*}
\end{equation}
The constraint \eqref{7a} implies also that
\begin{equation}
f \gg M_{Pl} \; .
\label{2+}
\end{equation}
This comes out from the estimate of the temperature $T_{end}$ at which the
oscillations of the pseudo-Goldstone field $\phi_2$ are converted into
usual particles. At this temperature the perturbations $\delta \phi_2$
are transformed
into adiabatic ones, and one requires that $T_{end} \gtrsim 100$~GeV
in order that the baryon asymmetry and dark matter may be thermally
produced afterwards. To estimate $T_{end}$ in our model, we notice that
the oscillations of $\phi_2$ begin at the time $t_{osc}$ when
$H(t_{osc}) \sim \mu$ where $\mu^2$ is the curvature of the potential
for the pseudo-Golstone
field. Again ignoring the dependence on $g_*$ we obtain the energy densities
of hot matter and the pseudo-Goldstone field at that time,
\begin{equation*}
\rho_{rad} (t_{osc}) \sim T^4 (t_{osc}) \sim \mu^2 M_{Pl}^2 \; ,
\;\;\;\;\;\;\; \rho_{\phi_2} (t_{osc}) \sim \mu^2 M^2 \; .
\end{equation*}
The oscillations end up when $\rho_{rad}/\rho_{\phi_2} = r \leq 1$,
so one finds
\begin{equation*}
\frac{T_{end}}{T_{osc}} \sim \frac{a(t_{osc})}{a(t_{end})}
= \frac{1-r}{r} \frac{\rho_{\phi_2} (t_{osc})}{\rho_{rad}(t_{osc})} \; .
\end{equation*}
Hence
\begin{equation*}
T_{end} \sim \frac{(1-r) M^2 }{r M_{Pl}^2} \sqrt{\mu M_{Pl}} \; .
\end{equation*}
Making use of \eqref{12a} we find
\begin{equation*}
T_{end} \sim \frac{h^2 f^2 r (1-r)}{\Delta^2 M_{Pl}^2} \sqrt{\mu M_{Pl}}
\; .
\end{equation*}
Finally, using \eqref{2*} and inserting $\Delta \sim 10^{-5}$
we obtain
\begin{equation*}
T_{end} \ll \left(\frac{f \mu^{1/2}}{M_{Pl}^{3/2}} \right)\; \mbox{GeV}
\; .
\end{equation*}
Since $\mu^2$ is the curvature of the scalar potential, one has
$\mu < M_{Pl}$. Therefore, high enough value $T_{end} \gtrsim 100$~GeV
can be obtained only for super-Planckian values of $f$. Together
with \eqref{2*} this implies that the coupling constant is extremely
small. In turn, the smallness of $h$ and eq.~\eqref{12a}
mean that the correct amplitude of
perturbations can be obtained only for huge number of
minima of the scalar potential, $N=M/F$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,772 |
Case Study: Using Lego To Implement Enterprise Risk Management
The risk management team in the Benelux region at the German life sciences multinational, Bayer, has spent the last year devising and piloting a new technique for risk assessments using a surprising product usually associated with children's toys. The team has been using the LEGO® SERIOUS PLAY® (LSP) approach, which applies bricks specially developed by the Lego Company for the business environment. The team believes it is among the first risk managers to use the LSP method in risk assessments. The function wanted to move away from traditional ways of identifying and analysing risk where information collated during the exercise is typically only provided by a minority of the participants. David Lannoy, risk and process manager in the Benelux, says that the old style of risk assessments usually entailed 20% of the participants delivering 80% of the comments on risk. Lannoy and Helene Peters, head of risk and process management in the Benelux, say LSP now ensures that 100% of participants in its risk assessments contribute from their perspectives.
Source: IRM's Enterprise Risk Magazine, Summer, 2018
Methodology: Lannoy and Peters put the rise in participation down to LSP's methodology. It comprises three steps: building an individual model that represents an employee's contribution to the organisation's objectives; joining that same Lego model to other models to create the overall business environment, a 'shared model'; and identifying the factors, 'agents', that could impact on the business model. If that sounds daunting, participants in the project team are asked to play with the blocks to build a simple model. This puts everyone at their ease and ensures people feel comfortable building with Lego bricks. Lannoy, who is visiting the IRM's London offices with Peters, presents me and Peters with some blocks. We both receive the instruction to building ducks and while both finished models look vaguely like ducks, they have been constructed differently. Lannoy says that the demonstration illustrates an important point – the individual contribution each team member brings to the process; the same instructions were interpreted and worked out differently, made visible through the 'ducks'. The next task participants are asked to carry out is to construct a more sophisticated model, such as creating their ideal holiday location from bricks. This helps develop confidence and sophistication with the blocks.
Assessment: Once the team is satisfied with their Lego building techniques, it's time to move on to the risk assessment. This starts with each member being asked to build a model representing how their work activities contribute to the objectives of the company, their division or their department. At the moment, Lannoy and Peters are focusing on the short-term objectives – spanning one to two years – in their exercises, though they hope eventually to build a construct around longer-term objectives. Once these models are built, each member presents it to the rest of the team and explains how it contributes to the company's objectives. As well as having the added value of performing a team-building exercise, members become aware of each other's perspectives and the way their roles contribute to the team's overall performance. "Yes, it's experienced as a teambuilding exercise but you are performing a risk assessment exercise throughout the whole day," Lannoy says. "You learn a lot about people around the table, how they perceive their job and their work environment." These perspectives are often missing from more traditional approaches. In addition, participants perform the Lego building exercises with a very positive frame of mind, focusing on what each of them does to make the organisation's objectives happen.
Questioning: While explaining their models to the team, other members of the group are encouraged to ask questions. For a sales function model, for example, the questions could revolve around what is driving the model: the quality of the product, or the customer relationship. Such questions and the chance to reflect on the answers are critical to the success of the exercise, says Lannoy. The team does not move to the next stage of the process until it is clear they all understand and accept the results, which underlines the fact that the full engagement of the whole team is key to the approach's success. The second phase of the risk assessment requires the team to work together to build a single shared Lego model by linking their creations together. The aim is to create a physical summary of the team's activities and objectives. Peters emphasises that by joining the individual parts together, the team presents a common, shared model where everyone in the project team feels represented and aligned on the objectives. It also captures all the key drivers of the business activities at the same time. In this shared view, key business drivers, such as customer focus, product quality or compliance, become clear. When the model is finished, each team member is asked to inspect it, reflect on whether everyone is represented and consider if it adequately explains the business model. Changes can be made with the agreement of each team member in a very hands-on experience. Peters says that the process is extremely valuable when the team is comprised of different staff functions who are less aware of each other's contributions to the business's operations or even have differing priorities; it creates a common ground and understanding, also for each other.
Identifying risks: The third phase of the process is to ask the team to identify the risks and opportunities in the business environment. However, Peters and Lannoy are careful not to use the terms' risks and opportunities in their sessions but to call them agents, according to the LSP methodology, or factors. This ensures that the process is kept neutral with no positive or negative connotations interfering with the creative process at this stage. When the team has identified the factors that affect their work they are asked to position them around the shared model depending on their frequency and potential impact on the business environment. Agents that occur frequently are placed near the Lego business model. Factors occurring infrequently are placed further away. As a final step, the team flags all the agents that according to them have a negative or potential positive impact on the business environment by applying red or green flags. Peters and Lannoy agree from their experiences with LSP that this part of the process is particularly successful at identifying business opportunities, which the traditional risk assessment methods often fail to do. This, they say, is also true for risks, with the LSP team builders identifying new risks of which they were previously unaware. A further advantage is an ease with which risk managers can prioritise the risks and opportunities simply by examining where the team has placed the factors on the business model landscape.
Creating relationships: So why have Peters and Lannoy taken a big leap of faith and asked colleagues working in the traditional roles of life sciences, legal and finance functions to spend a full day working with Lego bricks? Apart from the advantages already mentioned, they say Bayer actively encourages innovation in all spheres of the work. They say the activity brings out consensus-based decision-making and reinforces ownership of the business activities, risks and opportunities. There's also another factor – the move away from risk management being perceived as a controlling function and towards being considered a business partner or "coach". Lannoy says, "There's a trend in the profession with people not wanting to see the risk manager as a controller. They want to perceive the added value in the contribution and by doing this type of exercise, we create that relationship with the business". Peters and Lannoy were provided with the trust and resources from Bayer Benelux to devise the programme and pilot the first exercise last year. The initial project team to take part was a rather critical but open-minded group from the pharma division. Surprisingly, it has become their best promoter within the rest of the company. "They were positively surprised about the entire workshop and the outcome of the day," Peters says. "Throughout the whole day, the risk wasn't mentioned once, as it can be intimidating, but we still identified risks and opportunities related to their business objectives." Lannoy posted his experience with the pilot project on his LinkedIn site and was surprised to see what a stir it caused. As well as the praise for using such an innovative method, there was equal criticism. Lannoy says he was at pains to point out that the method is only for risk and opportunity identification and prioritisation. An evaluation of the risks and opportunities will still need to be done through an investigation of the impacts following research. This concluding bit is still part of the traditional risk assessment processes.
Improvements: So how can the programme be improved? Peters and Lannoy say getting a project team to commit to a whole day for the exercise is still a sticking point. But the pilot project was so successful that curiosity about the exercise has become infectious. They also say that the beauty of the method is its flexibility, which allows them to adjust their service to the participating project team's requirements. This serves to ensure the team feel they are actively choosing to sign up for a day's risk assessment with LSP rather than having it imposed on them. Further ahead, Peters and Lannoy are looking to adapt the programme to consider longer-term risks and opportunities. At present, it only focuses on the narrower range of one to two years. The ability to use this model for strategic risks and opportunities is very attractive. "Bayer has created an ecosystem where colleagues can learn from each other," says Peters. "We can cross-fertilise our risk management process and see where else it can benefit the company.
This article was originally featured in IRM's Enterprise Risk Magazine. Get the latest print edition today (India only)
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,772 |
Put Down Your Whip, also translated as Lay Down Your Whip (), is a 1931 Chinese street play written by Chen Liting during the Republican era, who drew inspiration from the earlier play Meiniang by Tian Han.
Originally an anti-government play, it was adapted to take on an anti-Japanese theme after growing Japanese aggression against China. It became the most influential street play during the Second Sino-Japanese War, and was performed countless times throughout China, and even in the White House for President Roosevelt by the actress Wang Ying. The future Madame Mao was also among its many performers.
Wang Ying's performance of Put Down Your Whip inspired Xu Beihong's eponymous painting, which in 2007 set an auction price record for Chinese paintings.
Plot
Original version
A young girl named Fragrance (Xiang Jie) and her old father are poor and homeless street performers. Fragrance sings the folk song "Fengyang Flower-Drum" and does acrobatics on the street. She performs badly because she is weak from chronic hunger. Angry with his daughter's poor performance, the old man raises his whip to punish her. A young man, who is an actor in disguise, charges out from the audience, shouting "Put down your whip!" He scolds the old man for abusing his own daughter. Unexpectedly, Fragrance defends her father, and recounts her family's plight: they are refugees who have escaped flood, exploitative landlords, and tyrannical government of their hometown. The spectators are deeply moved by her misfortune. At the end, the young man turns to the spectators and exhorts them to resist the oppressive Kuomintang government: "We must resist those who coerce us to live a life of starvation and homelessness."
Later development
After increasing Japanese encroachment into north China, and especially the December 9th Movement of 1935, the play was adapted to take on an anti-imperialist and anti-Japanese theme. Instead of escaping from government oppression and flood, Fragrance and her father now escape the brutal Japanese occupation of Manchuria; and instead of appealing to the audience to fight against the government, the young man urges his compatriots to unite and defend the country against Japanese invasion, or "we will soon meet the same fate as our countrymen in Manchuria." The theme song was also changed to the "September Eighteenth Melody" lamenting the September 18 Incident of 1931 when Japan invaded and occupied Manchuria. Other patriotic songs including the "March of the Volunteers", which later became the national anthem of the People's Republic of China, were also used. The location of Fragrance's hometown kept changing as new places became devastated by the Japanese invaders. After the 1937 Nanking Massacre, the fallen Chinese capital Nanking would become her hometown.
Reception
The play was written by Chen Liting, then a 21-year-old primary school teacher in Nanhui County outside of Shanghai, drawing inspiration from Tian Han's play Meiniang. Chen did not put his name on the script, however, because of the strong anti-government overtone in the original play.
The play was a failure when it debuted on 10 October 1931 in Nanhui. But after its adaptation to a patriotic anti-Japanese play, it became the most influential street play of the Sino-Japanese War, and was performed countless times throughout China during the war. The play was frequently staged by amateur performers as well as many famous stars. The great actress Wang Ying even performed an English version of the play in the White House for President Roosevelt, first lady Eleanor Roosevelt, and many diplomats. The future Madame Mao, then known as Li Yunhe, was also among its many performers.
Put Down Your Whip has been described in Chinese media as a "spiritual atomic bomb" against the Japanese invaders.
In art
In October 1939, Wang Ying performed Put Down Your Whip in Singapore. Artist Xu Beihong, who was a friend of the actress, painted a life-size portrait of her performing the play. In April 2007 the painting was sold in auction for US$9.2 million, setting a record for the highest auction price ever paid for a Chinese painting.
Artist Situ Qiao also saw Wang Ying's performance in Singapore and painted his version of the play in 1940, which has become the painter's most famous work.
References
1931 plays
Chinese Republican era plays
Second Sino-Japanese War | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,304 |
Q: Duplicating rows based on a multiplier Using R and having some trouble manipulating my data. I've identified bee collected pollen to types and their relative volumes ("adjusted_volume" below) (how much pollen on a slide). I'm now trying to calculate average pollen usage by bees at each of my 14 sites. My data looks like this:
head(pollen)
site treatment hive_code pollen_type adjusted_volume
A conventional 4 alnus_spp 248.5
B conventional 4 alnus_spp 71.0
B conventional 7 alnus_spp 35.5
My plan was to dcast and gather to get the amount of each pollen type per site...
data1 <- dcast(pollen, site + treatment ~ pollen_type, length)
data2 <- gather(data1, pollen_type, count, alnus_spp:vaccinium_corymbosum, factor_key=TRUE, na.rm=TRUE)
But that doesn't account for the differences in volume for each entry. I might be thinking about this the wrong way, but is there a way to multiply each row by the adjusted_volume number in the dcast function? So the first row would count as 248.5 alnus_spp at site A instead of just 1 record?
Thanks for your help in advance! And sorry if I'm going about this in a ridiculous way!
Edit:
This worked! Thanks all!
x <- ddply(pollen, .(site, pollen_type, treatment, hive_code), summarise, tot_pollen = sum(adjusted_volume))
> head(x)
> site pollen_type treatment hive_code tot_pollen
> A alnus_spp conventional 1 497.0
> A alnus_spp conventional 5 142.0
> A graminaceae_spp conventional 1 29.0
A: I think something like this might get at what you are looking for:
ddply(pollen, .(site, treatment, pollen_type), summarise, tot_pollen = sum(adjusted_volume))
This should summarize the volumes of pollen by site, treatment, and pollen_type.
Good luck!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,335 |
A "Cisco WebEx Network Recording Player Remote Code Execution Vulnerability" exists in Cisco WebEx Network Recording Player for Advanced Recording Format (ARF) and WebEx Recording Format (WRF) files. A remote attacker could exploit this by providing a user with a malicious ARF or WRF file via email or URL and convincing the user to launch the file. Exploitation of this could cause an affected player to crash and, in some cases, could allow arbitrary code execution on the system of a targeted user. Cisco Bug IDs: CSCve10584, CSCve10591, CSCve11503, CSCve10658, CSCve11507, CSCve10749, CSCve10744, CSCve11532, CSCve10762, CSCve10764, CSCve11538. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,258 |
Can he hide from himself?
Author: Ashok Malik
URL: http://www.dailypioneer.com/170285/Can-he-hide-from-himself.html
As Prime Minister Manmohan Singh's serial outbursts over the past week were remarkable not for what he said but why he said it. The 'war of words', to use journalism's oldest chestnut, began when the BJP categorised Mr Singh as a 'night watchman' and a 'caretaker', not the UPA Government's real leader. Mr LK Advani called him the "weakest Prime Minister" in Indian history and challenged him to a debate on governance issues.
True, the BJP's criticism was harsh, but it was not remarkably different from the everyday rhetoric of electoral politics. The party's slogan for the 2009 general election centres on the theme of a 'majboot neta' (strong leader) and to posit the rival candidate as 'majboor' (weak and vulnerable) was a pun only to be expected.
Initially, Mr Singh seemed to take it in his stride. He turned down the idea of a debate with Mr Advani, saying he was not as good a public speaker as the BJP leader. It was a clever, semi-sarcastic line, suggesting Mr Advani was a talker but Mr Singh was a doer.
The matter could have died down there but, as the BJP hammered away with its "weak Prime Minister" slogan, something inside Mr Singh snapped. His response began to get progressively vituperative and almost hysterical. At times, and this was unusual for Mr Singh, he resorted to non sequiturs and factual inconsistencies.
More substantively, Mr Singh began a personal assault on Mr Advani. He called him an "Iron Man" who "melted" under pressure, who couldn't take on terrorism as India's Home Minister, who was "weeping in a corner" when the Babri Masjid-Ram Janmabhoomi structure was demolished. Next, the Congress leadership claimed that, as Prime Minister, Mr Atal Bihari Vajpayee did not trust Mr Advani.
Finally, Mr Singh made his most dramatic statement on April 15, before a group of editors: "Any serious observer knows my remarks on Mr Advani are true. I owe it to myself and to the people of India to show where the shoe pinches. Enough is enough."
A facile explanation of all this would be to say, as some commentators have, that Mr Singh has finally come of age as a politician, that he has learnt the ways of the world and so on. This is nonsense. To be fair to the man, he is not given to wildcat exclamations and bitter, perverse words.
Like a good bureaucrat, Mr Singh is an artful survivor, perhaps the most accomplished in the labyrinths of Lutyens' Delhi. He has spent a career keeping mum when the boss is angry; retreating when the situation is not favourable; getting his way when it is. He does not believe in frontal combat. That is his strength as much as his failing.
In 2007-08, at the height of the India-United States nuclear agreement crisis, the Left and its fellow travellers often suggested Mr Singh was an American agent. His alleged devotion to the British Empire and to the American world order has also been commented upon by Mr Prakash Karat, the CPI(M) general secretary, in recent days.
None of this caused Mr Singh to take things personally. When he gave his famous interview to The Telegraph in August 2007, he was remarkably measured, pointing out that, "I don't get angry, I don't want to use harsh words." He was only arguing for an "honourable deal", he said, that would expand "India's development options, particularly in regard to energy security and environmental protection and … (wouldn't) affect our ability to pursue our nuclear weapons programme".
There was no "enough is enough". The shoe was not pinching because Communist spokespersons were calling the Prime Minister an American lackey. Yet, it did when the BJP called him "weak". Why? It cannot just be because Mr Singh knows there may be a Congress-Left tie-up after the election.
The answer is more complex and altogether simpler: It is because the BJP has spoken the truth. When the Communists accused Mr Singh of being an American stooge, he could shrug it off; he knew - and India knew - the Communists were talking rubbish. When the BJP accuses Mr Singh of being a weak Prime Minister, he cannot shrug it off; he knows - and India knows - the BJP is right.
Mr Singh's reputation is dear to him. He would not like to be reminded that he was not even found weeping in a corner when Sikhs were slaughtered on the streets of Delhi in October-November 1984. He did not take a stand.
Yet, even he cannot escape the feeling that he leaves his country in a mood of greater grimness and pessimism than when he became Prime Minister in 2004. His party may choose to blame it on the global economic slowdown but Mr Singh is intelligent enough to know that is only a fraction of the cause.
As Finance Minister in the 1990s, Mr Singh advocated fiscal prudence; as Prime Minister, he has watched helplessly as his Government has spent its way into a mess. On internal security, he may attack the BJP's Kandahar 'surrender' but he knows the Islamist threat perception that is the UPA's legacy is unprecedented. His administration has left India politically and legally crippled in the face of the terror challenge. Never mind the Press conference bravado.
Finally, there is the question of the Prime Minister's trust. Can Mr Singh put his hand to the Granth Sahib and swear he trusted his Telecom Minister, his Highways Minister, his former Health Minister, his former Home Minister or even the senior Cabinet Minister who was deputed to negotiate with the Left on the nuclear issue and who, according to leaks from Mr Singh's own PMO, was doing a double deal?
Mr Singh is trapped. He cannot look into the mirror and recognise the man he once was. He cheats his conscience. He can only squirm as the Gandhi inheritors compare him to the Mahatma and call him "Sher-e-Punjab". His party, his coalition and his political sponsors have made a laughing stock of him. After the election, he fears he will be dumped as expendable.
There is no point criticising Mr Singh. He deserves a sentiment far more devastating: pity.
(malikashok@gmail.com) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,344 |
4X game Humankind's next patch should arrive this month with balancing changes
Amplitude is aiming to have the next big Humankind patch ready before the end of October
Ian Boudreau
Published: Oct 7, 2021
Another patch is in the works for Civilization VI rival Humankind, Amplitude Studios' recently released 4X game. There's not a lot of information out in the wild about what the next update will include, but the developers say it will include some balance changes, which players should be pleased to learn.
Amplitude informally announced the projected schedule for the next Humankind patch during a livestream on September 30, which was then confirmed by a developer in a thread on the Humankind forums. "Regarding the next patch, we are aiming to release it at the end of the month," wrote user Daarkarr0w, a member of the dev team.
Another Humankind developer who posts under the name The-Cat-o-Nine-Tales provided some additional clarity. September's patch, they write, was primarily focused on rebalancing pollution so that it wouldn't end the game prematurely – although it also brought with it fixes to the naval combat system, which up to that point had not allowed players to lay siege to island cities. Both of those issues have been fixed, but Amplitude had to push some planned updates back to make that happen.
September's patch also addressed an issue that was leading to Game Pass players being "forcibly disconnected from the game" during multiplayer, and Amplitude says it's still working on the desync issues – although Cat-o-Nine-Tales says there's currently no estimate on when a patch will arrive for that.
However, this month's patch will also include some balance changes, possibly to address some of the incredibly powerful effects created by world wonders at the moment. We'll have to wait to see until the beta branch gets updated in the next several weeks.
Humankind Together We Rule review - filling in the blanks
Humankind developer "sure that there will be" a Humankind 2
Here's how Humankind's diplomacy works
First Humankind expansion adds its UN and six 'diplomatic' civs
Humankind could be the best 4X game, but has one big problem to fix
Humankind devs detail all six African DLC cultures for the 4X game
A PC gamer since the 1980s, Ian enjoys strategy games, RPGs, and FPS classics like Unreal and Quake. He's happiest commanding orcs in Total War: Warhammer or diarising his journey in Dark Souls. Prior to joining PCGamesN full time, he contributed essays and reviews to Game Informer, Vice, IGN, PC Gamer, Paste, and others.
Humankind Together We Rule review
The best 4X games on PC 2023
Best strategy games on PC 2023 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,523 |
"""Tests for tensorflow.ops.registry."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.python.framework import registry
from tensorflow.python.platform import test
class RegistryTest(test.TestCase):
class Foo(object):
pass
def testRegisterClass(self):
myreg = registry.Registry('testfoo')
with self.assertRaises(LookupError):
myreg.lookup('Foo')
myreg.register(RegistryTest.Foo, 'Foo')
assert myreg.lookup('Foo') == RegistryTest.Foo
def testRegisterFunction(self):
myreg = registry.Registry('testbar')
with self.assertRaises(LookupError):
myreg.lookup('Bar')
myreg.register(bar, 'Bar')
assert myreg.lookup('Bar') == bar
def testDuplicate(self):
myreg = registry.Registry('testbar')
myreg.register(bar, 'Bar')
with self.assertRaisesRegexp(
KeyError, r'Registering two testbar with name \'Bar\'! '
r'\(Previous registration was in [^ ]+ .*.py:[0-9]+\)'):
myreg.register(bar, 'Bar')
def bar():
pass
if __name__ == '__main__':
test.main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,595 |
Volunteers from the EM2 Locomotive Society spent the Easter weekend stripping Overhead Line Inspection Vehicle (OLIVe) No. DB998900 of charred interior parts following the arson attack last year at the Middleton Railway, Leeds.
The vehicle was then moved to the Keighley & Worth Valley Railway on May 2 to be repaired at Ingrow by the Vintage Carriages Trust, pictured en route near journey's end. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,523 |
Rebecca Tillett is an American college basketball coach who is currently serving as the head coach of the Saint Louis University women's basketball team. She was previously the coach of the Longwood University women's basketball team. She led both Longwood and Saint Louis to their first appearances in the NCAA Division I women's basketball tournament, in 2022 and 2023 respectively.
Career
Griffin graduated from the College of William & Mary in 1999; she was captain of the women's basketball team during her senior year. She began her career coaching high school basketball in Virginia, spending time at Jamestown High School, Osbourn Park High School, and Forest Park High School. In 2013, she took a position as an assistant coach of the Indiana University of Pennsylvania women's basketball team. After one season at IUP, she became an assistant coach at Navy, where she remained for the next four seasons.
In 2018, Tillett became the head coach of the Longwood women's basketball team, her first collegiate head coaching role. She coached at Longwood for four seasons. In the 2021–22 season, she led the team to a 22–12 record and won the Big South Conference tournament, granting Longwood its first appearance in the NCAA Division I women's basketball tournament. Longwood defeated Mount St. Mary's in the First Four before losing to NC State in the first round of the main tournament.
Tillett was hired by Saint Louis University to coach their women's basketball team in 2022. In the 2022–23 season, her first year as head coach, Saint Louis won the Atlantic 10 Conference basketball tournament, earning the team its first appearance in the NCAA Division I women's basketball tournament. She is the second coach in NCAA history, after Lisa Bluder, to reach the NCAA Division I women's basketball tournament with different schools in consecutive years.
Personal life
Tillett's parents are both teachers and sports coaches. Her father was a high school basketball and soccer coach, while her mother coached middle school soccer. Her brothers Nate and Daniel are both high school coaches as well, coaching boy's soccer and girl's basketball respectively.
Head coaching record
References
Living people
American women's basketball coaches
Basketball coaches from Virginia
Basketball players from Virginia
Longwood Lancers women's basketball coaches
Saint Louis Billikens women's basketball coaches
William & Mary Tribe women's basketball players | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,224 |
\section{Introduction}
Deep learning based approaches have recently prevailed in assisting medical image analysis, such as whole slide image classification \cite{zhang2018whole}, brain lesion segmentation \cite{zhang2018ms}, and medical image synthesis \cite{hou2019robust}.
U-Net \cite{ronneberger2015u}, as one of the most popular deep learning based models, has demonstrated its impressive representation capability in numerous medical image computing studies. U-Net introduces skip connections that aggregate the feature representations across multiple semantic scales and helps prevent information loss.\\
\noindent\textbf{U-Net Variants.} Recent works were proposed to extend the U-Net structure with varying module design and network construction, illustrating its potentials on various visual analysis tasks. V-Net \cite{milletari2016v} applies U-Net on higher dimension voxels and keeps the vanilla internal structures. W-Net \cite{xia2017w} modifies U-Net to tackle unsupervised segmentation problem by concatenating two U-Nets via an autoencoder style model. Compared to U-Net, M-Net \cite{mehta2017m} appends different scales of input features to different levels, thus multi-level visual details can be captured by a series of downsampling and upsampling layers. Recently, U-Net++ \cite{zhou2018unetpp} adopts nested and dense skip connections to represent the fine-grained object details more effectively. Moreover, attention U-Net \cite{oktay2018attention} uses extra branches to adaptively apply attention mechanism on the fusion of skipped and decoded features. However, these proposals may involve additional building blocks, which lead to a greater number of network parameters and thus an increased GPU memory. Unlike above variants, our BiO-Net improves the performance of U-Net via a novel feature reusing mechanism where it builds bi-directional connections between the encoder and decoder to make inference from a recursive manner.\\
\noindent
\textbf{Recurrent Convolutional Networks.} Using recurrent convolution to iteratively refine the features extracted at different times has been demonstrated to be feasible and effective for many computer vision problems \cite{han2018image,guo2019dynamic,wang2019recurrent,alom2018nuclei}. Guo \textit{et al}. \cite{guo2019dynamic} proposed to reuse residual blocks in ResNet so that available parameters would be fully utilized and model size could be reduced significantly. Such a mechanism also benefits the evolution of U-Net. As a result, Wang \textit{et al}. \cite{wang2019recurrent} proposed R-U-Net, which recurrently connects multiple paired encoders and decoders of U-Net to enhance its discrimination power for semantic segmentation, though, extra learnable blocks are introduced as a trade-off. BiO-Net distinguishes R-U-Net from its design of backward skip connections, where latent features in every decoding levels are reused, enabling more intermediate information aggregations with gradients preserved among temporal steps. R2U-Net \cite{alom2018nuclei} adopts a similar approach that only recurses the last building block at each level of refinement. By contrast, our method learnt recurrent connections in the existing encoder and decoder rather than recursing the same level blocks without the involvement of refined decoded features.\\
\noindent
To this end, we propose a recurrent U-Net with an bi-directional O-Shape inference trajectory (BiO-Net) that maps the decoded features back to the encoder through the backward skip connections, and recurses between the encoder and the decoder. Compared to previous works, our approach achieves a better feature refinement, as multiple encoding and decoding processes are triggered in our BiO-Net. We applied our BiO-Net to perform semantic segmentation on nuclei segmentation task and EM membrane segmentation task and our results show that the proposed BiO-Net outperforms other U-Net variants, including the recurrent counterparts and many state-of-the-art approaches. Experiments on super resolution tasks also demonstrate the significance of our BiO-Net applied to different scenarios.
\section{Methods}
\begin{figure}[t]
\includegraphics[width=\textwidth]{struct6.pdf}
\caption{Overview of our BiO-Net architecture. The network inferences recurrently in an O-shape manner. \texttt{CONV} represents the sequence of convolution, non-linearity and batch-norm layers. \texttt{DOWN} stands for downsampling (implemented by a \texttt{CONV} followed by max-pooling), while \texttt{UP} denotes upsampling (archived by transpose convolution).} \label{fig1}
\end{figure}
As shown in Fig \ref{fig1}, BiO-Net adopts the same network architecture as U-Net, without any dependencies on extra functional blocks but with paired bi-directional connections. It achieves better performance as \textit{t} increases with no extra trainable parameters introduced during its unrolling process. Moreover, our method is not restricted to U-Net and can be integrated into other encoder-decoder architectures for various visual analysis tasks.
\subsection{Recurrent Bi-directional Skip Connections} \label{bc}
The main uniqueness of our BiO-Net model is the introduction of bi-directional skip connections, which facilitate the encoders to process the semantic features in the decoders and vice versa.\\
\noindent\textbf{Forward Skip Connections.} Forward skip connections linking encoders and decoders at the same level can preserve the encoded low-level visual features $\textbf{f}_{enc}$, with their gradients well-preserved~\cite{ronneberger2015u,he2016deep}. Hence, the $l$-th decoder block could fuse $\textbf{f}_{enc}$ with its input $\hat{\textbf{x}}_{in}$ generated from lower blocks, and propagate them through the decoding convolutions $\texttt{DEC}$ to generate $\textbf{f}_{dec}$, which will be further restored to higher resolutions via $\texttt{UP}$ block. This process can be defined as:
\begin{equation} \label{eq1}
\textbf{f}_{dec} = \texttt{DEC}([\textbf{f}_{enc},\ \hat{\textbf{x}}_{in}]),
\end{equation}
where concatenation is employed as our fusion mechanism [$\cdot$]. The index notation ($l$-th) for encoders and decoders is omitted in this paper for simplicity purpose.\\
\noindent
\textbf{Backward Skip Connections.} With the help of our novel backward skip connections, which pass the decoded high-level semantic features $\textbf{f}_{dec}$ from the decoders to the encoders, our encoder can now combine $\textbf{f}_{dec}$ with its original input $\textbf{x}_{in}$ produced by previous blocks, and therefore achieves flexible aggregations of low-level visual features and high-level semantic features.
Similar to the decoding path enhanced by forward skip connections, our encoding process can thus be formulated with its encoding convolutions $\texttt{ENC}$ as:
\begin{equation} \label{eq2}
\textbf{f}_{enc} = \texttt{ENC}([\textbf{f}_{dec}, \
\textbf{x}_{in}]).
\end{equation}
The \texttt{DOWN} block feeds $\textbf{f}_{enc}$ to subsequent encoders for deeper feature extraction.
\noindent\textbf{Recursive Inferences.} The above bi-directional skip connections create an O-shaped inference route for encoder-decoder architectures. Noticeably, this O-shaped inference route can be recursed multiple times to receive immediate performance gains, and more importantly, this recursive propagation policy would not introduce any extra trainable parameters. Hence, the outputs of encoders and decoders equipped with our proposed O-shaped connections can be demonstrated as follows, in terms of their current inference iteration $i$:
\begin{equation}
\begin{split}
\textbf{x}_{out}^i &= \texttt{DOWN}(\texttt{ENC}([\texttt{DEC}([\textbf{f}_{enc}^{i-1}, \hat{\textbf{x}}_{in}^{i-1}]), \textbf{x}_{in}^i])),\\
\hat{\textbf{x}}_{out}^i &= \texttt{UP}(\texttt{DEC}([\texttt{ENC}([\textbf{f}_{dec}^i, \
\textbf{x}_{in}^i]),\hat{\textbf{x}}_{in}^i])).
\end{split}
\end{equation}
Compared to the vanilla U-Net, our BiO-Net takes both encoded and decoded features into consideration and performs refinement according to features from the previous iterations.
\subsection{BiO-Net Architecture} \label{arch}
We use the plain convolutional layers, batch normalization \cite{ioffe2015batch} layers, and ReLU \cite{nair2010rectified} layers in the network architecture. No batch normalization layer is reused.
\noindent The input image is first fed into a sequence of three convolutional blocks to extract low-level features. Note that there is no backward skip connection attached to the first stage block and, hence, the parameters in the first stage block will not be reused when recursing. The extracted features are then sent to a cascade of encode blocks that utilize max pooling for feature downsampling. During the encoding stage, the parameters are reused and the blocks
are recursed through the paired forward and backward connections as shown in Fig. \ref{fig1}. After the encoding phase, an intermediate stage of which contains convolutional blocks that are used to further refine the encoded features. Then, the features are subsequently passed into a series of decode blocks that recover encoded details using convolutional transpose operations. During the decoding stage, our proposed backward skip connections preserve retrieved features by concatenating them with the features from the same level encoders as depicted in Sec. \ref{bc}. The recursion begins with the output generated from the last convolutional block in the decoding stage. After recursing the encoding and decoding stages, the updated output will be fed into the last stage block corresponding to the first stage blocks. Similarly, the last stage blocks will not be involved in the recurrence.
\noindent We define several symbols for better indication: `\textit{t}' represents the total recurrence time; `$\times$\textit{n}' represents the expansion multiplier times to all hidden output channel numbers; `$w$' represents the number of backward skip connections used from the deepest encoding level; `\texttt{INT}' represents stacking decoded features from each iteration and feeding them into the last stage block as a whole and `$l$' represents the most encoding depth. Details can be seen in Fig. \ref{fig2}.
\section{Experiments}
\begin{figure}[t]
\includegraphics[width=\textwidth]{ablation.pdf}
\caption{Visualizations of setups in the ablation experiments. (a) BiO-Net with $\times$0.5. (b) BiO-Net with $w$ = 2. (c) BiO-Net with \textit{t} = 3 and \texttt{INT}. (d) BiO-Net with $l$ = 3.} \label{fig2}
\end{figure}
\noindent
\textbf{Datasets.} Our method was evaluated on three common digital pathology image analysis tasks: nuclei segmentation, EM membrane segmentation, and image super resolution on a total of four different datasets. Two publicly available datasets, namely MoNuSeg \cite{kumar2017dataset} and TNBC \cite{naylor2018segmentation}, were selected to evaluate our method for nuclei semantic segmentation. The MoNuSeg dataset consists of a 30-image training set and a 14-image testing set, with images of size $1000^2$ sampled from different whole slide images of multiple organs. We extract $512^{2}$ patches from 4 corners of each image, which enlarges the dataset by 4 times. TNBC is comprised of 50 histopathology images of size $512^2$ without any specific testing set. Both datasets include pixel-level annotation for the nuclei semantic segmentation problem. The second task we evaluated is EM membrane segmentation, where the piriform cortex EM dataset of the mice collected from \cite{lee2015recursive}, which contains four stacks of EM images with the slice image sizes of $255^2$, $512^2$, $512^2$, and $256^2$, respectively. Image super resolution is the last task we evaluated our method on, the dataset was constructed from a whole slide image collected by MICCAI15 CBTC. We sampled 2900 patches of size $512^2$ at $40\times$ magnification level, with the 9 : 1 split ratio for the training and testing set, respectively.\\
\noindent
\textbf{Implementation Details.} \label{imp}
We used Adam \cite{kingma:adam} optimizer with an initial learning rate of 0.01 and a decay rate of 0.00003 to minimize cross entropy loss in segmentation tasks and mean square error in super resolution tasks. The training dataset was augmented by applying random rotation (within the range [-15$^\circ$, +15$^\circ$]), random shifting (in both x- and y-directions; within the range of [-5\%, 5\%]), random shearing, random zooming (within the range [0, 0.2]), and random flipping (both horizontally and vertically). The batch size is set to 2 in both training and testing phases. Unless explicitly specified, our BiO-Net is constructed with an encoding depth of 4 and a backward skip connection built at each stage of the network. Our BiO-Net was trained by 300 epochs in all experiments, which were conducted on a single NVIDIA GeForce GTX 1080 GPU with Keras. Given the GPU limitation, we explore the performance improvement to its maximum possible temporal step at \textit{t}=3.
\begin{table}[t]
\caption{Comparison of segmentation methods on MoNuSeg testing set and TNBC.}\label{tab1}
\centering
\begin{tabular}{{l|c c|c c|c | c}}
%
\toprule
\multicolumn{1}{c}{} & \multicolumn{2}{c}{MoNuSeg} & \multicolumn{2}{c}{TNBC}& \multicolumn{2}{c}{}\\
\hline
Methods & IoU & DICE & IoU & DICE &\#params&model size\\
\hline
\hline
U-Net \cite{ronneberger2015u} \textbf{w.} ResNet-18 \cite{he2016deep}& 0.684 & 0.810 & 0.459 & 0.603&15.56 M&62.9 MB\\
U-Net++ \cite{zhou2018unetpp} \textbf{w.} ResNet-18 \cite{he2016deep} &0.683&0.811&0.526&0.652&18.27 M& 74.0 MB\\
U-Net++ \cite{zhou2018unetpp} \textbf{w.} ResNet-50 \cite{he2016deep} &0.695&0.818&0.542&0.674&37.70 M& 151.9 MB\\
Micro-Net \cite{raza2019micro} & 0.696 & 0.819 & 0.544&0.701&14.26 M&57.4 MB\\
Naylor \textit{et al}. \cite{naylor2018segmentation} & 0.690 & 0.816& 0.482 &0.623 &36.63 M&146.7 MB\\
M-Net \cite{mehta2017m} &0.686&0.813&0.450&0.569 &0.6 M&2.7 MB\\
Att U-Net \cite{oktay2018attention} &0.678 & 0.810 & 0.581 & 0.717&33.04 M&133.2 MB\\
\hline
R2U-Net, \textit{t}=2 \cite{alom2018nuclei} & 0.678 & 0.807 & 0.532 & 0.650&37.02 M&149.2 MB\\
R2U-Net, \textit{t}=3 \cite{alom2018nuclei} &0.683 & 0.815& 0.590 & 0.711 &37.02 M&149.2 MB\\
\hline
LinkNet \cite{chaurasia2017linknet} & 0.625 & 0.767 & 0.535 & 0.682&11.54 M&139.4 MB\\
BiO-LinkNet \cite{chaurasia2017linknet}, \textit{t}=2 (Ours) & 0.621 & 0.766 & 0.541 & 0.690&11.54 M&139.4 MB\\
BiO-LinkNet \cite{chaurasia2017linknet}, \textit{t}=3 (Ours) & 0.634 & 0.774 & 0.571 & 0.716&11.54 M&139.4 MB\\
\hline
BiO-Net, \textit{t}=1 (Ours)& 0.680 & 0.803 & 0.456 & 0.608 &15.0 M& 60.6 MB\\
BiO-Net, \textit{t}=2 (Ours)& 0.694 & 0.816& 0.548 & 0.693 &15.0 M& 60.6 MB\\
BiO-Net, \textit{t}=3 (Ours)& 0.700 & 0.821& 0.618 & 0.751 &15.0 M& 60.6 MB\\
BiO-Net, \textit{t}=3, \texttt{INT} (Ours)& \textbf{0.704} & \textbf{0.825}& \textbf{0.651} & \textbf{0.780} &15.0 M& 60.6 MB\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Semantic Segmentation}
\textbf{Nuclei Segmentation.} In this task, our method is compared to the baseline U-Net \cite{ronneberger2015u} and other state-of-the-art methods \cite{raza2019micro,naylor2018segmentation,zhou2018unetpp,mehta2017m,oktay2018attention,alom2018nuclei}. Following \cite{graham2018hover}, models were trained on the MoNuSeg training set only and evaluated on the MoNuSeg testing set and TNBC dataset. Dice coefficient (DICE) and Intersection over Union (IoU) are evaluated. As shown in Table \ref{tab1}, our results are better than others on the MoNuSeg testing set. Our results on the TNBC dataset are also higher than the others by a large margin, which demonstrates a strong generalization ability. Additionally, compared with other extensions of U-Net, and, our BiO-Net is more memory efficient. Qualitative comparison of our method and the recurrent counterpart R2U-Net \cite{alom2018nuclei} is shown in Fig. \ref{fig3}. It can be seen that our model segments nuclei more accurately as the inference time increases. In our experiments, BiO-Net infers a batch of two predictions in 35, 52, and 70 ms when \textit{t}=1, \textit{t}=2, and \textit{t}=3, respectively. Further evaluation of incorporating our method into another encoder-decoder architecture LinkNet~\cite{chaurasia2017linknet}, which is also shown in the table. Our BiO-LinkNet adds the skipped features element-wisely and, hence, shares the same number of parameters as the vanilla LinkNet.
\begin{table} [t]
\caption{Ablative results. The parameters are defined as depicted in Sec. \ref{arch}. IoU(DICE), number of parameters, and, model size are reported.}\label{tab2}
\centering
\resizebox{\linewidth}{!}{\begin{tabular}{{c| c c c |c c c | c | c}}
\toprule
\multicolumn{1}{c}{} & \multicolumn{3}{c}{MoNuSeg} & \multicolumn{3}{c}{TNBC}& \multicolumn{2}{c}{}\\
\hline
& \textit{t}=1 & \textit{t}=2 & \textit{t}=3 & \textit{t}=1 & \textit{t}=2 & \textit{t}=3 & \#params & model size \\
\hline
\hline
$\times$1.25 & 0.685(0.813) & 0.698(0.819)& 0.695(0.817)& 0.490(0.637)& 0.557(0.697)&0.623(0.758)&23.5 M & 94.3 MB\\
$\times$1.0 & 0.680(0.803) & 0.694(0.816) & 0.700(0.821)& 0.456(0.608)& 0.548(0.693)&0.618(0.751)& 15.0 M & 60.6 MB\\
$\times$0.75 & 0.676(0.800)& 0.678(0.805)& 0.691(0.815)&0.516(0.661)&0.571(0.710) &0.598(0.738)& 8.5 M&34.3 MB\\
$\times$0.5 & 0.668(0.792)& 0.680(0.806) & 0.691(0.814)& 0.491(0.644) & 0.543(0.679)&0.611(0.742) & 3.8 M& 15.8 MB\\
$\times$0.25 & 0.667(0.791)& 0.678(0.804)& 0.677(0.804)& 0.524(0.674)&0.535(0.678) &0.575(0.710) & 0.9 M& 4.0 MB \\
\hline
$w$=3 & 0.680(0.803) & 0.694(0.817) & 0.688(0.814) &0.456(0.608) &0.510(0.656)& 0.620(0.757) &15.0 M & 60.3 MB\\
$w$=2& 0.680(0.803) & 0.672(0.801)& 0.686(0.813) & 0.456(0.608) &0.527(0.679)& 0.601(0.742) & 14.9 M & 60.1 MB\\
\hline
\texttt{INT}& 0.680(0.803) & 0.689(0.812)& \textbf{0.704(0.825)}& 0.456(0.608) & 0.588(0.728)&\textbf{0.651(0.780)}&15.0 M &60.6 MB\\
\hline
$l$=3& 0.681(0.806) & 0.679(0.805)& 0.689(0.812) & 0.613(0.742) & 0.594(0.733)& 0.615(0.741) &3.8 M& 15.4 MB \\
$l$=2& 0.690(0.810) & 0.695(0.817) & 0.697(0.818)& 0.596(0.734)&0.647(0.775)&0.596(0.735)& 0.9 M& 4.0 MB\\
\bottomrule
\end{tabular}}
\end{table}
\begin{figure}[]
\centering
\includegraphics[width=\textwidth]{res1-2.pdf}
\caption{Qualitative comparison between our models and the R2U-Net \cite{alom2018nuclei} on the MoNuSeg testing set and TNBC. Red boundary boxes indicate the effects of integrating features from each iteration at the last stage block.} \label{fig3}
\end{figure}
\noindent Table \ref{tab2} demonstrates our ablation study by varying the setups as defined in Fig. \ref{fig2}. The results show that recursing through the encoder and the decoder with the proposed bi-directional skip connections improves network performances generally. Integrating decoded features from all inference recurrences yields state-of-the-art performances in both datasets. Furthermore, we find that when there are insufficient parameters in the network, increasing inference recurrence has little improvement or even makes the results worse. It is also interesting to observe that when constructing BiO-Net with shallower encoding depth, our models perform better on the two datasets than those with deeper encoding depth. \\
\noindent
\textbf{EM Membrane Segmentation.} We further evaluated our method by segmenting Mouse Piriform Cortex EM images \cite{lee2015recursive}, where the models are trained on stack1 and stack2, and validated on stack4 and stack3. The results were evaluated by Rand F-score \cite{arganda2015crowdsourcing}. As shown in Table \ref{tab3}, our method demonstrates better segmentation results with the proposed bi-directional O-shaped skip connections.
\begin{table} [t]
\caption{Comparison of different U-Net variants in EM membrane segmentation.}\label{tab3}
\centering
\begin{tabular}{{c | c c c || c |c c c}}
\toprule
Variants & stack3& stack4 & \#params & Ours& stack3& stack4&\#params\\
\hline
\hline
U-Net & 0.939&0.821&15.56M& BiO-Net, t=1& 0.941&0.827 &15.0M\\
Att U-Net & 0.937&0.833&33.04M &BiO-Net, t=2& 0.955 &0.871 &15.0M \\
U-Net++ &0.940&0.844&18.27M&BiO-Net, t=3&\textbf{0.958}&\textbf{0.887}&15.0M \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Super Resolution}
In addition to segmentation tasks, we are also interested in experimenting with our BiO-Net on a significantly different task: image super resolution, which has been studied actively \cite{chen2018efficient,sui2019isotropic}. In the super resolution task, low-resolution (downsampled) images are used as inputs to train the networks toward their original high-resolution ground truth, which can assist medical imaging analysis by recovering missing details and generating high-resolution histopathology images based on the low-resolution ones. Two state-of-the-art methods, FSRCNN \cite{dong2016accelerating} and SRResNet \cite{ledig2017photo}, are adopted to compare with our BiO-Net. The qualitative results along with the Peak Signal to Noise Ratio (PSNR) score over the entire testing set are shown in Fig. \ref{fig4}. It can be seen that, our method outperforms the state-of-the-art methods by a safe margin, which validates the feasibility of applying our BiO-Net on different visual tasks.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{res2-4.pdf}
\caption{Comparison of different methods in super resolution task. PSNR scores of the methods over the entire testing set are reported as well. The second row is projected from the yellow boundary box to have a closer view on the super resolution achieved in higher resolution.} \label{fig4}
\end{figure}
\section{Conclusion}
In this paper, we introduced a novel recurrent variant of U-Net, named BiO-Net. BiO-Net is a compact substitute of U-Net with better performance and no extra trainable parameters, which utilizes paired forward and backward skip connections to compose a complex relation between the encoder and decoder. The model can be recursed to reuse the parameters during training and inference. Extensive experiments on semantic segmentation and super-resolution tasks indicate the effectiveness of our proposed model, which outperforms the U-Net and its extension methods without introducing auxiliary parameters.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,247 |
{"url":"https:\/\/www.scm.com\/doc.2020\/AMS\/Tasks\/Geometry_Optimization.html","text":"# Geometry optimization\u00b6\n\nA geometry optimization is the process of changing the system\u2019s geometry (the nuclear coordinates and potentially the lattice vectors) to minimize the total energy of the systems. This is typically a local optimization, i.e. the optimization converges to the next local minimum on the potential energy surface (PES), given the initial system geometry specified in the System block. In other words: The geometry optimizer moves \u201cdownhill\u201d on the PES into the local minimum.\n\nGeometry optimizations are performed by selecting them as the Task. The details of the optimization can be configured in the corresponding block:\n\nTask GeometryOptimization\n\nGeometryOptimization\nConvergence\nEnergy float\nStep float\nStressEnergyPerAtom float\nEnd\nMaxIterations integer\nCalcPropertiesOnlyIfConverged Yes\/No\nOptimizeLattice Yes\/No\nKeepIntermediateResults Yes\/No\nPretendConverged Yes\/No\nEnd\n\nGeometryOptimization\nType: Block Configures details of the geometry optimization and transition state searches.\nConvergence\nType: Block Convergence is monitored for up to 4 quantities: the energy change, the Cartesian gradients, the Cartesian step size, and for lattice optimizations the stress energy per atom. Convergence criteria can be specified separately for each of these items.\nEnergy\nType: Float 1e-05 Hartree Energy convergence The criterion for changes in the energy. The energy is considered converged when the change in energy is smaller than this threshold times the number of atoms.\nGradients\nStep\nType: Float 0.01 Angstrom Step convergence The maximum Cartesian step allowed for a converged geometry.\nStressEnergyPerAtom\nType: Float 0.0005 Hartree Threshold used when optimizing the lattice vectors. The stress is considered \u2018converged\u2019 when the maximum value of stress_tensor * cell_volume \/ number_of_atoms is smaller than this threshold (for 2D and 1D systems, the cell_volume is replaced by the cell_area and cell_length respectively).\n\nA geometry optimization is considered converged when all the following criteria are met:\n\n1. The difference between the bond energy at the current geometry and at the previous geometry step is smaller than Convergence%Energy.\n2. The maximum Cartesian nuclear gradient is smaller than Convergence%Gradient.\n3. The root mean square (RMS) of the Cartesian nuclear gradients is smaller than 2\/3 Convergence%Gradient.\n4. The maximum Cartesian step is smaller than Convergence%Step.\n5. The root mean square (RMS) of the Cartesian steps is smaller than 2\/3 Convergence%Step.\n\nNote: If the maximum and RMS gradients are 10 times smaller than the convergence criterion, then criteria 4 and 5 are ignored.\n\nSome remarks on the choice of the convergence thresholds:\n\n\u2022 Molecules may differ very much in the stiffness around the energy minimum. Using the standard convergence thresholds without second thought is therefore not recommended. Strict criteria may require a large number of steps, while a loose threshold may yield geometries that are far from the minimum (with respect to atom-atom distances, bond-angles etc\u2026) even when the total energy of the molecule might be very close to the value at the minimum. It is good practice to consider first what the objectives of the calculation are. The default settings in AMS are intended to be reasonable for most applications, but inevitably situations may arise where they are inadequate.\n\u2022 The convergence threshold for the coordinates (Convergence%Step) is not a reliable measure for the precision of the final coordinates. Usually it yields a reasonable estimate (order of magnitude), but to get accurate results one should tighten the criterion on the gradients, rather than on the steps (coordinates). (The reason for this is that with the Quasi-Newton based optimizers the estimated uncertainty in the coordinates is related to the used Hessian, which is updated during the optimization. Quite often it stays rather far from an accurate representation of the true Hessian. This does usually not prevent the program from converging nicely, but it does imply a possibly incorrect calculation of the uncertainty in the coordinates.)\n\u2022 Note that tight convergence criteria for the geometry optimization require accurate and noise-free gradients from the engine. For some engines this might mean that their numerical accuracy has to be increased for geometry optimization with tight convergence criteria, see e.g. the NumericalQuality keyword in the BAND manual.\n\nThe maximum number of geometry iterations allowed to locate the desired structure is specified with the MaxIterations keyword:\n\nGeometryOptimization\nMaxIterations\nType: Integer Maximum number of iterations The maximum number of geometry iterations allowed to converge to the desired structure.\nCalcPropertiesOnlyIfConverged\nType: Bool Yes Compute the properties requested in the \u2018Properties\u2019 block, e.g. Frequencies or Phonons, only if the optimization (or transition state search) converged. If False, the properties will be computed even if the optimization did not converge.\nPretendConverged\nType: Bool No Normally a non-converged geometry optimization is considered an error. If this keyword is set to True, the optimizer will only produce a warning and still claim that the optimization is converged. (This is mostly useful for scripting applications, where one might want to consider non-converged optimizations still successful jobs.)\n\nIf the geometry optimization does not converge within this many steps it is considered failed and the iteration aborted, i.e. PES point properties block will not be calculated at the last geometry. The default maximum number of steps is chosen automatically based on the used optimizer and the number of degrees of freedom to be optimized. The default is a fairly large number already, so if the geometry has not converged (at least to a reasonable extent) within that many iterations you should step back and consider the underlying cause rather than simply increase the allowed number of iterations and try again.\n\nWhile a geometry optimization aims to find a (local) PES minimum, it may occur that it ends up finding a saddle point instead. The PESPointCharacter property keyword can be used to quickly calculate the lowest few Hessian eigenvalues to determine what kind of stationary PES point the optimization found. More information on this feature can be found on its Documentation Page.\n\nFor periodic systems the lattice degrees of freedom can be optimized in addition to the nuclear positions.\n\nGeometryOptimization\nOptimizeLattice\nType: Bool No Whether to also optimize the lattice for periodic structures. This is currently only supported with the Quasi-Newton, FIRE, L-BFGS and SCMGO optimizers.\n\nFinally the GeometryOptimization block also contains some technical options:\n\nGeometryOptimization\nKeepIntermediateResults\nType: Bool No Whether the full engine result files of all intermediate steps are stored on disk. By default only the last step is kept, and only if the geometry optimization converged. This can easily lead to huge amounts of data being stored on disk, but it can sometimes be convenient to closely monitor a tricky optimization, e.g. excited state optimizations going through conical intersections, etc. \u2026\n\n## Constrained optimization\u00b6\n\nThe AMS driver also allows to perform constrained optimizations, where a number of specified degrees of freedom are fixed to particular values.\n\nThe desired constraints are specified in the Constraints block at the root level of the AMS input file:\n\nConstraints\nAtom integer\nAtomList integer_list\nFixedRegion string\nCoordinate integer [x|y|z] float?\nDistance (integer){2} float\nAngle (integer){3} float\nDihedral (integer){4} float\nSumDist (integer){4} float\nDifDist (integer){4} float\nBlockAtoms integer_list\nBlock string\nFreezeStrain [xx] [xy] [xz] [yy] [yz] [zz]\nEqualStrain [xx] [xy] [xz] [yy] [yz] [zz]\nEnd\n\nAtom atomIdx\nFix the atom with index atomIdx at the initial position, as given in the System%Atoms block.\nAtomList [atomIdx1 .. atomIdxN]\nFix all atoms in the list at the initial position, as given in the System%Atoms block.\nFixedRegion regionName\nFix all atoms in a region to their initial positions.\nCoordinate atomIdx [x|y|z] coordValue?\nConstrain the atom with index atomIdx (following the order in the System%Atoms block) to have a cartesian coordinate (x, y or z) of coordValue (given in Angstrom). If the coordValue is missing, the coordinate will be fixed to its initial value.\nDistance atomIdx1 atomIdx2 distValue\nConstrain the distance between the atoms with index atomIdx1 and atomIdx2 (following the order in the System%Atoms block) to distValue, given in Angstrom.\nAngle atomIdx1 atomIdx2 atomIdx3 angleValue\nConstrain the angle (1)\u2013(2)\u2013(3) between the atoms with indices atomIdx1-3 (as given by their order in the System%Atoms block) to angleValue, given in degrees.\nDihedral atomIdx1 atomIdx2 atomIdx3 atomIdx4 dihedValue\nConstrain the dihedral angle (1)\u2013(2)\u2013(3)\u2013(4) between the atoms with indices atomIdx1-4 (as given by their order in the System%Atoms block) to dihedValue, given in degrees.\nSumDist atomIdx1 atomIdx2 atomIdx3 atomIdx4 sumDistValue\nConstrain the sum of the distances R(1,2)+R(3,4) between the atoms with indices atomIdx1-4 (as given by their order in the System%Atoms block) to sumDistValue, given in Angstrom.\nDifDist atomIdx1 atomIdx2 atomIdx3 atomIdx4 difDistValue\nConstrain the difference between the distances R(1,2)-R(3,4) of the atoms with indices atomIdx1-4 (as given by their order in the System%Atoms block) to difDistValue, given in Angstrom.\n\nNote that the above constraints do not need to be satisfied at the beginning of the optimization.\n\nBlockAtoms [atomIdx1 ... atomIdxN]\nCreates a block constraint (freezes all internal degrees of freedom) for a set of atoms identified by the list of integers [atomIdx1 ...\u00a0 atomIdxN]. These atom indices refer to the order of the atoms in the System%Atoms block.\nBlock regionName\n\nCreates a block constraint (freezes all internal degrees of freedom) for a all atoms in a region defined in the System%Atoms block. Example:\n\nSystem\nAtoms\nC 0.0 0.0 0.0 region=myblock\nC 0.0 0.0 1.0 region=myblock\nC 0.0 1.0 0.0\nEnd\nEnd\nConstraints\nBlock myblock\nEnd\n\n\nFor lattice optimizations, the following constraints can be used on the lattice degrees of freedom:\n\nFreezeStrain [xx] [xy] [xz] [yy] [yz] [zz]\nExclusively for lattice optimizations: Freezes any lattice deformation corresponding to a particular component of the strain tensor. Accepts a set of strain components [xx, xy, xz, yy, yz, zz] to be frozen.\nEqualStrain\u00a0 [xx] [xy] [xz] [yy] [yz] [zz]\nExclusively for lattice optimizations: Accepts a set of strain components [xx, xy, xz, yy, yz, zz] which are to be kept equal. The applied strain will be determined by the average of the corresponding stress tensors components.\n\nNote that in principle an arbitrary number of constraints can be specified and thus combined. However, it is the user\u2019s responsibility to ensure that the specified constraints are actually compatible with each other, meaning that it is theoretically possible to satisfy all of them at the same time. The AMS driver does not detect this kind of problems, but the optimization will show very unexpected results. Furthermore, for calculations involving block constraints the following restrictions apply:\n\n\u2022 There should be no other constrained coordinates used together with block constraints although this may work in many situation.\n\u2022 The user should absolutely avoid specifying other constraints that include atoms of a frozen block.\n\n### Restraints\u00b6\n\nNot all optimizers support constraints. An alternative is to use so-called restraints. These are not exact constraints, but rather a number of springs that pull the system towards the preferred constraints, see Restraints.\n\n## Optimization under pressure \/ external stress\u00b6\n\nPressure or non-isotropic external stress can be included in your simulation via the corresponding engine addons.\n\n## Optimization methods\u00b6\n\nThe AMS driver implements a few different geometry optimization algorithms. It also allows to choose the coordinate space in which the optimization is performed:\n\nGeometryOptimization\nMethod [Auto | Quasi-Newton | SCMGO | FIRE | L-BFGS | ConjugateGradients]\nCoordinateType [Auto | Delocalized | Cartesian]\nEnd\n\nGeometryOptimization\nMethod\nType: Multiple Choice Auto [Auto, Quasi-Newton, SCMGO, FIRE, L-BFGS, ConjugateGradients] Optimization method Select the optimization algorithm employed for the geometry relaxation. Currently supported are: the Hessian-based Quasi-Newton-type BFGS algorithm, the experimental SCMGO optimizer, the fast inertial relaxation method (FIRE), the limited-memory BFGS method, and the conjugate gradients method. The default is to choose an appropriate method automatically based on the engine\u2019s speed, the system size and the supported optimization options.\nCoordinateType\nType: Multiple Choice Auto [Auto, Delocalized, Cartesian] Optimization space Select the type of coordinates in which to perform the optimization. \u2018Auto\u2019 automatically selects the most appropriate CoordinateType for a given Method. If \u2018Auto\u2019 is selected, Delocalized coordinates will be used for the Quasi-Newton and SCMGO methods, while Cartesian coordinates will be used for all other methods.\n\nWe strongly advise leaving both the Method as well as the Coordinate type on the Auto setting. There are many restrictions as to which optimizer and coordinate type can be used together with which kind of optimization. The following (roughly) sketches the compatibility of the different optimizers and options:\n\nOptimizer Constraints Lattice opt. Coordinate types\nQuasi-Newton All, except strain constraints Yes All\nFIRE Fixed coordinates, strain constraints Yes Cartesian\nSCMGO No Yes Delocalized\nL-BFGS No Yes Cartesian\n\nFurthermore for optimal performance the optimizer should be chosen with the speed of the engine: a faster engine in combination should use an optimizer with little overhead (FIRE), while slower engines should use optimizers that strictly minimize the number of steps (Quasi-Newton, SCMGO). This is all handled automatically by default, and we recommend changing Method and Coordinate only in case there are problems with the automatic choice.\n\nThe following subsections list the strengths and weaknesses of the individual optimizers in some more detail, motivating why which optimizer is chosen automatically under which circumstances.\n\n### Quasi-Newton\u00b6\n\nThis optimizer implements a quasi Newton approach [1] [2] [3], using the Hessian for computing changes in the geometry so as to reach a local minimum. The Hessian itself is typically approximated in the beginning and updated in the process of optimization. It uses delocalized coordinates by default both for molecules and periodic systems. The molecular part is based mainly on the work by Marcel Swart [4]. Cartesian coordinates are used in the presence of an external electric field and\/or frozen atom constraints.\n\nThe Quasi-Newton (QN) optimizer supports all types of constraints and can be used for both molecular and periodic systems, including lattice optimizations. For cases where the optimization can be performed in delocalized coordinates, the number of steps taken to reach the local minimum is usually smaller than when optimizing in Cartesian ones. For fast compute engines, the overhead of the QN optimizer can become a bottleneck of the calculation, thus a more light-weight optimizer such as FIRE may give an better overall performance. In principle, a QN optimization in delocalized coordinates may run out of memory for a very large system (say over 1000 atoms) because of the SVD step. However, since it is going to be used for a moderate-to-slow engine we still recommend sticking to it for the benefit of fewer steps. Because of these properties the QN optimizer is the default in AMS for all kinds of optimizations with moderate and slow engines, such DFTB and ADF. It is also used as the optimizer back-end for the PES scan task, the transition state search as well as the calculation of the elastic tensor.\n\nDetails of the Quasi-Newton optimizer are configured in a dedicated block:\n\nGeometryOptimization\nQuasi-Newton\nMaxGDIISVectors integer\nStep\nEnd\nUpdateTSVectorEveryStep Yes\/No\nEnd\nEnd\n\nGeometryOptimization\nQuasi-Newton\nType: Block Configures details of the Quasi-Newton geometry optimizer.\nMaxGDIISVectors\nType: Integer 0 Sets the maximum number of GDIIS vectors. Setting this to a number >0 enables the GDIIS method.\nStep\nType: Block\nTrustRadius\nType: Float Initial value of the trust radius.\nUpdateTSVectorEveryStep\nType: Bool Yes Update TSRC vector every step Whether to update the TS reaction coordinate at each step with the current eigenvector.\n\nThe Quasi-Newton optimizer uses the Hessian to compute the step of the geometry optimization. The Hessian is typically approximated in the beginning and then updated during the optimization. A very good initial Hessian can therefore increase the performance of the optimizer and lead to faster and more stable convergence. The choice of the initial Hessian can be configured in a dedicated block:\n\nGeometryOptimization\nInitialHessian\nFile string\nType [Auto | UnitMatrix | Swart | FromFile | Calculate | CalculateWithFastEngine]\nEnd\nEnd\n\nGeometryOptimization\nInitialHessian\nType: Block Options for initial model Hessian when optimizing systems with either the Quasi-Newton or the SCMGO method.\nFile\nType: String Initial Hessian from KF file containing the initial Hessian (or the results dir. containing it). This can be used to load a Hessian calculated in a previously with the [Properties%Hessian] keyword.\nType\nType: Multiple Choice Auto [Auto, UnitMatrix, Swart, FromFile, Calculate, CalculateWithFastEngine] Initial Hessian Select the type of initial Hessian. Auto: let the program pick an initial model Hessian. UnitMatrix: simplest initial model Hessian, just a unit matrix in the optimization coordinates. Swart: model Hessian from M. Swart. FromFile: load the Hessian from the results of a previous calculation (see InitialHessian%File). Calculate: compute the initial Hessian (this may be computationally expensive and it is mostly recommended for TransitionStateSearch calculations). CalculateWithFastEngine: compute the initial Hessian with a faster engine.\n\nWhile there are some options for the construction of approximate model Hessians, the best initial Hessians are often those calculated explicitly at a lower level of theory, e.g. the real DFTB Hessian can be used the initial Hessian for an optimization with the more accurate BAND engine. Using the CalculateWithFasterEngine keyword can be used to automatically chose a fast engine at a lower level of theory. What the lower level of theory is depends on the main engine used in the calculation: DFTB with the GFN1-xTB model is used as the lower level of theory for relatively slow engines, e.g. DFT based engines. For semi-empirical engines like DFTB or MOPAC, the lower level of theory is currently UFF. If more control over the lower level engine is needed, the initial Hessian can be calculated with a user defined engine and then loaded from file, see this example.\n\n### FIRE\u00b6\n\nThe Fast Inertial Relaxation Engine [5] based optimizer has basically no overhead per step, so that the speed of the optimization purely depends on the performance of the used compute engine. As such it is a good option for large systems or fast compute engines, where the overhead of the Quasi-Newton optimizer would be significant. Note that is also supports fixed atom constraints and coordinate constraints (as long as the value of the constrained coordinate is already satisfied in the input geometry), as well as lattice optimizations (with strain constraints).\n\nFIRE is selected as the default optimizer for fast compute engines if it is compatible with all other settings of the optimization (i.e. no unsupported constraints or coordinate types).\n\nNote\n\nFIRE is a very robust optimizer. In case of convergence problems with the other methods, it is a good idea to see if the optimization converges with FIRE. If it does not, it is very likely that the problem is not the optimizer but the shape of the potential energy surface \u2026\n\nThe details of the FIRE optimizer are configured in a dedicated block. It is quite easy to make the optimization numerically unstable when tweaking these settings, so we strongly recommend leaving everything at the default values.\n\nGeometryOptimization\nFIRE\nAllowOverallRotation Yes\/No\nAllowOverallTranslation Yes\/No\nMapAtomsToUnitCell Yes\/No\nNMin integer\nalphaStart float\ndtMax float\ndtStart float\nfAlpha float\nfDec float\nfInc float\nstrainMass float\nEnd\nEnd\n\nGeometryOptimization\nFIRE\nType: Block This block configures the details of the FIRE optimizer. The keywords name correspond the the symbols used in the article describing the method, see PRL 97, 170201 (2006).\nAllowOverallRotation\nType: Bool Yes Whether or not the system is allowed to freely rotate during the optimization. This is relevant when optimizing structures in the presence of external fields.\nAllowOverallTranslation\nType: Bool No Whether or not the system is allowed to translate during the optimization. This is relevant when optimizing structures in the presence of external fields.\nMapAtomsToUnitCell\nType: Bool No Map the atoms to the central cell at each geometry step.\nNMin\nType: Integer 5 Number of steps after stopping before increasing the time step again.\nalphaStart\nType: Float 0.1 Steering coefficient.\ndtMax\nType: Float 1.0 Femtoseconds Maximum time step used for the integration.\ndtStart\nType: Float 0.25 Femtoseconds Initial time step for the integration.\nfAlpha\nType: Float 0.99 Reduction factor for the steering coefficient.\nfDec\nType: Float 0.5 Reduction factor for reducing the time step in case of uphill movement.\nfInc\nType: Float 1.1 Growth factor for the integration time step.\nstrainMass\nType: Float 0.5 Fictitious relative mass of the lattice degrees of freedom. This controls the stiffness of the lattice degrees of freedom relative to the atomic degrees of freedom, with smaller values resulting in a more aggressive optimization of the lattice.\n\n### SCMGO\u00b6\n\nThe SCMGO optimizer is a new implementation of a Quasi-Newton style optimizer working in delocalized coordinates. In the 2020 release of the Amsterdam Modeling Suite it is still considered experimental and therefore never selected automatically. However, for molecules and fully connected periodic systems it already shows a quite good performance, and could be a reasonable alternative to the classic Quasi-Newton optimizer.\n\nGeometryOptimization\nSCMGO\nContractPrimitives Yes\/No\nNumericalBMatrix Yes\/No\nStep\nEnd\nlogSCMGO Yes\/No\ntestSCMGO Yes\/No\nEnd\nEnd\n\nGeometryOptimization\nSCMGO\nType: Block Configures details SCMGO.\nContractPrimitives\nType: Bool Yes Form non-redundant linear combinations of primitive coordinates sharing the same central atom\nNumericalBMatrix\nType: Bool No Calculation of the B-matrix, i.e. Jacobian of internal coordinates in terms of numerical differentiations\nStep\nType: Block\nTrustRadius\nType: Float 0.2 Initial value of the trust radius.\nVariableTrustRadius\nType: Bool Yes Whether or not the trust radius can be updated during the optimization.\nlogSCMGO\nType: Bool No Verbose output of SCMGO internal data\ntestSCMGO\nType: Bool No Run SCMGO in test mode.\n\nNote that SCMGO also supports different initial Hessians, and uses the same options for the initial Hessian as the Quasi-Newton optimizer, see above.\n\n### Limited-memory BFGS\u00b6\n\nAMS also offers an L-BFGS based geometry optimizer. It usually converges faster than FIRE, but does not support constrained optimizations. For periodic systems it can be quite good for lattice optimizations. The new implementation has not been thoroughly tested yet, therefore never selected automatically. For large systems and fast engines you may want to disable symmetry: simply the detection of (non-existing) symmetry may be a huge overhead.\n\nGeometryOptimization\nHessianFree\nStep\nMaxCartesianStep float\nTrialStep float\nEnd\nEnd\nEnd\n\nGeometryOptimization\nHessianFree\nType: Block Configures details of the Hessian-free (conjugate gradients or L-BFGS) geometry optimizer.\nStep\nType: Block\nMaxCartesianStep\nType: Float 0.1 Angstrom Limit on a single Cartesian component of the step.\nMinRadius\nType: Float 0.0 Angstrom Minimum value for the trust radius.\nTrialStep\nType: Float 0.0005 Angstrom Length of the finite-difference step when determining curvature. Should be smaller than the step convergence criterion.\nTrustRadius\nType: Float 0.2 Angstrom Initial value of the trust radius.\n\nAMS also offers a conjugate gradients based geometry optimizer, as it was also implemented in the pre-2018 releases of the DFTB program. However, it is usually slightly slower than FIRE, and supports neither lattice nor constrained optimizations. It is therefore never selected automatically, and we do not recommend using it. Like L-BFGS, the conjugate gradients optimizer is also configured in the HessianFree block, see L-BFGS section above for details.\n\n## Troubleshooting\u00b6\n\n### Failure to converge\u00b6\n\nFirst of all one should look how the energy changed during the latest ten or so iterations. If the energy is decreasing more or less consistently, possibly with occasional jumps, then there is probably nothing wrong with the optimization. This behavior is typical in the cases when the starting geometry was far away from the minimum and the optimization has a long way to go. Just increase the allowed number of iterations, restart from the latest geometry and see if the optimization converges.\n\nIf the energy oscillates around some value and the energy gradient hardly changes then you may need to look at the calculation setup.\n\nThe success of geometry optimization depends on the accuracy of the calculated forces. The default accuracy settings are sufficient in most cases. There are, however, cases when one has to increase the accuracy in order to get geometry optimization converged. First of all, this may be necessary if you tighten the optimization convergence criteria. In some cases it may be necessary to increase the accuracy also for the default criteria. Please refer to the engine manuals for instructions on how to increase the accuracy of an engine\u2019s energies and gradients. Often this is done with the NumericalQuality keyword in the engine input.\n\nA geometry optimization can also fail to converge because the underlying potential energy surface is problematic, e.g. it might be discontinuous or not have a minimum at which the gradients vanish. This often indicates real problems in the calculation setup, e.g. an electronic structure that changes fundamentally between subsequent steps in the optimization. In these cases it is advisable to run a single point calculation at the problematic geometries and carefully check if the results are physically actually sensible.\n\nFinally it can also be a technical problem with the specific optimization method used. In these cases switching to another method could help with convergence problems. We recommend first trying the FIRE optimizer, as it is internally relatively simple and stable.\n\n### Restarting a geometry optimization\u00b6\n\nDuring a running optimization the system\u2019s geometry is written out to the AMS driver\u2019s output file ams.rkf after every step (in the Molecule section). This means that crashed or otherwise canceled geometry optimizations can be restarted by simply loading the last frame from there using the LoadSystem keyword, see its documentation in the system definition section of this manual:\n\nLoadSystem File=my_crashed_GO.results\/ams.rkf\n\n\nThis can of course also be used to continue an optimization but e.g. with tighter convergence criteria or a different optimizer, as it essentially starts a new geometry optimization from the previous geometry, and does not propagate any information internal to the optimizer (e.g. the approximate Hessian for the Quasi-Newton optimizer or the FIRE velocities) to the new job. As such it might take a few more steps to convergence than if the original job had continued, but allows for additional flexibility.\n\nReferences\n\n [1] L. Versluis and T. Ziegler, The determination of Molecular Structure by Density Functional Theory, Journal of Chemical Physics 88, 322 (1988)\n [2] L. Versluis, The determination of molecular structures by the HFS method, PhD thesis, University of Calgary, 1989\n [3] L. Fan and T. Ziegler, Optimization of molecular structures by self consistent and non-local density functional theory, Journal of Chemical Physics 95, 7401 (1991)\n [4] M. Swart and F.M. Bickelhaupt, Optimization of strong and weak coordinates, International Journal of Quantum Chemistry 106, 2536 (2006)\n [5] E. Bitzek, P. Koskinen, F. G\u00e4hler, M. Moseler and P. 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\section{\label{sec:level1} Introduction}
Unconventional superconductivity is defined by a pairing state which spontaneously breaks both gauge
and crystallographic space group or spin rotational symmetry. In this context the effects of uniaxial strain
can provide a unique probe of the pairing state symmetry, principally by altering the
crystallographic space group, for example from hexagonal or tetragonal to orthorhombic, in a controlled and
repeatable way. Group theory, combined with Ginzburg-Landau symmetry arguments can provide a
clear picture of the
effect of strain on different group representations ~\cite{Volovik_Split_Transition, Sigrist_group_theory, JamesGroupTheory, UPT3_Split}. Specifically a pairing state derived from a degenerate irreducible representation of the symmetry group,
will split into two non-degenerate irreducible representations as a result of the symmetry change induced by uniaxial strain.
This would then be observable by a splitting of the specific heat jump at T$c$ into a double transition. Furthermore, generic arguments about the coupling of the strain to the order parameter would imply that the splitting in T$_c$ will, for small enough strains, be a linear function of the strain.
Uniaxial strain may also be a useful probe of the pairing state
even when the pairing state corresponds to a non-degenerate irreducible
representation. External strain will lead to changes in the physical bond lengths in the crystal, which
in turn would lead to anisotropy in hoping parameters and hence changes in the Fermi surface shape. These effects
will be most pronounced at strains where the Fermi surface undergoes a Lifstitz change in topology, accompanied
by a van Hove Singularity (VHS) in the density of states ~\cite{VanHoveBandfillingTc, JamesVanHoveDoping, Friedel_1989, Markiewicz_1997}. The increase
in density of states at the VHS would normally be expected to lead to a significant peak in T$_c$ as a function of
strain, which would be observed for both conventional $s$-wave or non-degenerate $d$-wave or $s^{\pm}$-type
pairing symmetries.
The original motivation for the theoretical analysis of strain on unconventional superconductors came in the 1980s
in the context of heavy fermion systems, such as UPt$_3$ in which a split transition at T$_c$ was observed~\cite{Volovik_Split_Transition, UPT3_Split}. However, more recently the superconductor Sr$_2$RuO$_4$
has been the focus of attention, because of continued controversies about whether or not it is a chiral
$p$-wave triplet superconductor~\cite{Sr2RuO4Rev, KnightShiftPustogow}. Specifically a detailed set of uniaxial strain experiments \cite{HicksQuadraticStrain, SteppkeStrongTc,ResistivityVHS} did not show the expected linear splitting of T$_c$ which would have
been required by the degeneracy of the chiral $p$-wave state. Nevertheless a strong dependence of
T$_c$ on uniaxial strain was found, with T$_c$ rising from $1.5$K to about $3.5$K at a strain of $\approx -0.5$\%
\cite{SteppkeStrongTc,Tc_plot_not_hicks, ResistivityVHS}.
These experiments have contributed to a growing debate about the \textit{exact} nature of the superconducting state in Sr$_2$RuO$_4$~\cite{contreras2019anisotropic, UltrasoundEviTwo, HicksQuadraticStrain,SteppkeStrongTc, ThermoEvi, AlexPetsch, GrinenkoSplitTRSB, Kivelson_2020,Suh_2020,sophie2022}.
The first evidence that Sr$_2$RuO$_4$ is unconventional was provided by experiments showing the strong suppression of the superconducting state by the introduction of non-magnetic impurities \cite{Sr2RuO4Rev}. This is predicted by Anderson's theorem when applied to non s-wave superconductors \cite{AndersonTheorum}.
More specific evidence that Sr$_2$RuO$_4$ was a chiral triplet $p$-wave superconductor was provided by
the lack of change in spin susceptibility reported in Refs.~ ~\cite{TRSBSCinSrRuO4, PolarizedNeutron}. However more recent measurements of the $c$-axis spin susceptibility show a substantial change which does not support the predictions from the chiral triplet state
~\cite{KnightShiftPustogow, AlexPetsch}.
Independent support for a two component order parameter with degeneracy was provided by the observation of
time-reversal-symmetry breaking (TRSB) at T$_c$ ~\cite{TRSBSCinSrRuO4, KerrEffect}. However the strain experiments gave some indication
that under strain the transition temperature for TRSB might not be identical to T$_c$ \cite{GrinenkoSplitTRSB} at least in strained samples.
This has therefore led to a wide debate about possible pairing states linked to evidence for a two component order parameter model ~\cite{contreras2019anisotropic, ThermoEvi,GrinenkoSplitTRSB, UltrasoundEviTwo}, either with or without degeneracy
at T$_c$ in unstrained samples.
The aim of this paper is to examine the effects of uniaxial strain in a generic microscopic model of
unconventional superconductivity. The original symmetry arguments were made via Ginzburg-Landau theory \cite{JamesGroupTheory} which leads to some general predictions on the behaviour of the order parameter, but are generally not specific
enough to compare with experimental results. Ginzburg-Landau theory can identify
the leading symmetry breaking terms, linear in strain, but cannot predict non-linear higher-order changes. Hence a microscopic model is needed for comparison providing more detail.
Below we examine in detail the effect of strain in a simple negative $U$ model of superconductivity on a square lattice.
We treat strain as a small uniaxial perturbation introduced into the hopping parameters which changes the tetragonal lattice to an orthorhombic lattice breaking the x-y symmetry. For simplicity we consider a one band model. The band parameters
are chosen to be similar to those of the $\gamma$-band of Sr$_2$RuO$_4$ to allow some limited comparison to experiments.
But this is not intended to be a fully accurate model of the full band structure of Sr$_2$RuO$_4$ under strain.
In the following sections we first define our model of the strain dependent band structure, $\varepsilon_k$. We then examine changes in $T_c$ and the gap function corresponding to a variety of different superconducting pairing states, including
$s$-wave, chiral $p$-wave, $d_{x^2-y^2}$, $s^{\pm}$, $d+id$ and $d+ig$.
In cases where the order parameter degeneracy is split by the uniaxial strain we examine the splitting in the self-consistent
order parameter in the two channels. As well as changes to the order parameter symmetry,
strain can also have a significant impact on the transition temperature, $T_c$. There are several possible mechanisms but the simplest is the fact that strain can move the system through a Lifshitz transition in the Fermi surface topology accompanied by a van Hove singularity (VHS) in the density of states at the Fermi level. Other strain related effects on $T_c$ could also include softening of certain phonons related to crystal lattice instabilities, or changes to Fermi surface nesting, the spin-fluctuation spectrum, or moving a system away from or closer to a Mott instability by changing the effective electronic bandwidth. These latter effects are beyond the scope of this paper, where we concentrate on the effects of the Lifshitz transition assuming a constant effective pairing interaction.
\section{Theory}
\subsection{\label{sec:level2} Microscopic Theory}
Here, we present a simple model using a one band negative-U Hubbard model ~\cite{HubbardModel, Hubbard_model_book}, applied to a 2D square or rectangular lattice - upon which we allow electrons to sit on the lattice sites and hop between nearest and next nearest neighbours.
We take the standard form for the Hamiltonian\cite{JRobbingThesis}
\begin{equation}
H = H^0 + V^{(1)} ,
\label{eq:Hubbard_Hamiltonian}
\end{equation}
where $H^0$ is the kinetic term with hopping between electron sites on the lattice accounted for via the hopping integrals $t_{{\bf r},{\bf r}^{'}}$ ~\cite{BDG_book, PaulMillerThesis, JRobbingThesis}
\begin{equation}
H^0 = \sum_{{\bf r},{\bf r}^{'}, \sigma} t_{{\bf r}, {\bf r}^{'}}~c^{\dagger}_{{\bf r} \sigma}c_{{\bf r}^{'}\sigma}
~ \text{.}
\label{eq:H0_Realspace}
\end{equation}
On the square lattice we set $t_{{\bf r}, {\bf r}}=\varepsilon_0 $, $t_{{\bf r},{\bf r}^{'}}=-t$ for nearest neighbours and
$t_{{\bf r},{\bf r}^{'}}=-t'$ for next nearest neighbours. Then uniaxial strain along the $x$ direction modifies the hopping
to $t_x$ for nearest neighbour bonds along $x$ and $t_y$ for nearest neighbour bonds along $y$. The four next neighbour bond lengths remain equivalent so we retain a single hopping parameter, $-t'$. Transforming to $k$-space we have the Hamiltonian
\begin{eqnarray}
H^0 &=& \sum_{{\bf k}\sigma} \Bigg(\varepsilon_0 -2\big( t_x cos(k_x a) + t_y cos(k_y b)\big) \nonumber \\
& & ~~~~~~~ - 4t^{'}\big(cos(k_x a) cos(k_y b)\big) \Bigg) c^{\dagger}_{{\bf k} \sigma} c_{{\bf k} \sigma} \nonumber \\
& =& \sum_{{\bf k} \sigma} \varepsilon_{\bf k} c^{\dagger}_{{\bf k} \sigma} c_{{\bf k} \sigma},
\label{H0_full}
\end{eqnarray}
where $a$ and $b$ are the lattice constants of the rectangular unit cell and the single particle energy band dispersion is $\varepsilon_{\bf k}$.
Results are presented in units of the unstrained hopping parameters, t$_0$ which is evaluated at t$_0$ $=$ 81.62meV.
We assume the general interaction between two particles at positions ${\bf r}$ and ${\bf r}^{'}$, $V^{(1)}$, is given by a generalized Hubbard type interaction
\begin{equation}
V^{(1)} = \frac{1}{2} \sum_{{\bf r}, {\bf r}^{'} \sigma \sigma^{'}}V({\bf r}-{\bf r}^{'}) c^\dagger_{{\bf r} \sigma} c_{{\bf r} \sigma} c^{\dagger}_{{\bf r}^{'} \sigma^{'}} c_{{\bf r}^{'} \sigma^{'}}\ \text{,}
\label{eq:V1_Realspace}
\end{equation}
where the creation and annihilation operators ($c^{\dagger}_{{\bf r}\sigma}$ and $c_{{\bf r}\sigma}$) take their usual meaning \cite{JamesBook}.
Transforming to $k$-space the components of the effective pairing interaction,
given Cooper pairs with zero centre of mass momentum can be written as
\begin{equation}
V^{(1)}= \sum_{{\bf k},{\bf k}^{'}} V_{{\bf k}, {\bf k}^{'}}
c^{\dagger}_{{\bf k} \sigma} c^{\dagger}_{-{\bf k} \sigma^{'}} c_{-{\bf k}^{'} \sigma} c_{{\bf k}^{'} \sigma^{'}}\ \text{,}
\label{eq:V1_k_space}
\end{equation}
where
\begin{equation}
V_{{\bf k}, {\bf k}^{'}} = \sum_{r,r'} V({\bf r}-{\bf r}^{'}) e^{i ({\bf k} - {\bf k}^{'}) \cdot ({\bf r}-{\bf r}^{'})}\ \text{.}
\label{eq:Vkk_prime_space}
\end{equation}
In order to analyse different Cooper pair symmetries it is useful to redefine the pairing interaction
into symmetry distinct pairing channels. To do this
we treat $V_{{\bf k}, {\bf k}^{'}}$ as a linear operator which we define via an eigenvalue equation
\begin{equation}
\sum_{{\bf k}^{'}} V_{{\bf k},{\bf k}^{'}} \Gamma_i({\bf k}^{'}) = U_i \Gamma_i({\bf k}) \ \text{,}
\label{eq:v_kk_prime_eigenvalue}
\end{equation}
where $U_i$ is a eigenvalue corresponding to eigenvector $\Gamma_i({\bf k})$.
The properties of the group representations will ensure that the eigenvectors $\Gamma_i({\bf k})$
can be classified according to the different irreducible group representations of the appropriate symmetry group, in this case either square, or rectangular. The corresponding eigenvalue, $U_i$, is the effective pairing strength in that
symmetry channel.
We use the linear expansion of an operator to rewrite $V_{{\bf k},{\bf k}^{'}}$ as a sum of the basis eigenvectors
\begin{equation}
V_{{\bf k}, {\bf k}^{'}} = \sum_i U_i \Gamma_i({\bf k}) \Gamma_i({\bf k}^{'})\ \text{.}
\label{eq:linear_expansion_of_an_operator}
\end{equation}
Here, we used the fact that the eigenfunctions $\Gamma_i({\bf k})$ can be chosen as real valued functions for the cases of the $D_{4h}$ and $D_{2h}$ point groups, and assumed that the orthogonal eigenfunctions are normalized to obey
$ \sum_{\bf k} \Gamma_i({\bf k})^2 = 1 $.
In principle we have an infinite set of eigenvectors for a general operator, but assuming our original
pairing interaction $V({\bf r},{\bf r}^{'})$ was limited to either on-site, nearest neighbour or next nearest neighbor
bonds, we typically have only a unique eigenfunction or pair of degenerate eigenfunctions for each symmetry channel, as specified in the relevant character tables given in Appendix 1.
We solve the k-dependent BCS gap equation~\cite{BDG_book, SupMetalsandAlloys, schrieffer1999theory}. Using the eigenvector expansion, Eq. \ref{eq:linear_expansion_of_an_operator}, of the pairing interaction
the gap equation can be written in the form
\begin{equation}
\Delta_i = \frac{ U_i}{2} \left( \sum_{\bf k}
\Gamma_i({\bf k})
\frac{\Delta_{\bf k}}{E_{\bf k}} (1 - 2f(E_{\bf k})) \right) ,
\label{eqn:GapEquation}
\end{equation}
where the k-dependent gap function is
\begin{equation}
\Delta_{\bf k} = \sum_i \Delta_i \Gamma_i({\bf k}) .
\end{equation}
and the $\Delta_i$ are the order parameter amplitudes in each pairing channel~\cite{alexandrov2003theory,Tinkham}.
Here, as usual, $E_{\bf k} = \sqrt{\varepsilon_{\bf k}^2 + |\Delta_{\bf k}|^2}$ and $f(E_{\bf k})$ is the Fermi-Dirac function evaluated at eigenenergy $E_{\bf k}$.
Note that the factor of $\frac{1}{2}$ from Eq.~(\ref{eq:V1_Realspace}) has been absorbed into the constant $U_i$.
In the case of
a non-degenerate irreducible representation this gives simply $\Delta_{\bf k} = \Delta \Gamma({\bf k})$ where $\Delta$ is the numerical value of the gap function.
Furthermore, in the case of conventional local BCS singlet $s$-wave pairing
$V_{{\bf k},{\bf k}^{'}}=-U$
we have $ \Gamma({\bf k})=1 $ and
$\Delta_{\bf k} = \Delta$ implying that we recover the usual $s$-wave BCS gap equation \cite{schrieffer1999theory}. Note that the expression for $E_{\bf k}$ is valid for all
equal spin pairings presented here, either spin singlet
pairing (for which $\Delta_{\bf k}=\Delta_{-{\bf k}}$) or
opposite spin triplet pairing (with ${\bf d}_{\bf k}=-{\bf d}_{-{\bf k}} = \Delta_{\bf k}\hat{\bf z} $). More general types of triplet pairing, such as nonunitary states,
would require a more complex gap equation.
The self consistent equation (Eq.~\ref{eqn:GapEquation}) can be solved numerically on a grid in k-space making use of the group tables, table \ref{table:D2h} and \ref{table:D4h}, to choose the corresponding function $\Gamma({\bf k})$ given by table \ref{table:gap}~\cite{JamesGroupTheory}.
Physical insight into the effects of strain on $T_c$ can be
gained by considering the linearized gap equation in which a single order parameter
component $\Delta_i$ is non-zero, but infintesmimally small. Linearizing
Eq. (\ref{eqn:GapEquation})
we recover the usual BCS expression for $T_c$,
\begin{equation}
1 = \frac{U}{2} \int_{-\infty}^{\infty} N(\varepsilon) \frac{1}{\varepsilon} (1 - 2f(\varepsilon)) d\varepsilon\ \text{,}
\label{eqn:GapEquationTc}
\end{equation}
but where the effective density of states is weighted
by the eigenvector $\Gamma_i({\bf k})$ of the specific pairing channel
which becomes non-zero at $T_c$
\begin{equation}
N(\varepsilon) = \sum_{{\bf k}} \Gamma_i({\bf k})^2 \delta(\varepsilon-\varepsilon_{\bf k}) .
\label{eqn:DensityOfStates}
\end{equation}
The presence or absence of a van Hove singularity in this weighted density of states at the Fermi energy ($\varepsilon=0$) will then be a useful guide to
whether or not strain leads to an enhanced $T_c$ in the corresponding pairing channel $\Delta_i$.
\subsection{\label{sec:level2} Strain induced lattice anisotropy}
The remaining consideration is the inclusion of lattice anisotropy resulting from the uniaxial strain. In our approach it is most convenient to simply model this by adjusting the hopping integrals for the $x$ and $y$ oriented bonds in the crystal lattice, as resulting from the shrinking or increasing the physical distance of the bonds in real space.
To better model the physical system we note that the uniaxial strains $\epsilon_{xx}$ and $\epsilon_{yy}$ are not fully independent, but linked by the Poisson ratio. In compressing the $x$-axis an expansion of the y-axis will result from the Poisson ratio of the material (Sr$_2$RuO$_4$: $V_{xy} \approx -0.4$ \cite{HicksQuadraticStrain}).
Representing the changes to the square lattice parameter, $a$, as $\delta_x$ and $\delta_y$ respectively, we therefore assume $\delta_y = V_{xy} \delta_x$, and the corresponding changes to the $x$ and $y$ nearest neighbour hopping integrals will be given by
\begin{equation}
t_y-t_0= V_{xy} (t_x - t_0) ~ .
\label{eq:anisotropichopping}
\end{equation}
To avoid additional unknown parameters we assume, for simplicity, that the next neighbor hopping $t'$ is unchanged by strain. We also assume that the pairing interaction
$V({\bf r}-{\bf r}^{'})$ is unchanged by the strain. The overall strength of the pairing interaction in the unstrained lattice is tuned to give a $T_c$ of about $1.5$K, similar to that of Sr$_2$RuO$_4$, in each channel, $U_i$ considered below.
Finally, note that the strain induced changes to the hopping integrals alter the
band structure $\varepsilon_{\bf k}$ significantly, especially as the van Hove singularity is approached. The changed band energies can lead to an unwanted side effect, changes in the total number of electrons ($N_e$) given by
\begin{equation}
N_e = 2 \sum_{\bf k} f(\varepsilon_{\bf k})\ \text{.}
\label{NumberOfElectrons}
\end{equation}
Clearly the total number of electrons has to remain constant as a function of strain, and so we adjust the on-site energy $\varepsilon_0$ together with the hopping integrals $t_x$ and $t_y$ keeping $N_e$ constant and the chemical potential at zero for all values of the strain.
\section{Results}
\subsection{\label{sec:NormalState} Normal State}
\begin{figure}[t]
\includegraphics[width=42mm]{s_dos.jpg}
\includegraphics[width=42mm]{Lifshits_transistion.jpeg} \\
\includegraphics[width=42mm]{s_Heat_map.jpg}
\includegraphics[width=42mm]{s_wave_tc.jpg}
\caption{\label{fig:top_trans} a) The normal state dos presented at various lattice anisotropies.
b) The energy (red dots) for which the $VHS_{Lifshitz}$ occurs in the normal state as a function strain $t_x/t_0$. The green dots show the Fermi energy as a function of strain which changes marginally due to the self-consistent procedure defined by Eq.~\ref{NumberOfElectrons}. The crossing of both dependencies defines the strain ($t_x = 1.07 t_0$) for which the $VHS_{Lifshitz}$ crosses the Fermi energy.
c) The constructed Fermi Surface for an s-wave model at $t_x = t_0$ (red) and $t_x = 1.08 t_0$ (blue).
d) $T_c$ as a function of strain for conventional s-wave pairing. }
\end{figure}
First, we study the density of states (DOS) in the normal state as a function of strain induced lattice anisotropy (see Fig.~\ref{fig:top_trans}~(a)). The unstrained square lattice (blue line)
has two distinct van Hove singularities (VHS), one corresponding to the upper band edge
and the other associated with the band saddle point in $\varepsilon_{\bf k}$ at ${\bf k}=(\pi/a,0)$ and ${\bf k}=(0,\pi/a)$. As a function of the hopping anisotropy
the upper band edge singularity shifts only slightly relative to the chemical potential ($\varepsilon=0$),
but a more significant change is that the single VHS of the unstrained lattice splits into two distinct peaks associated with the separate band saddle points at ${\bf k}=(\pi/a,0)$ and ${\bf k}=(0,\pi/b)$, now no longer degenerate.
For all anisotropies we can identify two VHS, the one at lower energy gradually crossing the Fermi energy at higher tensions and a second one at high energies. The first gives rise to the Lifshitz transition and we will label it $VHS_{Lifshitz}$. For the uncompressed system its energy is at roughly $15$ $meV$ corresponding to the band at ($k_x, k_y$)$=$($\pi, 0$) in good agreement to experiment~ \cite{Fermi_level_value}. The second VHS moves upwards until eventually it merges with the band edge in our one band model, and this singularity will subsequently be labeled as $VHS_{\varepsilon_{max}}$.
Fig.~\ref{fig:top_trans}~(b) shows the energy of the lower of these two peaks
as it moves down in energy and
crosses the Fermi level at an anisotropy of about $t_x = 1.07 t_0$. This point can be identified as the Lifshitz transition in Fermi surface topology, changing from a closed to open
topology, as shown in Fig.~\ref{fig:top_trans}~(c).
As the Lifshitz transition is a topological change of the Fermi surface which can be caused by applying high pressure \cite{Lifshitz1960ANOMALIESOE} it is important to identify the precise tension for which this transition appears. We summarize this transition in Fig.~\ref{fig:top_trans}~(b), where $VHS_{Lifshitz}$ crosses the Fermi energy at about $t_x = 1.07 t_0$ inducing the Lifshitz transition. This is further underlined by Fig.~\ref{fig:top_trans}~(c)
indicating the opening of the spherical Fermi surface into the neighbouring Brillouin zone for a tension of $t_x = 1.07 t_0$ just above the Lifshitz transition. Strictly speaking, what we visualise in Fig.~\ref{fig:top_trans}~(c) is a contour plot of Eq.~\ref{eqn:GapEquation} for the case of an s-wave $\Gamma(k)=1$ state. This way we are able to visualize the regions of the BZ, which predominantly contribute to the superconducting gap. For an s-wave state on a spherical Fermi surface this procedure essentially traces the Fermi surface. In the following we will see how for different pairing symmetries distinct regions of the Brillouin zone will contribute.
\begin{figure*}[t]
\includegraphics[width=34mm]{d_wave_tc_inset.jpg
\includegraphics[width=34mm]{extended_s_wave_tc.jpg}
\includegraphics[width=34mm]{chiral_p_wave_tc.jpg}
\includegraphics[width=34mm]{chiral_d_wave_tc_inset.jpg}
\includegraphics[width=34mm]{d_plus_ig_tc.jpg}
\caption{\label{fig:Tc_all} $T_c$ as a function of hopping anisotropy
modelling uniaxial strain for
a) d-wave ($d_{x^2-y^2}$) pairing -
$\Gamma_{\bf k} = \cos{(k_xa)} - \cos{(k_yb)} $,
b) extended s-wave pairing $\Gamma_{\bf k} = \cos{(k_xa)} + \cos{(k_yb)} $, (c)
chiral p-wave pairing - $\Gamma_x({\bf k}) = \sin{(k_xa)} $, $\Gamma_y({\bf k})= \sin{(k_yb)} $.
d) $d+id$ pairing - $\Gamma_1({\bf k}) = \cos{(k_xa)} - \cos{(k_yb)}$,
$\Gamma_2({\bf k}) = \sin{(k_xa)}\sin{(k_yb)} $ and,
e) $d+ig$ pairing - $\Gamma_1({\bf k}) = \cos{(k_xa)} - \cos{(k_yb)}$,
$\Gamma_2({\bf k}) = ( \cos{(k_xa)} - \cos{(k_yb)} ) \sin{(k_xa)}\sin{(k_yb)} $, (omitting normalization constants for clarity).
Insets to (a) (d) and (e) are of the value of the zero temperature gap parameters in the two component calculations as a function of the anisotropy. For the pure d-wave, the inset shows a two component system modelling pure d-wave in the x (red) and y(blue) channels. For the $d+id$ and $d+ig$, the channels are color coded such that the $d_{x_2-y_2}$ channels are red, the the $d_{xy}$ and $g$ channels are blue in both the main figure and the insets.
}
\end{figure*}
\subsection{Superconducting $T_c$ and pairing symmetry}
In a first step we present the critical temperature $T_c$ as a function of hopping anisotropy for conventional s-wave pairing in Fig.~\ref{fig:top_trans}~(d). In this case the
BCS equation for $T_c$, Eq. \ref{eqn:DensityOfStates}, applies with the weighting function
$\Gamma({\bf k})=1$. So $T_c$ is determined by the full dos and therefore has a clear maximum at the point when the VHS crosses the Fermi energy, as seen in Fig.~\ref{fig:top_trans}~(b). The increase in $T_c$ near the peak has a maximum at about 2.25K, which is substantial, but not as large as observed in Sr$_2$RuO$_4$ ~\cite{SteppkeStrongTc}.
This discrepancy is to be expected since Sr$_2$RuO$_4$ is not a conventional BCS s-wave superconductor~\cite{Sr2RuO4Rev}.
Now we turn to consider the changes in $T_c$ for all unconventional gap symmetries considered here (see Fig.~\ref{fig:Tc_all}). We see that the d-wave models,
$d_{x^2-y^2}$, $d+id$ and $d+ig$ give the largest increases in $T_c$, as seen in
Figs. ~\ref{fig:Tc_all}~(a), (d) and (e), respectively.
For the conventional BCS s-wave pairing, T$_c$ is enhanced by about $50\%$ relative to the zero temperature. In contrast, the enhancement is about $150\%$ for the two d-wave symmetry models considered.
We can see qualitatively different behavior for the case of extended $s$-wave ($s^\pm$), where $T_c$ decreases smoothly with lattice anisotropy and there is no peak in $T_c$ at any value of lattice anisotropy.
The case of chiral $p$-wave pairing, shown in Fig.~\ref{fig:Tc_all}~(c) is also
qualitatively different. Importantly, and discussed above already, the uniaxial anisotropy breaks the symmetry between the p$_x$ and p$_y$ pairing leading to an enhancement of T$_c$ for $p_y$ pairing and a suppression of T$_c$ for the $p_x$ pairing.
Here, T$_c$ is initially changing only linearly with the changing hopping anisotropy. The linear splitting of the two degenerate $T_c$'s is expected on symmetry grounds~\cite{JamesGroupTheory} (also see Appendix 1), although Ginzburg-Landau symmetry arguments alone cannot explain the unequal slopes of the two $T_c$'s. In marked contrast, for the $d+id$ and $d+ig$ pairing states there is also a splitting of $T_c$ in two channels degenerate at zero strain
but in this case the splitting is quadratic, rather than linear, in strain. The T$_c$ increase in the y channel does not match the expected maximum found experimentally.
The difference arises because in the $p_x+ip_y$ case the zero strain degeneracy is required by symmetry, while in the $d+id$ and $d+ig$ cases the degeneracy is accidental (a point
discussed further in Appendix 1 below). In principle the symmetry breaking anisotropy
also mixes the $d_{x^2-y^2}$ and $s^\pm$ states but, as we show in the inset to Fig.~\ref{fig:Tc_all}(a), the difference between $\Delta_x$ and $\Delta_y$ (where $\Delta_{\bf k}=\Delta_x \cos{(k_x a)}- \Delta_y \cos{(k_y b)}$) is negligible even near the Lifshitz point, and so we can continue to regard the pairing state as having a dominant $d_{x^2-y^2}$ symmetry.
\begin{figure*}[t]
\includegraphics[width=42mm]{chiral_dx_dos.jpg}
\includegraphics[width=42mm]{ext_s_dos.jpg}
\includegraphics[width=42mm]{chiral_dy_dos.jpg}\\
\includegraphics[width=42mm]{chiral_px_dos.jpg}
\includegraphics[width=42mm]{chiral_py_dos.jpg}
\includegraphics[width=42mm]{g_dos.jpg}
\caption{\label{fig:dos-all} Normal state density of states at various anisotropies for a) d-wave ($d_{x^-y^2}$), b) d-wave ($d_{xy}$), c) chiral p-wave ($p_x$), d) chiral p-wave ($p_y$) and e) g-wave pairings. The hopping anisotropies shown are the same as given by Fig.~\ref{fig:top_trans}~(a).
}
\end{figure*}
\begin{figure*}[t]
\includegraphics[width=42mm]{dx_Heat_map.jpg}
\includegraphics[width=42mm]{ext_s_Heat_map.jpg}
\includegraphics[width=42mm]{dy_Heat_map.jpg}\\
\includegraphics[width=42mm]{px_Heat_map.jpg}
\includegraphics[width=42mm]{py_Heat_map.jpg}
\includegraphics[width=42mm]{dig_y_Heat_map.jpg}
\caption{\label{fig:Fermi-all} A snapshot of the change in the constructed Fermi surface - calculated from the k-space points that contribute to the self consistent integration of $\Delta$ from Eq.(\ref{eqn:GapEquation}) plotted at $t_x = t_0$ (red) and $t_x = 1.08 t_0$ (blue) for a) d-wave ($d_{x^-y^2}$), b) d-wave ($d_{xy}$), c) chiral p-wave ($p_x$), d) chiral p-wave ($p_y$) pairings) and e) g-wave. The red contour is the unstrained Fermi surface and the blue contour is the Fermi surface strained past the Lifshitz transition at $\frac{t_x}{t_0} = 1.08$. Coefficients a and b represent the atomic bond length in real space presented here to keep a square Brillouin zone. Unstrained, as expected, a=b. As soon as hopping anisotropy is applied this no longer holds and they are distinctly different.}
\end{figure*}
This different behavior is a consequence of the
fact that the weighted density of states in the $s^\pm$ channel has no VHS singularity at the Fermi level for any lattice strain, Fig.~\ref{fig:dos-all}~(b). There is no peak in $T_c$ because the weighting function $\Gamma({\bf k})^2$ in this channel is zero at the Lifshitz point.
To obtain more insight in the effects of changing symmetries on the DOS we present
in Fig.~\ref{fig:dos-all} the weighted dos (Eq.~(\ref{eqn:DensityOfStates})) for various possible gap symmetries. In that framework Fig.~\ref{fig:top_trans}~(a) represents the corresponding density of states for conventional s-wave symmetry $\Gamma(k)=1$. Among the other symmetries considered only the d-wave ($d_{x^2-y^2}$) symmetry induces two clearly defined VHSs, with one of them crossing the Fermi energy at the Lifshitz point. In the other cases shown, chiral p$_x$, p$_y$, d${_{xy}}$ and g-wave symmetry, the only visible VHS is pushed to higher energies essentially merging with the VHS arising from the band edge. A similar behavior is seen in the case of extended s-wave symmetry (Fig.~\ref{fig:dos-all}~(b)) where the only VHS present moves up towards the band edge, rather than crossing the Fermi level.
Fig.~\ref{fig:dos-all}(d)-(e) shows that the weighted dos for chiral p$_x$ and p$_y$ symmetry are affected strongly but in opposing ways. While the VHS energy moves up with the induced anisotropy for the p$_x$ symmetry, it moves down for the p$_y$. This breaking of the symmetry between the to possible states is a direct result of the breaking of the symmetry via the uniaxial strain we apply. The $d_{xy}$ and g-wave density of states, shown in Figs.~\ref{fig:dos-all}(c) and (f) are relatively unaffected with only minor changes around the Fermi-level resulting from the anisoptropy.
To clarify the origin of these differences in the weighted density of states
in a microscopic picture we show the regions predominantly contributing to the superconducting
gap equation
for all pairing symmetries in Fig.~\ref{fig:Fermi-all}. They are the equivalent to Fig.~\ref{fig:top_trans}~(c) using the respective basis functions for each pairing symmetry. As before the figures are contour plots of the integrand of Eq.~(\ref{eqn:GapEquation}).
From Fig.~\ref{fig:dos-all}~(a) it is clear that
the reason that the $T_c$ for the $d_{x^2-y^2}$ state is enhanced more strongly than for on-site s-wave is the increased weighting of the VHS peak which crosses the Fermi level at the Lifshitz point. In both the total and d-wave weighted densities of states, Fig.~\ref{fig:top_trans}~(a)
and Fig.~\ref{fig:dos-all}~(a),
the single VHS of the unstrained lattice splits into two
peaks, one of which eventually crosses the Fermi energy at the Lifshitz transition.
For the d-wave case the enhancement in the dos is larger than for conventional s-wave pairing, because
the weighting function $\Gamma({\bf k})^2$ for $d_{x^2-y^2}$ pairing has a maximum
in $k$-space at the point ${\bf k}=(\pi/a,0)$. We can see this directly in the
corresponding k-space plot for the d-wave case, Fig.~\ref{fig:Fermi-all}, compared
to the on-site s-wave case Fig.~\ref{fig:top_trans}~(c).
In contrast, there is no peak in $T_c$ at the Lifshitz point in any other pairing channel
shown in Fig.~\ref{fig:Tc_all} as a direct result of the absence of a VHS crossing the Fermi level
in the respective weighted densities of states in Figs.~\ref{fig:dos-all}~(b)-(f). For example, the extended s-wave pairing case
has no enhancement of the weighted density of states at the Lifshitz point
(Fig.~\ref{fig:dos-all}~(b)) because the corresponding weighting function $\Gamma({\bf k})$
has nodal lines which pass through the Lifshitz point, Fig.~\ref{fig:Fermi-all}(b).
Similar nodal lines exist for the $p_x$, $p_y$, $d_{xy}$ and $g$ weighting functions, visible in
Fig.~\ref{fig:Fermi-all}(c)-(f). The splitting in the
$T_c$ for $p_x$ and $p_y$ pairing can be understood from the changes in the
weighted density of states in these pairing channels,
with one of the two channels having a slightly
enhanced dos at the Fermi energy.
The small linear
increase in the upper $T_c$ in Fig.~\ref{fig:Tc_all}(c) is consistent with the
small enhancement in the dos shown in Fig.~\ref{fig:dos-all}(e).
The fact that the lower $T_c$ drops rapidly for even the very small strain values
shown in Fig.~\ref{fig:Tc_all}(c) is a consequence of the non-linearity in the coupled $p_x$, $p_y$ gap equation, Eq.~\ref{eqn:GapEquation}. This
difference can also be seen in the $p_x$ and $p_y$ k-space plots of Figs.~\ref{fig:Fermi-all}(d)-(e).
The much stronger splitting in the $T_c$ values for the $d+id$ case is a consequence
of the existence of the strong VHS singularity crossing the Fermi level in the $d_{x^2-y^2}$
pairing channel, which is totally absent in the $d_{xy}$ channel (Figs.~\ref{fig:dos-all}(a) and (c)). Again the nodal points of the $d_{xy}$ pairing function (Fig.~\ref{fig:Fermi-all}~(c))
coincide with the Lifshitz point, while the nodal lines of the $d_{x^2-y^2}$ gap are
far from the Lifshitz point, Fig.~\ref{fig:Fermi-all}(a). Therefore, $T_c$
strongly increases in the $d_{x^2-y^2}$ channel. The $d_{xy}$ $T_c$ was chosen to be
degenerate with the $d_{x^2-y^2}$ channel at zero strain, but that ``accidental'' degeneracy
is not preserved once the strain is applied. Based on the weighted density of states alone we would not expect such a strong decrease in $T_c$ for the $d_{xy}$ channel, but at temperatures below the upper of the two transition temperatures the non-linearity of the gap equation (Eq.~\ref{eqn:GapEquation}) means that an enhancement in pairing in one channel is accompanied by
a suppression of the pairing in the other channel. Very similar behavior is found in the $d+ig$ scenario~\cite{Kivelson_2020}, which also assumes an accidental degeneracy present in the unstrained samples, which is no longer the case in the presence of uniaxial strain. Interestingly, for our model parameters
the $d_{xy}$ and $g$ pairing channels eventually become the higher of the two $T_c$'s for the
$d+id$ and $d+ig$ states for strains beyond the VHS topological transition point, Figs.~\ref{fig:Tc_all}(d) and (e) .
\section{Discussion}
We now consider the above results in the context of the recent uniaxial strain experiments on
Sr$_2$RuO$_4$ ~\cite{HicksQuadraticStrain,SteppkeStrongTc,GrinenkoSplitTRSB,Tc_plot_not_hicks,ResistivityVHS}.
Firstly we note that the two dimensional one band Hubbard model has been utilised here due to its simplicity. Clearly a more realistic three dimensional 3-band model gives a more accurate overall picture for Sr$_2$RuO$_4$ ~\cite{ReenaPaper, JamesInterlaying}. Nevertheless, only one of the three Sr$_2$RuO$_4$ Fermi surface sheets, the $\gamma$ sheet, lies sufficiently close to the topological
Lifshitz transition to be tunable through the transition in experiment ~\cite{SteppkeStrongTc,ResistivityVHS}, and so it remains a reasonable assumption that a single band model can capture the relevant physics near to the
topological transition. The experiments also appear to be consistent with the simpler two-dimensional band model. In a full three dimensional band the logarithmic VHS in the dos is split into two cusp singularities~\cite{VanHove}. The fact that the overall band structure of Sr$_2$RuO$_4$ is highly two dimensional~\cite{Sigrist_group_theory} as well as the fact that experimental strain measurements~\cite{ResistivityVHS} seem consistent with a single Lifshitz point (as in 2d) rather than a pair (as in 3d) implies that the assumption of a two-dimensional band seems well justified.
In the full three band model there is
also a significant spin orbit coupling, SOC, associated with the Ru d-orbitals, which is not included in the one band model ~\cite{James_SO, S_O_Coupling_one}. SOC is likely to be most significant in the case of spin triplet pairing states, but even in the case of singlet d-wave pairing properties such as the low temperature Knight shift can be strongly influenced by the SOC interaction~\cite{ReenaPaper}. However the general symmetry principles (discussed below in Appendix 1) relating to the Ginzburg-Landau theory near to $T_c$ are not affected by SOC at all in the case of singlet pairing~\cite{JamesGroupTheory}. For the triplet case we can also omit explicit SOC from the model, provided that the effective SOC energy scale is stronger than the energy scale for pairing~\cite{James_SO, S_O_Coupling_one}. For example, this will be true if the
SOC simply locks the triplet ${\bf d}-{\bf k}$ vector to a specific orientation, such as $\hat{\bf z}$ as in the chiral $p_x+ip_y$ state examined above.
A further complication in a more realistic multiband model is that it would require changing many unknown parameters in the model corresponding to the change caused by uniaxial hopping anisotropy.
This contrasts with the simplicity of the one band model which
has the advantage of requiring only one significant parameter which we must take from experiment,
effectively related to the Poisson ratio of $x$ and $y$ uniaxial strains.
Under our assumption, the changes in the second neighbor hopping integral $t'$ are negligible compared to the changes in $t_x$ and $t_y$ arising from the uniaxial strain. Then the only
strain related variables become $t_x$ and $t_y$, which are related by the assumption
that both hoppings vary linearly in the corresponding strain and that the ratio
of these variations is given by the Poisson ration, Eq.~\ref{eq:anisotropichopping}.
We present our results in the previous section as functions of the hopping anisotropy $t_x/t_0$, rather than specific physical strain values of Sr$_2$RuO$_4$. Comparisons to the case of
Sr$_2$RuO$_4$ then simply need to map from our critical value for the Lifshitz transition ($t_x \approx 1.07 t_0$) to the experimental strain where the transition is observed, as about $\epsilon_{xx} \sim - 0.5 \%$~\cite{SteppkeStrongTc,ResistivityVHS}. With this comparison of scales the
ranges of lattice strains plotted in Figs.~\ref{fig:Tc_all}-\ref{fig:dos-all} coincide
with the ranges of compressive lattice strains from $0$ to $-1\%$ studied
experimentally~\cite{HicksQuadraticStrain}. For the anisotropy dependence of $T_c$ in the case of the extended s-wave pairing Fig.~\ref{fig:Tc_all}~(b) the figure has been cut of at the strain at which the gap function becomes zero.
Above we have laid out the assumptions for treating the potential $V(r,r^{'})$ as a tune-able constant. However it should be noted that this could be better treated including the position dependence since the bond length will change with strain. The strains used here are small and hence we have assumed the strain to have a low impact on the change of the potential. It would, of course, be straightforward to allow the pairing interaction to explicitly depend on the lattice strain, but at the expense of adding another unknown parameter into the model.
With the above caveats aside; we move onto the interesting results obtained by
comparing our results, Fig.~\ref{fig:Tc_all}, with the recent experiments.
We can confidently say that within the confines of this model, the $d+id$ and $d+ig$ pictures of superconductivity in $Sr_2RuO_4$ match well with experimental data in comparison to the behaviour of other pairings presented. Furthermore, the d-wave picture maps out very well in comparison to the experimental data presented ~\cite{Tc_plot_not_hicks, SteppkeStrongTc, ResistivityVHS} as shown in Figure \ref{fig:Tc_all}~(d)-(e). The conventional one component d-wave model also reproduces the increase in $T_c$ however it shows total suppression of $T_c$ after the Lifshitz transition which is not in agreement with experiment unlike the $d+id$ and $d+ig$ models. The proposed $d+id$ and $d+ig$ states also conform with experimental evidence that $Sr_2RuO_4$ is a spin singlet, two component order parameter system with a possible TRSB state ~\cite{VanHoveBandfillingTc,KnightShiftPustogow, AlexPetsch,UltrasoundEviTwo,ThermoEvi, GrinenkoSplitTRSB,TRSBSCinSrRuO4,KerrEffect}.
We note that there is a remarkably small discrepancy, of about $0.2$K, between the calculated value of $T_c^{MAX}$ as we move through the topological transition, Fig.~\ref{fig:Tc_all}(d)-(e) compared to experiment~\cite{Tc_plot_not_hicks,SteppkeStrongTc, ResistivityVHS}. This discrepancy
can most likely be attributed to the simplified choices of parameters we made in the tight-binding model
(for example the neglect of strain induced changes in $t'$ in $V({\bf r}-{\bf r}')$). Alternatively this discrepancy may point perhaps to something more profound requiring the full three band solution of the gap equation
or strong correlation effects related to the changed topology of the
full Fermi surface.
We note that the peak in $T_c$ is roughly similar for both the
pure $d_{x^2-y^2}$ pairing state, the $d+id$ pairing state and for the $d+ig$ pairing state, see Figs.~\ref{fig:Tc_all}~(d)-(e).
In the above cases the leading factor in the changes in $T_c$ appears to be arising from the approach of the topological VHS to the Fermi energy.
The VHS moves \textit{towards} the Fermi Energy causing an increase in the density of states at the Fermi Energy as shown in Fig.~\ref{fig:dos-all}(a). This corresponds to the maximum of $T_c$ at the Lifshitz point and then a decrease in $T_c$ once the VHS has moved through the Fermi Energy and the effective dos decreases. For the pure $d_{x^2-y^2}$ pairing state
we find that $T_c$ drops well below the original $1.5$K for anisotropies
corresponding to strains of about $\epsilon_{xx} \sim -1\%$, but this is not seen experimentally. In experiments the $T_c$ appears to level off at about $1.2$K
for strains of about $\epsilon_{xx} \sim -1\%$~\cite{Tc_plot_not_hicks}. This behavior is closer to what we find for the $d+id$ and $d+ig$ cases, shown in Fig.~\ref{fig:Tc_all}(d)-(e). As is clear
from that figure the large strain limit is dominated by $d_{xy}$ or $g$ pairing rather than $d_{x^2-y^2}$
as they cross over at some value of the strain after the Lifshitz transition. Exactly where this occurs will depend on the degeneracy (assumed accidental in our model) leading to the
$d+id$ or $d+ig$ pairing state. The $d+id$ and $d+ig$ pairing states also correspond to superconducting states
with TRSB, which would be consistent with $\mu$SR and Kerr effect experiments~\cite{TRSBSCinSrRuO4,KerrEffect}. However, note that the accidental degeneracy
required for these states must somehow be preserved under both isotropic strain and disorder
~\cite{UnsplitSCandTRSB}. This seems unlikely for an accidental
degeneracy, but might be possible if there is some higher "hidden" symmetry present beyond that simply required by the $D_{4h}$ lattice space group (for example $d_{x^2-y^2}$ and $d_{xy}$ are degenerate in hexagonal, $D_{6h}$, or cylindrical, $D_{\infty h}$, point groups).
A note of interest for the 'd-wave' models considered here
is that once the lattice is strained, the d-wave model has an automatic
anisotropy between $x$ and $y$ directions, which provides an implicit mixing
of two non degenerate order parameters $d_{x^2-y^2}$ and $s^\pm$. However the calculations shown in Fig.~\ref{fig:Tc_all}(a) that this mixing is almost negligible at the strains considered, and so it is still meaningful to consider a ``pure'' $d_{x^2-y^2}$
paring symmetry even in the slightly
strained lattice.
Note that the other model we considered here with TRSB the $p_x+ip_y$ state
has a degeneracy required by symmetry and so would not be ruled out by the
experiments of Grimenko {~\cite{UnsplitSCandTRSB}}.
The chiral-p dos shows the maximum of the dos for $p_y$ moving towards the Fermi energy and $p_x$ moving away from the Fermi energy, but neither $p_x$ or $p_y$ has a significant peak in the weighted
density of states, Fig.~\ref{fig:dos-all} and so there is no significant uplift in $T_c$
near the Lifshitz point.
Our calculated
$T_c$ splittings in Fig.~\ref{fig:Tc_all}(c) are a poor match to the experiments. Furthermore, they show linear not quadratic change in the higher $T_c$ value with strain, inconsistent with experiment ~\cite{HicksQuadraticStrain}, and secondly the two $T_c$ values would lead to a double
peak in specific heat, which is not seen experimentally~\cite{Tc_plot_not_hicks}. And finally, the T$_c$ past the Lifshitz transition is not in agreement with experiment.
Note that we have not explicitly calculated the different $d+id$ state with
TRSB symmetry ($d_{xz}+i d_{yz}$) since this requires a three dimensional
pairing model~\cite{ReenaPaper,Suh_2020}
and cannot be realised in the 2-dimensional Fermi surface model considered here
(there is a symmetry required
nodal gap in the $k_z=0$ plane). But based on the symmetry arguments of
Appendix 1 and the analogy to the $p_x+i p_y$ case we believe that this state would
also be a poor match to experiment, for example with a linear not
quadratic $T_c$ splitting for small strains.
In summary, despite its simplicity, the one band model has accurately reproduced experimental data and provided insight into what happens through the Lifshitz transition.
In Fig.~\ref{fig:top_trans}~(c) and Fig.~\ref{fig:Fermi-all} we can see the nodal lines on the Fermi surface for each type of pairing symmetry considered here. When the nodal lines match up with the opening of the VHS, we see a decrease in $T_c$ seemingly from the fact that there simply are not enough available electrons to contribute to the superconducting state in the crucial energy range. In contrast, the only pairing states which are expected
to give a major uplift in $T_c$ are either pure or mixed states with a dominant $d_{x^2-y^2}$
component at the maximum $T_c$. The simple model therefore
appears as a powerful tool but should be tested further, perhaps combined with density functional theory to model more precisely a full band system in a tetragonal and strained lattice.
It would also be useful to apply these same techniques to other unconventional superconductors,
since uniaxial strain provides a unique and direct probe of the pairing state symmetry.
\begin{acknowledgments}
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant S100154-102 for the University of Bristol, Faculty of Science.
This study was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol http://www.bris.ac.uk/acrc/.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,546 |
Dedication
For Kerensa
Contents
Dedication
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-One
Chapter Twenty-Two
Chapter Twenty-Three
Chapter Twenty-Four
Chapter Twenty-Five
Chapter Twenty-Six
Chapter Twenty-Seven
Acknowledgments
About the Author
Also by Sarah Fox
Copyright
About the Publisher
Chapter One
STANDING OVATIONS NEVER got old. Not for me, at least. As Maestro Hans Clausen flicked his baton to signal the end of Sergei Rachmaninoff's Symphony No. 1 in D Minor, the audience rose amid a thundering of applause. A thrill of happiness ran up from my toes, right out to the tips of my fingers. We'd pulled off a successful opening of another season for the Point Grey Philharmonic.
My fellow musicians and I stood as one to acknowledge the audience. I soaked in the appreciative applause that filled the theater, enjoying every roaring second. In time the noise died down and the audience members jammed up into bunches as they tried to file out of their rows and head for the lobby.
With the first concert of the new season truly over, I scooped up my folder of music and wended my way through chairs, music stands, and other musicians until I reached the wings of the stage. From there I made slow progress as I headed down a carpeted hallway with at least twenty other members of the orchestra, many of whom were walking slowly to chat with one another and created a human traffic jam.
Eventually I reached the musicians' lounge where we stored our instrument cases and other belongings during concerts and rehearsals. I tucked my violin and bow safely away in their case and placed it in my locker. I would take my instrument home with me later, but the night wasn't yet over.
"Ready to head to the reception?" Mikayla Deinhardt, my friend and stand partner, leaned against the neighboring locker.
I unfastened the clip at the back of my head and let my dark hair fall over my shoulders. "Almost." I tossed the clip onto the shelf in my locker and shut the door.
As I ran my fingers through my hair to make sure it was free of tangles, first violinist Janine Ko removed a hot pink handbag from her locker. Aggie, a viola player, and Melissa, a flautist, immediately zeroed in on her.
"Oh my gosh! That's gorgeous!" Aggie gushed.
"Is it a Michael Kors bag?" Melissa asked.
Janine beamed at them. "Yes."
The women continued to chatter excitedly about the handbag as I secured my locker door with a combination lock.
"Now I'm ready," I said to Mikayla. But as I turned around, a cascade of blond hair swatted me in the face.
Wincing, I stepped back and hit the bank of lockers. Mikayla grabbed my arm to steady me as I wavered off balance. I blinked and saw Elena Vasilyeva, the Point Grey Philharmonic's concertmaster, fixing her long and ridiculously gorgeous blond hair right in front of me.
I glared at the back of her head, my distaste for her stemming from far more than getting swatted in the face by her golden locks. Finished with her hair, she now stood with her hands on her hips, talking to two of her fellow first violinists.
"It's probably a knockoff," she said, her accented words disdainful. "There's no way she could afford a real one."
I realized that she was referring to Janine and her handbag. Unfortunately, Janine realized that too. Her smile faltered and she returned the handbag to her locker, turning her back on the rest of the room.
Anger bubbled up inside of me as Elena swept past Janine and out of the musicians' lounge, walking—as always—as if she were strutting along a catwalk in a fashion show.
I growled under my breath, my eyes following Elena until she disappeared from view.
Why did she always have to be so snooty? Even if I hadn't discovered that she was involved with the man I'd fallen for a few months earlier, I still wouldn't have liked her. She always acted as if she were superior to everyone else. And poor Janine. Elena's words must have hurt all the more because Janine idolized her. She always hung on the concertmaster's every word and tried to emulate her hairstyles and fashion choices. Sometimes I wanted to shake Janine. She'd be far better off just being herself.
"Forget about Elena," Mikayla said, noting my reaction and giving my arm a tug. "Let's go to the reception."
I let her pull me toward the door and we joined the trickle of musicians heading for the theater's swanky reception room. I did my best to push Elena from my thoughts, not wanting to let her ruin my night. The successful concert had left me with a happy buzz running through my body and I was looking forward to the next part of the evening. Although the Point Grey Philharmonic didn't follow every concert with a reception, doing so was a tradition for the opening of each season. For our benefactors and season ticket holders, it was a chance to mingle with the musicians and the board of directors. For me, it was a chance to partake of some free food and the occasional glass of champagne.
When we reached the reception room with its red carpet, floor-to-ceiling windows, and fancy arched ceiling, my eyes went immediately to the food tables. I let out a quiet sigh of disappointment when I realized that the spread didn't include finger sandwiches. Oh well. Mini sandwiches were my favorite party food of all time, but the tables still held an array of other tasty morsels I wouldn't hesitate to sample.
Several nonmusicians had already arrived and waiters dressed in black and white glided through the room, balancing trays of filled champagne flutes. I nodded a greeting at Dr. Daniel Beaufort, the vice chair of the PGP's executive committee, and aimed myself at the food tables, Mikayla at my side.
"I'm starving," I whispered to her, my eyes on the spread of catered food.
Mikayla grabbed my arm to halt my progress. Just in time, apparently. A portly, elderly man stepped into our path, hunched over a cane with a silver handle.
"Good evening, ladies."
With reluctance that I tried not to show, I tore my eyes away from the free food and focused on the man before us.
"Hello, Mr. Major," I said as Mikayla added a greeting of her own.
The man smiled, apparently pleased that we knew who he was. It would have been hard not to know, though. I'd never spoken to him before, but I knew perfectly well that he was the PGP's most generous individual financial supporter. For that reason it was probably a good idea not to brush him off in favor of an enthusiastic attack on the generous plates of food spread out behind him.
"How are you tonight?" I asked.
"Very well indeed. Particularly because of the delightful company."
As his watery, pale blue eyes raked over me and Mikayla, a smile that could only be described as lecherous pulled at his thin, dry lips. I wanted to gag, but managed to refrain for the sake of the orchestra's financial future.
"Would you ladies like to join me in a glass of champagne?" He gestured to the nearest waiter, who came over and presented us with a tray of champagne glasses.
"Of course," Mikayla said. She smiled at the elderly man, but I could tell she enjoyed his company about as much as I did.
We each took a champagne flute from the tray, and the waiter disappeared into the growing crowd.
Mr. Major raised his glass. "To beautiful music and . . ." His eyes roamed over our bodies again. " . . . to even more beautiful musicians."
I stifled another gag and managed a weak, insincere smile before sipping at my champagne. I would need something stronger if I was expected to spend much more time with the old sleazebag.
"Evening, ladies, Mr. Major." Maestro Hans Clausen appeared by my side, a charming smile on his face. "I hope you enjoyed the concert," he said to Major.
"As always."
Hans put a hand to the middle of my back as he addressed Major again. "I'm sorry to steal Midori and Mikayla away from you, but I need to have a quick word with them."
"Of course." Mr. Major raised his champagne flute to me and Mikayla once more as the maestro ushered us several feet away.
As we came to a stop next to a pedestal displaying a bust of Beethoven, I stepped to the side so the maestro's hand no longer rested on my back. "You need to speak to us?" I might have sounded suspicious, and I was. What could be so important that it couldn't wait until another time?
"Actually, no." Hans flashed his charming smile again. "But I thought you might need rescuing from Mr. Major. He's known for his . . . rather inappropriate interest in young, beautiful women."
"Thank you, Maestro," Mikayla said. "Much appreciated. He did have the effect of making me feel a desperate need for a thorough shower." Her eyes drifted to Dave Cyders, one of our bassoonists, where he stood across the room. "Will you excuse me?"
I wanted to grab her arm to hold her there but she was already gone, leaving me alone with the man I'd had a short-lived relationship with the previous spring. Until I'd discovered that he was a jerk and a liar. His good looks and charismatic smile had charmed me in the beginning, but finding out that he was carrying on with Elena at the same time had doused the flames of attraction with icy water.
Since then I'd managed to maintain a professional relationship with him for the sake of my career, but I still wasn't keen on spending time alone with him. Mikayla knew that full well, and yet she'd abandoned me for her bassoonist boyfriend. I shot a glare at her retreating back before returning my attention to Hans.
"I'm sure we could have handled Mr. Major on our own, but thank you anyway." I stepped toward the food tables, intending to distance myself from Hans, but he had other ideas.
"Midori." He put a hand to my elbow to stop me.
I sighed, perhaps somewhat dramatically, and turned back to face him. "I thought you didn't need to speak with us."
"Well, no." He rubbed the back of his neck with one hand, reminding me of how I used to like running my fingers through the blond hair at the base of his skull when we kissed. "But I was hoping to tell you something."
I waited.
"Elena and I have broken things off. For good this time."
I blinked at him. "I'm not sure how I'm supposed to respond to that. Sorry? Congratulations?"
"I just wanted you to know."
"I don't know why."
It was his turn to sigh, but that didn't move me, nor did the disappointment in his ice blue eyes.
"I thought we'd come to an agreement," I said. "Months ago."
"We did."
"Then let's stick to it. Besides, I'm seeing someone." I didn't wait around to see his reaction to that news. "Excuse me."
Relieved to have extracted myself from that conversation, I finally made it over to the food, grumbling to myself in my head as I went. Did Hans expect me to throw myself into his arms? There was no way that would happen. Besides, Elena had once told me that Hans always went back to her. Why would I believe things would be any different this time? Even if their breakup really was final, I'd never go back to someone who'd treated me as Hans had, even if I wanted to. Which I most definitely didn't. I'd well and truly moved on, and I wished he would do the same.
Pushing thoughts of Hans from my mind and focusing on the enticing spread of food, I bypassed the mini quiches and zeroed in on the colorful petits fours. I selected a chocolate one and took a nibble.
Divine.
"Those look delicious." Mikayla reached past me to snatch a petit four with pink and white icing.
I narrowed my eyes at her as she tasted the little cake.
"Mmm. They are." She took a second bite before noticing my glare. "What?" she asked once she'd swallowed.
"You totally abandoned me."
"I did, didn't I? Sorry."
"So much for loyalty between stand partners," I said melodramatically. "Leaving me to suffer in my time of need."
"I'm sorry," she said again. "I sensed incoming awkwardness and bolted." She finished off her petit four. "Are you going to tell me what he said to you?"
I let out a huff, but decided to let her treachery slide. "He wanted to tell me that he and Elena have broken up." I rolled my eyes. "Did he really think I'd care?"
"Do you?"
"Of course not."
"Good. You're much better off with Aaron," she said, referring to my boyfriend.
I washed down my last bite of cake with a sip of champagne. "I know." And I did. Aaron was gorgeous and sweet, with a British accent that made my knees weak, and he'd never given me any reason to believe he was anything but genuine. He was worth a hundred Hans Clausens.
"Speaking of Aaron," Mikayla went on, "when's he coming back from London?"
"Tomorrow." The word came out with a heavy sigh.
"Wow. Such enthusiasm."
I cringed. "I didn't sound enthusiastic?"
"Um. No." Mikayla eyed me over her champagne flute as she took a long sip. "What's going on?"
"Nothing."
That was the truth. At least, I thought it was. Aaron had spent the last three months in the UK and Europe, touring with his cousin's band, so I hadn't seen him in person for what felt like ages. Maybe I was worried that our relationship wouldn't be quite the same after such a lengthy time apart, but surely such a concern was unfounded. Wasn't it?
I decided a quick change of subject was in order. "How are things with you and Dave?"
Mikayla had been dating the bassoonist for over four months now.
"Great," she said. "But you're changing the subject."
Darn. I should have known she'd notice.
"We'll talk later," I said as I took a step away from her. For some reason that I couldn't quite pinpoint, the thought of discussing Aaron any further made my stomach twist into knots. "I'm going to speak to Ernest. He looks lonely."
I escaped from the questions I knew Mikayla wanted to ask and approached Ernest, a short and rotund French horn player in his late fifties. His normally pale face was flushed and he stood by himself at the edge of the room, one hand fiddling with the lapel of his tuxedo jacket as he stared through his thick glasses at the crowd of mingling people.
"Hi, Ernest."
He started when I addressed him.
"Oh. Hello, Midori." He cleared his throat and continued to tug at his lapel. "The concert went well, don't you think?"
"Very."
His gaze drifted back to the crowd in the middle of the room. I followed his line of sight. Mrs. Duffy—Mr. Major's daughter and the mother of one of my violin students—was helping her father into a wheelchair. He sat down heavily and Mrs. Duffy hooked his cane over one of the handles.
A middle-aged woman with glasses and dull, frizzy brown hair hovered behind the wheelchair and patted Major on the shoulder once he was seated. The elderly man must have grown tired of standing, but I doubted that he'd ever grow tired of creeping out women less than half his age.
I returned my attention to Ernest. His eyes were still fixed on Mr. Major and his expression had transformed in the past few seconds from bland to darkly angry.
The drastic, unexpected change startled and puzzled me. "Do you know Mr. Major?"
"What?" Ernest swiveled his head toward me, his thick glasses drawing my attention to his gray eyes and their staccato blinking. "No. I've never met the man."
"Oh." How odd. Why would he have such an intense dislike for a man he'd never met? Unless I was mistaken about whom he'd been focused on.
Ernest pulled a handkerchief out of his pocket and patted his perspiring forehead. "Excuse me."
He made a direct line to the nearest waitress and snagged a flute of champagne off her tray. My eyebrows shot up as he gulped down the entire contents in no more than a second. He abandoned the empty glass on a nearby table and moved through the crowd, patting his damp forehead again.
Weird.
Or was it? I'd never seen Ernest act like that before, but then again, I barely knew him and had never spent time with him outside of the orchestra. For all I knew he was odd on a regular basis.
Shrugging off Ernest's behavior, I decided to join some of my fellow second violinists who had gathered near one of the grand arched windows, the view nothing but darkness at this time of night. I threaded my way through the clusters of people, making sure to stay behind Mr. Major so he wouldn't see me and have a chance to run his sleazy eyes over me again. As I passed within a few feet of his wheelchair, Mrs. Duffy spread a small blanket over his knees.
"Are you warm enough, Dad?"
Major swatted her hand away. "Stop fussing. I don't need your incompetent brand of help."
My eyes widened at the rancor in his voice. So did Mrs. Duffy's. She choked back a sob and turned away from her father, quickly squeezing her way through the crowd.
I glared at the back of Major's head. What a mean old bastard.
He continued to grumble under his breath. The frizzy-haired woman patted his shoulder again and spoke to him in quiet, soothing tones.
I set my empty champagne glass down on a nearby table and searched the room for Mrs. Duffy. I spotted her just as she slipped out through a door at the far end of the room. Abandoning my plan to join my fellow violinists, I worked my way through the crowded room until I reached the far door. I pushed it open and slipped out into a corridor lined with the same red carpeting as the reception room.
There was no one in sight. I knew there was an exit around the corner, so it was possible that Mrs. Duffy had stepped outside to collect herself. I wasn't sure if I should continue to look for her to make sure she was okay. Maybe she'd prefer to be left alone. After all, I didn't know her particularly well. I'd taught her son, Jordan, violin for seven years, but had never talked to her for more than a few minutes at a time, and the topics of our conversations had always stayed confined to her son's progress or lesson schedules. Certainly we'd never discussed anything personal or established any sort of friendship.
I turned back to the door, intending to return to the reception room.
"What are you doing here?" a female voice asked.
I spun around, thinking the question had been aimed at me, but I was still alone.
"I need some cash," a man said.
"And you think I have extra lying around?" I recognized the female voice as belonging to Mrs. Duffy. "You know I'm having my own financial troubles since I left Gregory."
I paused with my hand on the doorknob. I knew this was a conversation that wasn't meant for my ears, but somehow I couldn't bring myself to go back into the reception room. I'd always been too curious for my own good.
Two quiet steps took me farther along the corridor, closer to the branch that led to the exit.
"Of course I know," the male voice said. "I need you to get some money off Dad for me."
"Kevin, you know I can't do that. If I even mention your name these days he goes through the roof."
The man let out a string of colorful swearwords, most of them unsavory descriptors aimed at Mr. Major Senior. "Can't you pretend it's for you? I'm desperate here, sis."
"I can't." Mrs. Duffy sounded close to tears. "He's not much happier with me than he is with you lately. He thinks I'm a failure since my marriage fell apart."
"Has he been bullying you again?"
Mrs. Duffy sniffled.
I jumped as a loud bang reverberated along the corridor.
"Kevin! Be careful!" Mrs. Duffy admonished in a hushed voice. "You almost put a hole in the wall."
"That damn bastard," Kevin spat. "Always trying to make everyone else miserable."
A door opened nearby and a draft of chilly air wafted along the corridor toward me.
"Where are you going?" Mrs. Duffy asked, her voice tight with worry.
"I've had enough of the old miser," Kevin said. "And I'm going to make sure we never have to deal with him ever again."
A door slammed shut, the noise jolting me into motion. Not wanting Mrs. Duffy to know I'd overheard the conversation, I slipped back into the reception room and pulled the door closed behind me.
Chapter Two
THE CHATTER OF dozens of happy voices was as soothing to my ears as a lullaby after the unsettling conversation I'd just overheard. Despite my initial curiosity, I no longer had any desire to know more about the dynamics of Mr. Major's family. Clearly they weren't a cheery, love-filled bunch, and that saddened me, particularly since I was quite fond of Major's grandson, Jordan. But perhaps the family was simply experiencing an unusual rough patch. The exchange between Mrs. Duffy and her brother led me to seriously doubt that, but what did I know? For Jordan's sake, I hoped his family life was better than what my recent eavesdropping had suggested.
As I made my way back through the reception room, I decided to switch my focus to something far more pleasant than Mr. Major and his family—free food. I sampled a few fancy hors d'oeuvres and another delectable petit four. I chatted with some of my fellow musicians as I ate, enjoying both the food and the company. A viola player who'd studied music at the University of British Columbia at the same time as me had run into our music history professor the day before. We reminisced briefly about his outfits, which were always comprised of khaki pants and one of the same five hand-knitted sweaters. Apparently, that hadn't changed since our graduation.
When the two of us had finished sharing our memories with the others, the group's conversation shifted to football, a subject I knew nothing about and had little interest in. Finishing off my last morsel of cake, I decided to follow it up with a cup of tea.
Detaching myself from the group, I headed for a hot-water urn set on one of the white-clothed tables. From a selection of pretty teacups set out on the table, I chose one decorated with red sweet peas and filled it with water. As I held my cup beneath the nozzle, Mr. Major wheeled himself toward me. Somehow I managed to keep my groan under my breath.
Lucky for me, it turned out that the man was more interested in somebody else and didn't notice me. I let out a sigh of relief as he wheeled past me to approach Dr. Daniel Beaufort, the PGP's vice chair, who was helping himself to a cup of coffee.
"Mr. Major," Beaufort greeted when he looked up and saw the other man. He didn't sound thrilled to be in Major's presence.
"Beaufort," Major returned, his voice holding a note of condescension. "I hope you've thought about what we discussed the other day."
I inched my way along the table, hoping to distance myself from the men and their conversation. Still, I couldn't help but overhear their next exchange, despite the fact that Dr. Beaufort lowered his voice to little more than a harsh whisper.
"Threatening me will be ineffective. It's also something I'd advise against."
Major sneered at Beaufort. "Something tells me you'll change your tune if the symphony and your career suffer because of your inability to listen to reason. Why don't we find out?"
"Do your worst," Beaufort snarled. "I have nothing to hide."
"Is that so?" Major's watery eyes glinted with malice. "I wonder what the police would find if I called them tonight."
A dark flush crept up Beaufort's tanned neck. "If you do that, you're the one who will end up looking foolish."
A cold smile pulled at Mr. Major's dry lips. "I doubt that."
Beaufort said something in return, but he'd lowered his voice further and I couldn't hear his next words. I was glad of that. I didn't want to spend any more of my evening listening in on other people's unpleasant conversations. What was wrong with everyone, anyway? The reception was supposed to be a pleasant, happy occasion. Maybe it was for the majority of attendees, but Mr. Major seemed to have a special knack for spreading animosity and negativity.
I resolved to steer clear of him for the rest of the evening. I didn't want anything more to do with his bad vibes or his sleaziness. Plus forcing myself to be polite around him might not be so easy now that I'd had a few glimpses of his true personality. The guy might be rich and he might be the symphony's most generous benefactor, but that didn't mean he should get away with treating other people like dirt. The problem was, it seemed like he did get away with it, and I knew all of us musicians would be expected to treat him with respect. After the last hour, I didn't want to be in that position, so I figured it was best to keep well away from him.
I watched out of the corner of my eye as Dr. Beaufort stalked away from Major, his expression clouded with dark anger. As I turned away from both men, I nearly collided with Mrs. Duffy.
"Oh, hello, Midori," she said as she brushed a strand of her brown hair off her face and tucked it behind her ear. She'd regained her composure and showed no signs of her earlier distress. "The concert was lovely."
"I'm glad you enjoyed it."
The frizzy-haired woman I'd seen hovering near Mr. Major earlier in the evening bustled up to us and put a hand on Mrs. Duffy's arm. "Sorry to interrupt, Andrea, but I think it might be time for your father to switch over to coffee." She glanced at me before leaning closer to Mrs. Duffy and whispering, "He's starting to slur his words."
Mrs. Duffy attempted to smile—at least I thought that was the expression she was going for—but it ended up looking more like a pained grimace. "Thank you, Marjorie. I'll be right there." She nodded at me. "Excuse me."
"Of course." I remained where I stood and watched as she accompanied Marjorie back to her father.
She snatched a half-empty champagne flute from her father's hand and passed it to Marjorie, who placed it on a table out of his reach. Mrs. Duffy grabbed a clean cup from the nearby table and headed for the coffee urn.
Gareth Hollingsworth, the chair of the PGP's executive committee, stepped in to meet her in front of the urn and took the cup from her, filling it with hot coffee as he spoke to her in a whisper. I had no chance of catching his words or reading his lips—not that I wanted to—because he had his back to me. Mrs. Duffy nodded at whatever he'd said, and he gave her the cup.
She returned to her father's side and handed him the coffee. He made a face of disgust but, to my surprise, didn't argue with her.
I realized why a second later. As soon as Mrs. Duffy and Marjorie had turned their backs to him, he slipped a silver flask out from beneath his jacket and added a dollop of something to his coffee.
Sneaky.
I felt bad for Mrs. Duffy. Dealing with her father couldn't be easy. But at least the evening was almost over. Hopefully she wouldn't have to put up with him much longer that night.
Weaving my way around several clusters of people, I found Mikayla with Bronwyn—another violinist—and a percussionist named Anton. Bronwyn had a large shoulder bag hooked over her arm and seemed tense.
"I need to get home," she said after I greeted the group. "The baby is sick and I don't want to leave my husband to cope on his own for too long."
We saw her off with goodbyes and well wishes for her sick baby, and then went back to chatting and—in my case—drinking tea.
Janine Ko and cellist Nina Kim joined our group, and our conversation soon turned to our plans for the weekend.
"My quartet is playing at a wedding on Sunday," Janine said, "but I'm hoping to walk the seawall tomorrow if the weather's nice."
"I didn't know you played in a quartet," I said.
"It's a new thing. A couple of friends asked if I was interested in forming a quartet and I figured it was a good way to earn some extra cash. I want to stay on top of my car payments and student loans."
"I hear you with the car payments," Mikayla said.
The conversation continued, but in my head I replayed what Janine had said. As mean as Elena's words were earlier, maybe she wasn't wrong. If Janine needed extra money to pay off her debt, it was unlikely that she could afford a genuine designer handbag. Although maybe such frivolous purchases were the reason why she needed the extra cash to make her loan and lease payments.
Deciding it didn't matter to me either way, I pushed those thoughts out of my head and tried to tune back into the conversation going on around me. Once I'd caught up, I added a few words of my own. But as I finished off my hot drink, I realized that my energy was draining out of me drip by drip. There was no clock on display in the reception room and I'd left my cell phone in my locker, but I figured it was late enough that I could quietly slip away and head for home. I'd had my fill of food, and the thought of a hot bath appealed to me far more than hanging around the theater and continuing to mingle. I liked most of my fellow musicians and considered some of them good friends, but it had been a long, busy week and I was ready to ease into its coda.
As a waitress with an empty tray passed by, I held out my teacup and saucer. The waitress paused and I placed the china on her tray before she continued on through the crowd.
Two other musicians joined our group but Mikayla pulled me aside and lowered her voice. "You didn't have any other awkward moments with the maestro this evening, did you?"
"No, thank goodness. One was more than enough. And I should be safe now because I'm going to head home."
She snagged my sleeve with her fingers. "But we never finished our talk about Aaron."
My stomach threatened to twist into knots again. "There's not much to talk about. Besides, I'm beat. I want to go home and get some sleep."
Although her eyes narrowed with suspicion, she let go of my sleeve. "Okay, see you next week."
"Bye." Relieved to escape, I waved to Anton and the others. Then I aimed for the nearest door.
I smiled at a few familiar faces but didn't stop to chat with anyone else, not wanting to delay my exit by getting pulled into a conversation. As I reached the edge of the reception room and approached the door, a man's voice rang out above the chattering of the crowd.
"Get away from me!"
I swung around as conversations broke off throughout the room. Everyone's attention, including mine, zeroed in on Mr. Major. He swore and threw his coffee cup at Mrs. Duffy. She jumped back with a gasp, barely managing to avoid the dregs of dark liquid that splashed toward her.
Her expression as shocked as everyone else's, frizzy-haired Marjorie moved in to placate the elderly man. He lashed out at her as soon as she drew near, hitting her hard in the face. This time several gasps sounded from the onlookers.
Major struggled out of his wheelchair. "You're trying to kill me, all of you! Don't think I don't know. I'll have you thrown in jail!" He staggered forward and the crowd parted to keep clear of him.
Dr. Beaufort dodged around several stunned musicians and season ticket holders to close in on the scene. Hans and the chair of the Point Grey Philharmonic's executive committee weren't far behind him. Almost without realizing what I was doing, I moved in closer myself.
"How about we calm down, Mr. Major," Beaufort said in a level voice.
Major swayed on his feet, patches of red on his perspiring face. He fixed his eyes on Beaufort, his nostrils flaring as he let out a wheezing breath. "Miscreant!"
He charged like a bull with several years of pent-up anger. One of his wild, swinging arms smashed several empty champagne flutes off a shocked waiter's tray. A woman screamed and Mrs. Duffy let out a choked sob.
Dr. Beaufort and Hans grabbed Major and attempted to restrain him. He struggled against their strong grip, his eyes crazed.
"Get your hands off me," he ordered, but the vigor had gone from his voice.
A second or two later, he ceased his fight against the two men holding him and blinked at them in confusion. His head swiveled around as he searched for someone in the crowd.
"Elspeth? Elspeth, where are you?"
Mrs. Duffy rushed up to her father, her face pale. "Dad? It's me, Andrea. Why don't you sit down?"
She motioned to Marjorie to fetch the wheelchair, but Major's knees buckled. Dr. Beaufort and Hans managed to keep him from falling, but he sagged between them, his eyes drifting shut.
"Let him down gently," Beaufort instructed as he and Hans lowered Major to the floor. "Give us some space, please."
I took a few steps back along with the rest of the onlookers.
"Call an ambulance," Dr. Beaufort said to Hans.
"What's wrong with him?" Tears ran down Mrs. Duffy's face.
Marjorie put an arm around her, but Mrs. Duffy didn't seem to notice.
Gareth Hollingsworth, the PGP's chairman, addressed the stunned crowd. "Ladies and gentlemen, in view of this medical emergency, I suggest we disperse so the paramedics will have room to work when they arrive. I appreciate everyone's attendance and thank you for your continued support of the Point Grey Philharmonic."
Murmurs ran through the clusters of elegantly dressed men and women as they migrated toward the room's two exits. As I moved with them through one of the doors, I spotted Ernest lurking outside the reception room. His angry, disgusted glare was back, and it was directed at the open door of the room we'd just left. I paused as he reached inside his tuxedo jacket and pulled out a piece of paper. His features still twisted with hatred, he crumpled up the paper and chucked it in a nearby garbage receptacle with startling vehemence.
For the second time that evening I found his behavior odd. What was with all his ill will and why was it so strong that it survived in the face of Major's obvious suffering?
I didn't have a chance to ponder those questions.
Mikayla appeared at my side and took my arm, leaning in close to whisper in my ear. "What do you think is wrong with him?"
I knew she wasn't referring to Ernest. "I have no idea," I said. "Do some people act like that when they're having a stroke?"
"I'm not sure."
As we waited for the backed-up crowd to file farther along the corridor ahead of us, I glanced over my shoulder and through the open door to the reception room. Although half a dozen people still gathered around Major, I caught a glimpse of him convulsing on the floor.
"Oh my God. That's awful." I gripped Mikayla's arm and we hurried away from the room as soon as we had a chance.
Mr. Major wasn't my favorite person by far, but I never would have wished for anything bad to happen to him. Seeing him convulse was upsetting and a bit frightening as well.
"I hope he'll be okay." Even to my own ears, my voice sounded doubtful.
How likely was he to survive whatever was happening to him? I didn't know, but I had a bad feeling that his chances weren't good. He wasn't exactly the picture of health before this terrible turn.
"Hopefully the ambulance will be here soon," Mikayla said as we retreated to the musicians' lounge where we'd left our belongings.
I unlocked my locker but simply stared at my coat and violin without removing them. "Do you think we should leave or stick around for a while?" I asked Mikayla.
The thought of going home without knowing if Mr. Major would be all right left me unsettled. I knew there was nothing I could do to help him, but I still wasn't sure if I could bring myself to leave right then.
"There's nothing we can do here," Mikayla said, echoing one of my thoughts. "I think we should go."
"You're probably right." I removed my coat from my locker and slipped it on over my black concert clothes.
"It's gone!"
The exclamation came from across the room. Even before I turned around I knew the voice belonged to Elena.
"What's gone?" cellist Johnson Lau asked.
"My brooch. My antique brooch."
"Are you sure you were wearing it tonight?" Melissa asked as she removed her flute from her locker.
"Of course I'm sure," Elena shot back. "I'm not a fool."
Melissa rolled her eyes, and I couldn't blame her.
"It must have fallen off at some point," Aggie said.
Elena's eyes roamed over the room. "I know it was still pinned to my dress when I came in here after the concert."
"Then it must be in here, the reception room, or out in the hall," Janine said. "I'll help you look."
I wanted to shake my head at my fellow violinist's eagerness. Even after Elena's cruel words earlier that evening, Janine still wanted to please her.
A couple of other musicians jumped in to help Janine search the room while Elena stalked around, flicking aside bags and other belongings that were strewn about on tables and couches.
"It's eighteen-karat gold," she went on as she picked a coat off a table with her thumb and forefinger, as if it were a piece of smelly garbage. Once she'd had a look beneath the garment she dropped it back to the table. "With a sapphire that exactly matches the color of my eyes."
I let out a resigned sigh and joined the search, Mikayla following suit. I didn't feel any particular need to be helpful to Elena, but I figured finding her brooch would be the best way to shut her up. Unfortunately, ten minutes of searching turned up nothing other than Aggie's favorite mechanical pencil, which she'd lost weeks ago.
Although Aggie was pleased, Elena was anything but. By that time we'd thoroughly searched the musicians' lounge and the hallway, but there wasn't a single karat of gold to be found, let alone eighteen plus a sapphire.
"It must have fallen off during the reception," Mikayla said once we'd all returned to the lounge. "I guess you'll have to wait until the paramedics are gone and then have a look in there."
Elena let out a huff and turned her back on the rest of us. She sat down on a chair in the corner, crossed her perfect legs, and focused all her attention on text messaging someone on her phone. No doubt the message had something to do with her precious brooch.
I shook my head and returned to my locker, more than ready to leave for home.
"Oh no," Janine said, opening her own locker as she threw worried glances over her shoulder at Elena. "She's so upset. I really hope she finds her brooch in the other room."
I didn't bother to comment, not wanting to waste another moment of my time on our ungrateful concertmaster. Although I could understand getting upset over a missing piece of jewelry, Elena hadn't exactly behaved graciously. She hadn't even thanked any of us for helping her search.
Knowing her as I did, I would have been willing to bet that she considered the medical emergency in the other room an annoying inconvenience, one that was keeping her from her brooch. She probably wasn't the least bit concerned with whether Mr. Major lived or died.
Sliding my folder of music into my quilted bag and gathering up my violin, I shut my locker and secured it with its combination lock. Mikayla and I said some subdued goodbyes to the other members of the orchestra and headed out into the hall. We were about to turn toward the stage door when I spotted Hans coming along the corridor from the direction of the reception room.
We stopped and waited as he approached. My stomach tightened as if a fist had closed around it. Hans's mouth was drawn in a grim line and his blue eyes telegraphed his bad news.
"Is Mr. Major all right?" Mikayla asked. Her voice betrayed the fact that she too had anticipated Hans's news.
He shook his head. "The paramedics arrived a few minutes ago, but it was too late. I'm afraid Mr. Major is dead."
Chapter Three
WHEN I ARRIVED home at my apartment later that night, the first thing I did was go straight to the bathroom and run a nice hot bubble bath. After a good long soak, I dressed in my comfiest cotton pajamas and snuggled up in an armchair with a quilt and a mug of hot chocolate topped with mini marshmallows. The hot bath had relaxed my body, but my mind was still wide awake and buzzing.
I stared into my mug of hot chocolate as the marshmallows slowly melted into a layer of white, gooey foam. Maybe it wasn't the best idea to drink a sugary beverage at such a late hour, especially when I was already too alert to allow sleep to come easily. But after contemplating my melting marshmallows for another second or two, I took a sip anyway.
No matter how much I tried to turn my thoughts to other matters, Mr. Major continued to occupy the prime real estate in my brain. Although I'd had the unfortunate and unpleasant experience of discovering a freshly dead body a few months earlier, I'd never actually seen someone die. I hadn't been present at the exact moment when Mr. Major drew his last breath, but I figured his convulsions had probably marked the start of his departure.
I couldn't shake the memory of the violent spasms racking his body as he lay there on the floor and I couldn't forget poor Mrs. Duffy's face as I'd last seen it. Perhaps that was what bothered me the most—knowing that Mr. Major's death would have an effect on his family, a family that included someone I cared about. My student Jordan was a good kid. After the scenes I'd witnessed that night, there was always a chance that Jordan didn't have a strong, positive relationship with his grandfather, but even if that were the case, I doubted it would be easy for him to hear the news.
With a heavy sigh, I finished off my hot chocolate and climbed out from beneath my quilt to take my empty mug to the kitchen. I brushed my teeth and got into bed, snuggling beneath the covers. Thoughts of Mr. Major, Mrs. Duffy, and Jordan continued to run through my head on repeat, but eventually my exhaustion got the upper hand and I slipped off toward sleep. As my mind grew fuzzy and my breathing deepened, one last thought chimed between my ears.
I wonder if Major's children are sad that he's dead.
BY THE TIME I cracked open my eyes the next morning my clock had already worked its way past nine o'clock. I allowed myself to stay snuggled beneath the covers, enjoying the fact that I didn't have to work that day.
For the past thirteen years of my life, starting in my mid-teens, I'd taught violin lessons on Saturdays. This was the first year I'd decided not to do so, cutting my work week back to only five days, except on the odd occasion when I had Saturday night concerts. I'd lost two students as a result, but the others had agreed to switch their lessons to weekdays. And I'd recently gained three new students, so I wasn't suffering financially because of the change.
After soaking in the luxury of lounging in bed for another half hour, I got up and switched my pajamas for jeans and a lightweight sweater.
My memories of Mr. Major in his last moments of life no longer had such a grip on me, and I was relieved about that. Yet the events of the previous evening still hovered at the back of my mind. As I sipped a cup of green tea and nibbled at a piece of toast, I recalled Ernest's strange behavior.
I didn't understand why he harbored such an intense hatred for Major, especially since he claimed he'd never met the man. But beyond that, I was puzzled by Ernest's mysterious piece of paper. What was it, exactly? And why had Ernest disposed of it with such anger at the moment he did?
As I finished my breakfast I tried to focus on other matters. I really did. Yet for some reason I couldn't forget about that pesky piece of paper. I blamed it on my inexhaustible curiosity. For as long as I could remember, my curiosity had been a force to be reckoned with. Whenever something piqued it, I couldn't simply turn my thoughts to other things. I knew that on occasion my curiosity bordered on nosiness, and more than once it had led me straight to trouble, but knowing that didn't make me any less inquisitive. I didn't like unanswered questions bouncing around in my head. They were too distracting.
I knew my interest in that piece of paper had reached a point where I wouldn't be able to forget about it, at least not anytime soon.
Unless I satisfied my curiosity.
After I tidied up my kitchen I stared at the hamper full of laundry waiting to be washed. I could stay home and take care of that chore or I could make a quick trip to the Abrams Center for the Performing Arts, home of the Point Grey Philharmonic, and see if Ernest's mysterious piece of paper was still around. I spent a full minute considering those options before I decided, with no shortage of reluctance, that I should be responsible and go with the laundry.
I gathered up my dirty clothes and a bottle of detergent and headed down the main hall to the laundry room. Every floor of my apartment building had its own laundry room, but with only one set of machines each. Before I reached the small room, I could tell someone had beat me to it that morning. Both the washer and the dryer rumbled away as they worked. I peeked into the room and checked the timer on the washing machine. I would have to wait awhile.
Perhaps I should have been disappointed, but instead I dumped my laundry inside my apartment and grabbed my purse and coat. Why sit around at home waiting for a chance to do laundry when I could pop out to the Abrams Center for a few minutes and take care of chores later?
As I wandered along the sidewalk to the nearest bus stop, I checked my phone for the first time that morning. Aaron had sent me a text message a few hours earlier as he waited at Heathrow Airport for his flight back to Vancouver.
I'll be on my way soon, his message read. Can't wait to see you!
Same, I texted back to him as I walked. I'll see you at the airport!
After the message was sent, I shoved my phone into my purse and dug out my bus pass. Guilt gnawed at me as I stood waiting at the bus stop. Aaron and I had started dating back in May and it was now September. Yet we'd only had time to go on half a dozen dates. Barely more than a month after he'd asked me out for the first time, he'd hopped on a plane for his hometown of London, England, and had remained overseas ever since.
His departure from the country didn't have anything to do with the state of our relationship. He'd arranged the trip long before he ever asked me out and had considered canceling so he could spend more time with me, but I'd told him that he should go. He hadn't seen his family in nearly two years and he had the chance to fill in for an injured drummer in his cousin's band during their summer tour of Great Britain and continental Europe. I knew he was excited about that, and I wasn't about to be the reason he missed out on such an opportunity.
So I told him I'd miss him but he should go. And he went.
We'd exchanged e-mails and text messages and Skyped a few times to keep in touch. Now I would finally get to see him in person.
I was looking forward to that. Really, I was. Just . . . not as much as I should.
Admitting that to myself made me cringe.
I'm sure it doesn't mean anything, I told myself. We didn't know each other all that well before he left, and we've been apart for a long time. As soon as I see him again, the butterflies and excitement will be back.
That sounded perfectly reasonable to me.
My guilt eased as I boarded a bus that had pulled up to the curb. I had nothing to worry about. I'd meet Aaron at the airport that afternoon and everything would be fine.
THE ABRAMS CENTER for the Performing Arts was located on the west side of Vancouver, and the second-story offices at the back of the building featured beautiful views of the water and North Shore Mountains. The theater stood out from the shops and other businesses located on the street because of its impressive size and eye-catching appearance. At least twice the width of any other building on the block, the front of the theater featured a white stone façade topped with a fancy cornice, and a large marquee advertised the latest concerts and shows.
Although not the swankiest theater in the city, the Abrams Center held its own and was a nice place to work, especially since the recent renovations that had spruced up the restrooms and upgraded the seating in the theater proper. Sometimes I almost had to pinch myself to make sure my life was real, that I truly did have my dream job of playing violin in a professional orchestra, giving concerts in a beautiful theater.
Aside from a brief phase of wanting to be a marine biologist, I'd always wanted to be a musician while I was growing up. My dream had come true and I'd spent so much time at the Abrams Center over the past several years that it was almost as familiar to me as my own apartment.
Leaving the street for a side alley, I made my way toward the black door located a stone's throw from the base of the building's fire escape. I knew the stage door was likely to be unlocked on a Saturday. The theater was used by other groups aside from the Point Grey Philharmonic, and there were events and rehearsals happening almost every day.
Once inside, I followed a corridor deeper into the building. Although I heard a murmur of distant voices at one point, I didn't meet a soul. Outside the large room where the orchestra had held its reception the other night, I approached the garbage can where I'd seen Ernest dispose of his paper.
I glanced around to ensure that no one would witness me digging through the trash, but the coast was clear so I pushed at the swing top with one finger and peered inside the can with no shortage of apprehension. I hoped that the trash hadn't been taken away in the past twelve hours so I could retrieve the paper, but, at the same time, I didn't want to have to poke through anything disgusting to find it. Luck was with me, however. The hallway trash can was clearly not a frequently used one, as there wasn't much inside of it. There at the bottom, resting on two pop cans, was a crumpled piece of paper.
Jackpot!
I checked my surroundings once more to make sure that I was still unobserved. I was. Wrinkling my nose, I reached one arm into the trash can. Down, down, down, until my fingers brushed against paper. I snatched up the thin bit of trash and pulled my arm out of the garbage receptacle. Triumphant, I uncrumpled the paper to reveal its secret.
It took me several seconds to register what was before my eyes. Someone—presumably Ernest—had cut out letters from magazines and pasted them into a startling message.
Archibald Major, you are scum. May you rot in hell.
I swallowed as an unpleasant sensation rolled through my stomach. Talk about disturbing. Prior to last night I would have had trouble picturing quiet, unassuming Ernest authoring the note. But after seeing the anger on his face as he glared at the elderly man at the reception, I could picture it all too easily.
Shuddering, I folded up the paper, hiding the ugly message from my eyes. I wanted to get rid of it, to toss it back in the garbage can where I'd found it, but I slipped it into my purse instead. Something told me I should hold on to it for the moment.
Wrinkling my nose again, I reached back into the trash can and retrieved the two pop cans. My hands were already dirty so I figured I might as well put the cans in the recycling bin where they belonged.
I tossed the cans into the bin located next to the garbage receptacle and slipped into the ladies' restroom down the hall to wash my hands. Free of icky trash can germs, I returned to the corridor and headed for the exit.
"Ms. Bishop?"
I turned back at the sound of the female voice. A woman in a navy blue business suit stood outside the door leading to the reception room. She wore her honey blond hair tied back, as she had every other time I'd seen her.
"Detective Salnikova? What are you doing here?" I hadn't seen the police detective since I'd helped solve the murder of a cellist several months earlier, and she was pretty much the last person I would have expected to encounter in the back corridors of the theater on a Saturday morning.
"Working, I'm afraid." She came a few steps closer. "How about you? It was my understanding that the orchestra wasn't rehearsing today."
"No, I'm not here for a rehearsal." I clutched my purse closer to me. I didn't have anything to feel guilty about, but somehow having Ernest's evil note in my possession left me feeling distinctly uneasy in the face of the police detective's gaze. "I came to pick something up."
"Were you at the reception last night?"
"Yes." As I recovered from the surprise of meeting Salnikova at the theater, I finally registered everything she'd said over the last half minute. "Wait. Are you here because of Mr. Major?"
"That's right."
"But you're a homicide detective."
A hint of a smile appeared on her face for a fleeting second. "I am."
Okay, so maybe she thought I was stating the obvious, but there was something that definitely wasn't obvious to me.
"Then why are you here? Mr. Major was old. Didn't he die of a stroke or something?"
All traces of amusement disappeared from the detective's face as she replied, "Not a stroke, no. We believe there could be foul play involved."
I stared at her for two full seconds before echoing, "Foul play?" I recalled Major's bizarre behavior in the moment before his convulsions. "Oh my God. Was he poisoned?"
Salnikova narrowed her eyes a fraction. "Why do you ask that?"
"He went all crazy before he collapsed. And as far as I know, nobody attacked him or anything, so if you suspect murder, then poison seems like a distinct possibility."
"We're still waiting for the autopsy and toxicology results," Detective Salnikova said.
I waited for her to continue, but she didn't. I should have expected that. She had an unfortunate habit of avoiding direct answers to my questions. Maybe it was her job to keep things mum, but that didn't mean it wasn't annoying. Still, in this case, I knew I was right. She suspected that someone had poisoned Mr. Major.
In other words, she suspected someone had murdered him, which explained her presence at the place where Major had died.
I swallowed and lowered my eyes to my purse. Getting Ernest in trouble wasn't something I wanted to do, but I couldn't in good conscience keep quiet about the note I'd retrieved from the trash can now that I knew the police were involved. I unzipped my purse and pulled out the rumpled piece of paper.
"There's something I think you should see."
Chapter Four
I WAS ABOUT to hand the paper to Salnikova when I stopped myself. "Oh." I lifted all but my thumb and forefinger away from the paper. "This could be evidence and I got my fingerprints all over it." When Detective Salnikova's eyebrows drew together, I rushed to add, "But I had no idea that there was a murder investigation so any thought about fingerprints or evidence never crossed my mind."
Detective Salnikova retrieved a pair of latex gloves from the pocket of her suit jacket and pulled them on. She took the note from where it dangled between my finger and thumb and unfolded it. I waited in silence as her eyes ran over the message formed by the cut-out letters. Mere seconds later she raised her gaze to meet mine.
"Where did you get this?" Her tone was serious, all business.
"From the trash." I pointed down the hall at the trash can with its fake gold trim and swing top.
Salnikova followed my line of sight but then her blue eyes zeroed in on me again. "Do you make a habit of digging through trash cans?"
"Of course not," I said, indignant.
"Then how did this come into your possession?"
"I saw someone throw it in the garbage can last night as we were leaving the reception. I was curious about it so I came back to see if it was still there."
"You saw someone throw a piece of paper into the garbage and that made you curious?"
Although she kept her tone mostly neutral, there was still an unmistakable hint of incredulity behind it. My cheeks grew warm as I realized how weak my explanation sounded.
"Okay, so there was more to it than that."
Salnikova raised her eyebrows, waiting.
"Ernest, the guy who threw it away, was giving Mr. Major—or someone in his vicinity—the evil eye during the reception. Then, after Mr. Major collapsed, he crumpled up the paper and tossed it in the trash, looking super angry. I couldn't help but wonder what that was all about."
"Is this Ernest a member of the orchestra?"
"Yes. He's a French horn player. He's been in the orchestra a long time, I believe. Since before I got hired, anyway."
"Do you know his last name?"
I dug through my memory but came up empty. "No. But it should be easy enough to find out."
"I'm sure it will be."
"I could ask someone, if you'd like," I offered.
"Thank you, but that won't be necessary." Salnikova nodded at the exit. "Were you on your way out?"
"Yes."
"Do you have some time? I need to track down and speak with possible witnesses and since you're one of them . . ."
"That's fine. I'm meeting someone at the airport later, but not until late this afternoon."
"Is there somewhere we can sit down and talk?"
"This way." I led her down another hallway as I dug my keys out of my purse.
When I found the one I wanted, I used it to unlock the door to the musicians' lounge. Aside from lockers, the room was also home to three couches, two tables, and some folding chairs. As I switched on the lights and sank down onto one of the couches, I wondered if I should be nervous about the upcoming interview.
Don't be ridiculous, I told myself. She just wants your witness statement. She doesn't suspect you of killing anyone.
At least, I hoped that was true. But what motive would I have for killing the old man? And although my fingerprints were all over the note, Ernest's would be too. That would help to corroborate my story. Besides, if I'd had something to hide I never would have shown Salnikova the note. She must have realized that.
My shoulders relaxed as Salnikova took a seat on the couch across from me. She produced a plastic evidence bag from inside her jacket and slid Ernest's note into it. After setting it on the cushion next to her, she removed her gloves and reached into another pocket to take out her notebook and pen.
As she opened the notebook to a fresh page, I recalled all the interactions I'd witnessed at the reception, as well as the conversation between Mrs. Duffy and her unseen brother. Now that I knew it was possible Mr. Major had been murdered, I viewed all of those incidents in a new, and rather disturbing, light.
Although I didn't want to get any innocent people in trouble, I also didn't want to hold back anything that could potentially lead the police to the murderer. So as Salnikova smoothed down the empty page of her notebook, I set my purse on the couch next to me and said, "I've got a lot to tell you."
Despite my word of warning, I think I surprised Detective Salnikova with how much I had to tell her about the previous night's reception. I mentioned Mr. Major's terrible attitude toward his daughter and his exchange of less-than-pleasant words with Dr. Beaufort, the executive committee's vice chairman. After that, I told her about the conversation I'd overheard between Mrs. Duffy and her brother outside the reception room, making sure to mention her brother's ominous statement about never having to deal with their father again. To wrap up, I described Major's odd behavior and collapse, followed by his convulsions.
Detective Salnikova listened carefully and made notes as I talked. Whether she thought I'd gathered so much information because I was remarkably observant or remarkably nosy, I didn't know for sure. My guess was the latter, particularly considering my involvement in the murder investigation back in the spring. But even if she did think I was a snoop, I was far more concerned with the possibility that there could be a murderer walking around free, one who might well have been in close proximity to me the night before.
"Do you really think Mr. Major was murdered?" I asked once I'd wrapped up my narrative and the detective had stopped writing. "Is it at all possible that he died of natural causes?"
"Nothing has been determined for certain yet." Salnikova tucked her notebook back in her pocket. "As I said, we're still waiting for the autopsy results."
"But you're here and you're asking questions," I pointed out. "Something must have raised a red flag."
Salnikova got up from the couch and retrieved the evidence bag holding Ernest's note. "I'm afraid I can't discuss the investigation."
The temptation to roll my eyes was nearly overwhelming, but I managed to keep them fixed straight ahead as I pushed myself up off the soft couch cushion. I should have known from past experience that I wouldn't get any real answers out of the detective, but trying had been instinctive. Her lack of an illuminating response didn't bother me as much as it might have, though, because I could guess the answer.
Dr. Beaufort was a surgeon and was right there with Major when he started acting crazy and collapsed. Maybe it was Major's behavior or some physical sign that had led him to suspect that something other than natural causes was to blame, but I was willing to bet it was Dr. Beaufort who had alerted the police to the possibility of foul play.
I left the musicians' lounge with Salnikova and locked it up behind us. I expected the detective to take her leave then, but she paused, and regarded me somewhat sternly with her blue eyes.
"I hope you're planning to stay out of the investigation into Mr. Major's death," she said. "The last time you got mixed up in police business, you almost came to serious harm. We wouldn't want that to happen again."
"I have absolutely no intention of getting mixed up in police business."
"I'm glad to hear it," she said, her gaze never wavering from my face. "Thanks for speaking with me, Ms. Bishop." With a nod, she left me there outside the door to the lounge, disappearing around a corner a few seconds later.
I frowned in the direction she'd taken, slightly miffed by her mild reproof. Okay, so there was some merit to what she'd said, but getting into dangerous situations wasn't the only thing I'd done during the previous spring's murder investigation. I'd also uncovered important information that had helped lead to a murderer.
Hooking the strap of my purse up over my shoulder, I left the theater through the stage door. As I traveled back home to my apartment, I thought about Mr. Major and who might have wanted him dead. Yes, I'd said only moments earlier that I wouldn't get mixed up in police business, but that didn't mean I couldn't think about the case. There was no harm in that.
Based on everything I'd witnessed the night before, I figured there might be a lot of people who weren't disappointed by Major's death. If his behavior at the reception was anything to go by, he'd had a knack for upsetting people, including his children, Dr. Beaufort, and Ernest.
In addition to that, he was a wealthy man. An extremely wealthy man. He was the PGP's most generous individual financial supporter, and I knew he owned a fancy home in the upscale neighborhood of Shaughnessy as well as other properties in the city. I'd even heard a rumor that he owned a private island somewhere in the Caribbean. I didn't know if that was really true or not, but either way, he had more millions than Mozart had symphonies.
By the time I reached my apartment, my mental list of people who might have wanted Major dead had only grown in length. If he was as unpleasant to his business associates and competitors as he was with his family (and I'd guessed that he was, if not more so) then maybe one of them would have been glad to see him go six feet under. I also wouldn't have been the least bit surprised if he'd crushed a few people on his path to wealth and success.
Aside from those of his family members I knew and a couple of other people present at the reception, I didn't have any actual names for my suspect list, just general profiles of the type of person who might not be sorry to see the last of Major. But my lack of specific suspects wasn't something to worry about. The police would be able to figure this case out on their own. Or so I hoped.
At any rate, I had other things to focus on. The washer and dryer were quiet and empty when I arrived home so I snatched the opportunity to get a couple of loads of laundry done. While my clothes rhythmically swished and tumbled in the machines, I scrubbed my bathroom and kitchen clean. Next I booted up my laptop and spent some time ordering new music books for several of my violin students.
Once my clothes were dry and folded, I only had a few minutes to spare. I checked my hair in the bathroom mirror and touched up my makeup before grabbing my purse and heading out the door. I took a bus to the train station and hopped onto the Canada Line. As the train whisked me off toward the airport, I put my earbuds in my ears and listened to a Piano Guys album. With the music keeping me company, I managed to make the trip without my thoughts straying too far into territory I wanted to avoid.
I cut off the music when I disembarked from the train, and stuffed my earbuds and phone into my purse. I immediately missed the calming effect the music had on me, but I wanted to be present in the moment. Aaron deserved my full attention. Besides, what reason did I have to be anything but calm? Once Aaron was there in front of me instead of on a computer screen, things would be as they were before he left. There was no reason why they shouldn't be.
The automatic doors parted before me and I stepped inside the airport. People moved to and fro, some towing suitcases past me out the doors, others hurrying to greet arriving travelers. I found some free floor space below one of the display screens and checked the arrivals. Aaron's plane had already landed.
I made my way around several people laden down with bags and suitcases and spotted Aaron by one of the luggage carousels. A smile tugged at my face at the sight of him. Along with it came a wash of relief.
See, of course you're happy to see him, I said to myself. You were worrying for nothing.
I picked up my pace, eager to reach him. When I got close, I put a hand on his arm to get his attention. As soon as he turned around and saw me, his whole face lit up.
"Hey, Midori." He gave me a quick kiss and pulled me into his arms. "It's good to see you."
My smile spread wider as I returned his hug. "You too."
After giving him a good squeeze, I stepped back, but apparently, that wasn't the end of our greeting. Aaron tugged me back toward him and kissed me deeply. I returned the kiss and for a moment almost forgot where we were, but when the luggage carousel eased into motion with a low rumble, I broke away.
"You don't want to miss your luggage," I said with a smile.
Aaron leaned in for another quick kiss and then took hold of my hand. We stood together, watching the assorted pieces of luggage trundle past us. He only dropped my hand for a moment when he stepped forward to grab his suitcase off the carousel. Then we were hand-in-hand again as we headed outside to catch a cab.
"So how was your flight?" I asked him once we were settled in the backseat of a taxi, my fingers still twined with his.
"Long, but all right." He gave my hand a squeeze. "Want to get something to eat?"
"Aren't you tired?"
"Nah. Pretty wired, actually. And starving."
I smiled, mostly because his London accent was like sweet music to my ears. "Why don't we drop off your stuff and have an early dinner then?"
"Sounds perfect." He grinned, bringing out his adorable dimples.
The last remnants of my anxiety and guilt ebbed away. As I'd hoped would be the case, I was happy to have Aaron back in the same city as me. With him seated next to me, all the worries I'd entertained over the past few days seemed silly. We hadn't had much of a chance to advance our relationship while he was away, but that would change now, and I looked forward to spending time with him. Starting right away.
After a brief stop at his apartment, we walked to an Indian restaurant and settled in for a delicious meal of curry and samosas.
"So what did I miss while I was gone?" Aaron asked once the waitress had set plates full of appetizing food on our table.
My thoughts immediately went to Mr. Major's death, but I decided that might not be the best subject to lead with. I didn't know if Aaron would appreciate discussing death over dinner. Luckily, I had no trouble coming up with something more pleasant to talk about.
"Did you hear about JT's new job?" I asked, referring to my best friend. Aaron was a drummer in the same band as JT, and I'd met Aaron at one of the band's rehearsals.
"Composing music for a TV show, right? I saw something about it on Facebook a few weeks ago."
I nodded as I scooped up a forkful of curry and rice. "Absolute Zero. It's a sci-fi show filming here in Vancouver."
"That's an awesome gig."
"It is."
Once I'd tasted my curry, I filled Aaron in on more of the details of JT's job, a smile on my face as I talked. Every time I thought about my best friend's music being used on a television show, I felt like I would burst with pride and excitement. Even so, I made sure I didn't spend the entire meal talking about JT's achievements.
"You must have some stories to tell," I said. "I know you've already told me a bit about touring with your cousin's band, but I want to hear more."
Aaron obliged, delving into stories about his time overseas. As we ate and chatted, I focused on nothing more than enjoying his company and his dreamy British accent. By the time we parted ways with a lingering kiss outside the restaurant, all thoughts of death and murder had long slipped from my mind.
Chapter Five
SUNDAY MORNING CAME with brilliant blue skies and bright warm sunshine. It was a perfect day to spend outdoors, enjoying the fabulous weather that hovered between summer and autumn. I was glad I'd planned to spend part of the day working in JT's garden, planting fall bulbs that would grow into colorful tulips and daffodils in the spring. Gardening was one of my favorite hobbies, but living in an apartment, I only had a couple of potted plants of my own. Fortunately, JT was more than happy to let me loose in his yard so I could add bursts of color during the spring and summer.
As soon as I'd dressed and had some breakfast, I set off to his house, enjoying the walk from the bus stop into the heart of JT's neighborhood. The air was crisp but not too cold, and the leaves on the stately trees lining the streets were in the midst of changing color. Soon summer would be a mere memory and autumn would be in full force. I didn't mind. I loved every season, for different reasons.
On my way through the neighborhood I passed several other pedestrians out enjoying the fresh morning air, some jogging and others walking their dogs. One woman who lived on JT's street was out tending her front garden, and I waved to her as I passed by. I loved this part of the city. It was peaceful and close to the forest, yet within walking distance of shops and restaurants. The houses were nice too, most of them two stories with decent-sized yards.
The only problem with the neighborhood was the high real estate prices. The prime location came at a hefty cost, and the only reason JT could live in the area was because he'd inherited his house, mortgage free, from his father. I knew I'd never be able to afford a house in the neighborhood, but I appreciated the time I got to spend there.
When I arrived at JT's white, two-story home, I dug through my handbag for my keys as I jogged up the front steps. Since I rented a room on the main floor of the house for use as my music studio, I had my own key to the front door. Once I'd fished it out of my bag, I let myself in and called out a greeting. I received nothing but silence in response.
Passing by the front room I used as my studio, I headed straight through to the back of the house. JT knew I was coming over, but had mentioned that he'd be out walking his dog, Finnegan, at some point in the morning. I figured that was the reason for their absence, and continued on out the back door and across the yard to a small shed. Inside I found gardening tools and the fall bulbs I'd purchased from a garden store the previous weekend.
After a brief battle with some clingy cobwebs, I picked up everything I needed and set off around the house to the front garden. JT had already turned the soil, making my job far easier. Within moments, I was digging holes for the bulbs, enjoying the light exercise and the fresh, gentle breeze that kept me from getting too warm.
Once I had several holes ready, I mixed in some plant food and set to work placing the tulip and daffodil bulbs a few inches apart. When spring arrived, the daffodils would bloom a bright shade of sunshine yellow, and the tulips would come out in hues of red, pink, and purple. With flowers planted on both sides of the front steps, JT's house would look extra welcoming in several months' time.
I got into a good rhythm of planting and the job went quickly. I was covering up the last of the bulbs with soil when JT turned up the walkway, Finnegan straining at the end of his leash.
"Hey, guys." I brushed dirt from my hands and crouched down to give the dog a hug.
His tail wagging enthusiastically, the collie malamute cross gave me a wet kiss on the cheek and then broke free of my embrace. JT unhooked the leash and Finnegan ran in a circle around me before bouncing over to sniff the trowel I'd left stuck in the dirt.
"Looks like you've been busy," JT said. "Want something to drink?"
"That would be great. Thanks."
"I'll be back in a moment."
He jogged up the steps and into the house. While he was gone, I smoothed out the dirt I'd shoveled over the bulbs and stood back to admire my work. When JT returned with two cans of root beer in hand, I went over to meet him. He passed me one of the cans and then sat down on the front steps.
"You planted all the bulbs you bought last week?" JT asked, eyeing the empty bag at the base of the steps.
"I did." I sat down next to him and popped open my can of root beer.
"Thanks. The garden will look great next spring."
"I hope so." I took a sip of my delicious, fizzy drink, savoring it before swallowing. "Did you guys have a nice walk?"
"We did." JT scratched Finnegan on the head as he trotted past, intent on exploring every corner of the front yard, despite the fact that he'd done so a thousand times before. "Any other plans for the day?"
"Not really." I turned my face up to the beautiful clear sky, enjoying the warmth of the sun as well as the hint of autumn crispness in the air. "Just relaxing, I guess."
"Sounds good to me." JT took a drink of his own root beer. "The guys are coming over later for an extra band practice. Feel free to stick around if you want."
"I might just do that." I hadn't been to one of JT's band practices for ages. Listening in might be fun. "Is Aaron coming? Or is he too jetlagged?"
"He texted this morning to say he's coming. Have you seen him since he got back?"
"I met him at the airport yesterday and we went out for dinner. Sounds like he had a good time over the summer."
"I bet he's happy to be back though," JT said. "He's really into you, you know."
"Really?" I shifted on the wooden step, my nerves suddenly on edge for some reason I couldn't pinpoint. I hurried to change the subject. "You'll never guess what happened at the concert the other night. Well, after the concert, to be exact. At the reception."
"Clausen professed his undying love for you?"
I grimaced. "Not quite," I said, remembering Hans's spiel about breaking up with Elena.
JT looked at me sharply. "Not quite? I was kidding."
"I know you were, but that wasn't what I wanted to talk about anyway."
"Hold on," JT said, his forehead furrowed with concern. "Did Clausen say something to you?"
I rolled my eyes. "Just that he and Elena have broken up. For some reason he thought I'd care. Which I don't," I added quickly.
"Good." JT relaxed. He'd never been thrilled about my involvement with Hans, and finding out that he was two-timing me hadn't helped matters. "So what else happened?"
"Archibald Major—one of the symphony's benefactors—died."
"Seriously? At the reception?"
"Yep." I took another sip of my drink and then frowned, remembering Major's last moments. "It wasn't very pleasant."
"What happened? A heart attack?"
"Actually, it might have been murder."
"Somebody killed him in the middle of the reception?"
"It's not like someone attacked him or anything, but murder is a possibility, at least. I know the police are investigating, but they're still waiting for the autopsy and toxicology results. But if homicide detectives are sniffing around, something made them suspicious. I'm pretty sure somebody poisoned him."
JT nearly choked on his root beer. "Poisoned?"
"Yep."
"Dori . . ." He seemed at a loss for words for some reason.
"What? I know it's awful but—"
He recovered enough to cut me off. "What if you'd been poisoned too? This is crazy."
His reaction caught me off guard, but it didn't take me long to figure out what was at the root of it—fear.
I patted his back. "Don't worry. I wasn't in any danger."
"How do you know that?"
"If the poison was in the food or the champagne, other people would have died or fallen ill too." That was true, I realized. I pondered the possible scenarios. "So, either someone put it in his champagne glass or coffee cup or . . ." I recalled what I'd witnessed the other night. " . . . his flask."
Yes, his coffee cup and flask were the most likely sources of the poison. Slipping it unnoticed into his champagne would have been tricky, and I guessed that poisoning any food he'd eaten would have been harder. But his coffee cup was another story, as was his flask. Major didn't fill his own coffee mug. Mrs. Duffy had done that for him, with some help from Gareth Hollingsworth, and their backs had been to the rest of the room at the time. I found it hard to imagine Jordan's mother killing her own father, even if he was unpleasant, but I couldn't ignore the fact that she'd had the opportunity.
And what about Mr. Hollingsworth? Could he have slipped something into the mug as he talked to her?
Maybe, but I couldn't think of any possible reason why he'd want Mr. Major dead.
I moved on to consider the flask. The poison could have been slipped into it before the reception. So the question was, who'd had access to his flask?
"Don't even go there."
JT's voice pulled me back to the present.
"Go where?"
"I can see the wheels turning in your head. You're trying to solve the crime. Don't you remember what happened the last time you got mixed up in a murder investigation?"
It was hard to forget. I'd almost been burned to a crisp twice, and those experiences continued to haunt me via the occasional nightmare.
"I'm not getting mixed up in anything."
"Are you sure?" The question was laced with suspicion.
"Positive. Do you know who you sound like?"
JT hesitated. "Your parents?" He clearly hoped that wasn't the answer.
"Gosh, no. I never told them about my involvement in the murder investigation last spring. No, you sound like Detective Salnikova. I ran into her at the theater yesterday and she went on about keeping my nose out of her investigation and leaving the sleuthing to the professionals."
"I like the sound of this Salnikova."
"I knew you would," I said. "But, really, JT, you have nothing to worry about." I didn't give him a chance to voice any more skepticism. "Do you have time for some lunch before band practice?"
"Sure."
JT helped me move the bulb food and gardening tools back to the shed. Once that was done and I'd washed the remaining dirt off my hands, we set off for a nearby bakery, where we bought sandwiches before returning to JT's house to eat. Not long after we finished our meal, JT's fellow band members began arriving. Rafael and Hamish showed up first, and soon wandered down to the basement with JT. I stayed behind in the kitchen, waiting for Aaron to appear.
He came in through the unlocked front door a minute or two later and I slipped off my stool at the breakfast bar so I could greet him. His dimples came out in full when he saw me. I gave him a quick hug and kissed him, but broke away when I heard someone coming up the stairs from the basement.
Aaron tried to pull me closer to him, but I resisted. "If Hamish sees, he'll make fun of us," I said by way of explanation. Teasing me was one of Hamish's favorite pastimes.
"Let him."
He kissed me, but I broke away again as the person coming up the stairs entered the kitchen. It was JT, rather than Hamish, but that didn't bring me any relief. Instead, my cheeks heated up and I suddenly didn't know where to look.
"Ready, Aaron?" JT asked with a grin.
"Sure." Aaron shot one last smile in my direction before heading down the basement with JT. Finnegan trotted after them, his fluffy tail wagging.
I almost followed, but hesitated, my cheeks still warm. I turned for the fancy coffeemaker that sat on the granite countertop, hoping to buy myself some time before heading downstairs. But as soon as I touched the machine, my stomach clenched up.
Maybe coffee was the last thing I needed. My nerves were jangling like wind chimes in a blustery storm, and I didn't think my stomach would be receptive to anything at the moment.
I hugged myself and closed my eyes.
What was wrong with me?
Nothing, I told myself. Nothing at all. Get yourself downstairs and enjoy the afternoon. Everything will be fine.
I did my best to believe that, forcing myself into action and descending the stairs to the basement. JT had a recording studio down there where he worked on his own compositions and helped other musicians record their albums. There was also a large room next to the studio where the guys were now set up with their instruments, ready to make some music. I made myself comfortable in a beanbag chair, and Finnegan curled up on the floor next to me. While the guys played the opening bars of one of JT's original songs, I stroked the sleek fur on the top of Finnegan's head, letting the familiar music wash over me.
Usually I found music calming and comforting, but for some reason my stomach wouldn't unclench and my nerves wouldn't stop their clashing and clanging. A minute or two later, Aaron looked over my way and flashed me a grin as he drummed out the song's catchy beat. I sent him a smile in return, filling it with as much enthusiasm as I could muster, but as soon as his attention strayed, my happy expression faltered and disappeared.
You don't feel the way you should about him.
The thought slipped into my head unbidden. I tried to push it away as soon as I was aware of it, but it wouldn't be ignored. It resounded in my head like the sonorous blast of a foghorn. I wanted to cover my ears to shield myself from the thought, but I knew that would draw the guys' attention without helping matters at all.
Sinking down deeper into the beanbag chair, I wondered why I was thinking such thoughts. Aaron and I had a great time together the evening before, so why question things now? I enjoyed his company, spending time with him.
But there's something missing.
That traitorous voice spoke up again.
You like him, but it doesn't go much beyond that.
I didn't want to believe the voice in my head. Aaron was an amazing guy and I wanted to think that our relationship had a chance to go somewhere incredible.
I tugged on my left earlobe as the guys tackled a second song. I wouldn't make any firm decisions yet, I resolved. Maybe we simply needed to spend more time together. We'd been apart so long, and I desperately wanted to believe the answer was as simple as that.
Somehow I managed to make it through the rehearsal without being lost in my thoughts the entire time. Afterward, Aaron offered to drive me home and I accepted, hoping some time alone with him would put my worries to rest.
But when he walked me from his car to my apartment building and kissed me goodbye, all my hopes for our relationship fluttered away like sheets of music scattered by a strong gust of wind. I had to work hard to keep my emotions off my face as we exchanged some parting words, but as soon as I was inside the lobby of my building with my back to Aaron, I closed my eyes with despair.
The truth rang clearly in my head. I could no longer ignore it, no matter how much I wanted to.
Even though Aaron was an amazing, gorgeous guy, whatever spark I'd felt between us in the beginning was now gone.
Chapter Six
WHEN I AWOKE the next morning it took me a moment to remember why my stomach was tied in a tight knot of unhappiness. As I left sleep farther behind me, it all came rushing back with the fury of a sudden, violent storm. With a groan, I rolled onto my stomach and pulled my pillow over my head. I didn't want to face reality. Unfortunately, my pillow wasn't enough to keep it at bay.
I let out another groan and threw off my covers, heading straight for the bathroom as soon as my feet hit the floor. As I showered and dressed, I hummed to myself, making my way through several tunes. That helped to keep my mind off of Aaron and the state of my feelings for him. I couldn't hum to myself all day though. Not unless I wanted people to think I was nuts. So when I left my apartment to shop for some groceries, my earbuds fed the music of Vivaldi's Four Seasons into my ears, providing me with the same distracting and calming effect.
On my way to the store I walked along at a leisurely pace, checking out the displays in the shop windows. When I reached a shoe store, I stopped, my eyes riveted on a pair of gray high-heeled boots in the window. I had a weakness for high-heeled boots and already owned a few pairs in different colors and styles. But I didn't have a tall gray pair like the one on display.
The boots were so beautiful that I didn't want to look away. Eventually, however, I stopped gawking at them long enough to step inside the store and ask the price. It was fairly hefty, as I'd expected, but not unaffordable. Still, I didn't want to spend that much money on a whim, no matter how much I wanted the gorgeous boots.
I decided to think about it for a while. If I still wanted them desperately in a couple of days, then I might treat myself to the purchase. After all, maybe some retail therapy was exactly what I needed. Although I doubted that even boots that beautiful could make me feel better about the state of my relationship with Aaron.
Leaving the shoe store behind, I continued on my way. Thoughts of Aaron still haunted me, but the music carried me through the morning as I stocked up on groceries and took care of some other errands. By the time I arrived at my studio shortly after noon, I only had fifteen minutes to kill before my first student of the day was scheduled to arrive.
After stowing my belongings in the small front room, I spent a minute or two greeting Finnegan. Once I'd given him a big hug and had received a sloppy kiss in exchange, I made my way toward the kitchen at the back of the house.
Several voices floated up the stairwell that led from the kitchen to the basement, and I figured JT was probably busy working. That was fine. If I'd had a chance to talk to him, he no doubt would have guessed that something was bothering me. That would have led to questions I didn't want to answer. Plus I had no desire to talk to JT about my romantic feelings (or lack thereof) for his friend and fellow band member.
So I didn't bother to interrupt him and stayed on the main floor, using his fancy coffee machine to make myself a vanilla latte. While I waited for my drink, I tried not to think about the first time Aaron had asked me out. He'd asked me right there in JT's kitchen, and at the time his interest had left me giddy with happiness. I wished I knew why those feelings had disappeared. Maybe knowing what had happened would help me find them again.
Pushing those thoughts aside and with my hot drink in hand, I returned to my studio and prepared for the arrival of my first student of the day.
Teaching was something I normally enjoyed, but that day I was extra eager to dive into several hours of lessons. While focused on my students, I didn't have time to dwell on other things. Like Aaron. Or deaths that might have been murder. Because of that, I managed to teach five students without having to deal with unwanted thoughts. But that respite wasn't to last.
As I waved goodbye to my last student of the day, Jordan Duffy walked up the concrete path to the house. For a second I thought he must have mistakenly believed it was Tuesday—one of his usual lesson days—but then I realized he didn't have his violin with him. His normally cheerful face seemed pulled down by what I guessed was grief, and he lacked the usual spring to his step.
I waited on the front porch as he approached. "Hey, Jordan," I said when he reached the bottom of the steps. "What's going on?"
"My mom's over on Dunbar Street buying some groceries. She sent me to ask if you could come to our place for my lessons this week. My grandfather died a few days ago and my mom says she's got too much going on to be driving me around town. I told her I could take the bus, but she wouldn't listen." When I didn't answer right away, he added, "She'll pay extra."
"I'm sorry about your grandfather," I said. "I was at the reception when he collapsed—I'm sure this is a tough time for your family. I can come to your place for a week or two. We'll just have to change the time of your lessons, because I can't be late for orchestra rehearsals."
"Can we keep them on Tuesday and Thursday? I have sports after school every other day of the week."
I considered that request. "That might be tricky. I'd have to come by after rehearsals and that would mean starting your lesson around nine p.m."
"Is that too late?"
"I suppose not, on a temporary basis, at least. That's not too late for you?"
"Nah. I never go to bed before eleven. Nine is fine."
I wasn't sure if his mom would be so agreeable. "Tell you what, run that by your mom and she can e-mail me with her final answer. All right?"
Jordan nodded, but he made no move to leave.
"How are you holding up?" I asked.
"All right."
Despite his response, he looked so sad that I didn't have the heart to send him away right then.
So instead I sat down on the top step and patted the spot next to me.
"Why don't we sit for a minute?"
Jordan climbed the steps and sank down next to me. "Did you know my grandfather was murdered?"
The question took me by surprise and a second or two ticked by before I responded. "I heard that foul play was suspected."
Jordan stared off toward the tree-lined street. "The police told us this morning that it's officially a murder investigation. My grandfather was poisoned."
Salnikova must have received the results she was waiting for.
I swallowed back a welling of sympathy for my student. It was bad enough that he'd lost his grandfather, but knowing someone had deliberately killed him must have made it worse.
"That's terrible, Jordan. I'm sorry."
His shoulders rose and dropped in a lifeless shrug. "It's not surprising, really. My grandfather was a total a—" He glanced at me and revised his description. "He wasn't a very nice guy."
Even though I'd gathered as much from Friday night's reception, I found it incredibly sad that Mr. Major's grandson held such an opinion of him. Did that mean the old codger hadn't had enough kindness in him to cultivate a positive relationship with Jordan? As sad as it made me to acknowledge it, I knew the answer was most likely yes.
"Did your grandfather have any enemies that you know of?" As soon as I asked the question I wanted to kick myself. Switching into amateur sleuth mode wasn't what Jordan needed from me.
He didn't bat an eye at the question, though. "Plenty. Not that I could name any of them, but I know he pissed people off on a regular basis, including my uncle Kevin."
My eyebrows rose an inch or two with his last words. "You think your uncle might have killed your grandfather?"
"No." Fierce intensity replaced the dullness in Jordan's blue eyes. "I don't think he did it. I know he did."
Surprise stuck my tongue to the bottom of my mouth. The little I knew about Kevin Major—gleaned from his conversation with his sister on Friday night—didn't give me any reason to doubt that he belonged on the suspect list. Clearly he harbored anger toward his father, and his final words before he stormed out of the theater that night could easily have been construed as ominous, if not threatening, but what startled me was Jordan's intense certainty about his uncle's guilt.
After several seconds I managed to shake off enough of my surprise to wiggle my tongue loose. "How do you know?"
Jordan glared out toward the street. "My uncle's a criminal. He's been in and out of jail my whole life and he always wanted money from my grandfather, but my grandfather refused to give him any. They had a huge fight a few days ago and I heard my uncle threaten to kill my grandfather."
Yikes. Maybe Jordan really had fingered the killer.
"Have you told this to the police?"
"I told them but I don't know if they took me seriously. I mean, who's going to listen to a fourteen-year-old?"
"Did you talk to a female detective? Detective Salnikova?"
"Yeah. You know her?"
"Sort of. I'm sure she'll consider everything you told her."
He shrugged again, clearly unconvinced. He sat up straighter and focused his eyes on me. "Maybe if you talk to her. She'll probably take you more seriously."
"I don't know . . ."
"Please, Midori. My uncle should be in jail."
It was hard to ignore the desperate plea in his blue eyes.
"I don't know what I could tell her that would do any good."
"Just make sure she investigates my uncle."
I tugged on my left ear as I thought over his request. There really wasn't any reason to say no. Salnikova might not listen to me, but I could at least talk to her to appease Jordan. There was no harm in that.
"All right," I said. "I can't promise that it will make any difference, but I'll talk to her."
Jordan's shoulders relaxed. "Thanks, Midori." He stood up. "I'd better go find my mom."
"See you tomorrow," I said as he headed for the sidewalk.
Once he'd disappeared from sight, I returned to my studio and checked my phone. Aaron had texted me, wanting to know when he could see me next. A big lump of dread lodged itself in my stomach. What was I supposed to say?
With relief I realized I wouldn't have a chance to see him for two days. I had another concert that night, and the next night we'd both be busy—Aaron with band practice and me with a rehearsal at the theater. That meant Wednesday was the earliest option. Guilt adding to the weight of the dread in my stomach, I sent him a quick reply to tell him that.
As I slipped my phone into my purse, JT tapped on the open door of my studio.
"Hey, do you have a few minutes?"
"Sure. Why?"
"I've got a pot of chili cooking. I thought you might want a bite to eat before your concert."
My stomach rumbled, reminding me of the fact that I hadn't eaten for hours. "Yum. Sounds good."
I followed JT down the hall to the kitchen, giving Finnegan a scratch on the head as the happy canine trotted along beside me. While JT ladled the chili into bowls, I perched on one of the stools at the granite breakfast bar.
"Thanks," I said when he set a bowl in front of me.
JT sat down next to me and Finnegan settled at our feet, watching with hopeful eyes for any tidbits of food that might tumble down to his level.
"I was thinking I should come to one of your concerts later this month."
I brightened. "Really? That would be nice. It's been a while since you last came to one."
"Almost a year, I think. Too long. My mom and stepdad would probably like to come too."
"Cool."
"Has Aaron been to one of your concerts yet?"
Despite the delicious smell of the chili, my appetite slipped away. "No, not yet."
"I bet he'd like to."
"You think so?"
"Of course. Like I said before, he's really into you."
I was so busy fighting my unpleasant feelings that I wasn't sure if I'd detected something odd in JT's voice.
"Something wrong?" he asked as I slid off my stool.
I shook my head and made my way around the breakfast bar. "Just getting a glass of water. Want one?"
"Sure."
I could sense his eyes on me as I filled two glasses with cold water. Sure enough, when I turned back to the counter, his brown eyes were focused on my face.
"Something is wrong."
Hiding things from him was next to impossible, but I really didn't want to discuss Aaron. I set down our drinks and slid back onto my stool, buying myself some time.
"You know my student Jordan?" Yes, I was deflecting his attention away from the subject of Aaron, but that was something I really needed to do at the moment.
JT thought for a second. "Blond hair? Teenager?"
"That's him. He was just here to ask about rearranging his lesson schedule. His grandfather was the guy who died at the reception on Friday night."
"Really? That's rough."
I chewed on a mouthful of chili and nodded. Once I'd swallowed, I said, "And it turns out he was definitely murdered. Poisoned."
Although I hadn't seen Jordan's uncle set foot inside the reception room at the theater, I wondered if Jordan could be right, if his uncle really had killed his grandfather. It wasn't something I could rule out, even though Kevin Major hadn't had access to Mr. Major's coffee that night. There was always the possibility that he'd had access to the flask.
"I can see why you're preoccupied then," JT said as he scooped more chili onto his spoon. "But you're still not launching an investigation of your own, right?"
"Of course not." I hopped down from my stool and took my dishes to the sink, trying to ignore the tiny voice in my head that questioned the veracity of that statement. But I really didn't have any intention of launching a full-scale amateur investigation. Although I decided it probably would be best not to mention the fact that I'd agreed to talk to Detective Salnikova on Jordan's behalf. JT might not take that the right way. If there was a right way to take it.
"Dor—"
"Thanks for dinner," I said, not giving him the chance to say anything more. "I'd better run or I'll be late."
I gave Finnegan a quick pat on the head and waved to JT as I fled down the hall to my studio. I didn't want to address the fact that he didn't believe me. Mostly because he was right not to. Even at that moment, thoughts of motives and suspects filled my head.
Maybe I wasn't mixed up in the investigation yet (not too mixed up in it, anyway), but there was no way I could promise that the situation wouldn't change. In fact, after seeing the heartbreaking mix of emotions in Jordan's eyes, I didn't think I'd be able to stay out of it. Whether Kevin Major was the murderer or not, Jordan needed closure. And if I could help him get it, that's what I'd do.
Chapter Seven
WHEN I REACHED the Abrams Center, I changed out of my jeans and into my black concert clothes. After shoving all my belongings in my locker, I scanned the musicians' lounge for Mikayla. I caught her eye and motioned to her to join me across the room. She said a few words to the people she was with and detached herself from their group. As soon as she was within reach, I took her arm and pulled her into a relatively quiet corner of the room.
Although I didn't want to talk about Aaron with JT, I did want to talk about him with Mikayla. I needed to.
"What's up?" she asked with concern.
I lowered my voice so no one else would hear me. "There must be something wrong with me."
"Okay," Mikayla said, drawing the word out as she raised one eyebrow. "Does this have something to do with Aaron?"
I was surprised that she'd caught on so quickly. "How did you know?"
"Maybe because you weren't excited about him coming back from London?"
Oh. Right.
My shoulders sagged. "I was hoping things would be better when I actually saw him."
"But I'm guessing they weren't."
"No." I leaned my back against the wall, tempted to let myself slide down to the floor in a heap. "What's wrong with me?"
"Nothing's wrong with you. You're just not feeling it. That's the way it goes sometimes."
"But Aaron's great. He's sweet and kind and gorgeous. And he has the dreamiest accent. I want to be crazy about him. I do. So why aren't I?"
Mikayla shrugged. "You can't force chemistry."
I closed my eyes, my heavy dread fusing into a hard, unbreakable rock of certainty. "Oh God. I have to break up with him, don't I?"
"Sounds like it."
I forced my eyes back open. "But I don't want to hurt him. I really, really don't."
"It's tough," Mikayla said, "but if it needs to be done, it needs to be done. Putting it off will only make it harder on both of you."
I groaned and dropped my head into my hands. I knew she was right.
Mikayla put her arms around me and gave me a hug. "Sorry, hon. I know that's not what you wanted to hear."
"No, but it's what I needed to hear. It's what I already knew but didn't want to accept."
I blinked back tears as she gave me another hug before letting go.
"You'll be okay," she assured me. "I promise. But it's best to get it over with so you can both move on."
"I know." I tried for a smile, though happiness eluded me at that moment. "Thanks, Mikayla."
She gave my arm a sympathetic squeeze before nudging me toward my locker. "We'd better get our instruments. It's almost time to head for the stage."
I retrieved my violin, glad to have an evening of music on the horizon to help calm me. While Mikayla had told me what I needed to hear, her words hadn't made me feel any better. If anything, I felt worse.
Now I knew for certain that I had to hurt Aaron, and it was impossible to view that scenario in any sort of positive light. I was a terrible person. I had to be. But at least I could lose myself in Rachmaninoff's music for the next two hours.
THE CONCERT WAS as much of a success as the first one of the season. More of a success, actually, considering that nobody died. Sure, Major had died at the reception after the first concert, but the incident had cast a shadow over the entire evening. This time, however, the post-concert mood was a happy one, thrumming with the energy of dozens of musicians pleased with their performance and the audience's reaction to it.
I basked in the thrill of the standing ovation all the way back to the musicians' lounge. It was only when I reached my locker that my mood sank back down to its pre-concert depth. The knowledge that I had to break up with Aaron the next time I saw him boomed inside my head, over and over like a deep drumbeat, impossible to ignore.
Weighed down by my thoughts, I tucked my violin in its case and loosened my bow. Bronwyn arrived at her locker, located next to mine, and chatted away with her stand partner as she clicked open her combination lock. I was reaching for my jacket when Bronwyn's shoulder bag fell from her locker and hit the floor. The contents spilled out onto the carpet, keys and tubes of lipstick mixed in with a compact and a package of Mentos.
But that wasn't all. A gold brooch with a gleaming sapphire had also spilled out of the bag. I stared down at the piece of jewelry, and so did several other people. Slowly, a hush settled over the lounge.
"What the . . ."
The words came from Bronwyn. I raised my eyes from the mess on the floor and took in the sight of her stunned, puzzled expression.
I wanted to say something, but didn't know what. The silence returned, but it seemed to thrum with tension. A shadow fell over the fallen shoulder bag.
Elena had arrived on the scene.
"My brooch!" She swooped down and snatched the jewelry from the ground. She pinned her fierce gaze on Bronwyn, her blue eyes full of icy fire. "You're a thief!"
Bronwyn's eyes widened. "No! I'm not. I swear!"
"Don't bother denying it." Elena held up the brooch. "Everyone here saw this fall out of your bag."
"No." Bronwyn said the word faintly, fear and shock written clearly on her face.
Elena ignored her denial. "You'll be thrown out of the orchestra for this. Maybe you'll even get tossed in jail. It would serve you right."
Spinning around on her heel, Elena marched out of the lounge. Janine hurried after her, but not before I caught a hint of a smirk on her face.
My eyes followed the two of them out of the room but then returned to Bronwyn. Her back was against the bank of lockers, her face alarmingly pale. Concerned, I grabbed one of her arms and Mikayla took the other. Together we guided Bronwyn over to one of the couches and got her to sit down.
Around us, people began talking again, but their voices were lowered to whispers and several pairs of eyes kept darting in our direction.
"I . . . I don't understand," she said, her face still pale. "I don't know how that brooch got in my bag." Her face lost more color. "Oh God. I can't go to jail. I've got a daughter to raise."
"You're not going to jail," I said firmly, sitting down next to her.
Mikayla sat on her other side. "Midori's right. We'll get this sorted out before it comes to that."
Bronwyn looked at us each in turn, her eyes desperate. "So you believe me?"
"Of course we believe you," I said. I'd known Bronwyn for years and she was one of the last people I could picture stealing jewelry, or anything else for that matter. But more than that, I knew her well enough to read her expressions, and her shock and puzzlement at seeing the brooch fall out of her bag had been one hundred percent genuine.
She relaxed slightly, but then she noticed the suspicious glances she was getting from other people in the room. "But nobody else does."
"Forget about them," Mikayla told her.
I nodded in support of her statement. "We're going to get this figured out and clear your name."
"But how?" Bronwyn asked, unshed tears glistening in her eyes.
I took her hand and gave it a squeeze. "By finding the real thief."
AFTER PROVIDING BRONWYN with a few more reassurances, Mikayla and I helped her put everything back into her shoulder bag and sent her home to her family.
"Do you really think it's possible to figure out who the real thief is?" Mikayla asked, not bothering to hide her doubt now that Bronwyn was gone.
"I sure hope it is." I considered what our first move should be. "I think we should talk to Maestro."
"No doubt Elena already has," Mikayla said.
"No doubt," I agreed. "Maybe Mr. Hollingsworth as well. Hopefully not the police, though."
After securing our lockers, we left the lounge. Fortunately, Hans hadn't left the theater yet and we found him in his office on the second floor. He was finishing up a phone call as I tapped on the open door.
Something like hope or expectation flickered in his eyes when he saw me, but then he spotted Mikayla next to me and his eyebrows drew together. "Are you here about Bronwyn?"
"Yes," I said.
"We're guessing Elena already spoke to you," Mikayla said.
"She did," he confirmed, and for a split second he looked a bit harried.
If I'd been standing in his office for a less serious reason, I might have taken some pleasure in the fact that he didn't seem to have enjoyed his conversation with Elena. As it was, I had more important things to focus on.
"We're not sure yet how the brooch ended up in Bronwyn's bag," I said.
"Isn't that rather obvious?"
"Elena thinks it is," Mikayla said, "but we don't."
"Bronwyn's not a thief," I added.
Hans raised his eyebrows. "The evidence suggests otherwise."
I tried to quell my rising frustration. "There has to be another explanation. Surely you won't kick her out of the orchestra without at least giving her a chance to prove her innocence."
Hans let out a sigh and ran a hand through his blond hair. "To be honest, I don't know what's going to happen. But it's getting late and there's not much we can do about it tonight anyway. I'm meeting with Mr. Hollingsworth tomorrow to discuss the matter."
I wasn't keen to leave things up in the air, but I could tell we wouldn't get any more from Hans that night. After Mikayla said good night to Hans and tugged on my arm, I followed her out of the office. With my spirits hovering about an inch above the floor, we returned to the lounge, where Dave and a handful of other musicians were still hanging around.
"Some of us are going out for drinks," Mikayla said as I retrieved my jacket from my locker and pulled it on. "Want to come?"
"No, thanks." I shoved my music folder into my quilted bag. "Between everything that's happened with Aaron and now Bronwyn, I'm not in much of a celebratory mood."
"I get that, but don't beat yourself up too much about Aaron. These things happen. As for Bronwyn . . . Hopefully things will turn out okay."
"Hopefully," I echoed.
She hesitated, probably because of my low spirits, but I didn't want to hold her back.
"Go on," I encouraged, doing my best to smile. "I'll be fine."
She gave me a quick hug and then grabbed her belongings and headed out of the lounge with Dave and three others. Once they were gone, I fished my phone out of my bag and checked for messages. I didn't have any.
A sudden sense of loneliness settled over me. What I really wanted at that moment was to hang out with my best friend, to share all my worries and guilt with him. At the same time, I still didn't want to talk to him about Aaron, and I couldn't do one without doing the other.
With my bag over my shoulder and my instrument case in one hand, I slammed my locker shut and snapped my combination lock into place. Even if I was willing to talk to JT about Aaron, it was too late to bug him. He wouldn't mind what time it was, but I knew he'd be working in the morning and I didn't want to keep him up late with my problems. That meant I'd have to handle all my pre-breakup emotions on my own. Good thing I had Smarties ice cream in my freezer.
On my way out of the musicians' lounge, I realized I wasn't the last to leave. Ernest was still by his locker. I paused, considering whether or not I should talk to him. It didn't take me long to come to a decision. Reversing my direction, I approached him and stood by his locker as he buttoned up his coat.
"Hi, Ernest. How are you doing?"
He made only the briefest of eye contact with me. "Fine, thank you."
"It's terrible what happened to Mr. Major the other night, isn't it?"
"Oh." His neck flushed and his eyes darted around the empty room. "Um. Hrm." He cleared his throat and returned his focus to his top two buttons.
His reaction only confirmed the suspicion I'd developed on Friday night—he didn't think it was so terrible that Major had kicked the bucket.
"Right," I said, trying to keep my voice light, "you didn't like him much, did you?"
Ernest's fingers slipped from his top button. He stared at me through his thick glasses, his gray eyes wide. "What . . . why . . . what makes you say that?"
"I saw the way you glared at him the other night. And then there was the note."
His eyes almost popped right out of his head. "Note?" Panic pushed up the pitch of his voice. "What note?"
"The one you threw away after Mr. Major collapsed. The police have it now."
His face flushed red and then drained of color. He shut his locker and fumbled with the lock until it was secure. "I have no idea what you're talking about."
"Ernest . . ."
"No. No. No." He clutched his instrument case to his chest. "I really must go."
He hurried away from me and out the door.
Poor guy. I'd freaked him out and didn't learn anything new in the process. Although if he was a murderer I didn't need to waste my time feeling sorry for him. Then again, if he was innocent I'd stressed him out and put him on Detective Salnikova's radar for no good reason. But of course I'd had to turn the note over to Salnikova. Withholding it would have been irresponsible, especially since I didn't know if Ernest was innocent or guilty. Jordan believed his uncle had killed his grandfather, but in my view Ernest still belonged on the suspect list, especially since he definitely had something to hide.
As I wandered out of the lounge and headed for the theater's nearest exit, I realized that Bronwyn's predicament and my encounter with Ernest had distracted me from my thoughts of Aaron, if only for a short while. Maybe that meant focusing on Major's murder and the jewelry theft would provide me with a good diversion over the next few days. Murder and theft weren't exactly pleasant subjects, but mulling over the crimes was still far less stressful for me than worrying about the sorry state of my love life and what the heck I would say to Aaron when I broke up with him.
Yes, I decided as I made my way along the darkened street to the nearest bus stop, I'd talk to Salnikova as I'd promised Jordan and I'd do a bit of digging into other matters to see what I could turn up. That way I'd be helping out my student and my friend as well as preventing myself from wallowing in guilt and dread.
It seemed like a perfect idea.
SINCE I HAD a few free hours the next morning before I was scheduled to teach my first student of the day, I hopped on a bus and headed for the police station. I hadn't enjoyed the best night's sleep, but I was alert enough to focus on what I hoped to achieve during my visit with Detective Salnikova. As Jordan had requested, I'd make sure she was well aware of Kevin Major's potential as a suspect in his father's murder. In addition to that, I hoped to glean whatever information I could from her about the progress of the investigation.
She wasn't likely to share much—if any—information with me, I knew. But I also knew there was always a chance I could learn something interesting if I kept my eyes and ears wide open while at the station. Maybe it was a long shot, but careful observation had led me to valuable information in the past.
Upon my arrival, I asked the man at the enclosed reception desk if I could speak with Salnikova. He directed me to take a seat while he checked if she was available. I hoped I wouldn't have to sit there for too long. I was all too familiar with how uncomfortable the chairs were.
A few minutes later, the man at the reception desk called for my attention.
"Ma'am?"
I got up and approached the desk, glad to leave the hard chair behind.
"I'm afraid Detective Salnikova isn't in at the moment and I'm not sure when she'll be back. Would you like to leave a message or speak to someone else?"
I tugged at my ear as I considered those options. "No, thanks. I'll call her later and set up an appointment when it's convenient for her."
The man nodded and turned his attention to the scruffy, middle-aged man who had just entered through the front door.
Disappointed that I hadn't accomplished anything during my visit to the station, I wandered across the reception area to the door. As I pulled it open, I heard a woman sob somewhere behind me. When I glanced over my shoulder, I let go of the door, forgetting my intention to leave.
A uniformed officer had entered the reception area from a back hallway. While the door was open for him to pass through, I caught a glimpse of another officer leading a crying woman along the hallway in my direction. Behind him followed Detective Salnikova, the person I'd come to see. But the detective wasn't the one who had caught my attention. Instead, my wide, shocked eyes locked on the woman in tears.
Mrs. Andrea Duffy.
Jordan's mother.
Chapter Eight
MRS. DUFFY?"
She raised her head when I called her name, but her tear-filled eyes barely registered my presence. I'd never seen her so distraught and disheveled, her light brown hair hanging in straggles and her makeup smeared. She hadn't even looked that bad when her father took ill at the reception.
I caught hold of the door as the uniformed officer with Mrs. Duffy guided her into a room halfway down the hall.
"Sorry, ma'am, you can't go back there," the officer next to me said as I made a move to leave the reception area for the back corridor.
I opened my mouth to protest, but Salnikova spoke up before I got a word out.
"It's all right, McGuire. She can come with me."
Relieved that she hadn't sent me on my way, I hurried to Salnikova's side. She ducked her head into the room where Mrs. Duffy had been taken and said a few words. I tried to peer in through the open door, but Salnikova put a hand on my back and guided me farther down the corridor.
"You've arrested Mrs. Duffy?" I still couldn't believe it.
"What are you doing here, Ms. Bishop?"
"I came to talk to you, but the man at the front desk said you weren't here."
"I just arrived. And Mrs. Duffy isn't under arrest. She's here to answer some questions she didn't want to answer at home in front of her son."
She offered that explanation as we reached an open area filled with several desks, a few of which were occupied by detectives in suits. The gray filing cabinets and light gray walls didn't exactly give the place a cheery atmosphere, but I figured most of the detectives' work was probably more grim than cheery. A lone potted plant stood in one corner, but a thick layer of dust left it looking almost as gray as the drab walls.
"Take a seat, Ms. Bishop." Salnikova's voice interrupted my study of the room.
"Could you please call me Midori?" I requested as the detective pointed me to a chair next to one of the desks. "Ma'am and Ms. Bishop are so formal, they don't sound like me."
"All right."
Her agreement brought a smile to my face even though I was still distracted by the presence of Jordan's mother down the hall. I sat in the offered chair and waited until Salnikova had settled behind her desk.
"Do you really think Mrs. Duffy could be the killer?" I asked. "I find that so hard to believe."
"I'm afraid I can't—"
"—discuss an ongoing investigation," I finished for her. I'd lost count of how many times she'd told me that in the past.
"That's right. Now, what did you want to see me about?"
It took me a second to shift gears. "Kevin Major. Jordan, his nephew, believes he killed Mr. Major. Maybe you don't agree, but I promised Jordan I'd talk to you and make sure you were at least looking into Kevin as a suspect. He has a criminal past and he did make that ominous statement the night of the reception, don't forget."
She ignored that last part. "And your connection to Jordan Duffy?"
"I'm his violin teacher."
"I see. Jordan raised his concerns with me the other day and I can assure you that we've been investigating a number of possible suspects."
"So you've talked to Kevin?"
"I have."
I waited but she didn't elaborate on her answer. She really wasn't going to give anything away.
A heavyset man with gray hair and a protruding stomach lumbered his way between the desks, heading in our direction and distracting me from my conversation with Salnikova. Although he wore cargo pants and a T-shirt rather than a suit, I recognized him right away.
"I was wondering where he was," I said as Salnikova's partner, Detective Bachman, drew closer to us.
Salnikova swiveled in her seat to see who I was talking about. "Mark? What are you doing here? You're supposed to be at home resting."
Bachman waved off his partner's concern. "I was going stir-crazy. Thought I'd come by to see how things are going with the Major case." His gray eyes slid to me and narrowed with recognition. "Don't tell me you're mixed up in this case."
"I'm not mixed up in anything," I said, unable to help sounding indignant. "I simply have a . . . peripheral involvement."
That wasn't too much of a stretch of the truth. At least, I didn't think it was. I hadn't yet done anything aside from retrieving Ernest's note and talking to Jordan and Detective Salnikova. That hardly qualified as getting mixed up in the investigation. JT and Bachman might not agree with me, but I was sticking to that opinion.
Bachman let out a dubious-sounding grunt and returned his attention to his partner. "I hear you've got the daughter in for questioning."
Salnikova glanced at me and got to her feet. "I'll just be a moment."
I sat back in my chair as the two detectives moved several feet away to converse in lowered voices. They both kept their backs angled toward me so I couldn't attempt to read their lips. I strained my ears and thought I heard the words "fingerprints" and "flask" but beyond that I couldn't catch anything.
Disappointed, I let my eyes wander over Salnikova's desk. A file folder sat in the middle of it. Although it appeared innocent enough, once I'd spotted it I couldn't forget about it. The thought that it might contain information about Major's murder burned in my brain.
I cast a glance in the detectives' direction. They were still in the midst of their hushed conversation and didn't seem to have the least bit of interest in me. A quick sweep of the open area told me that no one else was paying attention to me either. Biting down on my lower lip, I reached out toward the folder.
I knew I shouldn't touch it. I knew I could get in major trouble if anyone caught me snooping into its contents. But I couldn't resist. With another quick glance at Detectives Bachman and Salnikova, I lifted the folder open and craned my neck to peer at the top sheet of paper inside.
It appeared to be some sort of report. I scanned the top sheet, which mostly seemed to be filled with mumbo-jumbo. But I understood enough to conclude that it must be the toxicology report. And since Major's name was printed near the top, I knew I wasn't looking at the report for some other case.
Knowing I had limited time before Salnikova's return, I ran my eyes down the page for a second time, trying to retain whatever information I could. That wasn't much, but one word did jump out at me.
Brugmansia.
Letting the folder fall closed, I sat back and dug my phone out of my purse. Once online, I typed the familiar word into the search bar. As soon as the search results popped up, I knew why the word had rung a bell.
Angel's trumpet.
My mom had grown Brugmansia a few years back and the one thing I knew about it was that it was extremely poisonous.
Oh my God. That's what the killer used to poison Mr. Major.
As that thought echoed in my head I realized it didn't tell me much. It wasn't as if there was a limited class of people with access to angel's trumpet. It probably grew in many gardens around the city. Mrs. Duffy could have obtained some as easily as anyone else on my suspect list.
Although it was difficult for me to believe that Mrs. Duffy had killed her father, the police obviously had reason to think she at least had some pertinent information, otherwise they wouldn't have brought her in for questioning. I hoped it wasn't anything more than that because I didn't want to believe Jordan had a murderer for a mother. But the fact that she seemed so distraught had me worried.
Poor Jordan. He had to be devastated about the whole situation, especially now that the police had dragged his mom right into the middle of it.
Catching sight of Salnikova approaching out of the corner of my eye, I cleared away any evidence of the search on my phone and slipped the device back into the depths of my purse.
I smiled, hoping she didn't suspect that I'd snooped into the report. "Is Detective Bachman all right?"
"He's on medical leave," Salnikova replied as she sat down at her desk again. "He's recovering from some minor surgery, but he'll be fine."
She shifted the report off to one side. Tension I hadn't previously noticed eased out of my body when she made no sign that she suspected I'd touched it.
"That's good to hear," I said.
"Now, was there anything else you wanted to talk to me about?"
"No, I really came here for Jordan's sake. It must be hard for him to know you're questioning his mom."
A shadow of something close to regret passed across Salnikova's face. "It's a difficult time for him, of course."
"Hopefully he's hanging in there." I got up out of my chair. "I'd better get off to work. Thanks for taking the time to talk to me."
Detective Salnikova walked me out of the police station and I set off down the street toward the bus stop, my head buzzing. The police seemed to be working hard to solve Mr. Major's murder, and that was a good thing. It was never nice to think that a killer was roaming free. If they found the killer quickly, I'd no longer have to worry about solving the case to bring Jordan closure.
Of course, if Jordan's mother was a murderer rather than a person with pertinent information to share, then Jordan would be more distraught than ever and I wouldn't know how to help him. Hopefully the investigation wasn't going in that direction and never would, but as I once again remembered Mrs. Duffy's distress, I couldn't help but think that she was indeed a suspect in her father's murder.
AS SOON AS I finished teaching my last lesson of the day, I packed up my violin and set off from JT's house. Aaron was due to arrive for band practice in less than an hour and I didn't want to linger in case he showed up early. Maybe I should have stayed around instead of putting off our breakup until the next day, but I wasn't brave enough for that. I told myself it wouldn't be kind to break up with Aaron right before his band practice. An excuse, yes, but I let myself buy it.
On my way to the theater, I ducked into a small Japanese restaurant and ordered myself some dinner. I tried not to think about Aaron while I munched my way through a dynamite roll and washed it down with a cup of tea. I also tried not to think about him as I traveled the rest of the way to the theater and during the rehearsal. Although I wasn't entirely successful, the music was—as usual—a helpful distraction, as was Bronwyn's predicament.
I noted with a pang of worry that she wasn't present at the theater. During our break in the middle of the rehearsal, I leaned toward Mikayla and asked, "Have you heard anything from Bronwyn?"
"She texted me earlier today. She said she was too humiliated and embarrassed to show her face here tonight."
I frowned, feeling bad for my absent friend. "But she hasn't been kicked out of the orchestra yet?"
"Not to my knowledge."
My eyes settled on Hans where he stood speaking with two clarinet players. As their conversation drew to a close, I got up from my seat and wound my way around chairs and music stands, reaching Hans as the other musicians turned to walk away.
"Did you talk to Mr. Hollingsworth?" I asked without preamble.
Hans hesitated, but eventually responded in a lowered voice. "I did."
"And?"
His eyes scanned the stage around us, as if to make sure no one was listening in. "We think we can keep the theft quiet and avoid bad publicity."
"But what about Bronwyn? What's going to happen to her?"
Hans sighed. "Nothing has been decided for certain yet, but it doesn't look good for her. We can't have a thief in our midst."
"But she's not a thief!"
"Can you prove that?"
I frowned, not wanting to voice my answer.
"I admire you for sticking by your friend, Midori," he said. "But there's not much I can do for her in the circumstances." His eyes drifted away from me, his attention shifting. "It's time to resume the rehearsal."
Disappointed, I returned to my seat, wishing I could think of a way to help Bronwyn. But as much as I didn't want to admit it, I had no idea how to track down the real thief. I recalled that Bronwyn had carried her shoulder bag with her in the reception room when she came to say goodbye. That meant pretty much anyone present at the reception could have slipped the brooch into her bag.
But why had they done so?
Maybe someone had a grudge against her and wanted to get her kicked out of the orchestra. I found that hard to believe, knowing how nice Bronwyn was, but if the real thief was spiteful enough, it wasn't impossible.
I decided to consider that possibility further when I had a chance. In the meantime, I let myself get caught up in the rehearsal, and by the time it came to an end I had something else on my mind. It was the night of Jordan's first at-home lesson and I wanted to leave the theater as soon as possible so I wouldn't be late. Mrs. Duffy had e-mailed me the evening before, confirming that she was okay with the plan Jordan and I had come up with. She'd also provided me with the address of Mr. Major's house in Shaughnessy, where she and Jordan had been living ever since she and her husband had split up.
Soon after packing up my instrument and leaving the theater, I boarded a bus and headed toward the upscale neighborhood of Shaughnessy. I had to walk two blocks after disembarking from the bus, but streetlamps lit my way and I found the house without difficulty.
Before heading for the front door, I paused in front of Mr. Major's residence, taking in the sight of it. As I had expected, the house—a small mansion, really—was impressive. White and dark brown, it was built in the Tudor style, and each of its generous two stories had to be at least four times the size of my apartment. Probably more. I didn't doubt that the property was worth several million dollars.
I followed a curving driveway lined with low hedges and solar lights and approached the large front entrance. Several seconds after I pressed the bell, Jordan opened the door and stepped back to let me in the house. As we exchanged greetings, my gaze swept over the foyer. It was as impressive as the exterior of the house, if not more so. A crystal chandelier hung from the two-story-high ceiling and a grand staircase curved its way up to the second floor. Expensive-looking paintings hung on the white walls and the heels of my boots clicked against the beautiful hardwood floors.
Jordan shut and locked the door, but before he had a chance to do anything else, a phone rang somewhere on the second floor.
"Sorry," he said as he made a move for the stairway. "That's probably my aunt calling. I should answer it since my mom's not here. You can wait in the living room at the end of the hall. I won't be long."
As he disappeared up the stairs, I made my way along the hall. I paused at an open door on my right and took a quick peek into the room. It appeared to be Mr. Major's study. A stately wooden desk sat in the middle of the room and thick, leather-bound tomes lined a large bookcase, probably more for show than for reading, I suspected. The desktop was tidy, as was the entire room. Somehow it didn't surprise me that Mr. Major hadn't been the disorganized type.
Continuing on down the hallway, I found the living room Jordan had mentioned. At one end of the room was a massive stone fireplace, at the other, a wet bar with numerous bottles of alcohol in a cabinet with glass doors. Gorgeous antique furniture was placed here and there throughout the room and every piece of artwork was undoubtedly way out of my financial league.
I crossed the room to a set of French doors that led out to a large stone patio. Curious to know if the backyard was as luxurious as the house, I opened the doors and stepped outside. Beyond the patio, a neatly trimmed lawn stretched toward a rose garden. To the left was a lap pool, and to the right, a tennis court, both illuminated by bright floodlights. I didn't think I'd ever been in a backyard so large and I wondered how much anyone actually used it.
Deciding to head back inside, I turned around and stepped toward the open French doors. I stopped short when I noticed a large potted plant behind a lounge chair.
I recognized it right away. Its elegant, trumpet-shaped flowers were unmistakable.
Angel's trumpet.
Mrs. Duffy and anyone else living in the house had easier access to the plant than I'd imagined. I swallowed and tightened my grip on my instrument case. Even though the flowers were pretty, the plant seemed sinister to me. Tearing my eyes away from it, I stepped back inside the house.
As I pulled the door shut behind me, there was still no sign of Jordan. I considered taking a seat on an antique chesterfield, but the sound of rustling papers drifted toward me from the hallway leading toward the front of the house.
Wondering if Jordan had already come downstairs or if his mom had arrived home, I set down my violin and followed the sound along the hall.
"Jordan?" I called out. "Mrs. Duffy?"
The rustling stopped, replaced by silence that had a strange, heavy quality to it. I paused for a second but then continued on, the high heels of my black boots clicking against the hardwood floors and sounding unusually loud in the quiet house. I approached the open door to Mr. Major's study.
When I peered into the room, a flicker of movement drew my eyes to one of the floor-to-ceiling windows. A man in a dark suit slipped out the open window, dodged around the bushy plants outside, and fled toward the driveway.
"Hey!" I yelled.
The sound of my voice only sent the man running faster. I raced over to the window and stuck my head out into the cool night air, but the man had already disappeared into the darkness.
He'd made a swift escape, but he hadn't been fast enough for me to fail to recognize him. I'd caught a glimpse of his profile and that was enough for me to make a positive identification.
What he'd been doing in Mr. Major's house, I had no idea. But what I did know for certain was that the intruder had been none other than Dr. Daniel Beaufort, vice chair of the Point Grey Philharmonic's executive committee.
Chapter Nine
"MIDORI?" JORDAN STOOD in the study's doorway. His forehead creased as his eyes went from the open window to the large wooden desk, now covered in a mess of papers and file folders. "What's going on?"
Still stunned by Dr. Beaufort's presence in the house and his sudden flight, it took me a second to figure out where to start. "Was there supposed to be anyone else in the house?"
"No. Just us. Marjorie's gone out for the evening and my mom went to visit a friend after . . . after she took care of some things."
I knew those things involved the police. I wasn't surprised that she'd sought support from a friend after her interview, especially if she didn't want Jordan to see how upset she was.
"Why?" Jordan asked. "Was someone here?"
"Yes. I guess I disturbed him in the middle of . . . whatever he was doing. He took off out the window." I crossed the room to the large desk. All of its drawers were open and papers were strewn about on the floor as well as on the desk. Definitely different from the last time I'd looked in the room. "Do you know Dr. Daniel Beaufort?"
"No. Never heard of him." Jordan came over to join me by the desk. "Is that who was in here?"
I nodded. "He's a member of the Point Grey Philharmonic's board of directors. But what the heck would he be doing breaking in here?"
Jordan studied the messy surface of the desk. "He must have been looking for something. This desk was super tidy last time I looked in here. It always was. My grandfather liked everything perfect." He reached out for the nearest file folder.
I put a hand on his arm to stop him. "Hold on. Best not to touch anything. I'll call the police and tell them what happened. They'll probably want to come and take a look." I glanced around the office, noting a conspicuous absence. "Did your grandfather have a computer?" I didn't think Dr. Beaufort had been carrying a laptop when he took off, but I couldn't be completely certain.
"No," Jordan replied. "He refused to touch computers. He always wrote everything by hand. If he needed anything done on a computer, he got his secretary to do it at his office downtown."
So Beaufort hadn't made off with a computer. I wished I knew if he'd taken anything else.
As I retrieved my cell phone from my purse, my eyes roved over the papers scattered across the desk, hoping to find a clue as to what Dr. Beaufort was after. Without moving the pages I could only see small portions of each, and certainly not enough to glean any helpful information. Although I longed to shuffle through the papers for a better look, I knew I needed to follow my own advice and leave everything untouched until the police arrived.
Beside me, Jordan stuffed his hands in the pockets of his jeans, his shoulders slumping.
I put an arm around him. "Why don't we go sit in the other room while we wait for the police?"
He allowed me to guide him to the living room at the back of the house. He flopped down onto an antique chesterfield and stared at the unlit fireplace as I scrolled through my contacts until I found the number I'd stored last spring. I paced across the room as I put a call through to Detective Salnikova. Considering the hour, I wasn't sure if she'd pick up, but she did. After I explained to her about the intruder at Mr. Major's house, she instructed me to stay out of the study and wait for her arrival. I hung up and joined Jordan on the chesterfield.
"Detective Salnikova is on her way," I told him.
He nodded, but his eyes seemed distant.
"I'm sorry about everything you're going through, Jordan. And now with your mom . . ."
His eyes snapped toward me. "You know about that?"
"I saw her at the police station when I went to see the detective."
"Did you talk to her? Is she okay?" Desperation underscored his questions, and in that moment he seemed younger than his fourteen years. "She phoned me afterward, but she wouldn't say much."
"I only saw her briefly. She was upset but otherwise all right." I hoped that was the truth.
"She didn't do it, you know." Anger and certainty replaced the desperation in his voice. "There's no way she killed my grandfather."
I wanted to believe him for his sake, but I wasn't sure if I could. Even though I had trouble picturing his mother as a murderer, for all I knew at that point she might well have killed her father. At the same time, Dr. Beaufort's stealthy presence in the home complicated matters.
Instead of agreeing or disagreeing with Jordan, I asked him a question. "What do you think the police wanted to ask her?"
He shrugged and slouched against the back of the chesterfield again. "How should I know? It's not like they tell me anything."
I thought back to what little I'd overheard at the police station.
Flask.
Fingerprints.
Maybe they'd found Mrs. Duffy's fingerprints on Mr. Major's flask. And if the poison had been in the flask, then that was a link between Mrs. Duffy and the crime. But was it really? Mrs. Duffy was the victim's daughter and currently lived in his house. Was it so strange that she might have handled his flask at some point? I didn't think so, but I didn't know anything for sure.
"Jordan, did your grandfather often carry a flask with him?"
His eyebrows lifted a fraction of an inch. "Often? More like always."
"Was it common for anyone else to handle it?"
"Wait. Was that where the poison was? In his flask?"
"I don't know for certain, but I think so."
Jordan sat up straighter. "My mom didn't like my grandfather drinking so much. His doctor told him he should stop because of his health, but of course he wouldn't listen. I know there were a couple of times when my mom helped him out of his jacket and found the flask in his pocket. She took it out so he wouldn't drink in public, but she always paid for it later."
"How do you mean?"
Jordan's voice hardened and his hands clenched into fists. "He'd yell at her. He'd swear at her and call her names."
I tried to distract him from the unpleasant memory. "So there would be a perfectly reasonable explanation for her fingerprints being on the flask."
He nodded and his hands slowly unclenched. "Not that the police would care about that. They just want to arrest someone so it looks like they're doing their job. What do they care if it's not the right person?"
I wasn't sure if that was an entirely fair assessment of the situation, but I wasn't about to upset him by disagreeing with him.
"Okay," I said, thinking out loud, "so what about motive?"
"My mom would never kill my grandfather. She wouldn't."
"I'm not saying she did," I said quickly, hoping to defuse his growing anger. "I'm just trying to figure out what the police are thinking. Do you have any idea why they might think she would have wanted your grandfather dead?"
He thought for a moment. "Maybe for his money? Things have been really tough for my mom since she left my dad. She tried to get a job but nobody wants to hire her. And my grandfather wouldn't give her any money even though he has piles of it." Although Jordan's anger had diminished, a note of resentment crept into his voice. "He thought he was doing too much by letting us stay here. And he never let her forget that."
"But he left her a lot of money in his will?"
"I don't know, actually. Knowing him, he probably didn't leave us anything. But maybe the police think she expected him to even if he didn't?" He shrugged. "I don't know. Whatever they're thinking, it's stupid. My mom's not a murderer."
The doorbell rang but Jordan made no move to get up. He crossed his arms over his chest and glared at the fireplace. I understood that he wasn't happy with the police and figured he didn't need anything extra to deal with anyway, so I left him there on the chesterfield and headed for the spacious foyer. Once there, I unlocked and opened the door. Salnikova stood at the top of the wide stone steps, two uniformed officers behind her.
"Thanks for coming," I said as I stepped back and pulled the door open wider so they could enter the foyer. After shutting the door behind them, I led them down the hall to Major's study.
"This is where you saw the intruder?" Salnikova asked as she stepped into the room with the uniformed officers in her wake.
I remained in the doorway. "I was waiting for Jordan to come downstairs. I heard a noise and thought maybe it was him so I came to look. That's when I saw Dr. Beaufort take off through the open window."
"You recognized him?"
"Yes. Dr. Daniel Beaufort. He's on the Point Grey Philharmonic's board of directors."
"You didn't mention on the phone that you recognized him." Salnikova's voice held a note of rebuke.
Oops. "Sorry."
"And you're certain that's who the intruder was?"
"Absolutely."
"All right. Thank you, Ms. Bishop. I'll come talk to you and Jordan in a few minutes."
She turned her back to me and I sighed at her use of my last name and the obvious dismissal. As much as I wanted to stay outside the door and eavesdrop, Salnikova would probably note the absence of my heels clacking down the hallway. With a silent curse at my beautiful but nonstealthy boots, I returned to the living room where I'd left Jordan.
As I took in the sight of all the expensive furnishings and artwork, it occurred to me that Mr. Major had owned everything material that he could have possibly wanted—a mansion, impressive display pieces, a private island (if that rumor was true), and a heck of a lot more. But other areas of his life had been lacking, of that I was sure. I doubted the rooms of his house had seen much happiness or laughter over the years. That made me even sadder for Jordan. Had he left one unhappy home—disrupted by his parents' feuding—for another?
I forced myself to focus on the present, knowing I couldn't do anything to change my student's past.
"The police are taking a look around," I told him.
Jordan frowned. "Why are they so interested in a break-in when they aren't interested in arresting the right person for my grandfather's murder?"
I joined him on the chesterfield again. "I guess there's no real evidence linking your uncle or anyone else to the crime. At least not that they've found so far."
"Just because they haven't found it that doesn't mean it doesn't exist."
"I won't disagree with you there," I said. "And maybe they'll look into Dr. Beaufort now. The fact that he broke in here doesn't make him a killer, but it certainly makes him suspicious. Any idea what he could have been after?"
"None." Jordan got up and paced across the room to the fireplace. "But I know he's not the killer."
"Because you believe your uncle killed your grandfather?"
"I know he did."
I stifled a sigh, knowing a fourteen-year-old's certainty wasn't enough to get his mother off the detective's radar. "Okay, but do you have any proof? I know you heard him threaten your grandfather, but is there anything else to link him to the crime? And maybe he had motive, but what about the opportunity to poison your grandfather?"
"I don't know." Jordan resumed pacing. "He knows where my grandfather kept all his alcohol. He helped himself to it often enough. I'm sure he could have put poison in the flask as easily as anyone."
"Does your uncle live here?"
"Nah. He lives in some crummy apartment building downtown. But he shows up here from time to time. Mostly because he wants money. He doesn't have a key but the back door is often unlocked if there's somebody home. And Marjorie doesn't have the guts to turn him away if he shows up at the front door. Not that she'd be able to turn him away. He'd probably barge right past her."
My thoughts focused on the frizzy-haired woman who'd attended the reception. "Who is Marjorie, exactly?"
"My grandfather's slave." He caught sight of my dubious expression and amended his answer. "I don't know what to call her. I guess she was a housekeeper, cook, and a sort of companion for my grandfather. She looked after him, basically. She didn't seem to care that he bossed her around constantly. But then again, she was getting paid for it, unlike the rest of us."
But what if she did mind? Maybe deep down she'd hated her boss, her resentment building day by day and week by week until she snapped. "Do you know if she'll benefit from your grandfather's death in any way?"
"I don't see how. She's staying on here for a couple of weeks, but my mom already told her she'll have to look for a new job. So unless my grandfather left her a pile of money in his will, I'd say she's worse off now than she was when he was alive."
Perhaps she was worse off, but I still added her to my mental list of suspects. Maybe Mr. Major had hinted that he would leave her a substantial legacy in his will, whether he actually did or not. Or maybe it was worth it to her to be out of a job to be rid of her tormentor (if she thought of him as her tormentor). Sure, it would have been easier to quit her job to get away from him, but not everyone made their life decisions based on rational thought, particularly murderers.
"Besides, we're getting off track." Jordan's voice brought me back from my thoughts. "I need to find a way to convince the police to investigate my uncle."
"Why don't you talk to Detective Salnikova again?" I suggested. "Remind her that your uncle threatened your grandfather and make sure she knows he had access to the flask."
"Who the hell are you?"
I jerked around in my seat, startled by the new voice as well as its vehemence. A scruffy man in need of a shave and some clean clothes stood inside the French doors, now open. He glared at me with disturbing hatred, his beady eyes almost burning with it.
"Uncle Kevin," Jordan started. He had no chance to say more.
"Are you trying to get me in trouble with the police?"
The man I now knew to be Kevin Major advanced toward me, aggression radiating off his stocky body.
I jumped to my feet and stepped to the side, keeping the chesterfield between us as my heart beat like a racing metronome. "I'm not trying to do anything."
"Like hell you aren't!" He took another step toward the chesterfield. "I heard what you said. You're trying to pin the old man's murder on me." His hands clenched into fists. "But you know what? There's no way that's going to happen. No way in hell."
My breath caught in my throat. Kevin seemed to find my fear amusing, or at least pleasing.
He sneered at me and narrowed his eyes.
The next second, he lunged at me.
Chapter Ten
I DODGED AROUND the side of the chesterfield, barely staying out of Kevin's grasp.
"Leave her alone!" Jordan yelled, but Kevin only made another grab for me.
As I evaded his grasp once more, Jordan jumped between us. That, unfortunately, didn't deter his uncle. He snatched a small but sturdy statuette off an end table and drew his arm back before letting the piece of artwork fly. The statuette clipped Jordan on the shoulder.
I was ready to make a run for it, or maybe scream for Salnikova's help, because things were seriously out of hand. The commotion must have carried down the hallway because Salnikova appeared without any request from me, both uniformed officers at her side.
"What's going on in here?" the detective asked, her eyes going straight to Kevin, who held another statuette raised up in his hand, ready to take aim at me again.
His beady eyes took in the sight of the police and his nostrils flared. He let out an angry snarl and made a dash for the French doors, dropping the statuette as he ran.
"Hey!" Salnikova yelled after him.
She gestured at the uniformed officers and they both took off after Kevin. He disappeared into the darkness beyond the reach of the floodlights, and Salnikova followed as far as the patio. She pulled out her cell phone and spoke to someone in a low voice.
I wrapped my arms around myself, realizing only then that a slight shakiness had taken over my body.
"Are you all right?" I asked Jordan as he rubbed his shoulder.
"I'm fine." He gave his shoulder one last rub and dropped his hand. "Sorry about that. He's got a psycho temper. Can't you picture him killing someone?"
"Yes, I guess I can." What I didn't mention was that while I could picture Kevin Major using physical violence to harm or kill someone, I wasn't sure I could picture him using a more subtle method like poisoning.
I moved closer to the French doors as Salnikova ended her call and the two uniformed officers returned to the patio, out of breath and without Kevin.
"Sorry, Detective," the older of the two officers said as he huffed and puffed. "He vaulted over the fence and we lost him in the alley."
Salnikova stepped back in the house. "What was that all about?" She directed the question at both me and Jordan.
My student beat me to answering. "He overheard us talking about how he killed my grandfather and went all psycho. Like he does."
That wasn't entirely accurate, but correcting Jordan would only upset him, which I didn't want to do. Besides, I figured it was close enough to the truth for the moment.
"You two go back to the study," Salnikova directed the officers. As they headed off down the hall, she returned her attention to me and Jordan. "Is Marjorie Alberts in her suite?" she asked my student.
"Not right now," Jordan replied. "She's gone out to see a play or something."
"What about your mom? Will she be home soon?"
"She said she'd be home by eleven. And my aunt just arrived in town. She's on her way from the airport."
Salnikova nodded, apparently satisfied that he would have family with him before long. While Jordan was old enough to be on his own for a while, I too was relieved to know he wouldn't be alone. Especially since I'd scared off an intruder earlier that evening. I couldn't imagine Dr. Beaufort acting violently toward anyone, but before that night I also wouldn't have been able to picture him breaking into someone's house and rifling through their belongings. More concerning than Beaufort's intrusion, however, was Kevin Major's aggressive behavior. I didn't think I could count on him not taking out his anger on his nephew, and that left me more than a little worried.
"What about your uncle?" I asked Jordan. "If he comes back . . ."
"If he comes back," Salnikova interjected, "call 911 immediately."
Jordan didn't seem to share our level of concern. "I'll lock all the doors and arm the security system. I'll be fine."
I sure hoped that would be the case.
"We'll be another minute or two in the study and then we'll be on our way," the detective said.
She set off toward the front of the house, leaving me and Jordan alone. He locked the French doors and hit a button on a small remote control set on an end table. Blinds automatically lowered over the doors and all the windows.
"It's quite late now," I said. "Do you still want to go ahead with your lesson?"
"I'd rather not, if that's okay." Now that the surprise of Kevin's appearance had worn off, the sadness had returned to Jordan's eyes. "I'm sorry for wasting your time. I'm sure my mom will still pay you."
I nodded and collected my violin. My heart ached for Jordan but I knew there wasn't much I could do aside from trying to figure out a way to prove his mother's innocence, if she was indeed innocent. That wasn't something I could figure out right then, though. I needed more information and time to sort out what I already knew.
"Okay." I headed toward the foyer, Jordan falling into step with me. "I'll see you on Thursday."
"Hopefully by then my mom will have some good news."
"What do you mean?" I asked.
"Tomorrow morning my mom and I are going to see my grandfather's lawyer about his will. Maybe he didn't leave my mom anything, but I hope he did. She's so stressed about money these days."
The mention of the will reading piqued my interest, but I didn't let it show. I figured the police already had a copy of the will, and if Mrs. Duffy stood to inherit a substantial amount, maybe that was part of their reason for suspecting her. But she might not be the only beneficiary. I'd be interested to find out if Kevin or anyone else was about to come into some money. It wasn't any of my business, but that didn't mean I wasn't curious.
I paused by the front door as I remembered something. "Do you know who Elspeth is?"
"Elspeth? Why?"
"Your grandfather was asking for her right before he collapsed."
The sadness in Jordan's eyes intensified. "That was my grandmother's name. She died eight years ago."
"Oh. I'm sorry."
Jordan shrugged, but I regretted bringing the subject up.
"I'll see you in a couple of days," I told him as we parted at the front door.
The police were still busy in the study, light pouring out of the tall windows into the front yard. I walked along the driveway, down to the street, and followed the sidewalk from there. The streetlamps were lit and I was used to walking to and from bus stops at night since I didn't own a car, but still, I removed my cell phone from my purse and held it in my free hand, just in case I needed to make a quick call for any reason.
Thoughts of Daniel Beaufort and what he might have been searching for in Major's study occupied my thoughts as I walked along the quiet street. I recalled the less than amicable conversation between the two men at the reception on Friday night and wondered if Beaufort's search of the study was related in any way. I suspected that it was, but I hadn't overheard enough at the reception to give me any sort of clue as to what the problem between them was. And I wasn't sure if I'd be able to find out. Questioning Dr. Beaufort would be awkward and it was something I wanted to avoid if at all possible.
As I neared the bus stop, a twig snapped somewhere behind me. I spun around but saw no one on the street. Maybe it had only been a raccoon or other small animal. Most likely. Still, my heart beat faster and I tightened my grip on my phone. There were plenty of places for someone to hide. Large trees lined the street and bushes and stone walls surrounded several of the large residential properties. I thought a shadow moved near one of the trees. My heart jumped and increased its rate further, its beat like the percussion part of a wild, racing song.
The shadow didn't move again and I wondered if my eyes had merely played a trick on me. I didn't want to stick around to find out if that was or wasn't the case. Picking up my pace, I selected JT's number from the list of contacts on my phone. I put the device to my ear, but it rang three times and went to voice mail.
Darn it.
I hung up without leaving a message. He was probably in the middle of band practice. Not that I expected him to come rescue me from spooky shadows or anything, but I would have felt better hearing his voice and having him know where I was. The confrontation with Kevin Major had left my nerves more on edge than I'd realized.
With a small measure of relief, I reached the bus stop, situated in a pool of light beneath a streetlamp. I peered into the surrounding darkness in a constant search for any movement. Although I thought another shadow across the street moved in an unnatural manner, once again I couldn't be sure of what I'd seen. As I was thinking about texting JT with my location and concerns, a bus rumbled into sight and some of my nervous tension eased away.
When the bus stopped at the curb, I boarded it with an even greater sense of relief, glad to put some real distance between myself and the place where I'd last seen Kevin Major, and glad to be safely away from spooky, flickering shadows.
THAT NIGHT I expected to dream of dark shapes looming over me, or of an irate Kevin bent on silencing me forever. When I awoke the next morning, however, the only dream I could recall was about a cold ride on the back of a motorcycle on a dark, rainy night. I had no idea where that dream had come from.
What I did know was that I could no longer put off breaking up with Aaron. I didn't want to wait until later in the day as I had originally planned. Waiting while knowing what I had to do was only upsetting my stomach and making me fidgety. I had to get it over with. As soon as possible.
Still in my pajamas, I fetched my phone and sent him a quick text message.
Do you have time to meet for breakfast?
I showered and dressed while I waited for an answer.
Sure. When and where? was his response.
I sent him the name and location of a small café not far from my apartment and arranged to meet him there in twenty minutes. My stomach did a series of somersaults and I was tempted to dive back into bed and bury my head under the pillows. How could I do this to him? I really, really didn't want to. But I also didn't want to string him along and allow him to believe that our relationship was going somewhere when it wasn't.
After drying my hair and running my brush through it one last time, I set off to the café. When I arrived, I peered in through the large front window. Several patrons sat at the tables, but I didn't spot Aaron among them. Perhaps that was for the best. I'd prefer to talk to him outside of the café, rather than within earshot of all the morning diners.
A minute or two later, he came along the street toward me, smiling when he saw me waiting for him on the sidewalk. I tried my best to return the smile, but I was quite sure that it was more of a tremulous grimace.
"Hey," he greeted, giving me a hug and a quick kiss. "All right?"
I didn't know how to respond.
"What's wrong?" he asked when he saw my expression.
I took his arm. "Can we walk for a minute?"
"Sure." He fell into step next to me, but when I glanced up at him, his eyes held a hint of hesitation. "What's going on?"
Still holding his arm, I let out a deep sigh. "I'm so sorry, Aaron."
"Sorry for what?"
I couldn't bring myself to answer, but in the end I didn't have to.
He took in my expression and stopped in the middle of the sidewalk. "You're breaking up with me?"
I closed my eyes for a second. "Yes. I'm sorry."
He shook his head, dazed. "I don't get it. I thought things were good between us. Is it because I was away?"
"No. Maybe. I don't know." I sighed again. "All I know is that I don't feel the way I need to for us to stay together. I like you a lot. I really do. But . . ."
"Not enough," he finished for me, a note of hurt ringing behind his words.
I'm such a horrible person, I thought as I tried to keep myself from crying. Seeing the pain in his eyes as I broke the news to him was worse than I'd imagined. My throat burned from the effort of trying not to cry.
"I'm sorry," I said again, not knowing what else to say. My words sounded feeble and hung in the air between us.
Aaron scrubbed a hand down his face. "I guess I should have seen this coming."
"Why?"
"You didn't seem quite yourself the other day. It was like you were holding back or something. I thought maybe it was because I'd been away so long but . . ."
"I wish I felt differently, Aaron. I really do."
He stared past me, down the street. "So do I." After several impossibly long seconds, he met my gaze again. A wall had gone up, hiding his emotions. "I guess that's it then."
I swallowed and struggled to speak. "I guess so."
He leaned in and gave me a quick, light hug. "Take care of yourself, Midori."
"You too," I whispered as I returned the hug.
As soon as he let go of me, he was off down the street, moving away from me with long strides. I stood there in front of the café, watching him go, a single tear sliding down my cheek.
Chapter Eleven
I SKIPPED BREAKFAST. There was no way I could stomach any food after what I'd done to Aaron. So instead I walked. My last glimpse of Aaron's eyes haunted me and I couldn't shake the heavy dark cloud of sadness that hovered over me. I walked and walked without paying too much attention to where I was going. When I finally took in my surroundings I realized I'd automatically headed in the direction of JT's place.
By then I was already in his neighborhood, only a few blocks away from his house. My first student of the day wouldn't arrive for more than three hours so I sent JT a quick text message, letting him know that I was going to show up early. He never minded when I did that, but I always liked to let him know as a courtesy.
I'm at the park with Finnegan, he wrote back a few minutes later. But I'll see you soon.
I made a quick stop at a bakery to buy a sandwich and some potato salad for later in the day—hoping that I'd eventually have an appetite—and carried on to JT's place. When I arrived, I let myself into the silent house through the front door and went straight to my studio. After shedding my jacket, I fastened my hair into a messy twist at the back of my head and fetched my spare violin from the corner of the room. Since I hadn't planned on coming straight to my studio after breakfast, I didn't have my best violin with me, but that didn't matter. My spare would serve me just fine for teaching purposes.
I removed my bow from the case and tightened it, rubbing a bit of rosin on the hairs. Next, I tuned up my instrument. Then I took in a deep breath, let it out slowly, and lost myself in Jules Massenet's Méditation.
The intermezzo from Massenet's opera Thaïs never failed to speak directly to my soul. I drew the opening notes out of my instrument, allowing the peaceful melody to flow through me, to calm me. As I got deeper into the piece, it grew more passionate, and I poured all my emotions into the music, my eyes closed, nothing existing for me in that moment other than my violin and its song. When the melody grew peaceful and reflective again, I felt soothed, less burdened.
I was so absorbed in the music that I didn't hear JT come home. I only became aware of his arrival when Finnegan trotted in through the open door to my studio and rushed over to me for an enthusiastic greeting. Setting my bow on the ledge of the music stand, I knelt down to give him a one-armed hug. After giving my cheek a good lick, a satisfied Finnegan trotted back out of my studio and down the hall.
Giving up on the intermezzo, I tucked my violin and bow securely away in my instrument case and made my way to the kitchen. JT stood by the sink, chugging down a glass of water.
"Hey," he said when the last drop was gone.
"Did you guys have a good walk?"
"It was great, wasn't it, Finn?"
Finnegan sat in the middle of the kitchen and barked his agreement, his tail thumping against the tiles.
"Want something to drink?" JT offered.
"Sure. A vanilla latte, please." I leaned against the granite countertop as JT set about making my drink.
"Sorry I missed your call last night."
It took me a second to remember when and why I'd called. "No worries. It wasn't important. I forgot you'd be in the middle of band practice."
JT handed me the latte and we wandered out onto the back porch, where we settled on the top step.
"Speaking of band practice," JT said as Finnegan lay down at the bottom of the stairs, "you were all Aaron could talk about last night."
I didn't respond, not knowing what to say. Memories from my last conversation with Aaron came flooding back, obliterating the calm the music had brought me. All I could see was Aaron walking away from me, his shoulders weighed down by the hurt I'd caused him.
"Dori? What's wrong?" JT asked after several seconds of silence had ticked by.
I stared out at the grassy yard, taking a moment to rein in my emotions as much as possible before replying. "I broke up with Aaron this morning. It was awful. I mean, it's not like there was a big scene or anything, but I hated hurting him."
"I'm sorry, Dor. I never would have said what I did if I'd known."
"I know. Don't worry about it."
We sat in silence for a minute or two before JT asked, "What happened? I thought you really liked him."
"I did. I do. And in the beginning he gave me butterflies and all that but now . . . There's something missing. Something important." I released a growl of frustration. "Seriously, I don't get it. He's such an awesome guy. Why can't I be crazy about him?" I dropped my head into my hands. "Maybe I need to take some time away from dating so I can clear my head and figure out what it is I'm looking for."
"You'll find the right person, Dori."
"I hope you're right."
"I am."
I leaned against his arm and rested my head on his shoulder. Somehow sitting there with my best friend, sharing my thoughts with him, eased the storm of my unhappy emotions. He was such a steadying force in my life. I hoped he knew how much I appreciated that.
As we sat there together, watching Finnegan sniff at the bushes along the edge of the yard, a thought occurred to me. "How are things with you and Shauna?"
"We broke up a few weeks ago."
I raised my head from his shoulder. "What? I had no idea."
JT shrugged. "You couldn't have. I never said anything."
"But why not? You know I want to know what's going on in your life."
"I guess it didn't seem like a big deal. Things were never serious between us."
My stomach sank like a ship's anchor. "I should have asked you sooner."
He hadn't mentioned Shauna's name for weeks and I was well aware of that. But I hadn't asked.
"It doesn't matter, Dor. I promise."
"I'm a terrible friend." I didn't mean to say the words out loud, but I did.
"That's not anywhere close to the truth." He put an arm around my shoulders and gave me a squeeze. "Come on, don't be so hard on yourself."
I rested my head on his shoulder again. "I guess I'm not in the greatest mood today."
"Understandable."
Down in the yard, Finnegan let out a volley of loud barks and bounded toward the back fence.
"What's up, boy?" I called.
Finnegan didn't react to the sound of my voice. Instead, he kept his nose pointed at the fence, barking with unusual urgency.
JT and I stood up at the same time, descending the steps into the yard. As we approached Finnegan, his barking only grew more insistent. We leaned over the fence to check out the alley. A flash of movement to my left caught my attention. I leaned farther over the fence for a better look and caught a glimpse of someone darting out of the alley and onto the side street several houses down.
Finnegan gave one last bark and resumed sniffing at the bushes, his interest in the alley gone in a flash. JT turned back to the house and made his way across the yard. I trailed behind him, unease humming through my bones. Although I wasn't anywhere close to certain, I thought the man who'd made the swift departure from the alley looked an awful lot like someone I didn't want anywhere near me—Kevin.
AN HOUR OF practicing for the symphony's next concert and several hours of teaching my students helped to distract me from thoughts of Aaron. When I finished working for the day, JT was busy in his basement studio. I'd hoped to have his company over dinner but that wasn't to be, so I retrieved the potato salad I hadn't eaten at lunchtime and sat in the quiet kitchen eating my meal, Finnegan lying at my feet.
To prevent myself from replaying my breakup with Aaron like a broken record, I turned my thoughts to Bronwyn's situation. I felt like I was letting her down and I desperately wanted to find a way to help her.
Pausing between forkfuls of potato salad, I sent her a text message, asking if she'd ever had any conflicts with anyone in the orchestra. By the time I'd finished my salad and had washed it down with a glass of water, she'd responded.
Why? she wanted to know.
I'm wondering if someone wanted to frame you, I explained.
I can't think who, she wrote back a minute later. I've never had problems with anyone.
I tugged on my left earlobe as I considered her response. If none of the other musicians had a grudge against her, then I couldn't think of any reason why someone would purposely set her up to look like a thief.
Unless it was an accident.
I sent another quick message, asking Bronwyn if she'd ever left her bag unattended on the night of the reception.
She replied almost right away.
It was either locked in my locker or over my shoulder the entire evening.
Disappointed, I discarded the accident theory. It seemed highly unlikely that someone could have mistaken Bronwyn's bag for somebody else's if she'd never left it unattended while it was out of her locker. Unable to think of anything else remotely helpful at the moment, I sent Bronwyn a final message, telling her to try not to worry too much. That was probably a pointless suggestion, but I didn't know what else to say.
Leaving the kitchen, I wandered back to my studio and considered heading home, but realized I didn't want to go straight back to my apartment. Instead I decided to make a trip to the neighborhood library to pick up some books to keep me company later in the evening. I definitely needed a good story to distract me from unpleasant matters and I'd exhausted the supply of reading material I had at home.
Once I'd given Finnegan a hug goodbye and locked up JT's house behind me, I set off at a brisk pace toward Dunbar Street. Clouds had moved in over the past hour or so and the air held a promise of rain. The thought of having a good book to read appealed to me even more with worsening weather on the way. Curling up under a warm blanket and losing myself in a fictional world seemed like a good way to end a not so great day.
At the library I took my time browsing the shelves, choosing a historical novel and two mysteries, including the latest Richard Castle book. After checking out my selections at the circulation desk, I stepped outside to find rain pouring down from the dark sky, forming puddles that shimmered in the glow of headlights and streetlamps.
With my books tucked safely into my bag, I set off down the well-lit street. A bright flash of lightning lit up the sky, followed several seconds later by a deep rumble of thunder. I clutched the strap of my bag, my throat going dry. I hated thunderstorms. Ever since a childhood friend had died from a lightning strike at age eleven, I'd feared storms more than almost anything.
Hesitating under the awning of a closed bakery, I considered whether I should hop on a bus and head home or go back to JT's and wait out the storm there. I stepped out from under the cover of the awning and peered up the street. There were no buses in sight.
More lightning flickered across the sky and I jumped when another rumble of thunder—louder this time—boomed out over the noises of rain and traffic. By the time yet another fork of lightning cut a jagged path across the dark sky, I had already made up my mind. I wasn't about to stand around waiting for a bus, not when I could reach JT's house in five minutes.
Half walking and half jogging, I hurried around the corner onto a quieter street. As much as I disliked damp clothes, the rain seeping through my jeans and jacket was the last thing I was worried about.
The sky flickered again and this time the clap of thunder followed a mere two seconds later. I tried to pick up my pace, but could only go so fast in my high-heeled boots without risking a fractured ankle. Two blocks from JT's house, a shadowy figure darted out from behind a parked car. My already racing heart skipped a beat. The figure lunged toward me.
I screamed, but the rain and booming thunder swallowed up the sound. A rough hand clapped over my mouth and an arm locked around me from behind. I struggled and tried to scream again, but my attacker only tightened his grip.
"Shut up!" he growled in my ear.
I kicked him in the shin with the heel of my right boot and darted out of his grasp when he cursed with pain. I ran for the nearest house, but my attacker grabbed me from behind again and smashed me up against a large tree. He jerked my hands behind my back and kept me pinned against the trunk, my cheek mashed against the wet bark. When I whimpered, he tightened his grip on my wrists.
"I told you to shut up!"
Choking back a cry of pain and terror, I hoped and prayed that someone would look out their window and see us. But even if someone did look outside I wasn't sure they'd be able to see anything unless a flash of lightning illuminated us at the right moment.
I closed my eyes as my attacker leaned in close.
"Listen up. I didn't kill my father."
I registered his voice and his words.
Kevin Major.
"Got that? I didn't kill him. But if you don't stop interfering and telling the cops that I did, I won't hesitate to shut you up for good. Understand?" He didn't wait for me to answer. "I left a message with your boyfriend, but delivering it directly to you has been much more satisfying."
He jerked me back a few inches and slammed me against the tree again. I couldn't stifle a cry when my face hit the trunk. He yanked me away from the tree once more and I put my hands out in front of me, expecting another collision with the trunk, but this time Kevin wrenched me to the side before shoving me forward.
I crashed to the ground, my knees hitting wet grass, my palms slamming against the sidewalk. Pounding footsteps headed away from me. I raised my head and peered through the darkness and driving rain. Kevin fled away up the street, disappearing into the shadows. I gasped with relief and climbed to my feet, shaking so hard that I had to lean against the tree for support. When I was finally upright, I glanced around for my bag. It had fallen from my arm sometime during my struggle with Kevin. I spotted it on the sidewalk a few feet away.
Swiping it up off the ground, I fumbled around inside of it for my phone. My fingers brushed against the device, but my trembling hand wouldn't cooperate and it slipped out of my grasp, back into the depths of my bag.
I glanced over my shoulder, terrified that Kevin might return, but I was still alone. Only then did his words truly sink in, a swift rush of dread running through me.
I left a message with your boyfriend.
But how did he know about Aaron?
No, not Aaron.
Oh God. No, no, no.
Forgetting about my phone, I forced my shaking legs to run.
He didn't mean Aaron. He meant JT.
Chapter Twelve
I REMEMBERED THE person I'd seen in the alley that morning. It must have been Kevin. My eyes hadn't deceived me. He must have seen me with JT and assumed that he was my boyfriend.
I left a message with your boyfriend.
What did that mean? What had he done?
Please let JT be okay. Please, please, please.
I ran as fast as I could. My boots slipped on a scattering of wet leaves and I hit the ground again, this time knocking my knee against the pavement. I barely noticed the pain. I barely even noticed the storm still raging around me. All I could think about was JT. If anything happened to him . . .
No, no, no.
I slipped again but managed to stay on my feet. A few more seconds of running took me to the front of JT's house. The door stood wide open, light pouring out into the night.
My fear intensified with every gasping breath, building toward a panicked crescendo. I raced along the narrow walkway and up the steps to the front porch.
"JT!" I halted on the threshold.
A piece of paper fluttered in the door's mail slot, stuck halfway in and halfway out.
"JT? Finnegan?"
No one responded. I wanted to go inside, to search the house for my best friend and his dog, but fear held me to the spot like a cold and heavy suit of lead. What would I find? What had happened?
I had to go inside. If JT and Finnegan were hurt, I needed to get them help. Standing there frozen wouldn't do them any good.
As I forced one foot forward, a voice called out from behind me.
"Midori?"
I spun around and almost collapsed with relief. JT walked up the pathway to the house, Finnegan bounding along at his side. Even though my legs didn't want to work, I dropped my bag on the porch and ran down the steps. I crashed into JT, wrapping my arms around him and holding on to him as if my life depended on it.
"Dori?"
I could hear the concern in his voice, but I couldn't speak. When his arms went around me, I pressed my unharmed cheek against his chest, soaking in his warmth and solidity.
Finnegan bounced around us, the rain poured down, lightning flickered across the sky, and thunder continued to boom. Still, I stayed there, holding on to JT, allowing the steady rhythm of his heart to beat away some of my fear.
"Dori, what's wrong? Is it the storm? Let's get you inside."
He knew about my intense fear of storms and what it stemmed from, but he still sounded confused. I didn't blame him. As scared as I was of lightning, I'd never reacted to a storm like this before.
I shook my head as he guided me toward the house, one of my arms still around him. "Not the storm." My teeth chattered together as I finally became aware of how wet my clothes were. "The door was open. I thought something bad had happened to you."
"I must not have latched it properly. The wind probably blew it open. I went next door to help Mrs. Tilley. One of her windows was stuck open and the rain was coming in."
I shook my head again as he helped me up the stairs. I'd left out too much. "Jordan's uncle attacked me on the way here. He threatened me and said he'd left a message with you. I didn't know what he meant and when I got here . . ."
JT came to an abrupt stop on the porch. "Wait. Someone attacked you?"
"Jordan's uncle," I repeated.
"Did he hurt you? Are you okay?" His eyes searched me for injuries, zeroing in on my sore cheek now that some of the light spilling out of the house reached us. "Oh my God. Dori, what did he do to you?"
Hearing the fear and anger in his voice, I rushed to reassure him. "I'm okay. I promise. Just a few scratches."
"You're shaking like a leaf."
"I was scared." I pointed at the mail slot, my entire arm trembling. "He was here."
JT snatched the paper from the slot. I leaned against him, partly so I could read the message along with him and partly because my legs still didn't want to fully support me.
Kevin had scribbled out a message in blue ink, basically making the same threat as he had on the street, with a few swearwords added in.
Stop interfering or else, was the gist.
"What the hell, Dori? Why is this guy threatening you?"
"He thinks I told the police that he killed Mr. Major, but that's not really what happened."
"Is there a chance he came back here?"
"He took off in the opposite direction. I'm sure he's long gone."
JT retrieved my bag from the porch and guided me into the foyer. He set the paper on the hall table and shut and locked the front door as Finnegan trotted off toward the kitchen. "Still, maybe I should have a look around."
"I'm coming with you." There was no way I wanted to be left alone right then.
I grabbed on to his hand and together we moved through the house, checking each room.
"Finnegan doesn't seem concerned," I said when the dog joined us to check out JT's basement studio.
"No, that's a good sign." JT ducked his head into the last room in the basement. "All clear."
Finally, the last of my fear drained away and exhaustion moved in to replace it.
"You look like you're ready to collapse," JT said.
I was, but I didn't want to admit it.
"Come on." He gave my hand a tug. "Let's go back upstairs and I'll call the police."
The police.
Ugh. I didn't want to deal with that, not right then. But I knew it was necessary.
While JT made the phone call, I retrieved a small bag from my studio. Inside I kept a change of comfy clothing for evenings when I stayed after work to hang out with JT. After I'd changed into the wonderfully dry lounge pants and T-shirt, I bundled up my wet clothes and stuffed them in the dryer in the basement laundry room. When I returned to the main floor, I found JT in the living room, setting his cell phone on the coffee table.
"The police should be here before too long." He looked at me more closely as I stood in the doorway. "Are you sure you're okay?"
I nodded, unable to find my voice.
"You're really pale. You'd better sit down."
I walked over to the couch and sank down onto the cushions, tucking my legs up underneath me.
The concern hadn't left JT's eyes. "I'll get you something hot to drink."
He squeezed my shoulder as he passed by on his way to the kitchen. After he was gone, I could still feel the warmth where his hand had momentarily rested. I closed my eyes as a confusing rush of emotions coursed through me, tumbling over one another. Fear, joy, and sadness all mixed together, almost robbing me of my breath with their intensity.
I'd had a rough evening but my current state of emotional overload had nothing to do with Kevin Major or the thunderstorm that seemed to have worn itself out. No, my overwhelming assortment of feelings had to do with a sudden, sharp revelation.
I was in love with JT.
BY THE TIME a uniformed police officer arrived at JT's front door, I was halfway through a mug of hot chocolate. I'd hardly said a word to JT since my sudden recognition of my feelings for him, but he didn't seem to mind. He probably thought my near-silence was a result of the exhaustion that had crept up on me once I knew that he, Finnegan, and I were all truly safe.
I stole glances at him as he pulled the curtains across the bay window and again as he crossed to the foyer to answer the officer's knock on the door. I didn't understand how it could have hit me out of the blue like that. My feelings for him had evolved over time, slowly shifting from friendship to something different. I could see that now, looking back, but how had I not noticed earlier?
I hadn't wanted to, I realized. And I knew why. Being in love with my best friend scared the heck out of me. I didn't want to ruin our friendship or make things awkward between us. That would devastate me.
JT cared deeply about me, I knew that, but he'd never shown any sign of harboring romantic feelings for me. Had he?
No, I was quite certain he hadn't. So telling him about my revelation wasn't an option. I couldn't risk damaging what we had. I wouldn't risk it.
Thoughts about my relationship with JT distracted me to such a degree that I had a feeling he'd said my name more than once by the time he got my attention. When I finally managed to find my way back into the world around me, JT introduced the officer as Constable Dallinger.
At JT's invitation, the constable took a seat in an armchair, notebook and pen in hand. He appeared to be in his late twenties, probably within a year or two of my twenty-nine years. I did my best to focus on him instead of JT, who sat down next to me on the couch. My head was muddled, my body humming with all my different emotions, but I needed to concentrate.
Keeping my eyes on the dark-haired, blue-eyed constable, I told him about Kevin, starting with the confrontation at Archibald Major's house on the previous evening and then detailing the attack during the storm. Dallinger made notes and asked me several questions, which I tried to answer to the best of my ability. JT gave him the threatening note we'd found in the mail slot and he took it with him as he got up to go.
"Detective Salnikova is leading the investigation into the murder of Kevin's father," I told Dallinger as JT and I accompanied him to the foyer. "At least, I think she's leading it. She's involved with the case, anyway."
"I'll make sure she hears about this," the constable assured me. "And we'll be on the lookout for Kevin Major. But in the meantime, be careful. Try not to go anywhere alone, especially at night, and call 911 if you see any sign of him."
"I will."
JT opened the door for Dallinger, and I hugged myself against the cold, damp air that rushed into the house. The constable left after a few more parting words, and JT locked the door behind him, shutting out the post-storm chill.
I hoped Kevin wouldn't cause me any more trouble. He'd already caused more than enough, and I didn't want anything else to do with him. I suppressed a shiver as I recalled my terror from earlier that evening, but fortunately, JT's voice pulled me away from the unsettling memory.
"Do you want to watch a movie?" he asked.
"Sure." I stifled a yawn as I settled back onto the couch, eager for a distraction. "But nothing too long. I'd like to get home before midnight."
"Why don't you crash here for the night?"
I gave Finnegan's head a scratch as I considered the suggestion. "I guess I could do that. I'll have time in the morning to go home and get a change of clothes before I have to teach."
"Plenty of time, since I'll be driving you."
"JT, you don't have to."
"I'm not about to let that crazy guy have another chance to attack you."
"He won't. Not in broad daylight."
"You don't know that."
He was right. And if I were honest, I wasn't keen on the idea of going out on my own, even in daylight with plenty of other people around.
"All right. Thanks."
JT grabbed the TV remote from the coffee table. "And I'll drive you to and from your rehearsals and concerts until the police catch him."
"But what if you're working? I don't want you messing up your schedule for my sake."
"You mean for the sake of keeping you safe? I don't have a problem with that." He jabbed a button on the remote and the TV came to life.
A pleasant sense of warmth suffused my body and I had to blink against a sudden pricking of tears in my eyes. I wanted to hug him, but I was also scared to, worried that he might be able to detect my true feelings.
"Tell you what," I said when I could speak without getting choked up. "I'll see if I can get a ride to and from the theater with Mikayla. If I can't, you can drive me."
"Deal." He handed me the remote. "You choose the movie."
I scrolled through the options on the screen, on the lookout for something lighthearted and completely nonscary. After all, I'd had more than my fair share of scares for one night.
Chapter Thirteen
THE NEXT MORNING was uneventful, and for that I was grateful. JT drove me home so I could change and pick up my best violin, and after that he gave me a ride back to his place. I spent some time practicing for the upcoming concert, and by late morning it was time to teach one of my adult students. JT and Kevin played prominent roles in my thoughts during the early part of the day, but by focusing on music I was able to keep myself from getting overwhelmed.
In the middle of the afternoon I had an hour-long break since one of my students was off at school camp for the week. JT was working with some musicians down in his studio so I wandered out into the backyard with Finnegan at my heels. Although the September air was cool, the bright sunshine cut through the chill, warming my face.
I approached the back fence and leaned over to check out the alley, but aside from a squirrel darting up a utility pole and some crows on the telephone wire, it was all clear. That helped to calm my nerves, as did Finnegan's company. I knew he'd bark up a storm if a stranger like Kevin came near JT's property.
Enjoying the fresh air and sunshine, I wandered over to the apple tree in the corner of the yard. I reached up and snatched a red apple off one of the lower branches, biting into it with a crunch. It was juicy, sweet, and delicious.
While munching on the apple I surveyed the yard. JT kept the grass trimmed and the garden tidy, so everything looked neat and well cared for. I'd planted some flowers back in the spring and they'd added lots of bright color to the yard for the past several months. Thoughts of what I could plant the following spring had already entered my mind, and I considered expanding beyond flowers. There was a nice patch in the corner of the yard that would be perfect for a small vegetable garden. Maybe I'd try my hand at growing veggies next year, if JT liked the idea. Carrots and onions, and possibly some squash and green beans.
With my mind wandering away from garden planning, I turned my thoughts to the gray boots I'd seen at the shoe store. I still wanted to buy them as much as I had the day I'd first seen them, and considering the week I was having, buying something nice seemed like an attractive idea.
Yes, I decided after another moment, I'd go for it. As soon as I had a chance, I'd go back to the store and splurge on the beautiful boots.
My spirits buoyed, I fished my phone out of my pocket and checked for messages. Mikayla had replied to a text I'd sent her that morning, letting me know that she'd pick me up at JT's house later on and drive me to our evening rehearsal.
As I was about to return my phone to my pocket, it rang. I didn't recognize the number, but I tossed my apple core in the compost box and answered anyway.
"Hello?"
"Hi, Midori. It's Jordan."
"Oh, hi, Jordan. What's up?"
"I need to cancel tonight's lesson," he said. "I forgot that I'm supposed to go to a football game with my dad, and I don't see him much these days."
"I understand. That's not a problem."
After a slight hesitation, he said, "The police told me and my mom what my uncle did last night. Are you okay?"
"I'm fine," I assured him. "How are you holding up?"
"Okay, I guess. But things keep getting crazier and crazier."
"How do you mean?" I asked, curious.
"You should have seen me and my mom at the will reading this morning. We were totally stunned."
"Why?"
"It was nuts. Some things weren't so crazy, like my grandfather left money to the symphony and five thousand dollars to Marjorie. He left the house and a bunch of money to my mom—which is a relief—and he set up trust funds for me and my uncle. But get this—he also left a pile of money to his other daughter."
My eyebrows drew together as I absorbed that information. "I didn't know you had an aunt on your mom's side." As far as I knew Andrea and Kevin were Major's only children.
"I didn't either. Nobody did. Except my grandfather, apparently."
It took a second for the implication of what he'd said to register. "Wait—so she's your mom's half sister?"
"Yep."
"And your family didn't know about her until the will was read this morning?"
"Exactly."
Whoa.
"My grandfather must have had an affair way back when," Jordan continued. "My mom's kind of upset about it."
Understandably, I thought.
"So is this mystery daughter older or younger than your mom and uncle?"
"Younger. I guess my grandfather had an attack of guilt or something. I didn't know he was capable of feeling guilty about anything."
I ignored the harsh note that had entered Jordan's voice with his last sentence and focused instead on his other words. "You mean because he left her a bunch of money after never mentioning her?"
"Yeah, and there was something in his will about how he was formally acknowledging that she was his daughter. Whatever."
"Do you know her name?"
"Frances Barlow. The lawyer's going to try to track her down."
"That really must have been quite a shock, finding out about her."
"You're not kidding. Who knows what other secrets my grandfather was hiding."
Indeed. And could one of his secrets have led to his murder? Or was the motive simply financial? If that was the case, maybe this newly revealed daughter deserved a place on the suspect list. Perhaps Major's family hadn't been aware of her, but if she'd been aware of her biological father and somehow knew that she stood to inherit, she could have done away with him. That theory was a bit of a stretch, considering that I had no evidence she knew who her biological father was, let alone that she had enough of a relationship with him to know he would leave her money upon his death. But it was still something to consider.
I wondered if Detective Salnikova and her colleagues had given it any thought. If they'd seen the will, which I guessed they probably had, they would know about Frances, but I didn't know if they would view her as a viable suspect. Then again, I didn't know if she really was a viable suspect. And there was still Dr. Beaufort to consider.
Between my thoughts of JT and the murder, my head was spinning. Nothing made much sense to me at the moment and all I wanted to do right then was retreat back into the comfort and safety of my music. I ended my conversation with Jordan and returned to my studio. For the remaining minutes before my next student arrived, I immersed myself in Rimsky-Korsakov's music, pushing all my confusion aside, if only temporarily.
BY THE TIME Mikayla picked me up to drive me to the Abrams Center for our rehearsal, my mind, although not clear, was at least calmer. But Mikayla wasn't about to let me avoid the disagreeable subjects that had plagued my thoughts lately.
As she merged into the evening traffic on Dunbar Street, she asked, "What's going on? You said in your text message that you'd tell me later why you needed a ride tonight."
"Mr. Major's son attacked me last night."
Mikayla's brown eyes widened. "What? Are you serious?"
"Unfortunately."
"Oh my God. Are you okay?"
I touched the bruise on my cheek, glad I'd managed to conceal it with makeup. "I'm fine," I assured her. "Just a few scrapes and bruises. But it was scary and he's still out there somewhere."
"Wow. No wonder you don't want to be walking to and from the bus stop in the dark. But why the heck would he attack you? How does he even know you?"
I started off by telling her that Major had died as a result of foul play rather than natural causes. Once she'd expressed surprise about that, I recounted some of what had taken place at Major's house when I'd gone to teach Jordan's lesson, leaving out any mention of Dr. Beaufort. I wanted to know more about the doctor and his motives before I told anyone other than the police about him.
As I finished telling the story about Kevin Major, Mikayla shook her head in disbelief.
"Crazy. I hope the police find him soon and lock him up. But I can give you a ride any time. You know that, right?"
"Yes. Thanks."
Several raindrops hit the windshield and Mikayla flicked on the wipers. A few seconds ticked by before she glanced my way and said, "Not to dwell on unpleasant subjects, but did you do it?"
I didn't need any clarification. I knew she was asking if I'd broken up with Aaron.
"Yes. Yesterday morning."
"And?" she prompted as she made a left turn.
"And it was as bad as I expected. I felt like the worst person in the world. You should have seen the look on his face when he realized I was breaking up with him." I cringed at the memory.
"But it had to be done. Now you can both move on."
"I suppose."
I tugged at my left ear, my thoughts straying to JT. Part of me wanted to tell Mikayla all about my recent revelation. It would be a relief to share my feelings with someone, and I knew she'd be happy for me. But I also knew she'd push me to tell JT, to find out if there was any chance that he felt the same way. I wasn't sure if she'd understand my fear of ruining what he and I had.
In the end, that fear kept me quiet. It would be best for me to keep my feelings entirely to myself, safely locked away. I needed to find a way to go on as normal with JT, to ensure that nothing became awkward between us, and the best way to do that was to try and forget about the fact that I was in love with him. Maybe it wouldn't be possible to forget, but if I could at least push that knowledge to the back of my mind, perhaps everything would be all right. I had to hope that was the case.
Fortunately, Mikayla didn't catch on to the fact that something was bothering me and she went on to ask if I'd come up with any ideas about how we could help Bronwyn.
"Not yet," I said, wishing my answer could have been different.
"We might not be able to help her, you know," Mikayla said. "I don't like to say that, but I think we should be prepared for that possibility."
"I know." I hated the words as they came out of my mouth, but I knew Mikayla was right. I hadn't made any progress with finding the real thief or any sort of exculpatory evidence that would help Bronwyn. As much as I wanted to clear her name, I was no longer certain that I could.
Mikayla spent the rest of the drive chatting about her job as a high school orchestra teacher, and I appreciated the distraction from all my dispirited thoughts. When we reached the theater, I spent a few minutes talking to some of the other members of the orchestra before heading out of the musicians' lounge with my violin, bow, and folder of music. As I headed down the carpeted hallway that led from the lounge toward the stage, I stopped short.
Elena was walking toward me, cellist Johnson Lau at her side. Johnson smiled at me as they passed, but Elena completely ignored me. I expected no different from her, and her lack of acknowledgment didn't bother me in the least. I was far too focused on something else.
I turned around and watched them as they disappeared into the musicians' lounge. My eyes hadn't deceived me. There was no mistaking it—along with her designer jeans and expensive top, Elena was wearing the beautiful gray boots I'd admired at the shoe store.
Maybe they weren't the exact same pair from the same store, but that didn't matter. Even if the shop still had a pair in my size, there was no way I was going to hand over any sum of money to look similar to Elena. And not just because I didn't like her. If she ever caught me wearing boots like hers, she'd probably think I was copying her, trying to be like her. She was one of the last people I wanted to be like, but just knowing she would think that was enough to send my stomach into an unpleasant twist.
Still rooted to the spot, I glared at the door to the musicians' lounge. Trust Elena to ruin the one thing I was looking forward to that week. Although she'd had no clue that I wanted those boots, I still couldn't help but direct all my disappointment and frustration at her. If anyone else had walked into the theater in those boots, it wouldn't have been quite so bad. But of course it had to be Elena. That was the kind of week it was.
Letting out a quiet growl, I spun around and continued on down the hall, an invisible dark cloud hovering above my head. I'd barely made it half a dozen steps when a man's voice called out my name. I paused and turned back. Dr. Daniel Beaufort hurried along the hall to catch up with me. I stiffened, remembering that the last glimpse I'd caught of him was as he'd escaped from Major's study.
"Ms. Bishop, could I have a moment of your time?" Beaufort kept his voice low despite the fact that we were currently alone in the hallway.
"Sure," I said, although I had serious reservations about talking to him, especially without anyone else in sight.
Did he know I was the one who had disturbed him in Major's home, or had he fled before he'd had a chance to recognize me?
His next words assured me that it was the former.
"About the other night at Mr. Major's home . . ." He paused and waited as two cellists emerged from the musicians' lounge and made their way past us with their instruments. Once they were out of earshot he cleared his throat and continued. "I think there's been an unfortunate misunderstanding."
I couldn't stop my eyebrows from rising an inch. I wasn't quite sure how he could characterize it as a misunderstanding. "I take it the police talked to you."
He tugged at the left cuff of his dress shirt. "They did. A rather uncomfortable experience, I must say."
"It wasn't exactly comfortable for me to have to identify you as a thief," I pointed out.
"Thief?" He almost choked on the word. "I assure you, I'm not a thief. As I said, there's been a misunderstanding."
"Okay, so maybe you didn't take anything from Major's house," I said, although I didn't know if he had or hadn't, "but you were there without permission, going through his belongings. What was I supposed to think?"
Three more musicians made their way past us on their way to the stage.
Beaufort waited for them to pass and lowered his voice further. "If I'd been inside Mr. Major's house without permission, it would only have been for the sake of the orchestra."
"If? And for the sake of the orchestra?" I couldn't make much sense out of his words, aside from the fact that I'd gathered he wasn't admitting or outright denying that he'd broken into Major's house.
"It's rather a long story."
"One which you told to the police, I hope."
"Actually, that's why I wanted to speak with you."
Uh-oh. I didn't like the sound of that.
"I'm sure you understand that I didn't want to get charged with breaking and entering."
Then you shouldn't have committed the crime, I wanted to tell him. But I kept quiet and let him continue.
"So I denied that I was at Mr. Major's residence."
I stared at him. "But you were there."
He loosened his blue silk tie a fraction, as if to allow himself to breathe more easily. "Perhaps you were mistaken in what you saw?"
All the pieces clicked into place and I finally understood what he was getting at. "You want me to recant my statement? To tell the police that I'm not so sure after all that it was you I saw?"
Relief wiped the strained expression from his face. "For the sake of the orchestra, of course. I knew you'd understand."
But I don't understand, I wanted to say.
He didn't give me the chance. As Mikayla and several other musicians emerged from the lounge, Dr. Beaufort sent a quick nod my way and hurried off down the hallway. I stood there staring after him, completely befuddled.
"What was that about?" Mikayla asked when she reached my side, having noticed Beaufort's departure.
I fell into step with her and we headed toward the stage, my thoughts in a crazy whirlwind. "I couldn't tell you," I said as we walked, "because I'm not so sure myself."
Chapter Fourteen
THE REHEARSAL MEANT that I didn't have much time that evening to think about my encounter with Dr. Beaufort. Instead I spent the next two hours immersed in the music of Rimsky-Korsakov. After Mikayla dropped me off at home later that night, however, I couldn't think of much other than Beaufort. He even overshadowed my aggravation with Elena and thoughts of Bronwyn, Aaron, and JT.
How the doctor's evening of breaking and entering could have been for the sake of the Point Grey Philharmonic, I didn't know. What I did know was that his reasons for committing the crime didn't change the fact that it was a crime.
Beaufort hadn't seemed to notice that I'd never actually agreed to recant my statement, and I had no intention of doing so. I hoped that sticking to the truth wouldn't jeopardize my job with the PGP, but I doubted that it would. If Beaufort tried to somehow get me fired in retribution, all I had to do was reveal the story about his evening of crime, with the police to back me up. Dr. Beaufort was the one likely to get the boot if that happened, and I knew he'd be well aware of that.
All that aside, I couldn't help but wonder what was so important to Beaufort that it would drive him to risk his reputation, his position as vice chair of the PGP's executive committee, and his career by breaking into Major's house. Maybe Jordan could find out—if Dr. Beaufort hadn't taken whatever he'd been looking for. Jordan was, after all, living in Mr. Major's house and could snoop through his grandfather's belongings if he were so inclined. And I guessed that he might be so inclined, especially if there was a possibility that his snooping could prove that someone other than his mother was the murderer.
Why and how Beaufort would have gone about poisoning Mr. Major, I couldn't even guess. Well, I could guess, but at the moment my guesses would be nothing more than wild speculation drawn only from my imagination. But that didn't change the fact that Beaufort had behaved suspiciously, and in my mind his possible involvement in the murder needed to be investigated.
Glancing at the clock on my kitchen wall, I realized with a touch of disappointment that it was too late to call Jordan that night. Enlisting his help would have to wait until the next day. In the meantime, I decided to get some sleep. Maybe some of my confusion would disappear during my slumber. I doubted it, but I could always hope.
THE FIRST PERSON I talked to the next morning was Jordan. When I told him my idea, his initial response was less than enthusiastic, especially since I couldn't offer him a possible reason for Beaufort to want Archibald Major dead.
"But not knowing why he might have wanted him dead doesn't mean he didn't want him dead," I pointed out as I spoke to my student over the phone.
"True," Jordan conceded.
I dunked a teabag in a cup of hot water as I waited for him to think it over. I didn't have to wait long.
"All right, I guess I can take a look around. I'll text you if I find anything."
It took some effort, but I managed to suppress my exclamation of triumph. Instead, I removed the teabag from the water and dropped it in the kitchen sink for the time being. As I wandered over to my small dining table, Jordan continued on.
"And today will probably be a good day for it. My aunt's determined to get my mom out of the house for a while and with Marjorie gone, I'll be here on my own."
"Marjorie's gone? Gone where?"
"Off to a new job. She left first thing this morning."
"And you're not going to school?"
"Nah. I didn't feel like it today and my mom said that was fine. I'll go back to classes on Monday once the funeral is over with."
"When is the funeral?"
"Tomorrow afternoon."
I pulled out a chair and sat down at the table. "Is it open to the public?"
"Yes. You want to come?"
"If that's all right. I'd like to pay my respects." More than that, I wanted a chance to scope out the attendees and watch for any suspicious behavior, but I didn't mention that part.
"Sure. There's no reception after. Just a church service and a graveside service. I'll text you the details later. In the meantime, what exactly am I supposed to be looking for?"
I blew on my hot tea, rippling the surface. "I don't actually know. Anything that strikes you as unusual or suspicious. Possibly something connected to the Point Grey Philharmonic or Dr. Beaufort in some way."
"Okay. I'll see what I can find."
I could tell that he still had his doubts about the whole plan, but at least he'd agreed to it. After he assured me that he'd let me know if the search turned up anything, we ended the call and I sipped at my tea.
It was Friday and I had another concert that evening. I also had to teach most of the afternoon, so if I wanted to do any investigating I knew I had to do it that morning. The problem was that I didn't know how to move forward. Unless and until Jordan found something to shed more light on Dr. Beaufort's possible involvement, I didn't think there was much of anything I could do to work on that angle.
Actually, I didn't think there was much of anything I could do to work on any angles of the case. But I decided to give it a try and see what I could come up with.
Pushing my half-finished tea to the side, I fetched my laptop from across the room and set it on the table. Once the computer had booted up, I opened the Web browser and stared at the empty search bar, thinking.
While I doubted it would lead to anything helpful, I typed Dr. Daniel Beaufort's name into the search bar and pressed enter. I was entirely unsurprised when numerous results popped up on my screen. I scrolled down the page, scanning the list for anything that stood out. Most of the results were links to the Web site for the hospital where Beaufort worked as a surgeon, links to the PGP's Web site, or online articles about his medical work and involvement in charities.
I clicked on a link to one of the more recent articles and spotted Beaufort in a photo of a group of smiling, expensively dressed people at a charity benefit. He stood in the front row, flanked by a fair-haired man around his own age on his left and a woman with graying hair on his right. Archibald Major was also in the photo, standing three people away from Beaufort. I scanned through the article, but it didn't provide me with any pertinent information. So Beaufort and Major had helped to raise money for the same charity. I didn't think that was of any real interest.
Another photo near the bottom of the article caught my eye. It showed the quartet of musicians who had provided live music at the benefit. I recognized Janine Ko and recalled that she'd mentioned she was playing in a quartet for extra money.
Refocusing on my task, I hit the back button.
After returning to the search results, I clicked on a random link to see what would pop up. It was a short announcement of Dr. Beaufort's recent marriage to his long-time partner, Timothy Grimes. A photograph accompanied the announcement, showing Beaufort with the fair-haired man from the first photo accompanying the article I'd read only a minute earlier.
I hit the back button once more and scrolled through the results one last time. Beaufort was married, a skilled surgeon, and actively involved in charity work. Great, but none of that told me whether he was or wasn't a murderer.
Disappointed, I was about to give up on researching Beaufort when one word caught my eye.
Thefts.
I clicked on the search result containing that word and another article popped up on my screen. It referred to the same charity benefit as the first article I'd read, but touched on something other than the live music and the purpose of the event. Apparently, some jewelry had gone missing during the benefit and the police suspected that the thefts were the work of an experienced pickpocket. He or she had slipped away with two watches and a bracelet, all worth a good deal of money.
Bells dinged in my head. I read the article a second time. The charity event had taken place less than a month ago. Could the thefts that occurred at the benefit be related to the theft at the PGP's reception?
With a hum of excitement running through my bones, I considered the possibility. Similar objects were taken in both cases, and both events took place in Vancouver. If I could somehow link the two crimes, that would help Bronwyn. She wasn't at the charity benefit, so she couldn't have been responsible for those thefts.
At least, I didn't think she was at the charity benefit.
Deciding that I should make sure, I grabbed my phone and sent off a quick text message to Bronwyn, seeking confirmation that she hadn't been at the first event. While I waited for a response, I found a pen and a scrap of paper and made a list of everyone who had been at both the charity benefit and the PGP's reception. The list included Archibald Major, Dr. Beaufort, and Janine Ko. For the moment, I added Bronwyn's name to the end of the list with a question mark after it, hoping I could soon cross out her name. If for some reason she had been at the charity benefit, the case against her would be even stronger.
Pushing that thought aside for the time being, I considered the other names on my list. Archibald Major was rich and had no need to steal for financial gain. Even if he stole from people to satisfy a warped sense of fun, why plant the brooch in Bronwyn's bag? Could that also have been part of some twisted game? Maybe he simply wanted to cause trouble and watch it unfold. From what I'd learned about his personality recently, I wouldn't have put it past him. At the same time, I had nothing but speculation to go on. I would need evidence, and with Mr. Major now deceased, that could be difficult to come by.
Leaving Major's name uncrossed, I moved on to Dr. Beaufort. Again, he had no need for money, as far as I could tell. As a successful surgeon, he no doubt had a good income, but there was always a possibility that he had a tendency to spend beyond his means, or an expensive habit like gambling to support. But he seemed less likely than Mr. Major to set Bronwyn up as a thief. Unless . . .
I scooped up my phone and sent another message to Bronwyn.
Have you ever had any trouble with anyone on the PGP's board of directors?
I left it at that, not wanting to name any names yet.
Setting my phone aside, I stared at Janine's name. Memories flashed through my mind.
Janine mentioning her need for extra cash.
Her new designer handbag.
Maybe the handbag wasn't a knockoff as Elena had suggested. Maybe Janine was able to buy the real thing with money obtained through stealing. Or perhaps she'd stolen the bag.
My stomach sank as those thoughts went through my head. I didn't like the idea of any of my fellow musicians being a thief, but if I wanted to help Bronwyn I had to look at all the possibilities. Of all the people on my list, Janine seemed to have the strongest motive for stealing.
But I still couldn't figure out why she would have planted the stolen jewelry in Bronwyn's bag. Or could I?
Janine and Bronwyn had both studied music at the University of British Columbia at the same time, a few years before I went through the program. I knew from Bronwyn that Janine had been in the front row of the first violin section of the university's orchestra in the beginning, only two chairs away from the concertmaster. However, when the professor had tweaked the seating arrangements at the beginning of their third year in the program, he'd had Bronwyn and Janine switch places, basically demoting Janine. She hadn't been pleased at the time.
Maybe she blamed Bronwyn for the change. If that was the case, and she still held a grudge years later, it was entirely possible that she'd decided to get revenge by setting Bronwyn up to look like a thief so she'd get kicked out of the orchestra.
I continued to stare at Janine's name. Could she really be so devious?
It would explain so much.
But how could I prove it?
My phone buzzed, the sound cutting through my thoughts. Bronwyn had replied to my texts, answering both questions in the negative. So she hadn't been at the charity benefit, which was a good thing, and she'd never had any conflicts with anyone on the PGP's executive committee. As I crossed her name off my list, I decided to ask her another question.
How well do you get along with Janine Ko?
Her reply came back less than a minute later.
I don't think she likes me much, not since our university days, but she's never said anything outright. We've never argued or anything. Why? Did she put the brooch in my bag?
I don't know, I wrote back quickly. I'm looking into several possibilities.
Thank you, her next reply read. I don't know what I'd do without you and Mikayla.
I exchanged a couple more messages with her, telling her to hang in there, and then put my phone down.
As interested as I was in my new theory about Janine, I had other people I wanted to look into before I put my laptop away. This time I typed Andrea Duffy's name into the search bar and checked out the results. That search turned up very little on the right Andrea Duffy, and absolutely nothing of interest. I would have liked to look up Ernest, to see what—if anything—came up on him, but I didn't know his last name. There was a chance I could change that, though.
I logged into Facebook and checked the profiles of some of my friends from the orchestra. I didn't know if Ernest used social media, but I figured if he did we probably had friends in common.
Sure enough, we did.
It took a few tries, but I eventually found him on the friends list of one of the oboe players. His profile photo was clear enough to confirm that Ernest Pavlyuk was indeed the Ernest I knew. I couldn't see any information on his profile because of his privacy settings, but I typed his full name into the Web browser's search bar and checked out the results.
A few minutes later, I let out a frustrated huff. Again, nothing of interest. I'd learned that Ernest was an accountant, but nothing beyond that.
I decided to look up Marjorie Alberts next. Yes, she'd lost her job when Major died, and he'd only left her five thousand dollars in his will, but there were some crazies out there who would kill for that much, or even for less.
After I typed Marjorie's full name into the search bar, I spent several minutes scrolling through the results and following a handful of links. I soon came to the conclusion that none of the information available pertained to the Marjorie Alberts who had worked for Archibald Major. The women with the same name who showed up in the search results either lived in the wrong part of the world, were no longer living, were nowhere near the right age, or had completely different physical appearances. Not a single scrap of information related to the frizzy-haired Marjorie Alberts I was interested in.
Next, I tried Frances Barlow, the name of Major's newly revealed daughter. I found one person by that name who was approximately the right age. She even lived in the Lower Mainland of British Columbia, as evidenced by her involvement in a community theater group in Langley, a suburb of Vancouver. While it seemed like there was a good chance that she was indeed the Frances Barlow who stood to inherit under Major's will, that didn't do much for my investigation. None of the information I found on her seemed the least bit suspicious or illuminating.
I scanned my eyes over a photo of Barlow with her community theater group, taken to help promote a play presented back in the spring. The actors were dressed for their parts in the production of Cinderella. Frances Barlow, according to the caption, had the role of the wicked stepmother. I closed the Web browser.
My online research had proven at least somewhat helpful with respect to Bronwyn's predicament, but I couldn't say the same in relation to Mr. Major's murder. Tiring of my endeavor, I didn't bother typing in Kevin Major's name. I already knew he had criminal, violent tendencies as well as a motive to kill his father since he was in desperate need of money. Although I still wasn't convinced that he would have the patience or forethought to pull off a poisoning, I did like the idea of him being behind bars for a long, long time. And I didn't know him well enough to rule him out. Maybe he was capable of devious planning when he wasn't busy getting all physically aggressive and violent.
I sat back in my chair and stared at my computer screen with annoyance. I was no further ahead with figuring out Major's murder than I'd been when I woke up that morning. And although I now had a suspect for the jewelry theft, I wasn't sure what my next step should be in that respect. Confront Janine and see what she had to say for herself?
Possibly.
In the meantime, though, I decided to try to focus on something else. I wouldn't see Janine until that evening and I figured I'd leave my investigation into Major's death alone until Jordan came up with something of interest.
With several hours of teaching ahead of me and a concert that evening, I knew my day would be a long one. After organizing everything I needed, I left my apartment and headed for my studio at an unhurried pace, enjoying the pleasant autumn weather as well as a short respite from thoughts of murder, theft, and suspects.
Chapter Fifteen
MY LEISURELY WALK took me to Forty-First Avenue, where I purposely avoided looking at the shoe store where I'd seen the beautiful gray boots. Thinking about Elena strutting around in the same boots still irritated me, but I didn't want to focus on the negative while I was enjoying the fresh air and sunshine. Doing my best to keep Elena out of my thoughts, I boarded a bus that would take me to JT's neighborhood, leaving the shoe store well behind me. As I made my way toward an empty seat, I took my phone out of my purse to see if I'd heard from Jordan. I hadn't. I had, however, received a text message from JT, asking me if I was ready for him to pick me up and drive me to his place.
Oops. I'd completely forgotten that he didn't want me going anywhere by myself, even in daylight. I'd also forgotten about the danger posed by Kevin. That surprised me, but maybe it shouldn't have. With so much on my mind lately, it was a wonder I could remember anything at all.
I just got on the bus, I texted to JT, hoping he wouldn't be too upset with me.
Dori . . . was all he sent back.
Sorry! I totally forgot. But I'm fine. Don't worry.
If he was upset, he didn't let me know. In fact, he didn't send any sort of message back to me. I wasn't sure if that was a bad sign or no sign at all.
While the mere thought of another encounter with Kevin freaked me out, I didn't think I was in any danger while on a bus with several other passengers. Walking from the bus stop to JT's house might be another story, but at least it wouldn't be dark out this time.
In the end, however, it turned out that I didn't have to worry about walking on my own. When the bus pulled up to my stop, JT stood waiting for me on the sidewalk, Finnegan sitting at his feet. I smiled at the sight of him, although I noticed that his expression didn't match my own.
"I wish you'd take your safety more seriously," he said as soon as I stepped off the bus.
Finnegan strained at his leash to reach me.
"JT . . ." I said as I closed the short distance between us, crouching down to greet Finnegan.
"Seriously, Dori. I don't want that guy coming near you ever again."
"Neither do I, but . . ." I trailed off. I really didn't want to argue with him, and his expression told me loud and clear that he wasn't about to change his position on the matter. "Okay. I'll try not to be so forgetful next time." I poked him in the arm as we set off along the sidewalk. "Don't be grumpy, okay?"
The muscles in his jaw relaxed and he almost smiled. "All right." A second later his face turned serious again. "Aaron called me last night."
My face fell, right along with my stomach. "What about?"
JT hesitated. "He quit the band."
"What? Why?" My stomach dropped farther. "No. Not because of me?"
He hesitated again. The fact that he didn't want to answer was answer enough.
"But why?" I asked, dismayed. "I'm rarely at your band practices and I would have made sure to avoid them from now on."
"I guess he didn't want to risk running into you. He knows you spend a lot of time at my place. And maybe he thought it would be awkward since you and I are best friends."
"Ugh. I'm sorry, JT. I feel terrible." That was putting it mildly.
"It's not your fault," he assured me. "It was his decision."
"One he made because I dumped him."
"Don't worry about it."
"But you guys have a gig next weekend," I reminded him.
"We'll find a new drummer. And I know someone who can fill in temporarily until we do. So, really, don't worry."
"Ugh," I said again, because what else was there to say?
A change of topic was definitely needed.
"What are you working on these days?" I asked as we paused so Finnegan could sniff at the base of a fire hydrant.
"Mostly laying tracks for other peoples' albums." JT gave Finnegan's leash a gentle tug to hurry him along. "But I also finished a new track for Absolute Zero," he said, referring to the science fiction TV show he was composing music for.
"That's great. The series premieres next week, right?"
"Yep. Next Friday."
"And we're having a party?"
One corner of JT's mouth twitched upward. "If you don't mind my mom and stepdad being there. They're super excited about the whole thing."
"I am too." I hooked my arm through his. "And of course I don't mind. You know I love your parents."
"I don't want to make a huge deal out of it, though."
"But it is a huge deal."
"The show could get canceled after three episodes."
"Even if that happened—which it won't—it's still amazing that your music is going to be on TV."
"It is pretty cool," JT agreed, his smile growing.
"We can keep the party small if you want," I said. "But we are going to celebrate. No arguments there."
All traces of JT's earlier sternness had disappeared, leaving him relaxed and cheerful. "That's okay by me."
I gave his arm a squeeze and we continued walking along in companionable silence. I kept my arm hooked through his, not worried about anything right then, not even Elena, Kevin, or my feelings for JT. My mood was too cheerful for anything to drag it down. In that moment, with my best friend and my favorite dog at my side, everything was right in my world.
JORDAN DIDN'T FIND anything worth reporting that day. Or if he did, he didn't let me know about it. When I arrived at the Abrams Center before the evening's concert, I noted with relief that Bronwyn was present in the musicians' lounge. If she'd skipped out on the concert her position with the orchestra might have been in jeopardy, regardless of whether her innocence was ever proven in relation to the theft. She was clearly uneasy about being there, though. She stood at her open locker, her head down and her back to the rest of the room. As soon as I'd set down my violin, I went over to join her.
"I think I might know who the real thief is," I whispered.
Her head jerked up, surprise and cautious hope in her eyes. "Really? Who?" She dropped her voice. "Janine?"
"It's a good possibility."
"But why set me up? Does she really hate me that much?"
I shrugged. "Maybe. Some people can't let go of a grudge."
"But I never did anything to her. The seating change was the professor's decision, and that was years ago."
"I know, but resentment isn't always logical or time-limited."
Bronwyn glanced across the room where Janine was warming up by running through a melody on her violin. "Can you prove it was her?"
"Not yet," I said. "But I'm hoping that's going to change soon."
"Thank you, Midori. It means so much to me that you believe me."
I gave her a hug and then went to fetch my violin. Although I wanted to confront Janine, it wasn't the right time or place. Even though I strongly suspected she was the real thief, I didn't know for certain and didn't want to falsely accuse her in front of our fellow musicians. Hopefully I'd have a chance to speak with her after the concert.
As I set my violin case on a nearby table and undid the first clasp, I caught a flash of movement in my peripheral vision. I glanced up to see Bronwyn marching out of the lounge, her face set with grim determination. My eyes followed her line of sight just in time to see Janine disappear out into the hallway. Several feet behind her, Bronwyn picked up her pace.
"Oh no," I muttered under my breath.
Abandoning my violin, I rushed after Bronwyn, hoping I'd be in time to prevent a disaster.
Stupid, stupid, stupid, I berated myself silently. I never should have told Bronwyn about my suspicions.
I dashed out the door and around the corner, but then came to an abrupt stop. Bronwyn and Janine were already facing off.
"Did you frame me for the theft?" Bronwyn demanded.
I grabbed her arm. "This isn't a good idea."
Although I tried to lead her away, she stood her ground, her eyes still fixed on Janine.
"What are you talking about?" Janine said.
"The brooch," Bronwyn said. "Did you put it in my bag?"
"Of course not!" Janine narrowed her eyes. "Everyone knows you stole it. Trying to blame me is ridiculous."
"It's not ridiculous if you're the thief."
"Now you sound crazy."
I put my hand on Bronwyn's arm again. "Guys, let's not do this."
They both ignored me.
"It wasn't my fault you got bumped back to the second row in university, you know," Bronwyn said. "Why do you blame me for the professor's decision?"
"What are you talking about?" Janine's words were laced with disdain.
"Isn't that why you tried to set me up?"
"I didn't try to set you up!"
"Guys!" I stepped between them. "This isn't accomplishing anything."
Bronwyn glared over my shoulder at Janine. "Tell her, Midori. Tell her why you think she's guilty."
Janine narrowed her eyes at me this time. "You started this?"
"I'm just trying to help Bronwyn," I said, trying to keep my voice level. "I know she's not a thief so I'm trying to figure out who framed her."
"So you randomly throw the blame on me?"
"It's not random," Bronwyn said. "Right, Midori?"
I let out a heavy sigh. "No, I didn't pick you at random, Janine. I think the theft at the reception is related to thefts that occurred at a charity benefit a few weeks ago. The one your quartet played at. Some of the attendees had jewelry stolen, just like at the reception. You were at both events, you said yourself that you need extra money, and Bronwyn isn't exactly your favorite person."
"This is nuts." Janine put her hands on her hips. "First of all, I'm not a thief." She shot a baleful glare at Bronwyn. "Unlike some people." She plowed on before Bronwyn could object. "Second of all, even if I'd wanted to steal from people at the charity benefit, I wouldn't have had the chance. We musicians never mingled with the guests. We were either on a little stage in the corner or in a back room on our own." She aimed her dagger-eyes at me. "Ask the other members of my quartet if you don't believe me."
I opened my mouth, but Janine continued on.
"And finally, if I was going to steal jewelry because I needed the money, why the heck would I put it in Bronwyn's bag?"
"Revenge!" Bronwyn said.
Janine threw her hands up in the air. "Now you don't even know what you're accusing me of! Stealing for money or stealing to frame you so I could get revenge? Which, by the way, is ridiculous in itself. Like I'd waste my time on you."
Bronwyn took a step toward Janine and I hurried to put a hand out to hold her back.
"Okay, this really isn't accomplishing anything good," I said.
A light bulb seemed to go on in Bronwyn's head. "What about your new designer handbag?" she asked Janine.
Janine rolled her eyes in exasperation. "What about it?"
"If you're short on funds, how could you afford it?"
"So you're accusing me of stealing that too? It was a gift. From my boyfriend. You want to ask him?" she challenged, pulling her cell phone out of her pocket.
As much as I wanted to get the situation under control, I couldn't bring myself to tell her that wouldn't be necessary.
"You know what? I don't need to call him. Look at this." She ran her thumb over the screen of her phone, scrolling back through text messages. A second later she thrust the phone toward me and Bronwyn. "See?"
I took the phone and held it steady so I could read the messages. The one at the top of the screen was from someone named Andrew. Presumably, that was Janine's boyfriend.
Do you like your present? Andrew's message read.
I LOVE it!!! Janine had written back. How did you know I wanted this bag?
I saw you admiring it the other day.
I can't believe you remembered!!! And hot pink is my favorite color!!! I love you so much!!!
Janine had added a string of hearts after the last message.
I handed the phone back to her, not bothering to read any more of the messages.
"So you didn't steal the bag," Bronwyn said. "That doesn't mean you didn't steal the other things."
"I'm not so sure," I said before Janine had a chance to react.
"Who else would want to frame me?" Bronwyn asked.
"I don't know." I took her arm and gently pulled her a step back from Janine. "But I think we might be barking up the wrong tree here."
"I'll say." Janine shot more daggers at us with her eyes.
"I'm sorry," I said. "I was only trying to help Bronwyn. I never meant for things to get out of hand."
"Whatever," Janine said, her voice full of contempt. "Just don't you dare go spreading any more lies about me."
With that, she turned on her heel and marched off down the hallway, away from the musicians' lounge.
Beside me Bronwyn seemed to have deflated, her anger replaced by defeat. "You really believe she's innocent?"
"She seemed genuinely surprised by the accusations," I said. "Maybe she's just a good actress, but I have a feeling she's telling the truth."
Bronwyn's shoulders sagged. "I've made things worse, haven't I? She'll probably tell her friends I accused her of being a thief and then everyone will hate me."
"Everyone will not hate you," I said firmly. "And we're still going to figure this out."
Determined not to contribute to her sense of defeat, I didn't add that I had no idea how we would do that.
Chapter Sixteen
FORTUNATELY, NO ONE else in the orchestra seemed to catch on to the fact that there'd been a confrontation between Bronwyn and Janine out in the hallway. Janine wouldn't even look our way when she returned to the musicians' lounge a few minutes later, but at least she didn't seem to be spreading the tale of Bronwyn's accusations. Not yet, anyway.
I hoped the whole fiasco would blow over in time and that the tension with Janine wouldn't cause any more problems. I still wanted to kick myself for sharing my suspicions about Janine with Bronwyn. Clearly that had been a mistake. But what was done was done and all I could do now was move forward and hopefully not cause any more unrest among my fellow musicians.
Although there was still a chance that Janine was guilty and had lied to me and Bronwyn to protect herself, her denials had a ring of truth. If I really felt it was necessary, I could always track down the other members of her quartet and check the veracity of her claim that she hadn't had access to the guests at the charity benefit. But at the moment, I wasn't sure I'd bother. I had a feeling it would be a waste of time.
The fact that I still couldn't prove Bronwyn's innocence troubled me, but I knew I'd have to put the problem aside temporarily. Maybe after a good night's sleep I'd have some fresh ideas about how to help her. In the meantime, I needed to get my instrument and head for the stage.
Once I was settled in my seat next to Mikayla and had tuned my violin, I noted that Ernest was present on the stage. That led me to wonder how much Detective Salnikova and her colleagues had looked into him as a murder suspect. I was curious to know if they'd uncovered the reason for Ernest's hatred for Mr. Major, but I knew that wasn't something Salnikova would share with me. If I wanted to know the reason—which I did—I'd have to find out for myself. I wasn't sure how I would manage that, though. Ernest wasn't about to spill the beans to me when he wasn't even willing to admit that he'd authored the note I'd retrieved from the trash.
After the concert drew to a close and everyone on stage retreated toward the musicians' lounge, I considered trying to talk to Ernest again, in case I could get something more out of him. But he studiously avoided me, always sticking close to others so I couldn't catch him alone. Eventually I gave up on the idea of speaking with him that night and joined Mikayla, Dave, and several other musicians for a post-concert drink, hoping to enjoy myself far more than I had earlier that evening.
On the way to a nearby pub I considered filling Mikayla in on what had happened between Bronwyn and Janine, but we were walking in a group and I didn't want the others to hear about the incident. In the end I decided to tell her about it another time and made an effort to put all my worries aside for a while.
When we arrived at the pub it was fairly crowded, but our group managed to find two free tables to claim. Once we were all settled I ordered a strawberry daiquiri and sipped at it while enjoying the upbeat conversation with my colleagues.
While we chatted I discreetly observed Mikayla and Dave, noting the way they interacted with each other. When everyone else became absorbed in a discussion about the upcoming hockey season, I tapped Mikayla's shoulder to get her attention.
Leaning closer so only she could hear me, I said with a smile, "Dave's absolutely crazy about you."
Her eyes lit up and she glanced at her boyfriend. "You think so?"
I nodded. "Definitely. I can tell by the way he looks at you."
She smiled brightly. "I might be crazy about him too."
I matched her expression, but then my smile faded as my thoughts drifted to JT and how I wished I could have with him what Mikayla and Dave had together.
Mikayla must have mistakenly thought Aaron was the source of my sadness. "Don't worry," she said. "It'll happen for you too. I promise."
I tried to smile again, not wanting to pursue the topic any further. Fortunately, Mikayla steered our conversation in a different direction.
"Any word on the murder investigation?"
"Not really," I said, the lively chatter around us keeping our conversation relatively private. "I haven't heard anything new, but I think the poison might have been in Mr. Major's flask, and any number of people could have put it there. Several people had a motive to kill him, including members of his own family who will gain financially from his death."
"Wow." She brushed one of her corkscrew curls out of her face and took a long sip of her cocktail. Then she looked at me sharply. "Hey, how do you know all this?"
"His grandson is one of my students."
"Oh, right. Wow," she said again with a shake of her head. "So the police have no idea who did it?"
"I don't think so." I still didn't mention my own suspicions about Dr. Beaufort. As certain as I was that he was at the very least guilty of breaking and entering, I wasn't eager to damage his reputation.
Luckily, Mikayla didn't press me for more information. Dave pulled her into the conversation he was having with the rest of the people at our table, and I joined it as well. My drink was more than half gone by the time I caught sight of a familiar face across the pub.
Ernest.
He sat in a booth along one wall, taking occasional gulps of his drink while deep in conversation with Raul, one of the PGP's oboe players. A minute or so later, Raul got up and headed for the men's room, leaving Ernest alone. When I looked a little closer, I noticed that Ernest had an empty glass next to his half-full one. I couldn't help but wonder if the alcohol he'd consumed had loosened his tongue at all.
"I'll be back in a minute," I told my friends as I got up from the table, taking my drink with me.
I made my way across the pub and slid into the seat across from Ernest.
"Midori." He blinked startled eyes at me from behind his thick glasses.
"Hi, Ernest," I said with a smile that I hoped would put him at ease. "How are things?"
He grabbed a paper napkin from the table's dispenser and patted at his forehead. "Fine, fine."
"Have you had any more involvement with the police?"
His eyes grew bigger behind his lenses and he glanced around to make sure no one had overheard. "Must we talk about that?"
I leaned my arms on the table and lowered my voice. "It's that note of yours, Ernest. It makes you look suspicious."
"I didn't kill Archibald Major," Ernest said in a forceful whisper. "I'm not a murderer."
"But you're not sorry he's dead?" I guessed.
"Absolutely not." Anger gave his voice a sharp edge. "As I said in my note, I hope he rots in hell."
He'd finally admitted to composing the note. I'd never doubted that he had, but in my eyes the admission moved us forward a step. Hopefully I could keep us going in that direction.
"But why?" I asked. "How can you have so much hatred for someone you've never met?"
Ernest crumpled the paper napkin in his fist. "I never actually met him, but I've been aware of him almost my entire life."
I waited, hoping he'd elaborate. He craned his neck to survey the pub. I followed his line of sight and spotted Raul emerging from the washroom, but he ambled off toward the bar rather than in our direction.
Ernest took a sip of his drink and scowled into his glass, but after a few seconds had ticked by, he spoke again. "Archibald Major caused my mother's death."
"Your mother's death?" I asked, confused. "How so?"
He rubbed his nose and closed his fist around the paper napkin again. "They went to the same high school. My mother was fifteen to Major's seventeen when he got her pregnant. When she told him she was going to have his baby, he denied that it was his, even though she'd never been with anyone else." Ernest sniffed and dabbed at his perspiring forehead with the rumpled napkin. "He started spreading terrible rumors about her, saying that she was, well . . ." He cleared his throat. " . . . loose, if you understand me."
I did understand.
"She went to live with her aunt in Edmonton to get away from it all. She considered having an abortion or giving me up for adoption, but in the end she decided to raise me on her own. That required sacrifices, of course. She never finished high school and worked herself to an early grave at minimum wage jobs. And all the while, Archibald Major was sitting there in his mansion with all his millions." Ernest glowered at his drink before swigging back the last of it. "The bastard."
I stared at him as surprise and sympathy intertwined inside of me. "You're saying Archibald Major was your father?"
"In a biological sense, yes. In every other sense, absolutely not."
Wow. Jordan's mom and uncle had yet another half sibling. I wondered why Major hadn't mentioned Ernest in his will like he had with Frances. But maybe he'd never accepted that he was Ernest's father. Or—and this wouldn't have surprised me—maybe he didn't remember Ernest's mother's name and couldn't be bothered to do anything about it.
I pushed those thoughts aside. "Your mother told you the entire story?"
"Only once I found her crying while reading a newspaper article about Major's latest financial success. I was only fourteen at the time, but I could tell she was deeply upset. It took some prying, but I eventually got the story out of her."
I sat back in the booth, the weight of his narrative settling on my shoulders, heavy and poignant. "I'm so sorry, Ernest. How terrible for her. For both of you."
"Yes, well . . ." He picked up his glass before realizing it was empty. He set it back on the table and shoved it aside.
As bad as I felt for him and his deceased mother, I couldn't ignore the fact that his story gave him a motive for wanting Major dead. Revenge was a powerful driving force, and if Ernest's hatred for the man had festered for so many years, perhaps it had finally exploded in one deadly release.
"If you've known about him since you were fourteen, why write that note for him now?" I asked. "You've been in the orchestra for years. You must have known for a long time that Major was one of our benefactors. He's even been at the occasional reception in the past."
"That wasn't the first note," Ernest admitted. "I've left a few others for him over the years, usually on his car windshield, which was where I planned to leave the most recent one. Perhaps it seems like an odd thing to do, but I wanted him to know that there was someone out there who knew what he was truly like. I didn't want him as my father, I didn't want his money, but I needed to do something."
I supposed I could understand that, but I found it sad that Ernest had allowed Major to take up so much of his life with burning hatred. Maybe I would have done the same in his shoes—I didn't know—but it was still sad.
"I despised the man," Ernest said in a low voice. "But I swear to you, Midori, I didn't kill him."
I regarded him from across the table for a long moment, looking straight into his eyes. "I believe you," I said eventually.
And I did.
AFTER MY SOMBER but enlightening conversation with Ernest, I wasn't in the mood for hanging out with my friends any longer. As soon as I'd finished my drink, I said my goodbyes and took a taxi home to my apartment.
By the time I climbed into bed, exhaustion had taken a firm grip on me and I fell asleep within minutes of resting my head on my pillow. In the morning I stayed in bed until almost nine o'clock, lazing about and enjoying the fact that I had the day off.
It was a nice change having two days off each week, one I intended to enjoy to the fullest. My plan for the day was to go out for a walk along the beach if I could find a friend who was free and willing to go with me. I wanted some fresh air and a chance to clear my head. My conversation with Ernest had affected me more deeply than I'd expected.
From watching him at the reception, I'd surmised that Mr. Major wasn't a nice guy, but now, with a clear picture of what he was like, I realized that was an understatement. How many lives had he negatively affected during his seventy-something years? I didn't know, but I guessed the number was a high one.
No wonder he'd ended up murdered. He must have made enemies every which way he turned.
I hoped that some time at the beach would help to lighten my mood, but I tossed that plan aside as soon as I checked my phone and saw a text message from Jordan.
Found something! Want to come by and check it out?
My curiosity came alive, along with a spark of hope. I now knew Ernest's story and believed that he'd told me the truth when he said he hadn't killed Major. But I still didn't know Beaufort's story, and I wanted to.
Yes! I wrote back. In an hour?
His response came almost right away.
Sure. I'll be here.
Although I was tempted to ask him for more details right then and there, I decided I could wait. It wouldn't be long before I found out exactly what he'd discovered.
After munching my way through a piece of toast and washing it down with a cup of green tea, I took a quick shower and dressed for the day in jeans, a sweater, and high-heeled boots. Since the music books I'd ordered for my students had arrived by mail the previous day, I grabbed the one I'd bought for Jordan and slipped it into my bag. Remembering that I needed to keep myself safe from further attacks from Kevin, I called for a taxi, and a few minutes later I was on my way to the Major residence in Shaughnessy.
It didn't take long to arrive at my destination in the posh neighborhood, which was good news for my wallet. As I paid the cabdriver and climbed out of the taxi, Jordan called out to me from the bottom of the driveway.
"Hey, Midori." He jogged up to meet me, dodging around the taxi as it backed slowly down the driveway. "I just ran to the store. I really needed some sugar." He held up an empty candy bar wrapper before crumpling it and shoving it in the pocket of his jeans.
"I know that feeling," I said.
Jordan dug a key out of his other pocket and unlocked the front door for us.
"So what did you find?" I asked once we were in the spacious foyer, unable to contain my curiosity any longer.
"A letter." He shut and locked the front door behind us. "I left it in the living room."
He led the way along the main hallway toward the back of the house. As we approached the living room where my first confrontation with Kevin had occurred, Andrea Duffy's voice floated down the hallway toward us.
"I think I should come clean," she said, worry straining her voice. "They're going to find out sooner or later."
Ahead of me, Jordan slowed his steps, his back stiff.
After a short pause, Mrs. Duffy spoke again. "But the guilt is killing me. I don't know how long I can keep quiet."
Jordan and I reached the living room and my heels clicked against the hardwood floors. Mrs. Duffy spun around, her eyes wide and startled.
"I have to go," she said into the phone at her ear. She ended her call and gripped the phone with both of her hands, holding it to her chest. "Midori . . . hello. Jordan, I didn't realize you were back already."
"Hi, Mrs. Duffy," I said, keeping my voice light. That wasn't an easy task, considering the tension humming through the room.
"Midori came by to . . ." Jordan faltered, glancing my way.
"To deliver a new music book," I finished for him, retrieving the book from my bag and holding it up.
"Right. How nice. Well . . ." Mrs. Duffy glanced down at the phone in her hands and loosened her death grip on the device. "I'll set your suit out for you, Jordan. Be sure you're ready to leave for the church on time. Thanks for stopping by, Midori."
Even though I'd found Mrs. Duffy's phone call suspicious, or at least curious, I made sure not to let on to Jordan that I felt anything was amiss. His face was already troubled and I didn't want to upset him further.
"So," I said to him as I gave him the music book, "what kind of letter did you find?"
Jordan retrieved a piece of paper from an antique desk and handed it to me. It was a handwritten letter, signed at the bottom by Archibald Major.
As I scanned the slanted writing, Jordan asked me, "Do you think it might be what the guy was looking for in my grandfather's study?"
"Yes," I said as I finished reading the last line. "I think it very well could be."
Chapter Seventeen
AS I RODE home in another taxi, I read the letter through for a second time. Addressed to Mr. Hollingsworth, the chair of the PGP's executive committee, the handwritten letter set out an ultimatum. The first of the two paragraphs was a reminder to Hollingsworth of Major's support, financial and otherwise, of the orchestra over the years. The second paragraph contained the ultimatum and was the one that interested me the most.
I recently became privy to some rather disturbing information about Dr. Daniel Beaufort, the vice chair of the symphony's executive committee. Although I have been a staunch supporter of the Point Grey Philharmonic for almost two decades now, I regret that I cannot continue with my support if Dr. Beaufort is to remain on the board of directors. I do not wish to go into details as I believe that discretion is important for the sake of everyone involved. However, I hope you will accept that I would not have written this letter if my concerns were not well-founded. If Dr. Beaufort were to no longer be involved with the orchestra, I would of course reinstate my funding without hesitation.
Yours sincerely,
Archibald S. Major
Staring at the piece of paper in my hands, I realized that Jordan might well have uncovered a reason for Beaufort to want Major dead. With the elderly man out of the picture, Beaufort no longer had to worry about his secret coming to light and costing him his place on the board of directors. Except, if Major had threatened to send the letter if Beaufort didn't accede to his demands, then Beaufort would have known of its existence. He would have known there was still a chance that whatever he didn't want the world to know could be revealed if someone stumbled upon the letter.
This letter had to be what Beaufort was searching for in Major's study when I interrupted him. Whatever this disturbing information was that Major had become privy to, no doubt Dr. Beaufort didn't want anyone else finding out about it.
I recalled the snippet of conversation I'd overheard between Major and Beaufort at the PGP's reception. Perhaps Mr. Major had given Beaufort the opportunity to resign to avoid any fuss. Maybe he'd told him about the letter and the fact that it was all ready to send to Hollingsworth if Beaufort didn't resign.
If that was the case, Daniel Beaufort—acclaimed surgeon and well-respected member of the community—might have committed more than a break and enter. He might have committed murder as well.
That meant I needed to deliver the letter to Detective Salnikova. It would probably be a good idea to tell her about Beaufort's attempt to get me to recant my witness statement too. Maybe I should have done that already, but I'd had so much on my mind lately that it hadn't occurred to me.
I didn't have time to take the letter to the police station right then, however. Not if I wanted to make it to Mr. Major's funeral on time. Before parting with Jordan, he'd given me the address of the church where the funeral would be held and the time of the service. Although the start time of one o'clock gave me a chance to go home and change, it didn't leave room for any other excursions.
Besides, I figured there was a good chance that Salnikova would make an appearance at the funeral, probably for the very same reason that I wanted to go. There was always a possibility that the killer would show up. While I didn't expect the murderer to declare his or her guilt to everyone present, the gathering would provide a chance to watch for anything suspicious.
The taxi turned onto my street and I pointed out my building to the driver. As I paid my fare, I hoped the police would get Kevin off the streets sooner rather than later. Otherwise all the taxi fares would do some damage to my bank account.
On my way up the stairs to my third-floor apartment, my phone rang in my purse. I fished it out and saw JT's name on the display.
"Hi," I said into my phone as I reached the top of the stairs.
"Are you free today?" JT asked. "I thought maybe we could hang out and take Finnegan down to Spanish Banks."
I dug around in my purse for my keys with my free hand. "I was thinking of the beach today too, but I don't have time now. I'm going to Mr. Major's funeral at one o'clock."
"Alone?"
"I'll take a taxi," I said as I unlocked and opened my door. "There's no need to worry."
"Dori, this is the funeral for Kevin Major's father."
"Yes, but he probably knows the police are on the lookout for him and there's a good chance that a detective or two will be there. I doubt he'd want to show his face."
"Maybe not, but that doesn't mean he won't be lurking about somewhere nearby. And if he sees you, who's to say that he won't decide to come after you as you're coming or going?"
Okay, he had a point there.
I shoved the door shut behind me and dropped my purse on the kitchen table. "Are you saying I shouldn't go?"
"I'm saying I'm coming with you."
"You're choosing to go to a funeral for someone you've never met instead of the beach?"
"So are you."
I sank down onto one of my kitchen chairs and unzipped my left boot. "I met him. Once."
"Okay, but that's not why you're going, is it?"
I made a face as I kicked my foot free of my boot. JT knew me too well. "Don't try to talk me out of it," I said as I moved on to the zipper on my right boot.
"I won't. I know that's pointless. And it would also be pointless for you to try to talk me out of coming with you."
With a wiggle, my right foot came free of its boot. "I'm always happy for your company."
"Good. I'll pick you up in half an hour."
He hung up and I set my phone on the table next to my purse. It was time to find something to wear to Archibald Major's funeral.
FROM THE CHOICES in my closet, I selected a black dress and black tights. I switched my tall brown boots from earlier in the day for a pair of black ankle boots and fastened my hair in a twist at the back of my head. After touching up my makeup, I scrutinized myself in the mirror. So much black. Yes, I was on my way to a funeral, but I needed a little something to break up the drabness.
I searched through my belts and decided on a skinny silver one to add to my dress. Next, I chose a silver necklace with a small pendant in the shape of a treble clef. As I pulled it out of my jewelry box, my apartment's buzzer sounded. JT was right on time.
I buzzed him into the building and waited for him at my door, my necklace still in one hand. When he appeared before me, my breath caught in my throat for a beat or two of my heart. Handsome wasn't sufficient to describe JT. He always looked great but today, clean shaven and dressed in a dark suit, he could have been a movie star on the red carpet.
"Hey," he greeted. "Ready?"
I struggled to find my voice. "Almost."
On my way to retrieve my cell phone from the kitchen table, I tried to fasten my necklace around my neck. As I fumbled with the clasp, JT came up behind me.
"Here, let me." He took the necklace from my fingers.
My breathing nearly stopped as his hands brushed my neck. Despite the fact that I could hardly draw air into my lungs, I was disappointed when it only took him a few seconds to fasten the chain.
"There you go."
"Thanks." My voice came out only a notch or two above a whisper.
A sudden buzz of excess energy ran through me, accompanied by a desperate need to move. I turned around but JT was still in the same spot, and I found myself mere inches away from his face.
For the third time in the last few minutes, I had trouble breathing. I wanted to kiss him. Badly. I also wanted to run away. But I didn't do either because I couldn't seem to move.
A smile tugged at JT's mouth and my nerves almost short-circuited. Could he sense my nervousness? Did he know I wanted to kiss him?
Oh God. That would be humiliating.
Finally remembering how to move, I reached up to fiddle with JT's tie, even though it didn't need straightening. I didn't dare meet his eyes.
"That's better," I managed to say as I gave his tie a final pat.
Stepping away from him, I busied myself with slipping my cell phone into my purse, wishing I'd left my hair loose so it could hide my face.
JT held the apartment door open and I preceded him through it. As I locked the door behind us, I took a second or two to compose myself. I needed to get a grip. JT knew me so well that I couldn't afford to let my feelings show, even in the slightest. Somehow I needed to find a way to immunize myself against everything that attracted me to him.
Otherwise I risked messing up what was most important to me—our friendship.
WHEN JT AND I arrived at the downtown church, those who had arrived earlier than us had already filled two-thirds of the pews. We found a seat near the back and settled in with a few minutes to spare. The number of people present surprised me, considering that Mr. Major wasn't exactly the nicest of men. Perhaps they weren't all there to mourn him. It wouldn't have surprised me if many of the attendees were simply there for show, wanting to be respectful more for the sake of appearances than any real warmth of feeling toward the deceased.
In fact, it didn't seem as though anyone in the church was particularly distraught. I couldn't see many faces from my vantage point, but I also couldn't see any obvious sobbing or other signs of distress. My eyes roamed over the crowd during the service, hunting for familiar forms and searching for anything of interest.
I spotted the back of Salnikova's head several pews forward. If the man and woman sitting on either side of her were also with the police, I didn't recognize them. Jordan sat up at the front with his mother and another woman I assumed was his paternal aunt. Nobody else sat in their pew with them.
Poor Jordan. It had to be a tough day for him, even if he hadn't been bursting at the seams with love for his grandfather. It appeared that his father hadn't shown up to provide support. Although if his separation from Jordan's mother had been particularly acrimonious, perhaps that was for the best.
As I scanned the rest of the crowd, I picked out the familiar faces of Mr. Hollingsworth and a couple of other members of the PGP's board of directors. Dr. Beaufort, however, wasn't among them. I didn't count that as overly surprising, though. There was always a good chance that he was busy working.
I paused in my examination of the attendees as my eyes settled upon a blond head across the aisle and five rows up.
Hans.
It wasn't surprising that he was there. In fact, I probably would have expected his presence if I'd taken a moment to think about it beforehand. He was, after all, the face of the Point Grey Philharmonic. Since Mr. Major, despite his flaws, had been generous to the PGP during his life, it only seemed right that there were a few representatives of the orchestra in attendance.
As the service droned on, I shifted in my seat, growing bored with the prayers and dry stories about Major's primarily business-related life achievements. My attention wandered and found its way to JT. I became acutely aware of how close he was to me, the light touch of his arm against mine. His hand was mere inches from my own, and I itched to intertwine my fingers with his.
My gaze drifted up to his face and once there I couldn't seem to tear it away. He must have sensed my eyes on him because he looked my way. My cheeks suddenly warm, I refocused on the front of the church.
JT leaned closer. "Everything okay?" he asked in a whisper.
I nodded, trying to give him a hint of a smile. That was the best I could do, considering how distracted I was by his nearness.
From that point on, I kept my eyes straight ahead. A few minutes later, everyone stood for a final hymn and, thankfully, the service drew to an end.
JT and I were among the first people out of the nave, but I claimed a spot out in the narthex and watched as everyone filed past me. My foot nearly got crushed when a muscular woman in an ill-fitting pantsuit came dangerously close to stepping on me, so I inched back, closer to the wall, where I could watch the flow of people more safely. It took several minutes for everyone to clear out of the nave, but I was able to confirm two things I'd suspected during the service—everyone other than Andrea Duffy was dry-eyed and Marjorie was absent.
Maybe her absence wasn't so strange, though. Jordan had said that she had a new job, so maybe she hadn't been able to get the time off to attend the service.
As the crowd thinned out, leaving only a few people lingering in the narthex to chat with one another, JT nudged me in the arm.
"Are we done here?"
"Here, yes," I said, leading him out of the church. "But I want to go to the graveside service."
"Really?" He sounded less than thrilled by the idea.
I couldn't blame him. We'd already sat through a rather dull service in the church, and the thought of more prayers while watching Major's coffin get lowered into the ground wasn't overly appealing. Still, I didn't want to miss out on a chance to pick up a potential clue, and the graveside service would likely give me more of an opportunity to observe the faces of those in attendance than the church service had.
"You don't have to come if you don't want to," I said, even though I knew he wouldn't leave me to go on my own.
"I'm coming," he said, as I'd known he would.
We set off down the street to the spot where he'd parked his truck.
"Tell you what," I said as I buckled myself into the passenger seat, "I'll buy you dinner tonight to make up for your dreary afternoon."
"That's an offer I won't turn down." JT put his truck into gear and moved out into the street.
Although many of the cars that had parked near the church turned off in other directions, a small procession of vehicles headed for the cemetery, and JT pulled his truck in behind the line.
From my perspective, it was a good thing that the graveside service would have a smaller number of attendees than the church service. If only those closest to Mr. Major—and a few others like me and JT—were at the gravesite, it would be easier for me to keep an eye on all of them to watch for anything suspicious.
JT found a free space at the edge of the road bordering the cemetery and we followed a few other dark-clad figures along the graveyard's main path. Near the middle of the cemetery we had to leave the concrete path for the grass. JT gave me his arm as I attempted to navigate a small hill in my high heels, and I kept hold of it as we drew to a stop at the gravesite. I was glad to have him there with me, whether Kevin Major showed up or not. I was even more glad to find that my nervousness from earlier in the day had disappeared, leaving me once again comfortable with my best friend.
After a couple in their sixties joined the small crowd gathered around the gravesite, it appeared as though everyone had arrived. Moments later the service began, but my mind wasn't on the minister's words. Once again, I was busy scrutinizing those present.
Detective Salnikova was one of the people who had shown up at the cemetery, which didn't surprise me in the least. The dark-haired man who had sat next to her in the church pew was still at her side. He seemed to be close to Salnikova in age and I didn't doubt for a second that he was a detective as well.
Jordan stood with his mother, holding one of her arms while his aunt supported her from the other side. A few people I didn't recognize were present as well, mostly middle-aged or older. As the minister recited a prayer, my gaze wandered beyond the scene directly before me to the surrounding area. My eyes moved from gravestone to gravestone, from tree to tree, searching for any sign that Kevin might be lurking nearby.
Off in the distance a bulky figure hovered near a stone angel, facing in our direction. When I squinted, I thought I recognized the person as the muscular woman from the church. Seconds later she turned away and wandered off out of sight. I continued my surveillance but saw no one else other than an elderly woman setting a bouquet of flowers at the base of a tombstone. I couldn't look behind me without being obvious about it, but I hoped that part of the graveyard was similarly Kevin-free.
With my arm still looped through JT's, I glanced up at him. He too was checking out the area beyond the gravesite. I'd given him a brief description of Kevin earlier and I knew that was who he was watching out for. I squeezed his arm and gave him a grateful smile when he looked my way. It was sweet of him to watch over me, whether I needed it or not.
Knowing that JT was keeping an eye out for Kevin, I focused my attention on those gathered near us. I scrutinized facial expressions and body language. A hint of storminess clouded Jordan's face, but that could have been due to the police presence or simply the anger that came with dealing with all the recent drama in his family. Otherwise, nothing seemed odd or unusual. Nobody did anything suspicious and nobody seemed out of place.
Once the first handfuls of dirt had been tossed onto the coffin and the service had ended, I released my loose hold on JT's arm.
"I'm going to have a quick word with Jordan," I told him.
He nodded and I left him to approach my student. I had to wait a moment as others filed past with a word or two for Jordan and his mother, but eventually my turn came.
I gave him a quick hug. "Let me know if you need anything, okay, Jordan?"
"Thanks, Midori. I'll be at my lesson at the usual time on Tuesday."
"All right. I'll see you then."
I glanced at Mrs. Duffy, but she was busy speaking with a silver-haired woman so I didn't wait around to talk to her. Instead, I turned around, intending to make my way back to JT.
A step or two later, as I caught sight of my best friend, I hesitated. He stood close to where I'd left him, but he was no longer alone. Detective Salnikova was with him, and the two of them were deep in conversation. Maybe that shouldn't have worried me, but I couldn't help but wonder who had approached whom.
And why?
Chapter Eighteen
I DIDN'T HESITATE for long. I waited for a man and woman leaving the gravesite to pass me by and then picked my way around a weathered tombstone to approach JT and Salnikova. As I drew closer, JT said something I couldn't hear and Salnikova smiled in response.
She smiled.
I didn't think I'd ever seen her smile before. Not a full smile anyway.
And then JT grinned too.
What the heck was going on?
I wished I had super hearing so I could listen in on their conversation, but the wind was whipping their words off in the opposite direction. I picked up my pace in the hope of getting close enough to catch a few words before they spotted me, but I was out of luck. JT caught sight of me a second later and their mysterious conversation came to an end.
"Afternoon," Salnikova said to me as I approached.
"Detective," I returned. "Spot anything suspicious?"
Her usual poker face had replaced her smile from moments earlier. "I'm just here to pay my respects to the victim, as are you, I assume."
"Right."
Neither of us believed that for a second.
"Ready to go?" I asked JT.
"Sure," he said. "Nice to meet you, Detective."
"You too." Salnikova's smile reappeared, if only for a fleeting second.
I set off toward the edge of the cemetery at a good clip, leaving JT to catch up with me. Somehow the encounter between JT and Salnikova had thrown me off kilter, and it irked me that I didn't know what they'd talked about. The slightest hint of jealousy twinged in my stomach as well. Salnikova was pretty and not all that many years older than JT.
In my haste to leave my annoyance and jealousy behind, my left foot slipped on a damp leaf and I almost twisted my ankle.
JT caught my arm and steadied me. "Careful." He maintained a loose hold on my arm as we continued toward the main pathway. "Why are you in such a hurry?"
"No use sticking around."
"I take it you didn't pick up any clues."
"Nope. None."
We followed the concrete path to the main gates and headed along the sidewalk toward JT's truck.
"Sorry I wasted your time," I said as we climbed into his vehicle. "Unless," I added, watching him with suspicion, "it wasn't an entire waste of time for you."
"What does that mean?"
"Detective Salnikova."
"What about her?"
"Were you hitting on her?"
JT laughed, which only irked me more. "Why would you think that?"
"She smiled at you."
He fastened his seat belt with a snap. He'd stopped laughing but still seemed amused. "Police detectives aren't allowed to smile?"
I let out a dramatic sigh. "It's not that she's not allowed, she just doesn't. Not usually."
"You mean she doesn't smile at you. Probably because you're always mixed up in her investigations."
"I'm not mixed up in anything!"
"Relax, Dori," JT said with a grin as he put the key in the ignition and started the engine. "I wasn't hitting on her."
I did up my seat belt as relief eased away some of my frustration.
When JT left the truck in park instead of pulling out into the street, I glanced his way. His face had gone from amused to serious, his brown eyes on me. "Dori . . ."
Something outside the truck caught my eye, distracting me. After giving Jordan a quick hug, Mrs. Duffy separated herself from him and his aunt, heading on her own to a dark blue car.
"Jordan's mom came by herself?" I said, mostly to myself.
When she reached the driver's door, she paused, watching as Jordan and his aunt climbed into a silver car and drove off. Mrs. Duffy glanced around and only then climbed into the blue car.
"Strange," I said, still watching Mrs. Duffy's car through the windshield.
"Why is it strange?" JT asked. "Maybe she wanted some time by herself."
"Hm." I wasn't convinced. There was something odd about her behavior. "Let's follow her."
"Seriously? Why?"
"Because something's funny, that's why." I swatted his arm as Mrs. Duffy's car pulled out into the street. "Hurry up, before we lose her."
I could tell he thought I was nuts, but he put the truck into gear and pulled out into the street behind Mrs. Duffy's car, and that was all I cared about right then.
"She's probably going home," JT said.
"Nope." I pointed at the blue car as it turned left. "She's going the wrong way for that."
"Then maybe a reception. Is there a reception?"
"Nope," I said again. "Don't lose her," I added as a van turned onto the street ahead of us, moving into the space between JT's truck and Mrs. Duffy's car.
"Don't worry. I can still see her." He slowed the truck as the cars up ahead stopped at a red light.
"So," I said, my curiosity making a comeback as I relaxed into my seat, "if you weren't hitting on Detective Salnikova, what were you talking about?"
"You."
"Me?" I hadn't expected that. "But she smiled."
"You exasperate her sometimes," JT said as the light turned green, "but she likes you."
"She said that?"
"Not in so many words, but that's what I got from her."
The last remnants of my frustration drained away. I'd already known that I exasperated Salnikova on occasion, but it was nice to know she liked me despite that.
"Maybe behind her normally stoic expression she appreciates my investigative prowess."
JT fought a smile without success. "I wouldn't go quite that far."
I was tempted to stick my tongue out at him but—luckily for my dignity—I got distracted. "She's parking. We need to pull over."
"If we can find a spot."
"There!" I pointed to a free space by the curb between a BMW and SUV.
JT pulled into the opening and I unclipped my seat belt so I could lean forward to watch Mrs. Duffy more closely.
We'd stopped in a residential neighborhood on a street lined with large two-story houses and tidy lawns. Jordan's mother had parked a short distance ahead of us. She climbed out of her car and headed up the walkway to a stately house with a stone façade. While the house wasn't nearly as large or imposing as Mr. Major's mansion, it still had to be worth a pretty penny.
Up on the small front porch, Mrs. Duffy knocked on the front door and waited. Seconds later, a man opened the door. Only, it wasn't just any man. It was Gareth Hollingsworth.
Although I'd seen him at the funeral, he hadn't continued on to the graveside service. That had given him time to change out of his dark suit and into khaki pants and a polo shirt.
As soon as he saw who was on his porch, he opened his arms and Mrs. Duffy stepped into them.
Interesting.
Or was it?
I supposed it wasn't all that strange that they knew each other since Mr. Major was a common link between them. Plus they'd spoken briefly at the season's opening reception the previous week and had likely both attended similar functions in the past. Even with that in mind, I was still somewhat surprised that they knew each other well enough for Mrs. Duffy to seek comfort from him.
After they'd hugged for several long seconds, Mrs. Duffy tilted her head back. Mr. Hollingsworth said something to her and they kissed.
Not a quick, friendly peck but a deep, passionate, lovers' kiss.
Wow.
I stared at them, completely stunned.
When they eventually broke apart, Mr. Hollingsworth guided Mrs. Duffy into the house and shut the door behind them.
I sat back in the passenger seat, thoughtful and still surprised. "I guess she doesn't have any intention of getting back together with Jordan's dad."
"Please tell me you don't want to wait here until she comes back out," JT said. "Judging by that kiss, it could be a while."
"No, we can go," I said, still staring at the front door of Mr. Hollingsworth's house. My voice came out sounding vague, but that was because of all the thoughts churning around in my head.
JT wasted no time starting the engine and driving off down the street. "Was that worth our time?"
"Maybe."
"Really?"
"That wasn't just any Mr. Loverboy locking lips with Jordan's mother. He's the chair of the PGP's executive committee."
"So?" JT flicked on the truck's wipers as several fat raindrops splattered against the windshield.
"So . . ." I paused, not sure what to say next.
JT glanced my way, eyebrows raised.
"They were both at the reception when Mr. Major died. That could be significant."
"You were at the reception," JT pointed out. "That's not significant."
"But . . ." Again, I didn't know how to continue. I sighed. "Okay. It might not be significant. But it's important to look at every angle."
"For the police, sure, but for you?"
I smacked the palm of my hand against my forehead. "The police. Shoot."
"What?"
"I have a letter I need to pass on to Detective Salnikova. It could be pertinent to the case. But it totally slipped my mind when I saw her at the cemetery."
JT waged an unsuccessful battle against a smile. "Probably because you were too worried that I might be hitting on her."
"So not true," I said, even though he was right.
"Why does the idea bug you so much?"
"It doesn't." I unclipped my hair and let it fall loose over my shoulders. I hoped it would hide my ears. As warm as they felt, I was certain they had turned pink.
"Could have fooled me."
I retrieved my cell phone from my purse and busied myself with searching for Salnikova's number in my list of contacts. "It would just be weird, okay?"
When I came across the detective's name, I tapped it and put my phone to my ear. Tucking my hand out of JT's line of sight, I crossed my fingers. I needed Salnikova to answer my call so JT wouldn't be tempted to continue our conversation.
Luck was with me. The detective answered after two rings.
When I told her I had something for her, the short pause that followed seemed to echo with the exasperation JT had mentioned earlier.
But when she spoke, her voice was as neutral as ever. "I just arrived back at the station. Can you come by?"
I covered my phone with my hand. "Can we make a quick stop at the police station?" I asked JT.
"Sure."
"I'll be there within the half hour," I said into my phone.
Salnikova and I exchanged goodbyes and ended the call.
JT turned up the speed of the wipers as the rain increased its intensity. "You really think whatever you're going to give her could be important?"
"It's a letter written by Mr. Major," I said, fishing the piece of paper out of my purse to read it through once more. "And it could be someone's motive for murder."
JT waited for a pedestrian to cross the street before making a right turn. "In that case, I'm more than happy to help you get it out of your hands. Having someone's motive for murder in your possession isn't exactly the safest position to be in."
He was right. Dr. Beaufort didn't know I had the letter, but if that were to somehow change, I didn't want to know what might happen.
Chapter Nineteen
"WHAT DO YOU want for dinner?" I asked, deciding to turn the conversation in a more pleasant direction.
JT thought about the question. "I could go for a good lasagna."
"Mmm." I liked his answer. "Minerva's?"
"Sounds like a plan." He paused the truck at a stop sign. "But I'll need to stop off at home first. Finnegan will need a chance to get outside."
"Not a problem."
"What's this letter about, anyway?" JT asked as he eased the truck into motion again.
"It's addressed to Mr. Hollingsworth, the guy who was making out with Jordan's mother."
"The chair of the symphony's executive committee?"
"Exactly."
"So?"
"So, it's not whom it's addressed to that's interesting. What is interesting is the fact that the letter says Mr. Major will withdraw his funding if Dr. Beaufort doesn't resign as vice chair of the executive committee. Apparently, Major was privy to some disturbing information about Beaufort. The annoying thing is he doesn't get any more specific than that." I frowned, itching to know the doctor's secret. It really was annoying that Mr. Major hadn't left more clues in his letter.
"Okay," JT said. "So that's interesting. Sort of. But I'm assuming you meant the letter could be Dr. Beaufort's motive for murder. Was he even aware that Major knew . . . whatever it was he knew?"
"Definitely." I told him about the brief exchange I'd overheard at the symphony's reception. "And I think Beaufort either knew or suspected that Mr. Major had written a letter but hadn't yet sent it to Mr. Hollingsworth. I'm sure that's why Beaufort broke into Major's study."
"What? When did that happen?" He cast a sharp glance my way. "And how do you know about it?"
"I caught him in the act."
Before JT could become too alarmed, I gave him a quick, condensed version of the events that had transpired at Major's house the other night prior to Kevin's arrival.
JT pulled into a parking spot around the corner from the police station. "I don't know how you manage it," he said with a shake of his head.
"Manage what?" I asked.
"To find trouble whichever way you turn."
"It's not like I go looking for trouble. It just . . . happens."
He shut off the engine. "Are you sure about that?"
I thought about my tendency to let my curiosity guide me and decided I didn't want to answer the question. "This shouldn't take too long," I said instead as I unbuckled my seat belt.
Rather than sitting back to wait as I'd expected, JT undid his own seat belt and reached for the door.
"What are you doing?" I asked.
"Coming with you."
I may have let out a quiet sigh as I climbed out of the truck. As much as I loved JT—as a friend and otherwise—and as much as I appreciated the fact that he cared about my safety, it was getting a bit annoying to have him thinking he needed to accompany me everywhere. But I bit my tongue and didn't complain, instead dashing through the rain to the front door of the station.
We only had to wait a minute or two before Salnikova appeared in the reception area to lead us back to a small room with a table and chairs. She offered us coffee, but we declined. I figured that was the safest way to go, in case real police station coffee was anything like it was described in books and on television.
We all settled into chairs and Salnikova folded her hands on the tabletop. "What was it you wanted to show me?"
I unzipped my purse and fished out the letter Jordan had found. As I slid it across the table to Salnikova, I said, "This might be what Dr. Beaufort was searching for when he broke into Mr. Major's house."
Detective Salnikova's eyes scanned over the handwritten letter. "Where did you get this?" she asked once she was done reading.
"From Jordan. He found it among his grandfather's belongings."
I thought I detected a hint of suspicion in the detective's eyes, but, to my relief, she didn't ask if I'd put my student up to searching for clues.
"Oh, and Dr. Beaufort approached me before my concert last night. He wanted me to tell you I was mistaken about seeing him at Mr. Major's house."
"He what?" JT stared at me.
Salnikova reacted as well, her gaze sharpening. "Did he threaten you in any way?"
"No, but it was . . . uncomfortable. And confusing." I repeated what Beaufort had said to me about doing what he'd done—or not done, according to him—for the sake of the PGP. "But it makes more sense now that I've seen that letter. Sort of, anyway. Maybe he was trying to avoid a scandal that would, by association, reflect badly on the orchestra?"
A muscle in JT's jaw twitched "Why didn't you tell me about this?"
"It didn't occur to me to mention it earlier."
JT wanted to say more, but Salnikova jumped in before he had the chance.
"If Dr. Beaufort approaches you again, contact me immediately," she said. "But please do your best to avoid him for the time being."
"Do you think he might be dangerous?" I asked. "Could he be the murderer?"
"I'll look into this letter," Salnikova said. "And I'll look into whatever is behind it, if anything."
I was far from shocked that she stayed true to form and avoided answering my questions.
"I'm kind of surprised that you didn't arrest him for breaking and entering," I said.
"We couldn't. He had a solid alibi."
I couldn't believe it. "How is that possible? I saw him there and I know I wasn't mistaken."
"I understand that," Salnikova said, "but we didn't have enough evidence to charge him. He left no fingerprints and his presence elsewhere at the time was confirmed by three others."
That was frustrating. Whoever those three people were, they'd lied. I knew that for certain.
"But now you've got the letter," I pointed out. "And doesn't the fact that he wanted me to take back my statement seem suspicious to you?"
"It does. And as I said, I'll look into it."
"There's something else you should probably know." I hesitated, remembering what Hans had said about hoping to keep the theft quiet.
JT nudged my foot under the table. "Spill it."
I hesitated only a second longer, believing that Salnikova really should have the information. "There was a bit of a situation at the theater. Another situation in addition to Major's death, I mean. A brooch was stolen during the season's opening reception and I believe my friend was framed for the theft. I thought the real thief was another violinist, but now I'm not so sure. And after reading Mr. Major's letter I thought I should bring it up because I can't help but wonder if Dr. Beaufort is the real thief."
"Is the letter the only thing that makes you think that?" Salnikova asked.
"No," I replied. "Remember how I told you before that I'd overheard part of a conversation between Mr. Major and Dr. Beaufort at the reception? At the time it didn't really mean much to me, but now I'm sure it was related to whatever Mr. Major was referring to in the letter."
"What did they say?" JT asked.
I took a moment to think, wanting to be as accurate as possible. "Mr. Major asked if Dr. Beaufort had thought about what they'd discussed. Beaufort said that threats would be ineffective, but then Mr. Major said he thought Beaufort would change his mind about things if his career and the orchestra were to suffer because he wouldn't listen to reason. That's what makes me think the letter is related." I gave my left earlobe a tug before continuing. "After Dr. Beaufort said he had nothing to hide, Mr. Major said he wondered what the police would find if he called them in. I didn't hear much more, but Dr. Beaufort definitely wasn't pleased. And now that I think about it, maybe Mr. Major thought the police would find stolen items on Beaufort if they searched him."
Three ticks of silence went by before Detective Salnikova spoke. "It does sound as though that conversation and the letter are related, but tying both to the theft is really just speculation."
"Oh! I should have started out by mentioning the charity benefit."
"What charity benefit?" JT asked.
"There was a charity benefit a couple of weeks before the PGP's reception. Apparently, some jewelry was stolen there as well. The violinist I mentioned was playing in a quartet at the benefit, and that's one of the reasons why she was on my suspect list. But Mr. Major and Dr. Beaufort were there as well. That could be how Major discovered that Beaufort is a thief, if he is one. Maybe Major saw him stealing at the benefit and guessed that he might have done the same at the reception."
It made a lot of sense, now that I thought about it.
If Beaufort had indeed stolen the brooch, and had done so before his conversation with Mr. Major that night, maybe Major's threat to contact the police frightened Beaufort into getting rid of the stolen item. Bronwyn had her large shoulder bag with her, at least in the moments before she left the theater. Had a desperate Beaufort slipped the jewelry into her bag in case Mr. Major made good on his threat?
That theory made so much sense that I was convinced it was correct. But was Beaufort a murderer as well as a thief?
"I'll look into it," Salnikova said, pulling me out of my thoughts.
Before she had a chance to bring our meeting to an end, I jumped in with another question. "Who gave Dr. Beaufort his alibi for the night of the break-in at Mr. Major's house?" I couldn't help but be suspicious about that since I knew for a fact that it was false.
"I'm not at liberty to say."
Of course she wasn't.
I slumped back in my chair, frustrated.
"What about Kevin Major?" JT asked. "Any sign of him?"
"Not yet," Salnikova replied. "But he'll turn up."
"Hopefully before he has a chance to hurt Midori again," JT said.
Salnikova nodded her agreement with that statement. "If either of you see him, call 911."
She pushed back her chair and I knew our meeting was about to end. I wasn't ready for that to happen.
"Jordan and his family were sure surprised to find out that Mr. Major had another daughter," I said.
"Yes," Salnikova agreed as she stood up, "it must have been quite a shock."
"Did she know Mr. Major was her father? Because if she did and she also knew he'd decided to leave money to her, that makes her a suspect."
Salnikova tucked her chair under the table and rested her hands on its back rail. "Ms. Bishop, I've been on the police force for almost fifteen years now. Trust me when I say that I know how to do my job."
Next to me, JT fought back a smirk. My arm twitched, I wanted so badly to elbow him in the ribs.
"So you've tracked her down and spoken to her?" I guessed.
"I've followed that avenue of investigation, yes."
Her answer was far more informative than I'd expected.
"Did you know that Archibald Major was Ernest's father as well?"
"I'm aware of that, yes."
"How many kids did this guy father?" JT asked.
"Your guess is as good as mine," I said, before getting back on course. "I'm sure Ernest didn't kill Major, even though he's harbored a lot of anger toward him. He told me he didn't and I believe him. Maybe that sounds silly, but he seemed so sincere."
Salnikova blinked at me. Obviously my opinion didn't make much of an impression. I wondered if I should push my luck by asking another question.
JT didn't give me a chance. He got up from his chair and said, "We'll let you get back to work now, Detective."
As Salnikova nodded in acknowledgment, I swallowed a sigh and got to my feet as well.
Out in the corridor, a uniformed officer approached us.
"Detective," he hailed Salnikova. She met him halfway along the hall.
"We just got a call in from Surrey." The officer kept his voice low, but not quite low enough to keep his words from my ears. "They've got a preliminary ID on a body they found this morning. They thought you'd want to know about it right away."
"Why's that?" Salnikova asked.
"Because," the officer said, "the dead guy is Kevin Major."
Chapter Twenty
I HEARD THE words clearly enough but it took a second for them to register in my brain.
"Kevin Major is dead?" I joined Salnikova and her colleague in the middle of the hallway.
JT followed me. "Is that for real?" he asked.
"Who are you?" The uniformed officer looked from me to JT. He didn't seem impressed that I'd overheard, but it wasn't my fault he hadn't been more discreet about delivering his news.
"They're with me," Salnikova told him, with what I thought was a hint of a sigh behind her words.
The uniformed officer hesitated, but Salnikova nodded at him and he took a step back.
"I'll leave you to it, then," he said.
"Thanks for the heads-up," Salnikova said as he retreated down the hall.
"How did he die?" I asked, unable to suppress the questions bubbling up in the wake of the officer's news. "Was he killed or did something else happen?"
"At this point, I know as much as you do."
"I wouldn't wish death on anyone, including him," JT said, resting a hand on my shoulder, "but at least we don't have to worry about him coming after you anymore."
That was true.
"But remember what I said about Dr. Beaufort," Salnikova said to me.
"I will."
She nodded at the two of us. "If you'll excuse me, I have some things I need to attend to."
As much as I wanted to pester her with more questions about Kevin, I knew it would be pointless. She didn't have any answers and even if she did, she wouldn't share them with me. So instead, JT and I thanked the detective and went on our way.
AFTER A BRIEF stop at JT's house to let Finnegan out in the yard for a few minutes, we continued on to Minerva's restaurant, only a short drive away. We both ordered lasagna and dug in as soon as our steaming hot meals arrived.
"Kevin's death is probably enough to get his name off the list of suspects," I said as I waited for a forkful of lasagna to cool enough to eat.
"Not necessarily."
"Because the two deaths could be unrelated?"
JT nodded as he chewed and swallowed. "Kevin was a criminal, right? He probably had all kinds of unsavory associations. Even if his death wasn't natural or accidental, it might not have anything to do with his father's death."
"It might not," I conceded, although not without a generous dose of doubt. "But don't you think that's too coincidental?"
JT shrugged. "Not if he had a lot of enemies as a result of his criminal lifestyle."
"I suppose." Despite my words, he didn't have me convinced.
The timing really did seem too coincidental to me. Sure, it was possible that his death was unrelated, but I also thought it was highly unlikely.
If his death had indeed involved foul play, the question was why had someone killed him? Did he know who killed his father?
I wouldn't have put it past him to attempt to blackmail the murderer instead of turning them in. But there was really no way for me to know what had happened. Not yet, at least.
A plan took shape in my head, but didn't have the chance to fully form.
"Can we talk about something else?" JT asked, distracting me.
I didn't really want to change topics, but he'd put up with me and my investigating all day so I figured it was the least I could do. For a second I considered bringing up the recent annoyance of Elena and the gorgeous boots I'd drooled over, but I quickly changed my mind. Since he was a guy, I doubted JT would understand why such a thing would matter so much to me, even though he knew I had good reason not to like Elena.
While I savored another forkful of lasagna, I cast around for another subject that had nothing to do with crime or fashion. Before I could come up with one, JT took the lead.
"I talked to my mom yesterday. She and my stepdad definitely want to come to one of your concerts soon."
My face lit up. "Cool. There's one in mid-October I think they'd really like. We'll be playing some Mozart, Dvořák, and Prokofiev."
"Sounds good."
"And you'll come too?" I checked, hopeful.
"Of course."
My smile brightened. "Perfect. I'll get tickets for you guys." I enjoyed some more lasagna before saying, "Oh, I should talk to your mom about the Absolute Zero party."
JT regarded me with suspicion. "I thought we agreed to keep it small."
"We did. And it will be. But we'll still need food."
"We can just order some pizza."
I rolled my eyes. "Okay, fine. But we should also have cake."
"I'll never say no to cake."
"What kind?"
"Three guesses," he said. "First two don't count."
I grinned. "Chocolate, chocolate, and chocolate."
"You got it."
Still smiling, I refocused on my lasagna. Although I knew that thoughts of murder and secrets still lingered at the back of my mind, waiting for a chance to return to my mental spotlight, for the time being I was determined to concentrate on only one thing—enjoying dinner with my best friend.
BY MORNING THE rain had stopped and the sky had partially cleared. I spent the first half hour of my day talking on the phone with my parents, and the second eating a breakfast of toast and tea while reading the Richard Castle book I'd checked out of the library. Once I'd washed and dried my breakfast dishes, however, I had to admit defeat. I couldn't laze around and enjoy a quiet Sunday. I had to look into Kevin Major's death. As much as I wished I could focus on something else, my brain wasn't about to cooperate.
Using my phone to access the Internet, I searched through local news stories until I found a short piece on a body discovered in Surrey the day before. The article contained very little information, but it did state that a woman had found the body while walking her dog at the edge of a wooded area in her neighborhood. The article also mentioned the location of her neighborhood, and that was exactly what I needed.
As soon as I had my hair and makeup done, I grabbed my coat, stuffed my library book into my tote bag, and set off for the bus stop. Soon I boarded a bus that would take me on the first leg of my journey to Surrey, a suburb of Vancouver. The trip would require several transfers and a fair bit of time spent sitting on buses and the Skytrain, but with my book to keep me occupied, I didn't mind too much.
Although my thoughts did stray several times to the purpose for my trip, the fictional mystery set in New York City kept me distracted for good chunks of time. About two hours after I'd left my apartment, I put the book away, disembarked from the bus I was riding, and glanced down at the map displayed on my phone.
As soon as I had my bearings I set off on foot, heading for the neighborhood where Kevin's body had been found. While I walked, I took in deep breaths of the damp fall air, enjoying its freshness. A gentle breeze brushed against my face and played with the ends of my hair, lifting and twirling them in a subdued dance.
It was nice to be outdoors, even if I was heading for the scene of a possible murder. While I was sorry that Jordan's uncle had died, JT wasn't the only one relieved that he was no longer roaming the streets. I now had the freedom to walk around on my own without any out-of-the-ordinary dangers. That knowledge released tension from my shoulders that I hadn't realized was there.
After several minutes of walking, I silently thanked myself for having the forethought to wear flats. The on-foot portion of my journey had turned out to be longer than I'd predicted. If I'd worn heels my feet would have been killing me, but in flats I was able to enjoy the journey. Another minute or two later I spotted a wooded area at the end of a residential street and felt certain I'd found the right place.
I slowed my pace and followed the street to its dead end. A wide dirt path ran along the edge of the woods, providing the neighborhood's residents with a place to jog or walk their dogs. When I reached the path, I paused, not knowing whether to go left or right. The news article hadn't provided a more detailed location so I was on my own from there on out.
Unable to know which would be the correct direction, I decided to start by heading right. Less than two minutes later, I knew I'd chosen correctly. Up ahead, a piece of torn police tape dangled from a huckleberry bush, a sad and bedraggled marker of the place where Kevin's body had been discovered. I slowed my steps and stopped when I reached the bush. I stood facing the woods, noting all the details before me.
Deciduous trees with leaves in the process of changing color stood mixed in with conifers. Beneath the trees, numerous feet had trampled the underbrush, creating something of a pathway into the woods. From my vantage point, I could see that it didn't lead too far in, ending in a larger trampled area about twenty feet from the path.
If Kevin's body had been dumped, his killer hadn't gone to a whole lot of trouble to hide him. No wonder someone had discovered his corpse before too much time had passed.
I hesitated on the pathway, a battle brewing inside of me. On the one hand, my curiosity encouraged me to proceed into the woods, to check out the site where Kevin had been found. On the other hand, the knowledge that a dead body had been discovered mere feet away from me creeped me out and made me want to hightail it out of there.
But of course my curiosity won out. It nearly always did.
Glancing around to make sure I was unobserved, I drew in a deep breath to steady my nerves and stepped off the path. I picked my way through the trampled underbrush until I reached the small man-made clearing. Once there, I stood still and let my eyes do the work.
Although I spent several minutes looking at the ground and the surrounding bushes, there wasn't much to see. The footprints, smooshed wet leaves, and crushed underbrush only told me what I already knew—that several people had traipsed over the area recently.
A crow took flight from a branch over my head and I jumped at the sudden movement. I wrapped my arms around myself as a chill ran through my body, leaving me with goose bumps beneath my sleeves. The scene held nothing of interest for me to see and I found myself relieved by that. It gave me an excuse to turn around and get the heck out of the woods, away from the creepy vibes working their way into my bones.
Seconds later I emerged from the trees and returned to the pathway. A slight movement caught my eye and drew my gaze to the nearest house with its faded purple siding and graying trim. A side window looked out over the pathway where I stood, and I could have sworn that the curtain had twitched. I stared hard at the window but noticed no further movement.
If someone had watched me out of curiosity, I couldn't really blame them. After all, I had just emerged from what until very recently was the site of a police investigation. Still, the thought of someone keeping an eye on me only unnerved me further.
I shivered and rubbed my arms. It seemed as though my trip had been for nothing. I didn't know anything more than I'd known last night, and nothing about the neighborhood gave me a clue as to why Kevin might have gone there. If indeed he'd gone there of his own accord.
Some of the houses—like the faded purple one—could have used some TLC, but for the most part the neighborhood was well kept. It didn't strike me as a hive of criminality or a place where someone like Kevin would easily blend in, but who knew what lurked beneath the surface? There was always the possibility that one or more of the homes was a drug house or had rooms filled with stolen goods, but any secrets of that sort would remain hidden from me, along with the reason for Kevin's presence in the area, whether he'd arrived there alive or already dead.
In any event, I had no reason to stick around and no particular desire to either. I turned away from the purple house and headed for the nearest street. I'd only taken three steps when a woman's voice called out from behind me.
"Halloo!"
I paused and checked over my shoulder, wondering if the hail was aimed at me.
Apparently, it was.
A sixty-something woman with curly gray hair power-walked down the path toward me. She wore a pink and white sweat suit, and a little Yorkshire terrier trotted along at the end of a retractable leash. As soon as I looked in the woman's direction, she waved at me with great enthusiasm and kicked her already swift pace up another notch.
I retraced my three steps back to the path and waited as she approached, wondering what I was in for.
"Good morning," the woman called out in a cheery voice as she drew closer.
"Morning," I returned.
"I couldn't help but notice that you came out of the woods right where they found that body yesterday."
"Er . . . that's right." I waited for the interrogation to begin, expecting that the woman harbored suspicions about me and my presence near the woods.
"You're not from the neighborhood." It was a statement rather than a question. I had a feeling she probably knew every single one of the local residents, at least by sight.
"No," I said. "I was just taking a look around."
Contrary to what I expected, she nodded with understanding, a gleam of excitement in her hazel eyes. "I expected the body to draw some curious souls. It's rather thrilling, isn't it? A dead body in the woods."
"I suppose so."
"Thrilling" wasn't quite the word I would have used, but I wasn't about to contradict her. I was relieved that she seemed far more interested in gossiping than interrogating me, and I didn't want to risk changing that.
"All the police and media hoopla," she went on. "You should have seen all the officers and technicians that were here yesterday."
"It must have been quite a scene."
"Oh yes, it certainly was. We haven't had so much excitement in the neighborhood since Donna and Jim Baristo's marriage fell apart and she threw all his belongings out a window."
I did my best to appear interested, although I had a sudden picture in my head of me standing there for hours, listening to all the neighborhood gossip from the past ten years. Perhaps there was a way I could turn the conversation in a direction more to my own advantage.
"The reason I'm here," I said before she had a chance to delve further into neighborhood drama, "is because I knew the dead man."
The woman's eyes widened with shock, although their thrilled gleam didn't disappear entirely. "Oh my goodness. How terrible for you."
"I actually know his nephew better, but still . . . I wanted to come and see where he was found."
"Of course, of course."
She clicked her tongue at her terrier as he strained at the end of his leash, trying to reach the base of a Douglas fir. The little dog trotted back toward her in response, his leash retracting into its handle.
"You see," I continued, "it's been quite difficult for the family."
"Yes, yes, I imagine so."
"And the police aren't saying much. They can't, of course, at this point, but it's still hard not having any details."
The woman nodded in sympathy, drinking in every one of my words.
"We don't know if he died of natural causes or an accident or what," I said. "I thought I'd come by to see the spot where he died, but in the end that wasn't quite as helpful as I'd hoped."
"You poor thing." The woman clicked her tongue again as her dog lunged toward a squirrel darting up a tree trunk. "But I can tell you that he didn't die of natural causes and, in fact, this wasn't the place where he met his end."
"Really?" I asked, intrigued. "How do you know?"
"See that house?" She pointed to a blue and white one across the street from the faded purple house. "My best friend, Linnea, lives there." The woman beamed with pride. "She's the one who found the body."
Chapter Twenty-One
A CLEAR NOTE of excitement pealed through my head like the ringing of a stately church bell. I knew this woman was eager to share her information with me, unlike Detective Salnikova, who preferred to keep even the tiniest of scraps to herself.
"I'm Janet, by the way." The woman gestured over her shoulder with her thumb. "I live one street over. Linnea and I both walk this path several times a day. She has a toy poodle named Toby. He's having problems with his kidneys of late, poor little thing, but he still gets out for a short jaunt two or three times a day. And of course Linnea and I walk this way coming and going from each other's houses. It's nice to go by the woods here, to see the birds and squirrels. And this time of year, with the leaves changing color, it's really quite beautiful, don't you think?"
"Yes, it is." Her ability to rattle on almost overwhelmed me, but I pulled myself together and jumped in before she could ramble on further. "But you said Linnea found the body?"
"Yes. Although, to be precise, it was Toby who found the body. He was off leash and went for a little sniff. His kidneys might be a little wonky but his nose still works as well as ever. He darted through the trees, right to the body, and wouldn't come back. Linnea had to trek into the bushes to see what was up. And, my, did she ever get the shock of her life!"
"I'm sure." Again, her rapid wash of words had almost put me into a daze, but I tried once more to steer her back on course. "And Linnea could tell that the man didn't die of natural causes?"
"Oh, yes, indeed." Janet dug a small dog cookie out of one of her pockets and fed it to her Yorkshire terrier. "He had a nasty gash on his head, you see. Oh sure," she said, giving her hand a dismissive wave, "the police will have to do all their fancy tests before coming up with a definitive cause of death, but I'm telling you it was as clear as day to Linnea that the head wound was what killed him. She likes reading police procedurals, don't you know. Watches them on TV too. Blunt force trauma. That's what they call it."
"Right." I tried to filter out and absorb the pertinent information while keeping up with her rapid tempo. "But how did she know that he didn't die here in the woods?"
"The lack of blood, my dear." She nodded sagely, as if she had solved the entire case herself. "There was blood crusted on his head, face, and clothes, but none on the leaves or ground around him. And you know how head wounds bleed."
I did, from personal experience. At age seven I'd tripped in a friend's garden while playing tag and had cut my scalp open on the corner of a landscape tie. The amount of blood that came out of the wound had both frightened and impressed seven-year-old me.
"So he either died as a result of an accident or he was murdered," I said, more to myself than to Janet. "But either way, somebody moved his body from the scene of his death."
"Precisely." The excited gleam in her eyes took on a shade of curiosity. "Any idea who might have wanted him dead?"
"No clue," I said, though that wasn't quite true. I didn't have a name to give, but I still believed there was a good chance that his father's killer was also his own killer.
A hint of disappointment flitted across Janet's face. "Oh well. Perhaps the police will solve the case." She didn't sound as though she'd bet on it.
A glint of light flickered across my line of sight. My eyes went straight to the place I thought it had originated from—the side window of the faded purple house. A second quick flash confirmed that I'd pinpointed the source. Even so, I couldn't quite believe the implication of what I'd seen.
"Somebody's watching us with binoculars." Incredulity underscored my words.
Janet followed my gaze to the purple house. "Oh my. I suppose we can't blame them for being curious after yesterday's discovery."
"Do you know the people who live there?"
A slight frown turned down the corners of Janet's mouth. "Not really. The homeowner's a single woman. An older woman—probably her mother—comes by now and then but otherwise she keeps to herself."
I could tell that not knowing all the details about the life of someone in her neighborhood irked her. I fought back a smile and pulled my phone from my bag to check the time, more as an excuse to get on my way than because of any particular concern with the hour.
"It's been really nice talking to you," I said as I returned my phone to my purse, "but I need to be on my way."
"Of course, dear. It was nice talking to you too. I hope the poor dead man's family is able to find some peace before long."
"So do I," I said, although I knew many questions would need to be answered before that could happen. "Goodbye."
I waved to Janet and headed for the street, casting one last glance at the side window of the nearest house as I went. Perhaps whoever had watched us through their binoculars had simply done so out of curiosity, as Janet had suggested. But as I walked along the sidewalk, passing in front of the purple house, a shiver vibrated up my spine.
THE TRIP BACK home gave me plenty of time to think. With my library book forgotten in my bag, I sat in my seat on the Skytrain and went over everything I'd learned from Janet. Despite her tendency to chatter like an excited chipmunk, in among the irrelevant gossip she'd revealed a couple of nuggets of valuable information.
Kevin had sustained a significant head wound and he hadn't died there in the woods. When taken together, those two clues told me that someone had most likely murdered him. Sure, it was possible that he'd died as a result of an accident while in someone else's company and that person had panicked and dumped his body, fearing that they might be blamed for his death. But it struck me as far more likely that his death involved foul play.
The real question in my mind was why Archibald Major's killer would want Kevin out of the way.
To me, the simplest answer was that the killer had felt threatened by Kevin. If that was the case, then perhaps Kevin had known the killer's identity, or at least had known enough to worry the murderer. If he had indeed known something along those lines, it didn't surprise me that he hadn't gone to the police with the information.
Considering his history, Kevin probably didn't have the greatest relationship with law enforcement officers, and I didn't know if he had cared all that much about seeing his father's killer brought to justice. It was easier for me to envision him using his information for his own benefit. Perhaps that's what got him killed. If he'd attempted to blackmail the murderer, for instance, removing him from the picture might have seemed like a good idea, especially considering that the murderer had already killed once.
I knew my theory was formed mostly on the basis of speculation, but it made sense. It didn't answer all of my questions, however. I still wondered why Kevin's body had ended up in a patch of woods in Surrey. Jordan had mentioned that his uncle lived in downtown Vancouver, so Surrey wasn't all that close to home. Had he gone there for a particular purpose? Was he killed nearby or was his body transported a good distance after his death, perhaps by car? Was this neighborhood familiar territory for the killer?
So many questions. Even more questions popped into my mind when I considered my list of suspects in relation to this latest death. Although I could now attribute Andrea Duffy's suspicious phone call from the other day to her affair with Mr. Hollingsworth, I couldn't yet cross her off my mental list. It was still possible that she'd killed her father. I had a harder time picturing her as her brother's murderer, however, because I couldn't come up with a solid motive.
Even if Kevin's death meant that Jordan's mother would inherit more money, I didn't know how much she would care about that. She'd already gained millions by her father's death. And if Kevin's murder had come about as result of him figuring out the identity of his father's killer, I couldn't imagine him blackmailing or threatening his own sister.
The conversation I'd overheard between the siblings on the night of their father's death led me to believe that, while not of the most honorable character, Kevin was more likely to protect than deliberately harm his sister.
As for the other suspects on my list, I figured that any one of them could have killed Kevin in order to keep their identity as Archibald Major's murderer under wraps. Nonetheless, my top suspect at the moment was Dr. Beaufort. Whether he'd taken up theft for financial reasons or because he had psychological issues, I didn't know. But perhaps his motivation in that respect didn't really matter. I was convinced that his secret was related to the thefts and that his desperation to keep his habit quiet had driven him to break into Mr. Major's house. While he had less access to Major's flask than the other suspects, it wasn't inconceivable that he'd slipped poison into it. Perhaps he'd broken into Major's residence on a previous occasion, or maybe he'd used his skills as a pickpocket to get hold of the flask and return it unnoticed to Mr. Major's pocket.
Maybe that was a bit of a stretch, but I still liked Beaufort as a suspect for the murder in addition to the thefts. There was something fishy going on with him, that much was certain, and he wasn't above lying or burgling houses.
It still irked me that he'd provided the police with an alibi, casting doubt on my identification of him as the intruder in Major's study. I didn't know who'd provided him with that alibi, but whoever they were, they'd lied too. Unless Beaufort had an identical twin running about, I'd identified him correctly.
With a sigh, I stared out the window at the gray day. Despite the clues and information I'd gathered, I didn't seem much closer to uncovering the truth. I hadn't even narrowed down my list of suspects by much. I'd ruled out Ernest based on my gut instinct that he'd told me the truth at the pub the other night, and I now doubted that Kevin had killed his father, but I hadn't made much progress beyond that.
How disheartening. I couldn't solve the mystery and provide Jordan with closure if I kept coming up with more and more questions and very few answers.
As I transferred from the Skytrain to a bus, my mind drifted to thoughts of Aaron. I'd managed to divert my attention away from him for the last while, a fact I found comforting. I didn't want to dwell on the unhappy ending of our relationship. I wished he hadn't quit the band, but I had to admit things would be easier now that he had. With no chance of running into him at JT's house, I wouldn't have to worry about rushing away from my studio on band practice nights.
Scrolling through the list of contacts on my phone, I found Aaron's name and deleted his number. Somehow that added a sense of finality to the end of our relationship. While a tiny hint of guilt pinged in my stomach as I remembered the expression on his face when I'd last seen him, my sense of relief drowned it out.
I'd done what I'd needed to and now I could move forward. Whatever that meant.
I could see now that I never could have had the feelings for Aaron that I'd hoped, because my heart had belonged to someone else all along, though I hadn't known it at the time. That wasn't a thought I wanted to follow though. Not at the moment. It wouldn't take me anywhere new and I didn't want to worry about my relationship with JT. That would only make things more difficult.
With my bus ride only half over, I decided to dig out my library book to distract myself. But I hadn't even had a chance to open it when my phone rang in my hand. I didn't know who the caller was, but I answered anyway.
"Hello?" I said into the device.
"Midori? It's Andrea Duffy."
"Oh, hello, Mrs. Duffy."
Although her phone call caught me off guard, it didn't surprise me as much as the sound of her strained voice. When she sniffled, I realized that she either was or had recently been crying.
"Jordan asked me to call you to let you know that he won't be at his lesson on Tuesday. Actually, he wanted me to tell you that he won't be able to have any lessons in the foreseeable future."
My increasing concern kicked into overdrive. "Why not? Is he okay?"
"Not really," Mrs. Duffy replied after drawing in a shaky breath. "He's in police custody. He confessed to his grandfather's murder."
Chapter Twenty-Two
HER WORDS SHOCKED me into temporary silence. As she cried into the phone, I tried to wrap my mind around her revelation, but couldn't quite manage it. After several beats of silence on my part, I finally found my voice.
"I don't understand." I sounded as baffled as I felt.
"He didn't do it," Mrs. Duffy said as she sobbed. "He couldn't. Not my son."
"Of course not."
A muffled voice sounded in the background and Mrs. Duffy said, "I'm sorry. I have to go."
"Thank you for calling," I said quickly, getting the words in just before she hung up.
With the call over, I stared at my phone, still stunned.
Jordan, a murderer? No way.
Against my will, my brain analyzed the possibility.
My student certainly would have had access to the flask, and the presence of the Brugmansia plant out on his grandfather's patio meant he could easily have obtained the poison used by the killer. All he had to do was prepare the poison and slip it into the flask, ready for whenever his grandfather next took a swig or added a dollop to another drink. As for motive, maybe Jordan had grown tired of seeing his mother belittled by her father. Maybe he wanted her to have financial freedom as well as freedom from Archibald Major's bullying ways.
But although he had means, opportunity, and a possible motive, I still couldn't picture Jordan as a murderer. I'd known him since he was seven years old. It didn't fit. Jordan wasn't the type to harm anyone. I was sure of it.
So why had he confessed to the crime?
I didn't know, but I was determined to find out.
TEN MINUTES LATER I transferred to a different bus, one that would take me to the police station rather than my apartment. I doubted that my presence would please Salnikova or her colleagues, but I couldn't stay away. The thought of Jordan under intense interrogation or locked up with a bunch of hardened criminals chilled my blood. I didn't know what I could do to help him, but there had to be something.
As the bus rumbled along, numerous different thoughts clanged in my head, warring for my attention. It was like trying to think while stuck in a small room with several musicians, all playing different tunes, trying to outdo one another. I wanted to pull my hair out, but managed to stop myself. Instead I took a deep breath and did my best to focus. I needed to figure out the reason for Jordan's confession.
If he hadn't committed the crime—and I believed that he hadn't—then the only reason I could come up with for his confession was that he wanted to protect someone. And the only person I could imagine him wanting to protect to that degree was his mother.
Although Dr. Beaufort sat at the top of my current suspect list, perhaps Mrs. Duffy occupied the top spot on Jordan's. If that was indeed the case, I wondered if he merely suspected that his mother had killed her father or if he actually knew that to be the case.
As I'd already considered, Mrs. Duffy had the means and opportunity to poison the contents of Mr. Major's flask. She also had a motive, one that seemed far stronger than Jordan's. As soon as everything was finalized, she'd be a very wealthy woman. Her new money would give her the freedom to start a new life, free of her husband and any financial worries.
But did she really murder her father?
I didn't quite know how I could figure out the answer to that question, but I needed to find a way. The identity of the real killer had to come to light. Otherwise, Jordan could be at risk of getting locked away and having his life ruined over something he didn't do.
BY THE TIME I pulled open the front door of the police station, morning had long ago turned into afternoon and my stomach reminded me that I hadn't eaten for hours. I couldn't let something as trivial as mild hunger distract me from my mission, though. On a scale of importance, helping Jordan far outranked my need for lunch. Or dinner. Or whatever meal would be most appropriate considering the time of day.
A middle-aged couple and a thirty-something woman wearing too much makeup sat in the reception room's plastic chairs, but I didn't spare them more than the briefest of glances. I proceeded straight to the desk and asked the man behind it if Detective Salnikova was in and if I could talk to her.
Without answering my questions, the man asked me to sit down and wait. I did so, but found it hard to sit still. I imagined Jordan in a back room, Salnikova and perhaps another detective grilling him on the details of Archibald Major's murder. I didn't like the idea one bit. While I knew Jordan would have an adult with him because of his age, I hoped he had a skilled lawyer with him too. Although, if he continued to insist that he committed the crime, I didn't know how much good legal representation would do for him.
Maybe it wasn't such a bad thing for the police to interrogate him, if that's what they were indeed doing. If there were details of the crime that he clearly didn't know or got wrong, perhaps the police would doubt the veracity of his confession. But how many details were there to know? It didn't strike me as an elaborate crime and I didn't know how much Jordan knew or didn't know.
Oh no.
Dread lumped in my stomach.
I'd told Jordan about the flask. Why did I share that detail with him? Why didn't I keep quiet?
All along I'd only wanted to help Jordan, but now it seemed I might have done far more harm than good. If he told the police that he'd put poison in the flask, and if that was indeed where the poison was found, that would make his confession seem more truthful. The police probably thought the killer was the only person other than themselves who knew the poison had been in Major's flask.
I couldn't believe I'd been so stupid. I never should have involved Jordan in my investigation in any way whatsoever. My fingers tugged at my left earlobe as my concern for my student increased. The police couldn't charge him with murder. The mere thought was too terrible to think about.
Just as I was about ready to drop my head into my hands in despair, the door to the side of the reception desk opened and Salnikova motioned to me to join her. I jumped up and followed her into the hallway.
"I'm afraid I don't have much time to spare today," she told me as she led the way to her desk.
"I understand, but I had to talk to you about Jordan. Is he okay? You haven't charged him yet, have you?"
"Ms. Bishop."
For the first time I noticed dark rings under her eyes and I wondered how much sleep she'd managed to get during the past week. Not much, by the look of it. And here I was trying her patience yet again. But I couldn't let my guilt over that interfere with my quest to help Jordan.
"Okay, I get it," I said. "You don't want to tell me anything. Or you can't. But there's no way Jordan killed his grandfather. You have to know that."
"I don't know that." Salnikova sank down into her chair and gestured for me to take one of the empty seats on the other side of her desk. "And according to him, he did do it."
"But he's lying. He has to be. Jordan would never hurt anyone."
Salnikova sighed, a touch of sympathy showing in her eyes. "It's always hard to accept that someone you know could be a killer."
I shook my head, unwilling to follow her in the direction she wanted to take the conversation. "If he said he put poison in his grandfather's flask, that's my fault. I told him that's where the poison probably was."
Sharp scrutiny replaced the sympathy in Salnikova's eyes. "Why would you tell him that? What made you think that it was in the flask?"
I wasn't about to tell her that I'd eavesdropped on her conversation with Detective Bachman. "It makes the most sense," I said instead. "If the poison was in the coffee or the champagne, more people would have been poisoned. It could have been put in Mr. Major's coffee cup, but that would have been trickier to do with so many people around. The flask was the most likely place for it, so I asked Jordan who'd had access to it."
Maybe it was because she was tired, or maybe I'd simply crossed a line this time, but the detective didn't hide her annoyance with her usual impassive expression.
"You need to leave the investigating to the police, Ms. Bishop. I know you think you're helping, but you're not, especially when you interfere in the official investigation."
"I know I shouldn't have said anything. I never should have involved Jordan, but I never imagined that he would confess to a crime he didn't commit."
"Perhaps not, but I'm telling you now that you need to back off. I understand that you care about Jordan, but there's nothing you can do for him at the moment."
I didn't want to give in. "But he's lying. And I'm sure he's doing it to protect his mother. I don't know if she killed her father or not, but I think Jordan believes that she did. He's trying to deflect your suspicion away from her."
"As much as this might surprise you," Salnikova said, "my colleagues and I have considered that angle."
"You have? But then why is Jordan still in custody?"
"Because he still maintains he's guilty and, as of yet, we don't have enough reason to disbelieve him."
"But . . ." I didn't know what else to say. I couldn't help but feel that I'd failed Jordan somehow.
"Go home, Midori," Detective Salnikova said. "Let me do my job. If Jordan is innocent, that will be established in the end."
I wished I could be as certain about that as she was. What if Jordan's false confession was enough to get him convicted? And even if he didn't end up going to trial for something he didn't do, how long would he have to languish in a detention center before the justice system realized he wasn't a killer and set him free? I hated the thought of him locked away, but I no longer had any idea how to help him.
Not knowing what else to do, I followed Salnikova's lead when she got up from her chair. She escorted me back in the direction of the reception area. As we made our way down the hallway, a door to our right opened and a tearful Mrs. Duffy came out of an interview room.
"Midori?" she said when she saw me. "I didn't know you were here."
"I was worried about Jordan. Does he have a lawyer?"
"Yes, she's in with him right now." She dabbed at her watery eyes with a crumpled tissue. "I don't understand. He keeps insisting that he did it, but of course he didn't. He couldn't have. So why does he keep saying that he did?"
Beside us, Detective Salnikova cleared her throat. "Mrs. Duffy, I'm afraid Midori is on her way out."
As a fat tear rolled down Mrs. Duffy's cheek, I put a hand on her arm. "Why don't you step outside with me for a moment?"
Although I caught the flicker of disapproval that passed across Salnikova's face, I ignored it. If Mrs. Duffy wasn't the one under interrogation, I didn't see why I couldn't speak to her. The detective must not have had a good reason to prevent us from talking either, because she didn't voice her objections out loud.
Mrs. Duffy dabbed at her eyes again. "I'm not sure."
I tucked my arm through hers. "Just for a minute or two. Maybe the fresh air will do you good."
She didn't protest and allowed me to lead her through the reception area and out the front door. Salnikova returned to the back of the building and we stood out on the sidewalk, alone except for the occasional pedestrian passing by and the traffic on the street.
I knew Jordan's mother wouldn't want to be far from her son for long, so I wasted no time getting to the point. "Mrs. Duffy, I think Jordan might have confessed to protect you."
"Me?" She wiped her nose with her tissue. "I don't understand."
"I think he might believe that you killed your father."
"What?" Her watery eyes widened. "But I didn't. Why would he think that?"
"You've been hiding something from him, haven't you?" I already knew the answer to that, but even if it had been a mere guess, the shifting of Mrs. Duffy's eyes would have provided enough confirmation. "He knows you have."
"I'm in a relationship," she confided. "I was worried Jordan would be upset that I'm not interested in getting back together with his dad. Nothing I've been hiding from him has anything to do with my father's death."
"Maybe not, but Jordan doesn't know that. All he knows is that his grandfather was murdered and you've been acting strangely."
"Oh no." She leaned against the building and closed her eyes. "I never meant for any of this to happen. My poor Jordan." She stifled a fresh sob. "What do I do? How do I get him out of this?"
"Start by telling him the truth. Maybe once he knows he doesn't need to protect you, he'll start telling the truth too."
She nodded and straightened up. "Yes, it's time to tell him the truth. I just hope it's not too late for him, that he hasn't dug himself in too deep. And I hope he won't hate me for what I'm about to tell him."
"I doubt that he will," I said. "Remember, we're talking about the kid who confessed to murder likely to protect you."
Mrs. Duffy dabbed her eyes one last time. "You're right. And whatever it takes to get him out of this mess, it'll be worth it."
I gave her arm a reassuring squeeze.
"Thank you, Midori."
"You're welcome. And I'm sorry about your brother." While I was glad I no longer had to worry about Kevin, I still felt bad that Mrs. Duffy had lost another member of her family.
She nodded. "So am I."
She was about to turn away but I stopped her, still curious about something.
"Before he confessed, Jordan was adamant that Kevin had killed your father. Do you think he wanted suspicion to fall on his uncle for the same reason as he confessed—to protect you?"
"Possibly. He might have believed Kevin was guilty in the beginning, or he might have thought implicating him was a good way to shift suspicion away from me. Jordan's always known that his uncle was a criminal and he was terribly mad at Kevin for stealing his iPhone last month. It was more than a theft, you see. It was a betrayal."
I nodded with understanding. "I guess your brother was a good target for putting the blame on. After all, he did say he'd make sure you and he didn't have to deal with your father anymore."
Mrs. Duffy gave me an odd look. "How do you know about that?"
"That's a long story," I said quickly, not wanting to let on that I'd eavesdropped on her conversation with Kevin.
Luckily, she didn't pursue the matter.
"He never meant that he intended to kill our father. Kevin had his issues but he wasn't a murderer. Most likely he meant he was going to take part in a large-scale robbery or get more involved in the drug trade."
She let out a weary sigh and I knew it was time to let her go.
"Thanks for talking with me. Good luck with everything."
She flashed me a weak smile and disappeared back into the building.
I remained there on the sidewalk for another moment or two, barely aware of the traffic rumbling past me a few feet away. As much as I wanted to talk to Jordan myself, I knew that wasn't possible. All I could do was hope that I'd assessed the situation correctly and that Mrs. Duffy would be able to convince him that he had no need to lie. As to whether or not the police would let him go if he recanted his confession, I didn't know.
Again, all I could do was hope.
Chapter Twenty-Three
AS I WANDERED away from the police station I realized that I couldn't ignore my growling stomach any longer. Anxiety over Jordan's situation or not, I needed food if I didn't want to pass out. I took a bus to my neighborhood and decided on a small restaurant nestled between a bookstore and a children's clothing boutique. A handful of other patrons occupied four of the small tables, leaving five tables and the café's three booths free.
I tucked myself into one of the booths and a young waitress with spiky blond hair approached right away with a laminated menu and a bundle of cutlery in hand.
After exchanging pleasantries I ordered a glass of root beer and a BLT. The waitress bustled off and returned a moment later with my drink. As I waited for my sandwich, I sipped at my root beer and gazed at the painting on the wall next to my table. Although the picture of a small log cabin on the edge of a frozen lake was pretty enough, I didn't take in many of its details. Too many thoughts filled my head, keeping the majority of my attention elsewhere.
My brief talk with Mrs. Duffy outside the police station made me doubt her potential as a viable suspect. Her distress over Jordan's predicament and her surprise at the thought that he could believe her to be guilty seemed so genuine. Then again, that didn't necessarily mean that she hadn't killed her father. Being a concerned mother who didn't expect her son to think her capable of murder didn't preclude her from being a killer. Yet, if her concern for her son was as deep and genuine as I believed it to be, and if she had killed Mr. Major, wouldn't she have already confessed to the crime to get Jordan off the hook?
The waitress interrupted my thoughts by placing my BLT in front of me and asking if I needed anything else. When I answered in the negative, she left me alone again and I dug into my sandwich with gusto, my stomach practically purring with gratitude and appreciation.
Three bites in, I paused for a sip of root beer, and my thoughts returned once again to Mrs. Duffy. The fact was that I couldn't prove that she had or hadn't killed her father, and it was possible that Jordan would still be in jeopardy unless someone could establish that another person was the guilty party. Most likely that someone would need to be me since Jordan and his confession had the police currently occupied.
As I finished off my sandwich, I mulled over everything I knew about all the suspects still on my list, but that didn't get me very far. I didn't know enough to firmly implicate any one person in Archibald Major's murder, certainly not to the degree the police would require. There had to be something I'd missed or some clue I'd yet to find.
With my BLT all eaten, I ordered a slice of chocolate cheesecake for some additional brain fuel. After savoring the first delectable bite, I retrieved my phone from my bag and accessed the Internet. My previous online search hadn't revealed much about any of my suspects, but I didn't know what else to do. Maybe I could uncover a helpful nugget of information that I'd overlooked before, or maybe something I'd already seen would take on new significance. A long shot, perhaps, but worth a try since I had no better ideas at the moment.
I made sure to enjoy every forkful of my cheesecake, but between bites I scrolled through search results and scanned articles and Web pages. After my last bite of cake, I flicked through a few pictures I'd seen during my previous search. One photo in particular gave me pause.
As a high-pitched note of alarm sounded in my head, I enlarged the photo for a better look and zeroed in on a face I'd skimmed over previously. I could hardly believe my eyes. I peered more closely at the picture, but that didn't change anything. My eyes hadn't deceived me.
I read the caption beneath the photo. My heart rate upped its tempo and a mixture of worry and excitement jangled through my body. I tried to absorb the implication of what I was looking at. It was the clue I'd overlooked, the missing note that completed a murderous melody.
Although I still didn't understand quite how everything fit together, I knew enough.
I knew who'd killed Archibald Major.
After wrestling my wallet out of the depths of my tote bag, I pulled out a couple of bills and slapped them on the table as I slid out of my booth. Slinging my bag over my shoulder, I hurried out of the restaurant and onto the sidewalk. My phone still clutched in one hand, I set off at a good clip down the street to the bus stop.
All my hurrying was for naught, however. When I reached the stop and peered up the street, I couldn't see a bus anywhere. Impatience plucking at my nerves, I pulled up the bus schedule on my phone.
Darn. I'd temporarily forgotten it was Sunday. That meant the buses ran less frequently and I'd have to wait another twenty minutes before one showed up that would take me back to the police station. I couldn't wait that long. I needed to share my revelation with Detective Salnikova as soon as possible, and with all the nervous energy coursing through me, I couldn't stand around waiting for a bus to show up.
Giving up on public transportation, I decided to head home and speak to Salnikova over the phone. As I hoofed it around the corner and along a side street, I selected the detective's name from my list of contacts and put a call through to her number.
I turned another corner onto my street and stopped dead in my tracks.
Shoot.
I cut off my phone call mid-ring. As soon as I'd done that, I realized it was probably the last thing I should have done. I took a step backward, hoping I could sneak back around the corner and try calling Salnikova again.
"Ms. Bishop!"
Too late.
Dr. Beaufort, loitering outside the main door of my apartment building, spotted me before I could duck out of sight. He walked toward me with brisk strides and I gave up any hope of avoiding him. I was already keyed up and Beaufort's presence only served to send my anxiety level shooting toward the overcast and darkening sky. I kept my thumb hovering over my phone as he approached, ready to call for help.
"Dr. Beaufort, what are you doing here?" I asked, although I could guess the answer.
"I need to talk to you." Tension seemed to spark off his body. "Did you tell the police you were mistaken? Did you tell them I wasn't the person you saw at Major's house?"
"Um." I wondered if I should lie, considering how keyed up he was, but my momentary indecision was enough to give away the truth.
Beaufort's dark eyes flashed with either anger or desperation. Maybe both. "I thought we came to an agreement, an understanding. You need to talk to them as soon as possible."
"I never agreed to anything," I said as I edged past him, not liking the fact that he stood between me and my apartment building. "And I know I wasn't mistaken."
"I assure you that you were. I have people who are willing to vouch for my whereabouts on that evening, and they certainly won't say I was where you think I was."
My next words slipped out before I could stop them, ignited by a flicker of anger. "You might have people who will lie for you, but I still know what I saw. And besides, if you believe you have such a solid alibi, why are you worried about what I told the police?"
"It's awkward, don't you see?" The protruding tendons in his neck signaled his rising frustration. "The police are asking questions all over again. Even allegations and innuendo could be enough to damage my professional reputation, to harm the orchestra. Why can't you understand that?"
Again, I couldn't hold back my words. "If you didn't want to damage your reputation, you shouldn't have become a thief."
A dark flush rose up his neck and into his face. "I'm not a thief!" Despite the force he tried to put behind the words, his voice wavered. "How many times do I have to tell you that you were mistaken?"
"You could tell me a thousand times and it wouldn't make a difference. Besides, I'm talking about more than the break-in at Mr. Major's house. You're the one who stole the brooch at the opening night reception."
He stared at me with a mixture of surprise and horror, his eyes bulging in a frightening way. Although he opened his mouth to deny the accusation, no sound came out.
His reaction was enough to confirm my theory.
"I know it was you," I said, anger heating my words. "You took the brooch and when Mr. Major threatened to call the police, you decided you needed to get rid of it. So you slipped the brooch into Bronwyn Cassidy's bag. And now, thanks to you, people think she's a thief and she's in danger of getting kicked out of the orchestra."
Beaufort tugged at the collar of his shirt. "I didn't steal anything." His denial came out weak and strained, and I knew then that the rest of my theory was correct as well. After a second or two, he seemed to recover somewhat from his fright and surprise. He dropped his hand from his collar and glared at me, his nostrils flaring. "Mark my words, telling such lies will only lead to trouble."
I'd made my way past him as we'd spoken, but I still edged a little closer to the front entrance of my building. I really didn't like the way he was glaring at me. My heart danced a frantic jig in my chest and I had to tighten my grip on my phone to keep it from slipping out of my slick hand. I glanced around and a modicum of relief eased my panic down half a notch.
A man was walking along the other side of the street, a German shepherd with him on a leash. He seemed oblivious to my presence and predicament, but at least he was within shouting distance.
"I'd like you to leave, please," I said. Although my anger hadn't disappeared, my fear and anxiety had overpowered it and I had to struggle to keep my voice strong and steady.
Beaufort took a step toward me. "I don't think you get what's at stake here."
As much as I would have liked to hold my ground, instinct kicked in and sent me back two steps. "If you don't leave now, I'll have to call the police."
I had every intention of calling the police even if he left, but he didn't need to know that. The situation was already tense enough.
"You need to understand . . ."
He trailed off when I held up my phone. His flared nostrils twitched and his eyes hardened. My heart's dance increased its frenzied tempo and my pulse pounded in my ears like the beat of a bass drum. He raised his hand and I flinched, an automatic response. But instead of striking or grabbing me, he held his hand up in what I thought he meant as a placating gesture. The expression on his face and the undertone of anger beneath his next words didn't match the action.
"Fine. No need to involve the police. I only wanted to make you understand." He took one step back. "I hope you'll reconsider and do as I asked before. It would be best for everyone, believe me."
He turned and strode away down the street. I stumbled along the hedge-lined path to the covered entranceway of my building and sagged against the wall. My legs had taken on the consistency of wobbly Jell-O and threatened to give out on me at any second. In time the beating of my heart lessened its intensity and I drew in some deep, shaky breaths.
As soon as I thought myself capable of coherent speech, I hit Salnikova's name on my phone and remained leaning against the wall as the call went through. After three rings, the call went to voice mail.
No, no, no! I wanted to shout at the detective's recorded voice.
Of course she wasn't available. She was probably busy interrogating Jordan. Even so, I really wished she would have answered her phone.
As I waited for the tone so I could leave a message, I searched through my purse for my keys. Aside from getting in touch with Salnikova, the thing I wanted most in the world right then was to get inside my apartment and lock the door behind me. Before my fingers touched my keys, the beep sounded in my ear.
"Detective." My fear had left me short of breath and I had to pause after the first word. I gulped in a breath of air and pushed onward with my message. "I had a visit from Dr. Beaufort. He's not very happy with me and to be honest I'm a bit freaked out. But his visit isn't the only reason I'm calling. I found the missing link. Now I can prove that—"
Something hard hit the back of my neck. I gasped with pain and surprise as my body lurched forward and down. Twin knife stabs of agony pierced through my knees as they hit the ground. My phone skittered out of my grasp and across the pavement. I tried to reach for it but the pain in my neck intensified with overwhelming ferocity.
A cry escaped my throat and my vision blurred.
I collapsed the rest of the way to the ground.
Chapter Twenty-Four
I REMAINED CONSCIOUS, but pain and confusion clouded my mind. Someone loomed over me, I knew that much, but beyond that all my brain could compute was the fact that I was in danger. Blinking away the tears of agony that interfered with my vision, I reached one hand toward my phone. My fingers had barely made contact with the device when a booted foot kicked it out of my feeble grasp.
When I tried to turn my head to look up at my attacker, a fresh spike of pain stabbed through my neck, freezing my movements. Large, strong hands grabbed me by the shoulders and yanked me to my feet. I cried out as more pain tore through my neck. My attacker's arms wrapped around me and lifted me off my feet. I opened my mouth to scream, but a meaty hand clapped over my face before I could let out anything more than a whimper.
I kicked my feet and wriggled my body, struggling as much as I could. My heel hit a shin and my attacker grunted, but my efforts didn't seem to have any other effect. Still, I continued to struggle, fear and outright panic spurring me on despite the pain screaming through my body.
Within seconds of grabbing me, my assailant had me over by the curb and shoved me into the back of a black van. He tossed a wrench—probably the weapon he'd used on me—into the vehicle and I made a move to grab it. I didn't move fast enough. My attacker had crawled in behind me and now pinned me down face-first, a knee pressed into my back. I tried to scream again, but he stuffed a cloth in my mouth and secured it there with a bandana tied at the back of my head. I wanted to turn my face away, to shake my head in an attempt to make it more difficult for him to fasten the bandana, but I couldn't. My neck didn't want to move and the pain brought fresh tears to my eyes.
I flailed the rest of my body as much as I could, but my attacker was far stronger than me. He yanked my arms behind me and snapped a pair of handcuffs around my wrists. Keeping a knee pinned to my back, he snapped another set of cuffs around my ankles. He removed his knee and I rolled over, sobbing into my gag as my neck protested with a vengeance. A flash of surprise registered in my terrified mind. My assailant wasn't a man as I'd thought, but a large, muscular woman. A woman I'd seen before.
Not allowing my surprise to make me hesitate, I drew up my knees and aimed a double-legged kick at the brawny, dark-haired woman kneeling over me, but she ducked out of the way and my legs met nothing but air.
Before I had a chance to try again, the woman hopped out of the van—taking the wrench with her—and slammed the double doors. A door up at the front of the vehicle opened and the van rocked with the addition of new weight settling into the driver's seat. The front door slammed as well and the engine roared to life, the vibrations rattling through my bones and making me even more aware of my aching knees and throbbing neck.
With a rumble, the van pulled out onto the street and set off to who knew where. Despair hovered over me, a shadow ready to swallow me up in its darkness. What was happening to me? Who was the woman who had grabbed me?
In the moment when I'd first understood that someone had hit me from behind, I thought it was Dr. Beaufort. I figured he'd only pretended to leave and had doubled back to take out his anger on me. But now I was at a loss. The only time I'd ever seen this woman before was at Mr. Major's funeral.
What could she possibly want with me?
And how would anyone find me?
I hoped with all my might that someone had seen the woman grab me and shove me into the van. Maybe the guy walking his dog across the street. No, too much time had passed before my attacker struck me. The dog walker was probably long gone.
But apartments across the road looked out over the street. Maybe somebody saw something and called the police.
But what if they hadn't?
The looming shadow of despair edged closer to me.
No, don't think like that, I scolded myself. Find a way to save yourself. Survive. Escape.
I tried to piece my thoughts together, but it wasn't easy. Every time the van drove over the slightest bump, new shots of pain distracted me. The smell of motor oil reaching my nose didn't help much either. Together, the uneven motion of the van and the unpleasant odor made me queasy. I tried my best to breathe steadily in and out through my nose, knowing I couldn't risk being sick with a gag stuffed in my mouth. I didn't want to choke to death before I had a chance to escape.
Think, I told myself as I got a handle on both my despair and my nausea.
I wiggled my wrists and ankles, but the cuffs held tight and my abductor had left no room for me to slip out of them. Lying on my back, my bound hands squished beneath me, I lifted my head as far as my aching neck would allow. That was only about two inches, but it was enough to let me get a look at the van's back doors with their darkly tinted windows.
Although slim, there was a chance I could shift my way over to the doors and open them before the driver caught on. But even if she wasn't keeping a near-constant watch on me through the rearview mirror, what was I supposed to do once I got the doors open? Throw myself out into the oncoming traffic?
I didn't relish that idea. It seemed a more likely way to get killed than to stay put.
Even if I timed everything right so I opened the doors while the van waited at a red light, I'd likely land right in front of the next vehicle, unable to get to my feet quickly with my wrists and ankles bound. Again that seemed risky.
But wait. Maybe I didn't need to throw myself out of the vehicle. If I could get the doors open maybe I could signal to the driver behind us, if there was one. Surely the sight of a gagged and handcuffed woman in the back of a van would be enough to get them to call the police and take note of the van's license plate number.
At the moment, that seemed like my best option.
Biting down hard on my gag to stifle any cries of pain, I eased myself into a sitting position. Once I'd accomplished that, I sat still for a few seconds, breathing deeply through my nose as the aggravated pain in my neck and back slowly abated.
With my bound hands helping me to scoot along, I then shifted inch by inch toward the back doors. It didn't take long for me to get within reach of the doors, but then I had a decision to make. I wasn't sure if I should try to open the doors with my feet or shift around and use my hands.
My feet, I decided. I didn't like the idea of opening the doors with my back to them while barely balancing on my sore knees. Most likely I'd end up toppling out onto the street.
I couldn't turn my aching neck far enough to check if the driver was watching me so I simply hoped that she wasn't. Leaning back against my hands, I raised my feet up in the air and nudged one of the door handles with the toe of my shoe.
The van swerved with a squeal of tires. I slammed over onto my side with a shock of pain so intense that I cried out against my gag. The driver hit the brakes hard and my head smacked back against the floor of the van with another violent stab of pain radiating through my neck and back.
"What do you think you're doing?" My kidnapper loomed over me, her face blurred by my tears.
Even if I hadn't had a gag in my mouth, I wouldn't have been able to respond to her question. I could barely breathe through the terrible pain, and the only sound I managed was a pathetic whimper.
Unmoved by my extreme discomfort, the woman grabbed me under the arms and dragged me toward the front of the vehicle. My pain intensified to the point where I thought I might pass out. But I didn't.
When I was sitting with my back almost up against the console between the two front seats, she dropped me with a thud and climbed back into the driver's seat.
"You don't want to try anything like that again," she said as she restarted the van's engine. "Trust me."
As she set the van into motion again, I closed my eyes, tears rolling down my cheeks and pooling above my gag before getting absorbed by the fabric. I could hardly move my neck at all now, but the pain eased slightly as I held still and my tears stopped flowing.
I couldn't give in to the pain and fear. I had to think, to find a way to get out of this horrible predicament.
If only I knew why the woman had grabbed me. The fact that I'd seen her at the funeral made me strongly suspect that my abduction was somehow related to my investigation into Mr. Major's death. But who this woman was and why she would have felt threatened by my sleuthing, I didn't know. I couldn't even figure out how she could have been aware that I was looking into the matter.
But maybe the reason why she'd grabbed me didn't really matter. Whatever the reason for my abduction, I needed to escape.
Breaking my way out of the moving van or signaling for help no longer seemed like much of an option. The vehicle would have to stop at some point, though. My hands weren't of much use at the moment, but my legs could still kick. If I could manage to incapacitate my abductor once she had the back doors open, maybe I could buy myself enough time to get away.
Okay, so I wouldn't be able to run with my ankles cuffed together, but I could shuffle. And if there were pedestrians or passing cars nearby, maybe I could get someone's attention. Perhaps it wasn't the best plan in the world, but it was the only one I had.
Sitting there in the van, waiting as the dark-haired woman took me farther and farther away from home, I had a hard time not letting despair take hold of me. I drew my legs up closer to me and wished I could somehow get a message to Detective Salnikova or JT.
Or anyone.
But it was pointless to wish for that. My phone was all the way back at my apartment building, abandoned on the pavement.
I thought of JT and pictured his face with his eyes the color of sunlit root beer, but that only brought a terrible ache to my chest, one intense enough to rival the pain in my neck. I couldn't think about not seeing him again, couldn't even contemplate the possibility.
I had to focus. I had to be ready to make my move, whenever I might have the chance, no matter how desperate it might be.
The woman hadn't spoken since her threat after catching me at the back doors and I wasn't sure if that was good or bad. In some ways I wanted more information from her, but in other ways I didn't. If I didn't know how bad my situation truly was, I could still hold out some hope. And hope was all the fuel I had.
After what seemed like well over an hour, the van made a final turn and came to a stop, idling in place. By then, the light outside had faded and through the tinted back windows I could see a streetlamp glowing in the distance. I wondered why we had paused, but as soon as the question formed in my head I knew the answer.
A familiar rumbling sound came from outside, lasting several seconds. As the rumbling ceased, the van eased forward and the already dim light coming in through the tinted windows faded even more. The van stopped and the rumbling started up again. We'd entered a garage.
From my spot behind the front seat, I watched the garage door lower behind the van, knowing that the closing door lessened my chances for escape. Behind me, my abductor climbed out of the van and slammed the driver's door shut. Less than five seconds later, the back doors opened. The muscular woman climbed into the van and advanced toward me. I tried to shrink away from her, but I had nowhere to go.
She grabbed me by the arm and pulled, dragging me toward the open doors and out of the vehicle. I nearly toppled over when my bound feet hit the garage floor, but I leaned one shoulder against the van and found my balance.
As soon as I wasn't in any danger of falling, my eyes darted about, taking in my surroundings as quickly as possible. I needed to find an escape route so I'd know which way to flee when I made my move. There weren't many options available. Only one, actually—a side door to my left. Its dead bolt was in the locked position, but it was a more likely escape route than the large sealed door at the rear of the van.
My kidnapper slammed the van's door shut and grabbed my elbow, dragging me toward the side door of the garage. I decided to wait until she unlocked and opened it for me before making my move. That way I wouldn't have to fiddle with the dead bolt with my hands cuffed behind my back.
One big hand still gripping my upper arm, the woman turned the dead bolt and opened the door. I was ready to wrench myself out of her grasp, to drive my shoulder into her before attempting to flee. But as the door opened, I realized that my way to potential freedom wasn't clear.
Another woman stood on the other side of the door, illuminated by the yellow glow of an overhead light. Her dead-straight, dyed red hair hung to her shoulders and her blue eyes held no warmth. As soon as I saw her, I knew for certain that my abduction was related to Archibald Major's murder case.
I knew that because the woman standing before me was Mr. Major's killer.
Chapter Twenty-Five
"NICE JOB, BERNICE. Stash her in the basement for now," Frances Barlow said before leading the way along a narrow concrete path to the back of a house.
The burly woman maintained her grip on my upper arm and dragged me along behind Frances. While I now knew that my abduction had to do with my investigation into Archibald Major's murder, I didn't know why Andrea and Kevin's half sister wanted me out of the way. How could she have known that I was a threat, that I had information linking her to one or both of the murders?
Seconds later, I had an answer to that question. Although darkness had fallen during my trip in the van, a porch light illuminated enough of the house for me to notice something of interest—purple siding.
I knew my location. I was barely a stone's throw from where I'd spoken to Janet the day before, from where her friend's dog had found Kevin's body. Either Frances or Bernice—presumably her daughter—must have been the one watching me and Janet with binoculars. Plus Bernice could have seen me at the graveyard, as I'd seen her. So they'd known about my snooping and maybe thought I was getting too close to the truth for comfort. Or maybe they just didn't want to give me the chance to get close to the truth. Either way, I'd obviously worried Frances.
All the bits and pieces of information in my head were like various parts of a musical score, coming together to create a symphony. More importantly, knowing my location gave me more hope. If I could only get away from my captors, I knew where I could go for help.
Unfortunately, getting away from Frances and her daughter would be no small feat. I considered trying to make a break for it right then and there, but I could only shuffle along, and it was two against one. Plus Bernice's grip on my arm was so tight that I had no chance of wrenching out of it.
As she pulled me toward a dark stairwell, I tried my best not to panic. She put an arm around my waist and half carried me down a short set of stairs and through a door held open by her mother. She continued to half carry and half drag me along a short hallway.
My eyes darted about, trying to assess my new environment, but there wasn't much to see in the dim yellow light of the hall. As we passed by an open door I caught a glimpse of a room with shelves holding an array of trophies. Pictures hanging crookedly on the walls in the hallway showed a younger version of Bernice accepting the prizes while dressed in what looked like wrestling gear.
A championship wrestler. No wonder she'd had no trouble overpowering me. Not that I would have been much of an opponent for anyone with decent-sized muscles.
The thought had barely passed through my head when she shoved me into a dark room. One of my captors flicked a light switch, and I blinked as the room became bright. As my eyes adjusted to the light, I took in as much of my surroundings as I could in those first few seconds. I stood in a small powder room in desperate need of renovations. Discoloration marred the white porcelain of the pedestal sink, and the scarred beige linoleum flooring featured a large dark stain.
Most importantly, I noted that the room had no window.
"Get rid of the gag," Frances ordered.
I couldn't help but flinch when Bernice stepped closer and reached toward my head. She untied the knot in the bandana and yanked the cloth from my mouth.
After coughing a few times I sucked in a deep breath and glared at Frances. "What do you want with me?"
A cold smile turned up the corners of her mouth. "I want you out of the way, of course." Her chilling smile disappeared. "But first you'll tell me how much you know."
Despite the aching and trembling of my body, I made sure my voice came out strong and steady. "I know you killed your father."
Her eyes flashed with a disturbing glint, but I pressed on, a flicker of anger now fueling my words.
"I know you've been involved in community theater for years." I recalled the picture I'd studied earlier that day, and the familiarity of one of the people shown in it. I'd missed that detail during my first examination of the photo, but it had ultimately revealed the truth to me. "You disguised yourself with a wig and glasses and insinuated yourself into Mr. Major's life as his companion and housekeeper, Marjorie."
Bernice spoke up, a note of pride in her voice. "My mom's great with disguises."
A murderer and her proud daughter. Great.
I'd have felt more comfortable passing the time with snooty Elena. But since I didn't have a choice about my company at the moment, I needed to focus on staying alive.
"Mr. Major never caught on?" I asked, hoping to keep them talking so they wouldn't harm me.
"Of course he never caught on." Frances nearly spat the words out. "He'd never so much as seen a picture of me. Never wanted anything to do with me. The only reason I bothered with the disguise was so the police wouldn't put two and two together if they ever decided to question me."
"Which they did, right?" I said. "Once they knew you stood to inherit under Major's will."
Bernice made a sound halfway between a snort and a chuckle. "They asked their questions and they went on their way. Mom had them completely fooled. They have no idea that she and Marjorie are the same person."
The pride she had for her murderous mother turned my stomach, but I didn't have time to focus on my nausea.
Frances glared at me. "Or do they? How much have you told them?"
"Everything. They're probably on their way here as we speak."
I hoped my falsehood would scare them into fleeing, leaving me to find my way to safety. That hope was dashed a half second later.
"You're a terrible liar," Frances said, a cold, satisfied smile curling her lips. "Perhaps you should have tried acting classes."
Her daughter let out another snorting chuckle. "Too bad you won't have the chance."
An icy trickle of fear worked its way down my spine. "But why kill your father? And why go to such lengths?"
Once again, Frances's eyes flashed with a chilling, unnerving glint. "Don't you get it? My whole life, he denied that I was his daughter. I wanted what was mine, what I deserved."
"You mean money?" I guessed, willing to bet she didn't mean a father's love.
"Of course I mean money." She scowled. "But I couldn't just kill him outright. That wouldn't do me any good."
"Because you weren't mentioned in his previous will." I thought I was catching on.
"Damn right she wasn't," her daughter put in. "The bastard."
"So you needed him to write a new will. But how did you get him to do that?"
Frances's scowl transformed back into a creepy, cold smile. "I doted on the old crank. Agreed with everything he said and got him to like me. Then I told him my sad story about how my biological father had abandoned me, denied that I even existed, leaving me and my mother to fend for ourselves. It took time, but eventually I ignited a tiny spark of guilt inside of him. One day he confessed to what he'd done, to abandoning the daughter that resulted from his affair. I encouraged him to make things right by remembering her in his will." Her unsettling smile broadened. "And he did. I waited a little longer after that and then put the rest of my plan in motion."
"By putting Brugmansia in his flask."
Frances shrugged one shoulder. "It was so convenient, with the plant growing right there on the patio. He couldn't have made it much easier for me if he'd tried."
I shivered at her complete lack of remorse. "What about Kevin?"
"Troublemaker," Bernice muttered. "I took care of him."
Frances patted her on the arm. "That you did. And you did a good job too."
Bernice beamed at her mother's praise and I shivered again.
"Kevin decided to check out his half sister," Frances explained. "Tracked me down and followed me from my house to Bernice's house here. The real problem was that he wasn't quite as stupid as you might think."
"He recognized you?"
"I caught him peering into my car when it was parked out front of Bernice's house. I'd left my wig on the backseat and when he got a good look at me, he put two and two together."
"So you killed him."
"I killed him," Bernice corrected me. "Right here. And I dumped him in the woods later that same night. Which is what I'll do with you too, you nosy busybody."
I would have bristled at her name calling if her threat hadn't muted me with fear. The stain on the floor beneath my feet was a bloodstain. My stomach clenched.
"Or close enough," Frances amended her daughter's statement. She took a step backward, out into the hallway. "Come along, Bernice. We need to plan exactly what we want to do with her. I'm thinking the river might be better than the woods this time."
Bernice speared me with her dark eyes once more before stepping out of the powder room and shutting the door. I listened as something heavy scraped across the floor out in the hallway. The powder room door rattled in its frame as whatever my captors had pushed or dragged bumped up against it.
"That should hold her," Frances said, her voice only slightly muffled by the closed door.
Two sets of footsteps moved along the hall away from me, fading completely a few seconds later.
Once left in silence, I sagged against the stained sink, my head swimming with all the new information. Frances's story was so similar to Ernest's. Both had harbored anger and resentment toward their biological father. But there was a vital difference between the two half siblings—Frances was a killer and Ernest wasn't.
But I couldn't waste any time thinking about the differences between Ernest and Frances, or all the details I now knew about the two murders. I needed to save myself.
Quickly.
Chapter Twenty-Six
I DIDN'T KNOW how much time I had before Frances and Bernice decided how to get rid of me, and I didn't even want to imagine what their decision might be. Instead of allowing myself to dwell on such frightening thoughts, I focused on my most immediate problem—how to get out of the powder room.
After shuffling over to the door, I turned my back to it so I could reach the doorknob with my hands. I turned it and gave a push with my shoulder, but as I'd expected, the door didn't budge more than half an inch. I leaned more of my weight into the door but with the same result. Whatever Bernice and Frances had pushed up against the door, it was far too heavy for me to shift it.
I blew out a breath of frustration. At least they hadn't replaced my gag before locking me in the room. I considered screaming for help, but quickly tossed that idea aside. Considering that I was locked in a windowless basement room, the only people likely to hear my screams were Frances and Bernice. The last thing I wanted to do was spur them into getting rid of me sooner than planned.
There had to be another way out of the powder room. I considered breaking through the drywall as I'd done once in the past to escape a raging fire, but that would be too noisy and this time I had nothing but my bare hands with which to break through the wall. The lid from the toilet tank wasn't an option since it was missing along with the toilet seat.
Carefully, so as not to hurt my neck too much, I tipped my head back and examined the ceiling. I almost smiled when I realized that it was a drop ceiling. Maybe that was my way out.
I bit down on my lower lip as I considered my options. If I could climb up on the pedestal sink, I'd be able to reach the ceiling quite easily. But not with my hands behind my back. I lowered myself to the floor—doing my best to avoid the gruesome dark stain—and tried to wiggle and wriggle my hands down behind my legs to my feet. It took a lot of stretching and fierce protesting from my neck, but, despite nearly popping my shoulder out at one point, I finally got my cuffed hands to the front of my body.
It took me a few seconds to catch my breath after that struggle, but then I pushed myself to my feet and continued to put my plan into action. Holding on to the edge of the sink, I jumped up onto the rim of the toilet and from there crawled and shifted until I had my feet in the sink. Carefully, I raised myself into a standing position, my bound hands pressing against the wall to keep me steady.
Once I felt stable enough to let go of the wall, I raised my hands and pushed at the ceiling panel above my head. As I'd hoped it would, it lifted easily. I nudged it up and to the side so it slid over top of its neighbor, leaving a large square hole in the ceiling. The hole extended over the wall and into the hallway, providing an escape route. I gave the metal framework above my head an experimental tug. There was no way it would hold my weight. I'd have to climb straight from the sink to the top of the wall. I wouldn't have a lot of room to squeeze through at the top, but at least the joists ran perpendicular to the wall, giving me a chance to escape that way.
I took two or three seconds to simply breathe and settle my nerves, but I didn't waste any more time than that. The wall was about a foot away from the sink so I leaned forward until I could grab the top of it. Then I jumped up off the sink toward the wall. I bashed my head on the metal framework and my neck let me know of its displeasure, but I managed to hook one elbow up over the top of the wall.
I hung there for a moment, wondering if I'd fall to the floor and have to try again. Maybe my arms weren't strong enough to pull me up. They were already burning. I made a mental note to work out more in the future so I'd have stronger arms to help me out of my next predicament.
Who are you kidding? I said to myself. Work out more? You don't work out at all.
Okay, so that was true, but it wasn't the right time to dwell on my lack of discipline in the exercise department. Determined not to fall, I planted my feet against the wall and hoisted myself up far enough to hook my other elbow over the top. Then I gritted my teeth and heaved myself up farther, working my head and torso through the gap at the top of the wall.
It wasn't a comfortable position and I never would have scored any points for style, but at least I made it far enough to hook one of my knees up and over the edge. I almost panicked for a second when I thought I was stuck between two joists, but with a few wiggles and a lot of determination, I unwedged myself and got both my legs over the wall so I was looking back into the powder room. I edged back an inch or two and gravity did the rest.
My hands scraped against the top of the wall as I fell and I stifled a cry as all the injured parts of my body protested, but I landed on my feet, relatively unscathed. More importantly, I'd escaped.
Well, sort of.
At least I was out of the powder room. Fluorescent lights glowed overhead, allowing me to see that I now stood in the hallway, near the door to the powder room, which had been blocked by a sturdy wooden chest of drawers. I heard a low rumble of voices coming from somewhere above me and knew my captors weren't far away. All the more reason to get moving.
I squeezed past the chest of drawers and shuffled along the hallway, moving as quickly as I could. When I reached the door leading to the outside, the knob turned easily in my hands. I opened the door carefully, hoping it wouldn't creak or groan too loudly.
It didn't.
After stumbling out into the stairwell, I shut the door behind me, making as little noise as possible. Darkness filled the stairwell with deep shadows, but I put my cuffed hands out in front of me and shuffled forward until my toes hit the first step. I twisted around and sat down, bumping myself up one step at a time until I reached the top. I teetered as I straightened up again, but I didn't fall, and as soon as I regained my balance I peered into the shadows around me.
A meager amount of light reached me from the lights above the back porch and the side door of the garage. I didn't want to pass through the light to escape into the back alley, so I followed a concrete path around the side of the house.
Moving wasn't easy with my ankles cuffed together and I certainly couldn't go at a fast pace, but I got into a good shuffling rhythm and reached the front gate in less than half a minute. I glanced over my shoulder, my heart booming in my chest. Nobody was behind me.
Knowing that could change any second, I unlatched the gate and passed through it, shutting it behind me so my escape route wouldn't be immediately obvious once my captors discovered that I'd slipped out of my powder room prison. From the gate, I cut across the lawn and headed for the street. I probably looked like a stiff penguin as I hurried across the grass, but that was the least of my worries.
I focused on the house across the street, the one Janet had pointed out to me as belonging to her friend Linnea. If I could get over there and knock on the door, I could ask Linnea to call for help. I hopped down the curb and hustled across the street, hopping back up to the grass when I reached the other side.
Still shuffling, I cast a glance over my shoulder. A shadow moved by the gate I'd escaped through, sending my already racing heart into overdrive.
No, no, no!
Bernice was hot on my trail.
"Good gracious, what's happened to you?"
A woman stood on Linnea's front porch, the door open behind her. She wore a coat, and a toy poodle strained at the end of a leash. She stared down at me with a mixture of surprise and concern.
"Please, call the police!" I shouted to her in desperation.
I shot another glance over my shoulder. Bernice hesitated on her front lawn. She'd spotted her neighbor.
"Please!" I beseeched Linnea.
She whipped a phone out of her pocket. "Of course." She peered across the street as she punched numbers into her phone. "Is that man after you?"
In the dark it wasn't surprising that she'd mistaken Bernice's muscular form as that of a man. I didn't waste time correcting her.
"Yes!"
As Linnea put her phone to her ear, Bernice spun around and hightailed it back through the gate, disappearing into the shadows. I sagged against the porch steps with relief.
No doubt Bernice would alert her mom that the jig was up and they'd take off into the night. But at the moment I didn't care about them getting away. All I cared about right then was the fact that I was free and my kidnapper had given up the chase.
THE FIRST POLICE car arrived in less than two minutes. After some quick explaining on my part and the arrival of another two police vehicles, one officer helped me out of my cuffs while several other officers converged on the house across the street.
I knew they most likely wouldn't find Frances and Bernice there. The two women had probably hopped in Bernice's van and made a dash for the highway. But I'd given the police Bernice's name and they'd likely be able to look up her van's license plate, so all was not lost.
"Let's get the poor girl in the house," Linnea said to the officer at my side. "She's shivering."
She was right, I realized. I hadn't noticed before but my whole body trembled with a mixture of cold, relief, and exhaustion. The officer took my arm and helped me up the stairs and into the house.
He offered to call an ambulance, but I declined. Although sore and bruised, there was nothing seriously wrong with me. I was, however, glad to get inside and escape the chilly night air.
"You're Linnea, right?" I checked, just to be sure, after the officer had left me alone with the gray-haired woman. When she looked puzzled, I added, "I met Janet the other day. She told me that your dog found the body in the woods."
Linnea's face relaxed and she smiled. "That's right."
The toy poodle sniffed at my feet as I sat down on the couch.
I scratched him on the head. "And you must be Toby."
The dog sat down and leaned against my legs, apparently enjoying the attention.
"Sorry about your kidneys, little guy."
Linnea grabbed a blanket from the back of a nearby chair and wrapped it around my shoulders. "Let me get you a cup of hot tea."
"Thank you," I said gratefully. I pulled the blanket more tightly around me and twisted in my seat so I could look out the front window.
Aside from a couple of officers and a few onlookers milling about in the blue and red light cast by the police vehicles, there wasn't much to see. I kept my eyes on the scene outside anyway, unable to tear my eyes away. Linnea returned a few minutes later with a steaming cup of orange pekoe tea for me. After thanking her again, I sipped at the tea, grateful for its comforting warmth.
I soon had to turn my attention away from the window when the police officer who'd helped me into the house requested my statement. Holding tightly to my cooling cup of tea, I supplied him with all the details I could remember, and also filled him in on the two murders. He assured me that Detective Salnikova would be notified and went outside to join his colleagues.
Although exhaustion had infiltrated every bone in my body, I was too wired to close my eyes and relax. I considered borrowing Linnea's phone to call JT, but I decided to wait. As much as I wanted to talk to him, I didn't want to worry him. It would be better to tell him the entire story once it was all over, once the police had Bernice and Frances in custody. Then I could assure him that I was fine, that everything was fine.
After a time, I got up from my spot on the couch and paced Linnea's living room, the blanket still tucked around my shoulders. I only stopped moving when someone entered the house through the front door, left unlocked to allow the officers easy access.
A shadow moved in the foyer and Detective Salnikova stepped into the living room.
"Midori," she greeted. "I hear you've had quite the evening."
"Is there any news?" I asked, jumping right to the question clamoring in my head.
She nodded, and something close to a smile touched her lips. "Frances and Bernice Barlow were pulled over on the highway a few minutes ago. They're both in custody."
I sank back down onto the couch, my legs no longer willing to support me.
It was over.
Chapter Twenty-Seven
AFTER I'D SPENT a few more minutes with Salnikova, Linnea loaned me her phone so I could get in touch with JT. I gave him the shortest version possible over the phone, but that was still enough to freak him out. He calmed down—somewhat—after I repeatedly assured him that I was fine, but it took all my powers of persuasion to keep him from rushing out of his house and coming to pick me up. Salnikova had already told me that an officer would drive me home and I didn't want to wait the time it would take JT to get to Surrey before heading back into Vancouver.
"I'll meet you at your place then," he finally conceded. "I want to see with my own eyes that you're okay."
With a welling of warmth in my chest, I agreed to that arrangement and hung up. I waited for Salnikova to finish speaking with a uniformed officer on the front porch before leaving the house to join her in the chilly night air.
"All set to go home?" the detective asked me.
"Yes," I said, rubbing my arms to ward off the cold. "Except I wanted to ask you about Jordan. What will happen to him now that you have the real killers in custody?"
"He'll be released," Salnikova assured me. "It might take some time to get everything sorted out, but I'm sure he'll be home before long."
That was a relief. Yes, Jordan would still have to deal with the aftermath of the murders and he'd have to adjust to his mom's new relationship, but at least he wouldn't be behind bars. Plus I knew he was a resilient kid. He'd be all right in time.
After I ducked back inside to thank Linnea for her help, Salnikova led me down the front steps to a police cruiser parked at the curb. She introduced me to a female officer by the name of Jenkins and then left me to join two other detectives across the street.
The presence of the numerous emergency vehicles and the recent arrival of crime scene technicians had drawn a good-sized crowd of curious onlookers, but I had no interest in joining them in their gawking. For a change I simply wanted to leave the professionals to their work and go home.
Officer Jenkins held open the door to her cruiser and I climbed into the vehicle. Although we exchanged a few words at the beginning of our journey into the city, we soon lapsed into silence, my head resting against the back of my seat and my eyes drifting closed. I only opened them again when Officer Jenkins parked the vehicle.
She offered to escort me across the street to my building, but I told her it wasn't necessary. From inside the vehicle, I could see JT pacing back and forth in front of the entryway. Tears prickled at my eyes but I also smiled. I thanked Officer Jenkins for the ride and climbed out of the cruiser.
JT stopped his pacing when I shut the car door. I checked both ways and darted across the street. An expression of immense relief crossed his face as I approached.
When he opened his arms, I stepped right into them. My neck ached and exhaustion thrummed through my entire body, but in that moment I didn't care. All that mattered to me right then was the fact that I was alive, safe, and with my best friend.
THE MORNING OF the following Friday found me in my kitchen, carefully placing one cake layer over another, a spreading of chocolate icing between them. Once the top layer was in its proper place, I stepped back to anxiously examine my handiwork. So far, so good. The cake looked fairly even and no large chunks had crumbled away.
I wasn't much of a baker and usually figured it was best to leave the creation of tasty treats to those much more skilled. But that night I'd be at JT's house with his family to watch the first episode of Absolute Zero and I wanted to take something to the party. Despite my lack of confidence in the baking department, I'd decided to tackle the chocolate cake JT had requested. I could have picked up a cake from a bakery, but I wanted to put more effort into my offering.
Still, the task was a daunting one for me, even with a recipe and detailed advice provided by Bronwyn, an avid baker. Only time would tell if my decision to take on the challenge was a good one or a terrible one. If the cake turned out to be a complete disaster and tasted terrible, I could always make a quick run to the bakery and pick up a replacement, but I didn't want to have to do that.
Drawing in a deep breath, I grabbed the bowl of chocolate frosting I'd prepared minutes earlier and began the process of icing the top and sides of the cake. My movements were cautious and a few crumbs came loose from the cake here and there, but I managed to get the entire cake covered without running out of frosting. I turned the cake on its plate, inspecting it from every angle, and was relieved to see that it didn't look too bad. It was a far cry from a professional job, but not bad for a beginner.
After dipping a cake crumb in the bit of frosting left in the bowl, I tasted it and smiled. It was absolutely delicious. JT would be happy, and knowing that made me happy.
Or happier, to be more precise. The week had already given me plenty of reasons to be cheerful. I'd helped solve two murders, Frances and Bernice were safely behind bars, and Jordan was free. Another plus was that my conversation with Beaufort outside my apartment building had led to further investigation of the doctor and—in the face of increased pressure from the police—one of Beaufort's friends had admitted to giving him a false alibi.
On top of that, Beaufort's vehicle was caught on video by a traffic camera only a few blocks away from Major's residence within minutes of the call I'd placed to Detective Salnikova on the night of the break-in. Beaufort's credibility had further been damaged when the police discovered that he was in possession of jewelry stolen from the charity benefit and other past events. It seemed the doctor was something of a kleptomaniac.
As it turned out, my latest theory was indeed correct. Archibald Major had become aware of Beaufort's thefts at the charity benefit and had begun his attempts to get Beaufort to resign from the PGP's executive committee soon after.
Although Major wasn't around to see it, his goal had finally been achieved. Since Beaufort had been charged with criminal offenses, he'd resigned from the executive committee, and his medical career was likely in jeopardy as well. As far as I was concerned, Beaufort had brought his current situation upon himself and I was glad the police finally had some evidence to back up my witness statement. More than that, I was relieved that the truth about the theft of Elena's brooch had come to light and Bronwyn's name had been cleared. Janine still wasn't happy with either of us, but Bronwyn's place in the orchestra was secure, and that was what mattered most.
As for myself, I'd needed painkillers and an ice pack for a couple of days after my run-in with Bernice and her mother, but my injuries had now healed and I was back to playing my violin without any pain or stiffness.
I hadn't even needed to replace my phone. After convincing JT that I really was fine, I'd found the device right where I'd last seen it—near the bushes by the front of my building. I wasn't even upset about Elena and the gray boots anymore. While out shopping the other day I'd come across another pair of tall gray boots and had fallen more deeply in love than I had with the other pair. These boots had decorative stitching that the others had lacked, and they provided the perfect final touch to the outfit I was wearing to the party that night. Elena was welcome to the other pair.
Knowing that time was getting on, I carefully set the cake in a box and hooked the strap of my quilted tote bag over my shoulder. I eyed the box and then my violin, wondering how I'd manage to juggle both while walking and riding the bus. In the end, I decided to leave my violin behind and use the spare one I kept at JT's place while teaching that afternoon.
Locking up my apartment behind me, I set off for JT's house, excited for the party and the premiere of Absolute Zero. There were a couple of dicey moments on the way there when the cake shifted perilously inside the box, but when I arrived at my destination, the cake was still intact. Relieved, I tucked it safely away in JT's refrigerator and turned my focus to my students.
The hours seemed to pass slowly that afternoon, but eventually I was done working and the other party guests began to arrive. As JT had requested, the party would be small—just a couple of friends, JT's parents, and his aunt and cousin.
When I set the cake out on the table along with the munchies and goodies the other guests had brought, JT came up behind me and reached a finger toward the chocolate frosting.
"Hey!" I swatted his hand away. "You have to wait."
"I've been waiting all afternoon," he said. "Every time I opened the fridge, that thing was tempting me. I say we dig in."
Finnegan, hovering near the food-laden table with hopeful eyes, gave an enthusiastic bark.
"See? Finn agrees, even though he can't have any."
That got another bark out of Finnegan.
"Who wants cake?" JT called out to the other guests, all of whom were gathered down the hall in the living room.
"We haven't even ordered the pizza yet," I protested, although not without a hint of a smile.
JT sank the knife through the chocolate frosting. "Cake, then pizza, then more cake. It's a party, isn't it?"
My smile grew. I couldn't argue with that plan.
All the guests converged on the kitchen and soon each person had a plate of cake in hand. Although I wanted everyone to enjoy my chocolate creation, I was most anxious about JT's reaction.
As everyone set about eating, I kept my eyes on my best friend, watching and waiting. I didn't have to wait long. After he swallowed his first generous forkful of cake, a grin lit up his face.
"Wow. This is delicious, Dori."
The others agreed and I happily started in on my own slice.
"You know," JT said after he'd finished off his cake less than a minute later, "now that I know you can bake, I'll be bugging you to make me cake all the time."
"I don't mind," I said, a big smile on my face.
And I didn't. Not at all.
Acknowledgments
THIS BOOK AND series never could have become a reality without the invaluable support and assistance of my agent, Jessica Faust, and my editor at HarperCollins, Rebecca Lucash. Thank you! I'm also forever grateful to Nicole Bates, Sarah Blair, and Jessica Dainty Johns for reading early drafts of the manuscript and providing me with feedback, support, and encouragement.
About the Author
SARAH FOX was born and raised in Vancouver, British Columbia, where she developed a love for mysteries at a young age. When not writing novels or working as a legal writer, she is often reading her way through a stack of books or spending time outdoors with her English springer spaniel. Visit her Web site at www.authorsarahfox.com
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Also by Sarah Fox
Dead Ringer
Copyright
This is a work of fiction. Names, characters, places, and incidents are products of the author's imagination or are used fictitiously and are not to be construed as real. Any resemblance to actual events, locales, organizations, or persons, living or dead, is entirely coincidental.
DEATH IN A MAJOR. Copyright © 2016 by Sarah Fox.
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.
EPub Edition JANUARY 2016 ISBN: 9780062413017
Print Edition ISBN: 9780062413048
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1. Dedication
2. Contents
3. Chapter One
4. Chapter Two
5. Chapter Three
6. Chapter Four
7. Chapter Five
8. Chapter Six
9. Chapter Seven
10. Chapter Eight
11. Chapter Nine
12. Chapter Ten
13. Chapter Eleven
14. Chapter Twelve
15. Chapter Thirteen
16. Chapter Fourteen
17. Chapter Fifteen
18. Chapter Sixteen
19. Chapter Seventeen
20. Chapter Eighteen
21. Chapter Nineteen
22. Chapter Twenty
23. Chapter Twenty-One
24. Chapter Twenty-Two
25. Chapter Twenty-Three
26. Chapter Twenty-Four
27. Chapter Twenty-Five
28. Chapter Twenty-Six
29. Chapter Twenty-Seven
30. Acknowledgments
31. About the Author
32. Also by Sarah Fox
33. Copyright
34. About the Publisher
1. Cover
2. Contents
3. Startup Page
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Exploring Alberta's Heritage Park, Calgary
Situated in the heart of southwest Calgary, Heritage Park is one of North America's largest and most popular living history museums. Chronicling the epic 100 year history that made Western Canada what it is today, this unique park leads visitors on a captivating journey through time, with the help of authentic exhibits, lively costumed interpreters, numerous artefacts, demonstrations, activities and rides, along with fantastic shopping and dining options. Whether you're eager to learn more about the history of the early West or simply in search of some good old-fashioned entertainment, this premier tourist destination is sure to delight the entire family.
Journey through western Canada's historic past
Heritage Park is home to a wealth of historical and architectural attractions that bring Western Canada's past to life in vivid detail. Many of the buildings are originals with authentic facades, having been relocated to the park from across the province and further adorned with classic furnishings and genuine artefacts. There are four distinct areas to explore, each one spanning a key period in the province's history from the 1860's up to the 1950's. These chronicle Calgary's era of early settlement and agriculture, through to the development of the railway system, right up to the advent of the automobile and the birth of modern Canada.
Historical attractions and exhibits are brought to life through numerous inventive means, providing a real feast for the senses and a truly immersive experience. Park interpreters sport authentic period costumes and put on a variety of real life demonstrations. Watch as they tend farm animals, churn butter, carry out repairs on antique vehicles or stage live musical performances at the bandstand.
Embark on a self-guided walking tour, taking in the most popular locations and sights. The park boasts well over 100 exhibits, and here are just some of the highlights you won't want to miss.
First Nations Encampment
The earliest section of the park is dedicated to the heritage of the semi-nomadic plains Indians that first settled in southern Alberta. Take the opportunity to tour inside a wonderful tipi, hear age-old stories about First Nations life, try your hand at making bannock or beaded crafts, and even take part in a ceremonial drumming circle.
Pre-railway Settlement
From the mid-1800's, whiskey traders and early ramshackle settlements started to appear in the area, often accompanied by the ruthless exploitation of First Nations people. In 1873, the first police force was set up to tame the lawlessness and encourage more permanent settlement. Visit the historical building where the Berry Creek North West Mounted Police set up home during this era. Most early inhabitants during this time were farmers or hunters who joined forces as homesteaders to gain legal title to the lands. Explore some of the beautifully preserved heritage buildings of the time, including private homesteads, a farm implements shed and an authentic miner's cabin.
Prairie Railway Town
The completion of the Canadian Pacific Railway 1885 precipitated a huge increase in immigration at the turn of the century. Heritage Park features a representation of a typical prairie railway town dating from around 1910. Learn about the antique steam train system that spurred this development and explore several original train stations. Tour renovated historical buildings, ranging in style from elegant heritage homes, such as Prince House, to the more basic sod and log shacks. Also keep an eye out for the Dingman well that first struck oil in 1914, igniting Alberta's oil boom, as well as the Wainwright hotel and Little Synagogue, along with several old-fashioned entertainment venues.
Gasoline Alley Museum and Heritage Town Square
The 1920's to 50's era was a time of rapid development in Western Canada and one of the park's best-known attractions, Gasoline Alley, offers the chance to learn more about the technology that was so instrumental in spurring this far-reaching change. The museum houses one of the world's largest collections of rare vintage vehicles and artefacts relating to the auto industry. The gleaming colours of the vehicles are complimented by hordes of memorabilia harking back to the golden era of the automobile, along with interactive exhibits, crafts and story-telling that brings this momentous historical period to life right before your eyes.
Outside the museum, be sure to spend time exploring the 1930's Heritage Square. Open year-round, this town square features several charming shops and numerous enticing eateries, along with an interpretive microbrewery where you can learn about the craft brew process from the professionals. Period appropriate shops are dotted throughout the park, including rare antique stores and classic toy shops, where you can pick up a unique memento of your visit.
The park offers plenty of fun-filled ways to get around the sites and learn more about the various transportation revolutions of the period. A streetcar service, based on Calgary's former system, shuttles visitors to and from the visitor entrance. Once in the park, enjoy a relaxing horse-drawn wagon ride to get a superb overview of all the main sites.
Ride a real steam train around the entire park, passing authentic old train stations that have been relocated here from across Alberta. Or take to the water on Glenmore Reservoir in an old paddle wheeler, canoe or sailboat. The paddle steamer, S.S. Moyie, is a replica of North America's oldest steamship, one of many that serviced Western Canada's rivers and lakes from the 1860's and well into the 20th century.
The park's antique midway features numerous working historical amusement park rides. Take a spin on the old Ferris Wheel and marvel at the 1921 "Whip", the world's first thrill-based ride. Also not to be missed is the beautifully restored Bowness Carousel, dating from 1904, along with several old-fashioned games booths.
For a quick way to refuel, the park's authentic concessions offer plenty of history-inspired treats boasting flavours that have stood the test of time. Visit Alberta Bakery and savour the smell of freshly baked breads, tarts and pastries. Kids will love picking out old-fashioned penny candy from the grocery store or feasting on traditional candy floss at the amusement park. Enjoy a delicious cone from Vulcan ice cream parlour in a shady spot under a tree, or learn how to make old-fashioned ice cream for yourself with park staff.
For something more substantial, Heritage Park is home to several restaurants emphasising old Canadian classics with a modern twist. Stop by Club Café for a warming savoury soup or hearty all-day brunch. For a true taste of the West, head to Gunn's Dairy Barn – once a fully-functioning animal barn and now converted into a rustic dining area with a fun atmosphere. It offers up a classic menu of hamburgers, hot dogs, fries, sandwiches and salads, guaranteed to satisfy the whole family. If you're looking for a sit down meal with a difference, Selkirk Grille is a must. You'll be treated to an inspired menu bursting with fresh, organic, locally grown food that delights the taste buds. Standout options include the Prince Edward Island chorizo poached mussels for seafood fans, or the Alberta elk striploin for meat lovers. Pair your meal with a choice from the excellent wine list and enjoy in elegant surrounds accompanied with vintage décor, traditional music and impeccable service.
Top 10 restaurants in Calgary Read more
Experience WinSport, Canada Olympic Park during the summer Read more
Top 10 things to do in Calgary Read more
Self-Drive Family Adventure in Alberta
Banff and Lake Louise Winter Experience
Experience Calgary Stampede and The Rockies | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 457 |
Today, the Alabama Senate adjourned for spring recess with both Senate Bill 14 and Senate Bill 304 on the third reading calendar. The Senate will return for session on April 5, 2016. Consideration on both of these important bills is anticipated shortly after the Senate reconvenes in April. Over the next two weeks it is important that you contact your state Senator and politely urge him or her to support both SB 14 and SB 304. Please click the "Take Action" button below to contact your state Senator!
SB 14 is important legislation that would recognize a law-abiding gun owner's ability to possess a concealed handgun in a vehicle without first obtaining a government-issued permit. There are currently 25 states that already allow law-abiding gun owners to carry a loaded handgun in their vehicle for self-defense without government-mandated permitting and taxation. If an individual can lawfully own and possess a firearm, they should not have restrictions placed on their ability to exercise their fundamental right to self-defense outside of their home and in their vehicle.
SB 304 seeks to solve the significant disparity in concealed weapon license (CWL) fees between counties. SB 304 will accomplish this fix by allowing law-abiding individuals to apply for a permit in any county in the state.
Contrary to some claims, SB 304 is not the initial step in a scheme to eliminate the sheriffs' involvement in the CWL process. It was drafted and introduced in acknowledgement of one simple truth: citizens of Alabama should not be held to disparate standards in exchange for the ability to lawfully exercise the same fundamental right.
A CWL issued in Jefferson County ($7.50 a year) confers the exact same right across the state as a CWL issued in Baldwin County ($25.00 a year). Yet, over a 5-year period those rights cost approximately $85.00 more in Baldwin County. The permittee receives no additional benefit for this more expensive permit. As a result, the counties that charge excessive fees are simply generating revenue from a small group of law-abiding residents. The NRA believes that CWL fees should only reflect the actual cost of fulfilling all statutorily mandated responsibilities associated with the permitting process; any financial obligation in excess is a revenue generating tax. Your fundamental right to self-defense should not be held hostage in exchange for excessive taxes.
This legislation would not change existing background check requirements. Upon issuance of the CWL, the Sheriff of the county in which the permit is issued would be required to notify the Sheriff of the permittee's county of residence that a permit has been issued. Both the Sheriff of the issuing county, and the Sheriff of the county of residence would subsequently have authority to revoke the permit for any valid reason.
Once again, please click the "Take Action" button above to contact your state Senator in support of these critical pro-gun bills! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,000 |
\section{Introduction}
\label{sec-introduction}
The concept of \emph{domination} and its variations is one of the most active area of research in graph
theory because of its application in facility location problems, in problems involving finding a set of
representatives, in monitoring communication or electrical networks, and in various other areas of practical
applications (see \cite{Haynes1,Haynes2}). Over the years, many different variants of domination have been
introduced and studied in the literature. The concept of $(i,j)$-set is a very interesting and recent
variant of domination \cite{12set,YangWu}.
\subsection{Definitions}
\label{ssec-definitions}
For a natural number $m$, let $[m]$ denote the set $\{1,2,\ldots,m\}$.
Let $G=(V,E)$ be a graph. For $v \in V$, let $N_G(v)=\{u | uv \in E\}$ denote the open neighborhood of
$v$ and $N_G[v]=N_G(v) \cup \{v\}$ denote the closed neighborhood of $v$. The degree of a vertex
$v \in V$, denoted by $d_G(v)$, is the number of neighbors of $v$. Let $\Delta_G$ and $\delta_G$ denote the
maximum and minimum degree of $G$. (We will remove the subscript $G$ where it is obvious from the context).
Let $G[S]$ denote the subgraph induced by the vertex set $S$ on $G$.
A \emph{tree} is a connected graph which has no cycle. A tree is called a \emph{rooted tree} if one of
its vertices, say $r$, has been designated as the \emph{root}. The \emph{level} of a vertex is the number
of edges along the unique path between it and the root. A set $S \subseteq V$ of a graph $G=(V,E)$ is an
\emph{independent set} if no two vertices in $S$ are adjacent. If every pair of distinct vertices in
$K\subseteq V$ are adjacent in $G$, then $K$ is called a \emph{clique}. A graph $G$ is \emph{chordal} if
every cycle in $G$ of length at least four has a chord, that is, an edge between two non-consecutive
vertices of the cycle. A graph $G=(V,E)$ is called a \emph{split graph} if $V$ can be partitioned into
two sets, say $S$ and $K$, such that $S$ is an independent set and $K$ is a clique of $G$. Note that trees
and split graphs are chordal graphs. A \emph{claw} is basically a $K_{1,3}$, a complete bipartite graph
having one vertex in one partition and three vertices in the other partition. A vertex $u \in V$ is said
to be dominated by a vertex $ v \in V$ if $u \in N_G[v]$. A set $D \subseteq V$ is called a \emph{dominating set}
of $G$ if for every vertex $v \in V\setminus D$, $|N_G(v)\cap D|\geq 1$. The cardinality of a minimum dominating
set of $G$ is called the \emph{domination number} of $G$ and is denoted by $\gamma(G)$. Note that, a dominating
set $D$ dominates each vertex of $V\setminus D$ at least once. If, for some positive integer $i$, a dominating set
$D_i$ dominates each vertex of $V\setminus D_i$ at least $i$ times, then $D_i$ is called a $i$-dominating set.
A \emph{restrained dominating set} is a set $D_r\subseteq V$ where every vertex in $V\setminus S$ is adjacent to
a vertex in $S$ as well as another vertex in $V\setminus S$. The cardinality of a minimum restrained dominating
set of $G$ is called the \emph{restrained domination number} of $G$.
\subsection{Short review on $(i,j)$-set}
\label{ssec-short-review-1,j-set}
A set $D\subseteq V$ of a graph $G=(V,E)$ is called a \emph{$(i,j)$-set} if for every
$v\in V\setminus D$, $i\leq |N_G(v)\cap D|\leq j$ for nonnegative integers $i$ and $j$, that is, every
vertex $v\in V\setminus D$ is adjacent to at least $i$ but not more than $j$ vertices in $D$. The concept
of $(i,j)$-set was introduced by Chellali et al. in \cite{12set}. Clearly, it is a generalization of the
classical domination problem. Like domination problem, in this case, our goal is to find a $(i,j)$-set
of minimum cardinality, which is called the \emph{$(i,j)$-domination number} of $G$ and is denoted by
$\gamma_{(i,j)}(G)$. The decision version of $(i,j)$-set problem is defined as follows.
\vspace{10pt}
\noindent\underline{\textbf{$(i,j)$-Set problem ($(i,j)$-SET)}}
\begin{description}
\item[Instance:] A graph $G=(V,E)$ and a positive integer $k\leq |V|$.
\item[Question:] Does there exist a $(i,j)$-set $D$ of $G$ such that $|D|\leq k$?
\end{description}
In domination, we are interested in finding a set $D$ which dominates all the vertices of $V\setminus D$
at least once. But in some situation, we need to dominate each vertex at least $i$ times and at the same
time, dominating a vertex more than $j$ times, might cause a problem. Basically, we are interested in
finding a $i$-dominating set with a bounded redundancy. In these type of situations, we need the concept
of $(i,j)$-set. Also, $(i,j)$-set is a more general concept which involves
\emph{nearly perfect set} \cite{nearlyperfect}, \emph{perfect dominating set} \cite{perfect} (also known as
\emph{$1$-fair dominating set} \cite{fair}) etc. as variants. There is a concept of \emph{set restricted domination}
which is defined as follows: for each vertex $v\in V$, we assign a set $S_v$. A set $D_S$ is called a set restricted
dominating set if for all $v\in V$, $|N_G[v]\cap D_S|\in S_v$. Note that if $S_v=[j]$ for all $v\in V$,
we have a $(1,j)$-set. In that sense, $(1,j)$-set is a particular type of set restricted dominating set.
The concept of $(i,j)$-set has been introduced recently in 2013. Unlike other variations of domination, it
has not been well studied until now. As per our knowledge, only two papers have
appeared on $(i,j)$-set \cite{12set,YangWu}.
The main focus of \cite{12set} is on a particular $(i,j)$-set, namely $(1,2)$-set. In \cite{12set}, the
authors have made a simple observation that for a simple graph $G$ with $n$ vertices,
$\gamma(G)\leq \gamma_{(1,2)}(G)\leq n$. They have studied some graph classes for which these bounds are
tight. They have shown that $\gamma(G)= \gamma_{(1,2)}(G)$ for claw-free graphs, $P_4$-free graphs,
caterpillars etc. The authors have constructed a special type of split graph that achieves the upper bound.
But there are some graph classes for which $\gamma_{(1,2)}(G)$ is strictly less than $n$. These graph classes
involve graphs with maximum degree $4$, graphs having a $k$-clique whose vertices have degree either $k$ or
$k+1$ etc \cite{12set}. In \cite{12set}, the authors have studied the $(1,3)$-set for grid graphs and showed
that $\gamma(G)=\gamma_{(1,3)}(G)$. Using this result, they also showed that domination number is equal to
restrained domination number. From complexity point of view, it is known that $(1,2)$-SET is NP-complete for
bipartite graphs \cite{12set}. A list of open problems were posed in \cite{12set} indicating some research
directions in this field. In \cite{YangWu}, the authors showed that some graphs with $\gamma_{(1,2)}(G)=n$
exist among some special families of graphs, such as planar graphs, bipartite graphs. These results answers
some of the open problems posed in \cite{12set}. They also showed that for a tree $T$ with $k$ leaves, if
$deg_G(v)\geq 4$ for any non-leaf vertex $v$, then $\gamma_{(1,2)}(T)=n-k$. Nordhaus-Gaddum-type inequalities
are also established for $(1,2)$-set in \cite{YangWu}.
The main focus of \cite{12set} and \cite{YangWu} is $(1,2)$-set. In this paper, we study the more general
set, namely $(1,j)$-set. Apart from the open problems mentioned in \cite{12set}, a bound on the
$(1,j)$-domination number for general graphs is important. A general bound and its construction forms a
major thrust of this paper, which is presented in Section $2$. In Section $3$, we tighten the hardness
result by showing that $(1,j)$-set problem is NP-complete for chordal graphs. In Section $4$, we propose
two polynomial time algorithms that calculate minimum $(1,j)$-domination number for trees and split graphs,
that solves an open problem mentioned in \cite{12set}. Finally, Section $5$ concludes the paper.
\section{Upper bounds}
\label{sec-upper-bounds}
In this section, we shall prove an upper bound on the $(1,j)$-domination
number, i.e. $\gamma_{(1,j)}(G)$, of any graph $G=(V,E)$, having bounded minimum
and maximum degree, for `sufficiently large' $j$.
In~\cite{AlonSpencer}, Alon and Spencer describe a similar
upper bound on the domination
number $\gamma(G)$, using probabilistic methods. Their strategy, (a classic
example of the `alteration technique' in probabilistic methods), was to select
a random subset $X$ of vertices as a partial dominating set, and then to
include the set $Y$ of vertices not dominated by $X$, to get the final
dominating set. However, such a strategy is \emph{a priori} not applicable for
$(1,j)$-domination, because including or excluding vertices from the dominating
set could change the number of dominating vertices adjacent to some vertex.
Instead, we shall use a one-step process, and analyze it using the
Lov\'{a}sz Local Lemma and Chernoff bounds to ensure that the conditions
for $(1,j)$-set holds.
Our proof also implies a randomized algorithm, using the Moser-Tardos constructive
version of the Local Lemma~\cite{MoserTardos}, which would give a polynomial-time
algorithm for obtaining a $(1,j)$-dominating set.
We first state two well-known results, Chernoff bound and Lov\'{a}sz Local Lemma, in a form suitable
for our purposes. These results can be found in any standard text on
probabilistic combinatorics, e.g.~\cite{AlonSpencer}.
\begin{theo}[Chernoff bound]
Suppose $X$ is the sum of $n$ independent variables, each equal to $1$ with
probability $p$ and 0 otherwise. Then for any $0 \leq \alpha$,
$$
\prob{X > (1+\alpha) np} < \exp({-f(\alpha)np}),
$$
where
$f(\alpha) = (1+\alpha)\ln(1+\alpha)-\alpha$.
\end{theo}
\begin{lemma}[Lov\'{a}sz local lemma] \label{lemmalovaszlocal}
Let $\mathcal{A} = \{E_1,E_2,...,E_m\}$ be a collection of events
over a probability space such that each
$E_i$ is totally independent of all but the events in
$\mathcal{D}_i \subseteq \mathcal{A}\backslash \{E_i\}$. \\
If there exists a real sequence $\{x_i\}_{i=1}^m$, $x_i \in [0,1)$, such
that
\begin{eqnarray*}
\forall i \in [m]\mbox{, } \prob{E_i} &\leq& x_i\prod_{j:E_j\in \mathcal{D}_i}
(1-x_j)\mbox{, then} \\
\prob{\bigcap_{i=1}^m \overline{E}_i} &\geq& \prod_{i=1}^m (1-x_i) > 0.
\end{eqnarray*}
In particular, if for
all $i$, $|\mathcal{D}_i| = d$ and $\prob{E_i} \leq p$, then, if $ep(d+1) \leq 1$,
then
$$
\prob{\bigcap_i \bar{E_i}} > \exp\left(-\frac{m}{d+1}\right).
$$
\end{lemma}
Before stating the main theorem, we need some definitions:
Given $\alpha \in \Re^+$, let
$$
f(\alpha) \stackrel{\rm def}{=} (1+\alpha)\ln (1+\alpha) -\alpha.
$$
Also let $s(\alpha) \stackrel{\rm def}{=} \min\{1,f(\alpha)\}$ and for $\Delta \in \mathbb{Z}^+$, let
$$
g(\Delta) \stackrel{\rm def}{=} \ln (2e(\Delta^2+1)) =
1+\ln 2+ 2\ln \Delta + o_\Delta(1),
$$
where $e$ is the base of the natural logarithm.
\begin{theo}\label{thmubd}
Given $j\in \mathbb{Z}^+$, let $\alpha >0$ be the maximum real number
such that
$$
j+1 \geq (1+\alpha)\frac{\Gamma\, g(\Delta)}{s(\alpha)} \; \; \mbox{where}
\; \; \Gamma = \frac{\Delta}{\delta}.
$$
Then, if such an $\alpha$ exists,
$$\gamma_{(1,j)}(G) \leq (1+o_\Delta(1))\frac{g(\Delta)}{s(\alpha)\delta}n
\leq (1+o_\Delta(1))\left(\frac{1+\ln 2 + 2\ln \Delta}{s(\alpha)\delta}\right)n \, .$$
Further, there is a randomized algorithm to obtain a $(1,j)$-dominating set of
size at most $\frac{ng(\Delta)}{\delta s(\alpha)}$ that has expected runtime $O(n)$.
\end{theo}
\begin{proof}
Let $D \subset V$ be a subset of vertices obtained by tossing a coin for each vertex $v\in V$
independently and randomly with probability $p = \frac{g(\Delta)}{\delta s(\alpha)}$ and choosing
$v$ if the coin comes up Heads. We shall show using the Local Lemma, that the subset $D$ is a
$(1,j)$-dominating set with non-zero probability.
For each vertex $v \in V$, let $E_v$ be the event that $v$ is not $(1,j)$-dominated by
$D$, i.e. that $v \not\in D$, and $|N(v) \cap D|\not\in [j]$. We need to show
that
$$
\prob{\bigcap_{v \in V}\bar{E_v}} > 0.
$$
In order to use the Local Lemma, consider the dependency graph formed
by having the set of events $\{E_v\}_{v\in V}$ as vertices. Events $E_u$, $E_v$ $(u,v \in V)$
are dependent if and only if their outcomes depend on at least one common coin toss. Then clearly, the
events $E_u$, $E_v$ will be dependent if and only if
$$
N[u] \cap N[v] \neq \emptyset.
$$
This is possible only if the vertices $u$ and $v$ are at a distance at most
$2$ from each other in the graph $G$. Hence, the degree of the dependency graph is at most
$\Delta^2$. Now applying the symmetric form of the Local Lemma, we get that
$$
\prob{\bigcap_{v\in V} \bar{E_v}} > 0 \quad \mbox{if} \quad \prob{E_v} \leq \frac{1}{e (\Delta^{2}+1)}.
$$
We also
need to bound the size of the selected subset $|D|$. However, this can easily be obtained
by applying a Chernoff bound to the output of the Local Lemma.
The proof of Theorem~\ref{thmubd} is therefore completed with the following 2 claims:
Let $X \stackrel{\rm def}{=} |D|$. With $p$, $\alpha$ and $E_v$ defined as above,
\begin{cl} \label{clmlllreqmt}
For all $v \in V$,
$$
\prob{E_v} \leq \frac{1}{e(\Delta^2+1)}.
$$
\end{cl}
\begin{cl} \label{clmchernoffoverall}
For any $\varepsilon >\frac{1}{\sqrt{\Delta}}$, there exists $\Delta_0 \in \mathbb{Z}^+$, such
that for all $\Delta\geq \Delta_0$, we have
$$\prob{ \Big( X < (1+\varepsilon)np \Big) \bigcap \left(\bigcap_{v\in V} \bar{E_v}\right)} > 0.$$
\end{cl}
Thus the set $D$ is of size at most
$$
(1+o_\Delta(1))n\left(\frac{1+\ln 2+2\ln \Delta}{\delta}\right),
$$
and every vertex in $V\setminus D$ has at least $1$ and at most $j$ neighbours in $D$. The bound on
$\gamma_{(1,j)}(G)$ follows.
We will elaborate on the randomized algorithm, via Moser-Tardos's local lemma implementation,
for obtaining such a $(1,j)$-dominating set in
Remark~\ref{remark-2} at the end of this section.
\end{proof}
It only remains to prove the Claims~\ref{clmlllreqmt} and~\ref{clmchernoffoverall}.
\begin{proof}[Proof of Claim~\ref{clmlllreqmt}]
Given any vertex $v\in V$, define $X_v = |N(v)\cap D|$, and let $F_v$ denote the event that $X_v \not\in [1,j]$.
Then we have that
\begin{eqnarray*}
\prob{E_v} &=& \prob{E_v|F_v}.\prob{F_v} + \prob{E_v|\bar{F_v}}.\prob{\bar{F_v}} \\
&=& (1-p)\prob{F_v} + 0
\end{eqnarray*}
We shall prove the stronger condition : $\prob{\bigcap_{v\in V} \bar{F_v}} >0$.
Now,
$$ \prob{F_v} = \prob{X_v < 1} + \prob{X_v > j} .$$
Observe that $X_v$ has the binomial distribution $\bin{d(v)}{p}$.
Note that, if $j > d(v)$, then the event $F_v$ occurs only when $X_v = 0$ and for $j \leq d(v)$, the event
$F_v$ can occur when $X_v=0$ or when $X_v > j$. Therefore,
\begin{equation}
\prob{F_v} = \left\{ \begin{array}{ll}
(1-p)^{d(v)} & \mbox{ if } j > d(v) \\
(1-p)^{d(v)} + \prob{X_v > j} & \mbox{ if } j \leq d(v)
\end{array} \right.
\end{equation}
By the premise of the Theorem,
we get that
$$ j+1 \geq (1+\alpha)\frac{\Gamma g(\Delta)}{s(\alpha)} \geq (1+ \alpha)\frac{g(\Delta)}{s(\alpha)},$$
and hence, substituting the value of $p$, we get
$$
f(\alpha)d(v)p \geq \frac{f(\alpha)\, g(\Delta)}{s(\alpha)} \geq g(\Delta),
$$
since $f(\alpha) \geq s(\alpha)$.
Substituting in the expression for $f(v)$ gives
$$ (1-p)^{d(v)} = e^{d(v)\ln (1-p)} \leq e^{-d(v)p} \leq \frac{1}{2e(\Delta^2+1)},$$
since $d(v) \geq \delta$.
To compute $\prob{X_v \geq j+1}$, we use the Chernoff bound:
\begin{eqnarray*}
\prob{X_v \geq j+1} &\leq& \prob{\bin{d(v)}{p} \geq j+1} \\
&\leq& \prob{\bin{d(v)}{p} \geq (1+\alpha)d(v)p} \\
&\leq& \exp(-f(\alpha)d(v)p) \\
&\leq& \frac{1}{2e(\Delta^2+1)}
\end{eqnarray*}
where the last inequality follows from the choice of $p$.
Therefore, we get that
\begin{eqnarray*}
\prob{F_v} = \prob{X_v < 1} + \prob{X_v > 1}
\leq \frac{2}{2e(\Delta^2+1)}
\end{eqnarray*}
and hence that $\prob{F_v} \leq \frac{1}{e(\Delta^2+1)}$.
\end{proof}
\begin{proof}[Proof of Claim~\ref{clmchernoffoverall}]
To show that $\prob{A \cap B} > 0$ where $A,B$ are events in a probability space,
it suffices to
show that
$$
\prob{\bar{A} \cup \bar{B}} \leq \prob{\bar{A}} + \prob{\bar{B}} <1,\; \mbox{i.e.,}\;
\prob{B}-\prob{\bar{A}} >0.
$$
Taking $A$ to be the event $(X < (1+\varepsilon)np)$ and $B$ to be $\left(\bigcap_{v\in V}\bar{E_v}\right)$, we
shall first upper bound $\prob{A}$, and then use the lower bound on
$\prob{B}$ from the Local Lemma.
Using Chernoff bound, we get that
\begin{eqnarray*}
\prob{X \geq (1+\varepsilon)np}
&\leq& \exp{\left(-\frac{\varepsilon^2np}{3}\right)} \\
&\leq& \exp{\left(-\frac{\varepsilon^2ng(\Delta)}{3\delta}\right)}
\end{eqnarray*}
Now, from the Local Lemma, we get that
\begin{eqnarray*}
\prob{\bigcap_{v\in V}\bar{E_v}} &>& \left(1- \frac{1}{\Delta^2+1}\right)^{n} \\
&\approx& \exp\left(-\frac{n}{\Delta^2+1}\right).
\end{eqnarray*}
Let $\varepsilon = \frac{\sqrt{c\delta}}{\Delta}$, where $c>0$ is any constant. Then we get that for sufficiently
large $\Delta$,
\begin{eqnarray*}
\prob{B}- \prob{\bar{A}} &\geq& \exp\left(-\frac{n}{\Delta^2+1}\right) - \exp\left(\frac{-\varepsilon^2ng(\Delta)}{3\delta}\right) \\
&>& 0
\end{eqnarray*}
since for $\varepsilon = \frac{\sqrt{c\delta}}{\Delta}$, we get that
$$
\frac{\varepsilon^2ng(\Delta)}{3\delta} \geq \frac{cn\ln\Delta}{3\Delta^2} > \frac{n}{\Delta^2+1}.
$$
\end{proof}
In particular, taking $\alpha=e-1$ and $G$ $d$-regular, we get:
\begin{coro}\label{cor-1,j-dom-d-regular}
If $G$ is a $d$-regular graph, and $j > eg(d)$ then
$$
\dom{1}{j}{G} \leq (2+o_{d}(1))\frac{n \ln d}{d}.
$$
\end{coro}
\begin{remark}\label{remark-1}
We remark that from the known lower bounds on the domination number of random graph, our results
can be seen to be tight up to constant multiplicative factors, since $\gamma_{(1,j)}(G) \geq \gamma(G)$.
For instance, the result of Glebov, Liebenau and Szab\`{o}~\cite{GlebovLiebenauSzabo}, implies
that there exist graphs $G$ on $n$ vertices such that their domination number
$\gamma(G) \geq \frac{n\, \log d}{d}$.
\end{remark}
\begin{remark}\label{remark-2}
Elaboration on the Moser-Tardos's (MT) implementation:
We set up a SAT formula for domination, where each vertex $v_i \in V$ corresponds to a variable $x_i$ and
there is a clause $C(v)$ corresponding to each neighbourhood $N(v)$. $x_i= \mbox{``true''}$ means the vertex $v_i$ is
selected in the dominating set. Clause $C(v)$ is said to have failed if it is not satisfied
in the given assignment. In addition, for each vertex $v \in V$ having degree $d(v) > j$, there is a unique clause for
every subset of $N(v)$ of size $j+1$, which fails only if all the vertices in the corresponding subset of $N(v)$ are
selected in the dominating set. Now we run the MT algorithm on this formula,
(i.e. take a random assignment where
each variable is set to true independently with the probability $p$ used in the proof; choose an arbitrary
failing clause and randomly reset all variables inside the clause; repeat until all clauses are satisfied).
The LLL condition guarantees that the MT algorithm will terminate in expected time linear in the number of clauses, i.e. $O(n\Delta^{j+1})
= O(n^{j+2})$, and when this happens, the Chernoff bound guarantees that with high probability, not more
than $(1+o_\Delta(1))\frac{g(\Delta)n}{s(\alpha)\delta}$ many variables will be set to $\mbox{``true''}$.
\end{remark}
\section{NP-complete for chordal graphs}
\label{sec-NP-complete-chordal-graphs}
In this section, we show that $(1,j)$-SET is NP-complete when restricted to chordal graphs. Note that,
for $j=1$, the problem is basically perfect domination problem, which is known to be NP-complete for
chordal graphs \cite{perfectchordal}. For $j\geq 2$, we prove the NP-completeness by using a reduction
from Exact $3$-Cover problem (EX$3$C), which is known to be NP-complete \cite{garey}.
\vspace{10pt}
\noindent\underline{\textbf{Exact $3$-Cover problem (EX$3$C)}}
\begin{description}
\item[Instance:] A finite set $X$ with $|X|=3q$, where $q$ is a positive integer and a collection $C$ of $3$-element subsets of $X$.
\item[Question:] Is there a subcollection $C'$ of $C$ such that every element of $X$ appears in exactly one element of $C'$?
\end{description}
\begin{theo}\label{thm-NP-harness-chordal-graphs}
$(1,j)$-SET is NP-complete for chordal graphs.
\end{theo}
Clearly $(1,j)$-SET for chordal graph is in NP. We describe a polynomial reduction from
EX$3$C to $(1,j)$-SET for chordal graphs. Given any instance $(X,C)$ of EX$3$C, we obtain a
chordal graph $G=(V,E)$ and an integer $k$ such that EX$3$C has a solution if and only if $G$ has
a $(1,j)$-set of cardinality at most $k$.
Let $X=\{x_1, x_2, \ldots, x_{3q}\}$ and $C=\{C_1, C_2, \ldots, C_t\}$ be an arbitrary instance of EX$3$C.
The vertex set of the newly formed graph $G=(V,E)$ is
formed as a disjoint union of $V_1, V_2$ and $V_3$, that is,
$$
V = V_1 \sqcup V_2 \sqcup V_3.
$$
For each $C_p\in C$,
$p\in [t]$, we have a claw centered at a vertex $u_p$ and let $v_p, y_p$ and $z_p$ be the pendant
vertices of that claw. The set $V_1$ is given by
$$
V_1=\bigcup_{p=1}^t \left\{u_p, v_p, y_p, z_p \right\}.
$$
Also, we have a set
$V_2$ of $3q$ vertices $x_1, x_2, \ldots, x_{3q}$, each corresponding to an element of $X$. Furthermore, for
each $i\in \{1,\, 2, \ldots,\, 3q\}$, we add a gadget $G_i$, as shown in Figure~\ref{fig1}. The gadget $G_i$ is
basically a forest of $q$ number of rooted trees of depth 2 rooted at the vertices $w_1^i, w_2^i, \ldots w_{q}^i$
as shown in Figure~\ref{fig2}. In the gadget $G_i$, each $w_1^i,\, w_2^i,\, \ldots,\, w_{q}^i$ has $j$ children and
each of these $j$ children has $2$ more children. The set $V_3$ is given by
$$
V_3=\bigcup_{i=1}^{3q} V(G_i),
$$
where
$V(G_i)$ is the vertex set of $G_i$. Now we add the edges between $x_i$ and $v_p$ if the element corresponding
to $x_i$ is in $C_p$. Note that degree of each $v_p$ is $4$ for all $p\in \{1,\, 2,\, \cdots,\, t\}$. Also, we add
edges between every pair of distinct vertices of $\{x_1,\, x_2,\, \ldots,\, x_{3q}\}$, making it a clique. Finally,
for each $i\in \{1, 2, \ldots, 3q\}$, we add the edges $x_iw_1^i,\, x_iw_2^i,\, \ldots,\, x_iw_q^i$. The construction
of $G$ from the instance $(X,C)$ of EX$3$C is illustrated in Figure~\ref{fig1}. Clearly, the graph $G$ is a
chordal graph. Let $k=t+q+3jq^2$.
Theorem~\ref{thm-NP-harness-chordal-graphs} directly follows from the following result.
\begin{lemma}
EX$3$C has a solution if and only if $G$ has a $(1,j)$-set of cardinality at most $k=t+q+3jq^2$.
\end{lemma}
\begin{proof}
Suppose the instance $(X,C)$ has a solution $C'$. Since each
element of $X$ is covered by exactly one element of $C'$, $|C'|=q$. For each gadget
$G_i$, $i\in [3q]$, let $S_i$ be the set of all children of $w_1^i,\, w_2^i,\, \ldots,\, w_{q}^i$.
Clearly, each $S_i$ contains $jq$ vertices. We form a set $D$ as follows:
$$
D = \left\{u_i|~1\leq i\leq t\right\} \bigcup \left\{v_p|~C_p\in C' \right\} \bigcup
\left(\bigcup_{l=1}^{3q} S_l \right).
$$
Since $|C'|=q$, $D$ contains $t+q+3jq^2$
vertices. One can easily check that $D$ forms a $(1,j)$-set of $G$.
\begin{figure
\begin{center}
\includegraphics[width = 16cm]{ChordalNPC}
\end{center}
\caption{Reduction from EX$3$C to $(1,j)$-SET}
\label{fig1}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width = 13cm]{Gadget-2}
\end{center}
\caption{Gadget $G_i$ corresponding to $x_i$}
\label{fig2}
\end{figure}
Conversely, suppose that $G$ has a $(1,j)$-set $D$ of cardinality at most $k=t+q+3jq^2$.
First observe that since $D$ is a dominating set, $D$ must contain at least $t$ vertices from the
set
$$
V_{4} \stackrel{\rm def}{=} \{ y_{1}, \, \dots, \, y_{t}\} \cup \{ z_{1}, \, \dots, \, z_{t}\} \cup \{ u_{1}, \, \dots, \, u_{t}\}
$$
to dominate the pendant vertices
$$
\{ y_{1}, \, \dots, \, y_{t}\} \cup \{ z_{1}, \, \dots, \, z_{t}\}.
$$
Similarly, for a fixed $i$ and $r$, consider the tree $T^{i}_{r}$. To dominate the
pendant vertices of the tree $T^{i}_{r} = (V^{i}_{r}, E^{i}_{r})$ we need to select at least $j$ vertices from the set
$V^{i}_{r} \setminus \{w^{i}_{r}\}$.
Summing up over all $i$ and $r$, we get that $D$ contains more than $3jq^{2}$ vertices from the set
$$
V_{5} \stackrel{\rm def}{=} \bigcup_{1\leq i \leq 3q, \, 1\leq r \leq q} \left( V^{i}_{r} \setminus \{w^{i}_{r}\} \right).
$$
Observe now that the cardinality of $D$ is at least $t+3q.jq=t+3jq^2$.
Now to complete the proof we will only have to show that $V_{2} \cap D = \emptyset$.
Since if this is the case then each $x_i$ has to be dominated by either
some $w^{i}_{l} \in G_i$ or a $v_s \in V_1$, $s\in [t]$.
We have to dominate the $3q$ vertices of $V_2$ using at most $q$ vertices,
since we have used up the other $t+3jq^2$ vertices. Since each $w^{i}_{l}$ dominates only one $x_i$, while each $v_i \in V_1$
dominates $3$ $x_i$'s, this is possible only if there exist $q$ vertices $v_{i_1},\ldots,v_{i_q}$, which can dominate the $3q$ vertices $x_i\in V_2$.
Now define $C'$ to be the sets corresponding to these vertices, i.e. $C'= \{C_{i_1},\, \ldots,\, C_{i_q}\}$. Clearly $C'$ is an exact cover of $X$, and has
only $q$ sets.
Till now we have only used the fact that $D$ is a dominating set but for showing $D \cap V_{2} = \emptyset$
we will be crucially using the fact that $D$ is a $(1,j)$-set.
To reach a contradiction let us
suppose some $x_i \in D$. Then each $w^{i}_{r}$, $r\in [q]$ is $1$-dominated by $x_i$,
and either has to be in $D$ or can have at most $j-1$ other neighbours that are in $D$.
In either case, we get that for each tree $T^{i}_{r} \in G_i$, $|T^{i}_{r} \cap D| \geq j+1$.
Hence, $|G_i \cap D| \geq jq +q$. This implies that $|D|\geq t+1+(3q-1)(jq)+(j+1)q = t+1+3jq^2+q$,
which contradicts the assumption that $|D|\leq k$.
Therefore $D\cap V_{2} = \emptyset$.
\end{proof}
\begin{remark}
Following observations directly
follow from the NP hardness reduction:
\begin{enumerate}
\item
The only possibility of dominating $V_4$ by $t$ vertices is to take $\{u_1,\, u_2,\, \dots,\, u_t\}$ and
this set also dominates the set $\{v_1,\, v_2,\, \dots,\, v_t\}$.
\item
The only possibility of dominating $V_5$ by $3jq^2$ vertices is to take $\bigcup_{i=1}^{3q} S_i$
and this set dominates each $w^{i}_{l}$ exactly $j$ times. Note that
$S_i$ is the set of all children of $w_1^i,\, w_2^i,\, \ldots,\, w_{q}^i$ and
each $S_i$ contains $jq$ vertices.
\end{enumerate}
\end{remark}
\section{Polynomial time algorithms}
\label{sec-polynomial-time-algorithms}
\subsection{Tree}
\label{ssec-tree}
To design an efficient algorithm for finding $(1,j)$-domination number of a given tree $T$, we need the concept
of a more generalized set, namely $M$-set of an $M$-labeled tree. In fact, we design a dynamic programming
algorithm for finding the minimum cardinality of an $M$-set of an $M$-labeled tree. First let us define
an $M$-labeled tree and an $M$-set.
\begin{defi}
A tree $T$ is called an \emph{$M$-labeled tree} if each vertex $v$ is associated with two nonnegative integers
$M_a(v)$ and $M_b(v)$ such that $M_a(v)\leq M_b(v)$. A subset $S\subseteq V$ of an $M$-labeled tree $T=(V,E)$
is called an \emph{$M$-set} if $M_a(v)\leq |N_T(v)\cap S|\leq M_b(v)$ for every $v\in V\setminus S$. The minimum
cardinality of an $M$-set of an $M$-labeled tree $T$ is called the \emph{$M$-domination number} of
$T$ and is denoted by $\gamma_M(T)$.
\end{defi}
Note that if all the vertices of an $M$-labeled tree $T$ can be labeled as $M_a(v)=1$ and $M_b(v)=j$,
then an $M$-set of $T$ is nothing but a $(1,j)$-set of the underlying tree.
The main idea of the dynamic programming algorithm is to choose a specific vertex $u$ from $T$. Any minimum
$M$-set of $T$ should either contain $u$ or does not contain $u$. So the problem of finding the minimum
cardinality of an $M$-set of $T$ boils down to finding two parameters: $(i)$ $\gamma_M(T, u)$, the minimum
cardinality of an $M$-set of $T$ that contains the specific vertex $u$ and $(ii)$ $\gamma_M(T, \bar{u})$,
the minimum cardinality of an $M$-set of $T$ that does not contain the specific vertex $u$.
Suppose $uv$ is an edge of the $M$-labeled tree $T$. Let $H_1$ and $H_2$ be the subtrees of $T$ rooted at
$u$ and $v$ respectively. Note that $H_1$ and $H_2$ are $M$-labeled trees and the labels of the vertices of
$H_1$ and $H_2$ remain the same as they are in $T$. Our aim is to use the parameters
$\gamma_M(H_1, u)$, $\gamma_M(H_1, \bar{u})$, $\gamma_M(H_2, v)$, and $\gamma_M(H_2, \bar{v})$
(with suitable labeling $M$) to find $\gamma_M(T, u)$ and $\gamma_M(T, \bar{u})$. The following lemma
shows how the values of $\gamma_M(T, u)$ and $\gamma_M(T, \bar{u})$ are obtained.
\begin{lemma}\label{lemtreealgo}
Let $uv$ be an edge of an $M$-labeled tree $T$ and $H_1$ and $H_2$ be the subtrees of
$T$ rooted at $u$ and $v$ respectively. Then the following statements hold.
\begin{itemize}
\item[(a)] $\gamma_M(T,u)= \gamma_M(H_1,u)+ \gamma_{M'}(H_2)$, where the label $M'$
is same as $M$ except $M'_a(v)=\max \{M_a(v)-1, 0\}$ and $M'_b(v)=\max \{M_b(v)-1, 0\}$
\item[(b)] $\gamma_M(T, \bar{u})= \min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \}$,
where the label $M'$ is same as $M$ except $M'_a(u)=\max \{M_a(u)-1, 0\}$ and $M'_b(u)=\max \{M_b(u)-1, 0\}$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) Let $D$ be a minimum cardinality $M$-set of $T$ containing $u$. Let $D_1= V(H_1)\cap D$ and
$D_2= V(H_2)\cap D$. Clearly $D_1$ is an $M$-set of $H_1$ containing $u$. Now, $D_2$ may or may
not contain the vertex $v$.
\begin{description}
\item[Case $v\in D_2$:] In this case, $D_2$ is an $M'$-set of $H_2$ containing $v$.
\item[Case $v\notin D_2$:] In this case, $D_2$ is an $M'$-set of $H_2$ not containing $v$.
\end{description}
Since $\gamma_{M'}(H_2)=\min \{\gamma_{M'}(H_2, v), \gamma_{M'}(H_2, \bar{v})\}$,
we have $\gamma_M(H_1,u)+ \gamma_{M'}(H_2) \leq \gamma_M(T,u)$.
On the other hand, let $D_1$ be a minimum cardinality $M$-set of $H_1$ containing $u$ and $D_2$ be a
minimum cardinality $M'$-set of $H_2$. Let $D=D_1\cup D_2$. Clearly, whatever be the case
($v\in D_2$ or $v\notin D_2$), we can verify that $D$ is a $M$-set of $T$ and $u\in D$. Hence,
$\gamma_M(T,u)\leq \gamma_M(H_1,u)+ \gamma_{M'}(H_2)$.
Thus we have, $\gamma_M(T,u)= \gamma_M(H_1,u)+ \gamma_{M'}(H_2)$.
(b) Let $D$ be a minimum cardinality $M$-set of $T$ not containing $u$. Let
$D_1= V(H_1)\cap D$ and $D_2= V(H_2)\cap D$. Now, $D_2$ may or may not contain the vertex $v$.
\begin{description}
\item[Case $v\notin D$:] In this case, $D_1$ is an $M$-set of $H_1$ not containing $u$ and $D_2$ is an
$M$-set of $H_2$ not containing $v$. Hence, $\gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v})\leq \gamma_M(T, \bar{u})$.
\item[Case $v\in D$:] In this case, $D_1$ is an $M'$-set of $H_1$ not containing $u$ and $D_2$
is an $M$-set of $H_2$ containing $v$. Hence, $\gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v)\leq \gamma_M(T, \bar{u})$.
\end{description}
So we have, $\min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \} \leq \gamma_M(T, \bar{u})$.
On the other hand, for showing
$\gamma_M(T, \bar{u})\leq \min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \}$,
we have the following two cases:
\begin{description}
\item[Case $\min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \}= \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v})$:]
Let $D_1$ be a minimum cardinality $M$-set of $H_1$ not containing $u$ and $D_2$ be a
minimum cardinality $M$-set of $H_2$ not containing $v$. Let $D=D_1\cup D_2$. We can easily
verify that $D$ a minimum cardinality $M$-set of $T$ not containing $u$. So,
$\gamma_M(T, \bar{u})\leq \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v})$.
\item[Case $\min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \}=\gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v)$:]
In this case, similarly we can show that $\gamma_M(T, \bar{u})\leq \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v)$.
\end{description}
Hence in both the cases, $\gamma_M(T, \bar{u})\leq \min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \}$.
Thus we have, $\gamma_M(T, \bar{u})= \min \{ \gamma_M(H_1, \bar{u})+ \gamma_M(H_2, \bar{v}), \gamma_{M'}(H_1, \bar{u})+ \gamma_M(H_2, v) \}$.
\end{proof}
Based on the above lemma, we have the following dynamic programming algorithm for finding
$\gamma_M(T)$ for an $M$-labeled tree $T$. Note that, a tree with a single vertex forms the base case at
which $\gamma_M(T)$ can be easily computed depending upon the $M$ label.
\begin{algorithm}\label{algotree}
\KwIn{A $M$-labeled tree $T=(V,E)$.}
\KwOut{A minimum cardinality of an $M$-set of $T$, i.e., $\gamma_M(T)$.}
\Begin{
Select a vertex $u$ from $V$;\\
Select an edge $uv$ from $E$;\\
Calculate $\gamma_M(T, u)$ and $\gamma_M(T, \bar{u})$ according to Lemma~\ref{lemtreealgo};\\
$\gamma_M(T)= \min\{\gamma_M(T, u), \gamma_M(T, \bar{u})\}$;\\
\Return $\gamma_M(T)$;
}
\caption{Min\_M-set\_Tree}
\end{algorithm}
The correctness of Algorithm \ref{algotree} is based on Lemma~\ref{lemtreealgo}.
Since the dynamic programming runs over the edges of the given tree, Algorithm~\ref{algotree} take linear time.
Also, as noted earlier, if we initialize the $M$-label as $M_a(v)=1$ and $M_b(v)=j$ for all $v\in V$,
then $\gamma_{(1,j)}(T)=\gamma_M(T)$. Hence we have the following theorem.
\begin{theo}
The $(1,j)$-domination number of a given tree can be computed in linear time.
\end{theo}
\subsection{Split graph}
In this subsection, we design an algorithm which finds $(1,j)$-domination number for a given split graph $G$
in polynomial time. This algorithm is important because most of the domination type problems like
domination~\cite{domsplit}, total domination~\cite{tdomsplit}, $k$-tuple domination~\cite{kdomsplit} etc.
are NP-complete for split graphs.
Let the vertex set $V$ of a split graph $G=(V,E)$ is partitioned into a clique $K$ and an
independent set $S$, i.e.,
$V=K\cup S$. Also assume that $|K|=n_1$ and $|S|=n_2$. Note that in finding a minimum $(1,j)$-set, $j$ can be
considered as a constant. Now if $n_1\leq j$, then we are done. Because, in that case, we consider all possible
subsets of $K$ and based on the neighborhood set of these subsets we can find a minimum cardinality $(1,j)$-set.
Since $j$ is a constant, the number of subsets of $K$ is bounded by a constant (this constant is huge, $2^j$).
This implies that, in this case, we can find a minimum $(1,j)$-set in polynomial time. Hence we assume that $j< n_1$.
The idea of the algorithm is based on a simple fact that if a $(1,j)$-set, say $D$, contains more than $j$ but
less than $n_1$ vertices from $K$, then there exists a vertex in $K\setminus D$ which is dominated by more than
$j$ vertices, which is a contradiction to the definition of $(1,j)$-set. Hence we have the following observation.
\begin{obs}\label{obsalgosplit}
Every $(1,j)$-set of a given split graph $G$ contains only $i$ vertices from $K$ where $i\in \{0,1,2,\ldots, j, n_1\}$.
\end{obs}
Now, for each $i=0, 1, 2,\ldots, j$ and $n_1$, we find a minimum cardinality $(1,j)$-set $D$ of $G$ such that
$|K\cap D|=i$. Finally we pick the minimum cardinality $(1,j)$-set among these $j+2$ types of $(1,j)$-sets of $G$.
Hence the main task in this algorithm is to find a minimum cardinality $(1,j)$-set $D$ of $G$ such that
$|K\cap D|=i$ for each $i=0, 1, 2,\ldots,j$ and $n_1$. The following lemma gives a complete characterization
of these $j+2$ types of $(1,j)$-sets.
\begin{lemma} \label{lemsplitalgo}
Let the vertex set $V$ of a connected split graph $G=(V,E)$ is partitioned into a clique $K$ and an independent
set $S$, i.e., $V=K\cup S$ and $|K|=n_1$ and $|S|=n_2$. Let $D$ be a minimum $(1,j)$-set of $G$. Then the
following statements are true.
\begin{itemize}
\item[(a)] If $K\cap D=\emptyset$, then $d_G(v)\in \{n_1, n_1+1, \ldots, n_1+j-1\}$ for all $v\in K$.
In this case, $D=S$ is the only $(1,j)$-set of $G$.
\item[(b)] For all $i\in [j-1]$, if $K\cap D=\{v_1, \ldots, v_i\}=K_i$, then
$d_{G[K\cup S_i]}(v) \in \{n_1-1, n_1, \ldots, n_1+(j-i)-1\}$ for all $v\in K\setminus K_i$, where $S_i= S\setminus N_G(K_i)$.
In this case, $D=K_i\cup S_i$ is a minimum cardinality $(1,j)$-set of $G$ containing $K_i$.
\item[(c)] If $K\cap D=\{v_1, v_2, \ldots, v_j\}=K_j$, then $S\subset N_G(K_j)$. In this case,
$D=\{v_1, v_2,\ldots, v_j\}$ is a $(1,j)$-set of $G$ of minimum cardinality.
\item[(d)] If $K\subseteq D$, then $S_2\subseteq D$, where $S_2=\{u\in S|~d_G(u)\geq (j+1)\}$. In this case,
$D=K\cup S_2$ is a $(1,j)$-set of $G$ of minimum cardinality.
\end{itemize}
\end{lemma}
\begin{proof}
(a) In this case, since $S$ is an independent set, $D=S$. Again, since $K$ is a clique, $d_G(v)\geq n_1-1$
for all $v\in K$. If $d_G(v_i)= n_1-1$ for some vertex $v_i\in K$, then
$$
N_G(v_i)\cap D = \emptyset.
$$
This
is a contradiction to the definition of $(1,j)$-set. Again if $d_G(v_t)\geq n_1+j$ for some vertex $v_t\in K$,
then
$$
|N_G(v_t)\cap D|\geq (j+1).
$$
This will force $v_t$ to be in $D$, which is a contradiction to
$K\cap D=\emptyset$. Thus, we have
$$
d_G(v)\in \{n_1,\, n_1+1,\, \ldots,\, n_1+j-1 \} \quad \forall v \in K.
$$
(b) In this case, clearly $(K_i\cup S_i)\subseteq D$. Again, since $K$ is a clique, $d_{G[K\cup S_i]}(v)\geq n_1-1$
for all $v\in K$. If $d_{G[K\cup S_i]}(v_t)\geq n_1+(j-i)$ for some vertex $v_t\in (K\setminus K_i)$, then
$|N_G(v_t)\cap D|\geq j$. This will force $v_t$ to be in $D$, which is a contradiction to $|K\cap D|=i$. Thus
$$
d_{G[K\cup S_1]}(v) \in \{n_1-1, \, n_1,\, \ldots,\, n_1+(j-i)-1\} \quad \forall v\in K\setminus \{v_1\}.
$$
In this case,
clearly $D=K_i\cup S_i$ is a minimum cardinality $(1,j)$-set of $G$ containing $K_i$.
(c) If possible, let $u\in S\setminus N_G(K_j)$. Clearly $u\in D$. Since $G$ is connected, there exists at least
one vertex $v$ in $N_G(u)\setminus K_j$. Now for that vertex $v$,
$$
|N_G(v)\cap D|\geq j.
$$
This will force $v$ to
be in $D$, which is a contradiction to $|K\cap D|=j$. Thus $S\subset N_G(K_j)$. Clearly in this case, $D=K_j$ is
a $(1,j)$-set of $G$ of minimum cardinality.
(d) The proof is trivial and hence omitted.
\end{proof}
Based on the above lemma, we have Algorithm \ref{algosplit} that finds a minimum cardinality $(1,j)$-set
of a given split graph $G$.
\begin{algorithm}[h]\label{algosplit}
\relsize{-1}{\KwIn{A split graph $G=(V,E)$ with $V=K\cup S$.}
\KwOut{A minimum cardinality $(1,j)$-set $D$ of $G$.}
\Begin{
\If{$d_G(v)\in \{n_1, n_1+1, \ldots, n_1+j-1\}$ for all $v\in K$}{
$D_0==S$;}
\Else
{$D_0==\emptyset$;}
\ForEach{$i\in [j-1]$}{
\ForEach{$i$ element subsets $K^i_t$ of $K$}{
\If{$d_{G[K\cup S^i_t]}(v) \in \{n_1-1, n_1,\ldots, n_1+(j-i)-1\}$ for all
$v\in K\setminus K^i_t$, where $S^i_t= S\setminus N_G(K^i_t)$}{
$D^i_t==K^i_t\cup S^i_t$;}
\Else
{$D^i_t== \emptyset$;}
}
$D_i==D'$, where $D'$ be the minimum nonempty set among all $D^i_t$, for all $t$;
}
\ForEach{$j$ element subsets $K^j_t$ of $K$}{
\If{$S\subset N_G(K^j_t)$}{
$D_j==K^j_t;$
}
\Else
{$D_j==\emptyset$}
}
$D_{j+1}==K\cup S'$, where $S'=\{u\in S|~d_G(u)\geq (j+1)\}$;
$D==D_p$, where $D_p$ is the minimum cardinality nonempty set among all $D_i$, $0\leq i\leq (j+1)$.
\Return $D$;\\
}
\caption{Min\_(1,j)-set\_Split} }
\end{algorithm}
The correctness of Algorithm~\ref{algosplit} is based on Observation~\ref{obsalgosplit} and
Lemma~\ref{lemsplitalgo}. Next we analyze the complexity of Algorithm~\ref{algosplit}. Note that we can
compute $D_0$ and $D_{j+1}$ in $O(n)$ time. For each $1\leq i\leq (j-1)$, the set $D_i$ can be computed in
$O(n^i+ i n^i \log n)$ time because in each case, we have to check all possible $i$ element subsets of $K$,
i.e., $O(n^i)$ subsets and after that we have to assign the minimum cardinality subset to $D_i$, which takes
$O(i n^i \log n)$ time. Hence computing all the sets $D_i$ for $0\leq i\leq (j+1)$ can be done in polynomial
time. For $D_j$, we have to check all $j$ element subsets of $K$, i.e., $O(n^j)$, and since $j$ is a constant,
it takes polynomial time to compute $D_j$. Also finding the minimum cardinality nonempty set in line $19$ takes
polynomial time. Hence, Algorithm~\ref{algosplit} can be done polynomial time. Thus, we have the main theorem
in this subsection as follows.
\begin{theo}
For any fixed $j$, the cardinality of a minimum $(1,j)$-set of a given split graph can be computed in polynomial time.
\end{theo}
\section{Concluding remarks}
In this paper, we have obtained an upper bound on $(1,j)$-domination number. We have shown that $(1,j)$-SET
is NP-complete for chordal graphs. We have also designed two algorithms for finding $\gamma_{(1,j)}(G)$ of a
tree and a split graph. In~\cite{12set}, the authors constructed a special type of split graph $G$ with $n$
vertices for which $\gamma_{(1,2)}(G)=n$. Lemma~\ref{lemsplitalgo} gives a more general type of split graph
for which $\gamma_{(1,j)}(G)=n$. It actually characterizes the split graphs with $n$ vertices having
$\gamma_{(1,j)}(G)=n$. The characterization is as follows:
\begin{coro}
Let $G$ be a split graph with $V=K\cup S$ and $|K|=n_1$, $|S|=n_2$. Then $\gamma_{(1,j)}(G)=n$ if and
only if the following conditions hold.
\begin{itemize}
\item[(i)] There exists at least one vertex $v$ in $K$ such that $d_G(v)\geq n_1+j$.
\item[(ii)] For all $i\in [j-1]$ and for each $i$ element subset $K_i=\{v_1, \ldots, v_i\}$ of $K$,
there exists some vertex $v_t\in K\setminus K_i$ such that $d_{G[K\cup S_i]}(v_t)\geq n_1+(j-i)$, where $S_i= S\setminus N_G(K_i)$.
\item[(iii)] For each $j$ element subset $K_j$ of $K$, $S\not\subset N_G(K_j)$.
\item[(iv)] For every $u\in S$, $d_G(u)\geq (j+1)$.
\end{itemize}
\end{coro}
Condition $(i), (ii)$ and $(iii)$ actually force all the vertices of $K$ in a minimum cardinality $(1,j)$-set
$D$ and condition $(iv)$ forces all vertices of $S$ in $D$. Using this type of split graphs, we can construct
a graph $G$ (not a split graph) having $\gamma_{(1,j)}(G)=n$. The construction is as follows: Let
$$
G_1=(V_1, E_1),\, G_2=(V_2, E_2),\, \ldots,\, G_p=(V_p, E_p)
$$
be $p$ split graphs having partitions
$$
V_1=K_1\cup S_1,\, V_2=K_2\cup S_2,\, \ldots,\, V_p=K_p\cup S_p.
$$
The vertex set of the constructed graph $G=(V,E)$
is given by
$$
V=\bigcup_{i=1}^p V_i
$$
and the edge set is given by
$$
E = \left( \bigcup_{i=1}^p E_i \right) \bigcup E',
$$
where $E'$ is an
arbitrary edge set between the vertices of
$$
K_1,\, K_2,\, \ldots,\, K_p.
$$
We can easily verify that $\gamma_{(1,j)}(G)= |V|$.
But characterizing the graphs with $n$ vertices having $\gamma_{(1,j)}(G)=n$
seems to be an interesting but difficult question.
Also, it would be interesting to study the open problems mentioned in~\cite{12set}.
\subsection*{Acknowledgements}
Kunal Dutta and Arijit Ghosh are
supported by the Indo-German Max Planck Center for Computer Science (IMPECS).
Subhabrata Paul is supported by the Indian Statistical Institute, Kolkata.
\bibliographystyle{alpha}
\addcontentsline{toc}{section}{Bibliography}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 586 |
\section{Background}
\subsection*{Higher Order Mutants (HOM)}
Mutants made with mutation operators can be classified by the number of mutation operators applied to the original program. First Order Mutant (FOM) is mutant generated by applying mutation operator once. (definition borrowed) Mutants generated by applying mutation operator more than once are called Higher Order Mutants.
For HOM $h$, consider set of FOM $F = \{f_1, f_2, \dots, f_n\}$ where each mutant $f_i$ is generated by applying one of mutation operators used to generate $h$. We call the mutants the constituent FOMs of $h$ and say $h$ is consisted of $f_1, f_2, \dots, f_n$. Specifically, second order mutants (SOM) are mutants generated by applying mutation operator twice in original program.
Through lots studies it has been shown that Higher Order Mutation Testing can be used to overcome the \emph{equivalent mutant problem}: The process of deciding whether two mutants are equivalent or not is undecidable problem. Offutt's (reference) empirical study shows that $10\%$ of the first order mutants were found to be equivalent while only $1\%$ of the higher order mutants were. Therefore using HOMs, we are able to solve the equivalent mutant problem to more extent.
Also, the concept of Strongly Higher Order Mutant (SSHOM) encourages us to use HOMs. SSHOM was introduced by Yue Jia. SSHOM are by definition, mutants that are harder to kill than its constituent FOMs.
Let $h$ be a HOM constituted of $\{f_1, f_2, \dots, f_n\}$. Let $T$ a set of test cases and $T_f \subseteq T$ a set of test cases that kills a mutant $f$. We call $h$ is a subsuming mutant if:
\begin{equation}
|T_h| < |\bigcup_{i} {T_{f_i} }|
\end{equation}
Applying more strict conditions, mutant $h$ is a strongly subsuming mutant (SSHOM) if:
\begin{equation}
T_h \subset \bigcap_{i} T_i
\end{equation}
Every test case that kills $h$ should kill all of its constituent first order mutants.
Since $h$ is harder to kill than its constituent FOMs, all constituent mutants of $h$ can be replaced by a $h$ thus decreasing the test effort. For a HOM to be strongly subsuming (subsuming), multiple mutation operations must mask each other. This is because there must exist a test case that all its constituent FOMs gets killed but the higher order mutant lives.
Now for a set of mutants $M = \{m_1, m_2, \dots , m_k\}$ strongly subsuming rate (subsuming rate) of M is calculated as follows.
\begin{equation}
SSR(M) = \frac{|SSHOM(M)|}{|M|}
\end{equation}
Where $SSHOM(M)$ is the set of mutants in $M$ that are strongly subsuming.
\subsubsection*{Diversity of Mutants}
Another important feature to look for when sampling mutants is the diversity. Since the purpose of each mutant are made to mimic real faults of developers, we always want more diverse mutants. To define the diversity metric, we first introduce the kill vector of a mutant.
For a mutant $m$ of original program $P$ with test suite $T$, a kill vector $v \in \mathcal{R}^|T|$ of $m$ is defined as follows.
\begin{equation*}
v_i =
\begin{cases}
1
&
\text{if $T_i$ kills $m$;}
\\[10pt]
0
&
\text{otherwise.}
\end{cases}
\end{equation*}
where $T_i$ refers to $i$-th test case of $T$. ($1 \le i \le |T|$)
The diversity of set of mutants $M = \{m_1, m_2, \dots, m_k\}$ is then defined as follows.
\begin{equation}
dScore(M) = \frac{|\{v_1, v_2, \dots \}|}{|M|}
\end{equation}
where $v_i$ is the kill vector of $m_i$.
Note here that higher $dScore$ is achieved if certain set of mutants contain more mutants with distinct kill vectors. The reason we define diversity with kill vectors is because two mutants with same kill vectors are not distinguishable from the point of test suite. Yet if the two mutants have different kill vectors, we know for sure that the two mutants have different faults.
\section{Conclusion}
\label{sec:conclusion}
We propose a new approach of sampling higher order mutants by using Causal
Program Dependence Analysis (CPDA). Specifically, we show that causal effect
can effectively guide the generation of SSHOMs. We compare four different
SSHOM generation heuristics and the SSHOMs generated by them. The quality of
mutant set is measured in terms of two metrics, number of SSHOM and
diversity. Our results show that MWM and Dsort heuristics
can effectively sample SSHOMs. For future work, we plan to add more benchmark
programs, and investigate more sophisticated heuristic design that considers factors other than causal effects simultaneously.
\section{Experimental Setup}
\label{sec:experimental_setup}
\subsection{Research Questions}
\subsubsection*{RQ1. Causal Effect and SSHOM} Does high Causal Effect lead to
strongly subsuming second order mutant? To answer this question, we first
calculate causal effects between program elements of studied programs. We then
distribute the pairs with non-zero causal effect values into ten equal size
buckets; we also group all pairs with causal effect value of 0 in a separate
bucket. Subsequently, we randomly select five program element pairs from
each of the 11 buckets, and generate 100 second order HOMs from each pair. We
then calculated the number of SSHOM made from each bucket. We repeat this ten
times to remove the sampling bias.
\subsubsection*{RQ2. SSHOM Heuristics} How do different heuristics compare to
each other in terms of the number of SSHOMs generated, as well as their
diversity? We implemented all introduced heuristics and calculated the rate of
SSHOMs, as well as the diversity metric. We generate 1,000 HOMs for each
heuristic, and repeat the process five times to remove the sampling bias.
\subsubsection*{RQ3. HOM Survival Rate} Which algorithm achieves the highest
survival rate? We compared the survival rate of HOMs generated with each
heuristic, along with the FOMs. The purpose of RQ3 is to see to what extent higher order
mutants can survive compared to current First Order Mutation testing. For
FOMs, we sample 1,000 mutants generated by MUSIC~\cite{Phan2018aa}.
\subsection{Diversity Metric}
We measure the diversity of mutant set by comparing the set of test
cases that kill each mutant . For a mutant $m$ of original program $P$ with test
suite $T$, a kill vector $v \in \mathcal{R}^{|T|}$ of $m$ is defined as
follows.
\begin{equation*}
v_i =
\begin{cases}
1
&
\text{if $T_i$ kills $m$;}
\\[10pt]
0
&
\text{otherwise.}
\end{cases}
\end{equation*}
where $T_i$ refers to $i$-th test case of $T$. ($1 \le i \le |T|$)
The diversity of set of mutants $M = \{m_1, m_2, \dots, m_k\}$ is then defined as follows.
\begin{equation}
dScore(M) = \frac{|\{v_1, v_2, \dots \}|}{|M|}
\end{equation}
\noindent where $v_i$ is the kill vector of $m_i$, and the numerator
represents the number of distinct kill vectors. A higher $dScore$ is
achieved if the mutant set contains more mutants with distinct kill
vectors. The reason we define diversity with kill vectors is because two
mutants with same kill vectors are not distinguishable from the point of test
suite~\cite{Shin2016qy}.
\subsection{Benchmark Programs}
We study two C programs: a toy example called Bill's Car, and \texttt{schedule} from the SIR benchmark~\cite{Do:2005zp}.
\subsubsection{Bill's Car}
Algorithm~\ref{alg:billscar} shows Bill's Car, which is a C program that calculates parking fees. The fee depends on the
day of the week, the car type, and minutes stayed in the parking zone. There
are three kinds of vehicle types. Senior, car, and truck. The fee structure, as
well as discounts based on the day of week, means that the program contains nontrivial dependence structure.
\begin{algorithm}[ht]
\caption{Pseudo code of Bill's Car\label{alg:billscar}}
\scriptsize
\DontPrintSemicolon
\SetKwProg{Main}{Function}{:}
\Main{Main(vehicle, minutes, day)}{
\eIf{$vehicle = senior$}
{$fee = 0$}{
\eIf{$vehicle \neq car$ \&\& $vehicle \neq truck$}
{$Invalid Vehicle$}{
\eIf{$vehicle = car$}{
$cost = Compute Car Fee(minutes)$}{
$cost = Compute Truck Fee(minutes)$
}
\eIf{$cost = -1$}
{$fee = -1$}{
\eIf{day = Thursday}
{$cost = 0.9 \times cost$}{
\If{day = Saturday}
{$cost = 1.1 \times cost$}
}
$fee = cost$ \\
$Print Fee(vehicle, day, minutes, fee)$
}
}
}
}
\SetKwProg{Car}{Function}{:}
\Car{ComputeCarFee(duration)}{
$hours = duration / 60$ \\
\eIf{$hours \le 2$}
{$fee = 0$}{
\eIf{$hours \le 5$}
{$fee = 0.5 \times (hours - 2)$}{
\eIf{$hours \le 15$}
{$fee = 0.5 \times 3 + 0.25 \times (hours - 5)$}{
$fee = -1$
}
}
}
$return$ $fee$
}
\SetKwProg{Truck}{Function}{:}
\Truck{ComputeTruckFee(duration)}{
$hours = duration / 60$ \\
\eIf{$hours \le 1$}
{$fee = 0$}{
\eIf{$hours \le 3$}
{$fee = 1.0 \times (hours - 1)$}{
\eIf{$hours \le 15$}
{$fee = 1.0 \times 2 + 0.75 \times (hours - 3)$}{
$fee = -1$
}
}
}
$return$ $fee$
}
\end{algorithm}
The test suite for Bill's Car is consisted of 101 test cases and is constructed in
a combinatorial manner. It covers three vehicle types, three categories of
days of week, and 11 different time intervals (0 to 1,000 minutes at intervals
of 100): this results in $3 \times 3 \times 11$ test cases. We add two edge
cases, one with invalid car type and the other with missing arguments (only car type specified).
We use 6,400 mutants (100 mutants for each of 64 program elements) to build CPDA.
\subsubsection{Schedule}
Schedule is a C program from the Software-artifact Infrastructure
Repository~\cite{Do:2005zp}. It is a schedule that calculates an ordering of
given tasks. We use the coverage-extended test suite 456, which contains 81
test cases.
To build CPDA, we use 940 mutants (10 mutants for each of 94 program elements).
\subsection{Implementation and Environment}
Every first order mutants are made by C mutation testing tool,
MUSIC~\cite{Phan2018aa}. Second order HOMs are generated by independently
mutating program elements in the given pair, and combining the mutated lines.
We use the \texttt{networkx} Python library to compute maximum weight matching.
Experiments for Bill's Car was ran on Ubuntu 18.04, Intel(R) Core(TM) i7-10700
CPU @ 2.90GHz with GeForce RTX 3070. Experiments for schedule was ran on
Ubuntu 16.04.5, Intel(R) Core(TM) i7-6700 CPU @ 3.40GHz with GeForce GTX 1080.
\section{Future Works}
\section{Heuristics for SSHOM Generation}
\label{sec:heuristics}
In this section, we introduce heuristics to effectively sample second order mutants (SOM).
\subsection{Heuristics for selecting SOM}
In order to search for efficient methods of sampling second order mutants we
came up with four heuristics: \emph{Random}, \emph{Prop},
\emph{Dsort}, and \emph{MWM}. The purpose of each heuristic is to 1) sample as
much SSHOM as possible with fixed number of mutants, 2) sample diverse
mutants. Therefore, each heuristic tries to find the best pair of program
elements to generate the second order mutants along with the number of mutants
to generate per pair.
While we investigate through second order mutants, it can easily be extended
to select three for more mutation locations utilizing the causal effect (e.g.,
choosing $n$ locations whose sum of the pairwise causal effect is high.)
All heuristics other than Random approach make use of the dependency
calculated in terms of causal effect. Therefore, modeling CPDA and
calculating the causal effect comes firsthand for the three last heuristics.
The four heuristics are as
follows.
\subsubsection{Random}
Among all possible pairs of program elements, we choose a random pair. We repeat
this process independently for number of total mutants, while allowing
duplicates. If a certain pair is selected $k$ times, we create $k$ mutants by
mutating the pair of program elements.
\subsubsection{Prop (Proportional)}
We first weight each pair of program elements with their causal effect
values. Subsequently, we choose a pair with a probability proportional to its
weight. Let $\mathit{PP}$ be the set of all pairs of program elements. Let $CE(p_i)$ be the causal effect value of $p_i \in \mathit{PP}$. The probability of $p_i$ being selected, $P(p_i)$ is:
\begin{equation}\label{eq:proportional}
P(p_i) = \frac{CE(p_i)}{\sum_{p_j \in \mathit{PP}} {CE(p_j)}}
\end{equation}
Note that, according to Equation~\ref{eq:proportional}, a pair with causal
effect value of 0 cannot be selected. We repeat this process of choosing a
pair for number of total mutants. Similar to Random, if a pair of program
elements is selected $k$ times, we build $k$ mutants by mutating the pair.
\subsubsection{Dsort (Descending Sort)}
With Dsort, we sort all pairs of program elements according to their causal
effect values, and choose the top $n$ pairs. Subsequently, we distribute the
number of mutants to generate, $k$, equally to the chosen $n$ pairs. In our
evaluation, we set $n$ for Dsort as the number of pairs selected by the MWM heuristic.
\subsubsection{MWM (Maximum Weight Matching)}
While Dsort picks program pairs by considering solely causal effect,
MWM considers the diversity of the mutant set. Since the purpose of each
mutant is to mimic real faults of developers a good mutant set should contain
diverse mutants rather than mutants with similar faults.
To sample diverse set of mutants, we utilize \emph{maximum weight matching} from graph theory.
A set $M$ of independent edges in a graph $G = (V, E)$ is called a
matching~\cite{Diestel:2016aa}. A maximum weight matching $M$ of graph
$G = (V, E)$ where every edge $e \in E$ have weight $w_e$ is set $M$ of
independent edges maximizing the sum of weights of edges in $M$.
To perform maximum weight matching with respect to causal effects in the program,
we modify the causal structure $G$.
Whenever there is a path from $v_1$ to $v_2$, we add a directed edge from $v_1$ to $v_2$.
Subsequently, we weight all edges: the weight
of $e$ from $v_1$ to $v_2$ is the causal effect $v_2$ gets from $v_1$.
After modifying the causal structure $G$ we compute the maximum weight
matching of $G$. The aim is not only to select pairs with high dependency, but
also to select a diverse set of program pairs across the entire program.
Similar to Dsort, total number of mutants to be made is then distributed
equally to all program pairs, making same number of second order mutants per
pair.
\section{A Motivating Example}
\label{sec:motivating_example}
Consider a HOM $h$ that is composed of two FOMs, $f_1$ and $f_2$. In order for
$h$ to be a SSHOM, there must exist at least one test, $t$, such that $t$
fails due to $f_1$ as a FOM, which is masked by $f_2$, so that $t$ does not
fail under $h$. For the masking to take place, it is
reasonable to assume that the locations of $f_1$ and $f_2$ are connected by
program dependence: otherwise, a mutation in one of the location is not likely
to mask the mutation in another.
\begin{algorithm}[ht]
\caption{Example Program\label{alg:example}}
\DontPrintSemicolon
\SetKwProg{Main}{Function}{:}
\Main{Main (b: int)}{
$a = 1$
$a = a + 1$
\If{$b~\%~2 = 0$}
{$a = a \times 2$}
$c = 100$
$return$ $a$
}
\end{algorithm}
Algorithm~\ref{alg:example} contains a concrete motivating example. We are
going to use Expression Replacement mutation operator to generate HOMs. Let
Line 2 be one of the constituent FOMs: for example, we mutate it to $a = 2$.
Now, let us consider the program element in Line 3. Whenever $a$ in Line 2
gets mutated, the effect is delivered to program element in Line 3. In other
words, program element in Line 3 is \emph{frequently} dependent on program
element in Line 2. In contrast, the variable $a$ in Line 5 is affected by
$a$ in Line 2 only when $b$ is even. The element in Line 5 is, therefore,
\emph{less frequently} dependent on element in Line 2 when compared to
$a$ in Line 3. Finally, program element in Line 6 is not dependent on $a$
in Line 2.
Now consider mutating either Line 3, 5, or 6, to mask the effect of the mutation
in Line 2 towards Line 7.
Mutating program element in Line 3 has highest chance of masking the first
mutation, since the effect of the mutation in Line 2 is always delivered to
Line 3. Mutating Line 5 \emph{can} mask the fault, but with a smaller
probability, since the effect of the mutation in Line 2 is only delivered when
$b$ is even. Finally, no mutation in Line 6 can mask the mutation in Line 2,
since it cannot change the value of $a$.
Through the motivating example, we can observe that the more \emph{frequently}
dependence relation occurs between two program elements, the higher the
chance of fault masking will be. The fact that we need program dependence
relationship between two program elements for fault masking to happen may be a
trivial observation. However, what Causal Program Dependence Analysis allows
us to reason about is the relative likelihood of a dependence relationship
actually affecting the value of a specific program element. Unlike traditional
dependence analysis whose outcome is binary (either dependent or not), CPDA
allows us to reason about the degree of dependence quantitatively. We will
present a brief introduction to CPDA in Section~\ref{sec:cpda}.
\section{Results}
\label{sec:results}
\begin{table}[t]
\caption{Causal Effect and Average Number of SSHOMs per Bucket}\label{tbl:buckets}
\centering
\scalebox{0.8}{
\begin{tabular}{r||rrr|rrr}
\toprule
& \multicolumn{3}{c|}{Bill's Car} & \multicolumn{3}{c}{Schedule} \\
Buc. & Pairs & CE Range & Avg. SSHOMs & Pairs & CE Range & Avg. SSHOMs\\ \midrule
0 & 2,094 & 0 & 0.1 & 1,392 & 0 & 0.6 \\
1 & & 0.004 - 0.144 & 0.0 & & 0.008 - 0.021 & 0.0 \\
2 & & 0.145 - 0.229 & 0.0 & & 0.021 - 0.030 & 0.1 \\
3 & & 0.229 - 0.297 & 0.0 & & 0.030 - 0.051 & 1.8 \\
4 & & 0.297 - 0.349 & 0.0 & & 0.051 - 0.080 & 3.7 \\
5 & & 0.349 - 0.397 & 0.0 & & 0.080 - 0.133 & 0.1 \\
6 & & 0.397 - 0.437 & 1.1 & & 0.133 - 0.167 & 5.4 \\
7 & & 0.437 - 0.486 & 1.7 & & 0.167 - 0.218 & 0.0 \\
8 & & 0.487 - 0.553 & 0.7 & & 0.222 - 0.322 & 1.9 \\
9 & & 0.553 - 0.669 & 1.1 & & 0.326 - 0.495 & 4.1 \\
10 & & 0.669 - 1.000 & 12.9 & & 0.495 - 0.997 & 11.8\\
\bottomrule
\end{tabular}}
\end{table}
\subsection{RQ1: Causal Effect and number of SSHOM}
Table~\ref{tbl:buckets} shows the results of the bucketing analysis for RQ1.
While Bill's Car had more program element pairs with zero causal effect,
Schedule had more program element pairs with positive causal effects.
For Bill's Car, $2.25\%$ of the total pair of program elements turned out to
have causal effect value over $0.5$ while for Schedule $18\%$ of pairs were.
The range of causal effect values for each bucket tends to increase.
Average range of first three buckets for the two benchmark programs was
$0.056$, while the average range of final three buckets was $0.214$.
This suggests that the space of program pairs get sparse as the causal effect
value goes up.
\begin{figure}[ht]
\centering
\subfigure[Bill's car\label{fig:SSR-bill}]{\includegraphics[angle=0, width=0.23\textwidth]{figure/billcar_rq1.pdf}}
\subfigure[Schedule\label{fig:SSR-sch}]{\includegraphics[angle=0, width=0.23\textwidth]{figure/schedule_rq1.pdf}}
\caption{Boxplot of Number of SSHOMs per Bucket\label{fig:boxplots}}
\end{figure}
Figure~\ref{fig:SSR-bill} and \ref{fig:SSR-sch} shows the number of SSHOM for
the two benchmark programs. From all SSHOM made from all trials, $73.71\%$
SSHOM were from the top bucket for Bill's Car while for Schedule, $40.83\%$
SSHOM were. Specific number of average SSHOM made per bucket for each trial (5
pairs from bucket with 100 HOMs each) is shown in Table~\ref{tbl:buckets}. We
were able to observe that the top bucket does significantly better job of
generating SSHOM than other buckets.
Since number of program element pairs in the top bucket are much smaller than
total pairs ($6.85$\% for Schedule, $2.40$\% for Bill's Car), we can
significantly reduce the search space for second order HOMs by focusing on
the pairs in the top bucket. Based on these results, we conclude that mutating
program elements with high causal effect can lead to the generation of
SSHOMs with higher probability.
\begin{figure}[ht]
\centering
\subfigure[Bill's Car - Number of SSHOMs\label{fig:billssr}]{\includegraphics[angle=0, width=0.24\textwidth]{figure/billcar_rq2_ssm.pdf}}
\subfigure[Bill's Car - Unique SSHOMs\label{fig:billssmdiv}]{\includegraphics[angle=0, width=0.24\textwidth]{figure/billcar_rq2_ssm_div.pdf}}
\subfigure[Schedule - Number of SSHOMs\label{fig:schedulessr}]{\includegraphics[angle=0, width=0.24\textwidth]{figure/schedule_rq2_ssm.pdf}}
\subfigure[Schedule - Unique SSHOMs\label{fig:schedulessmdiv}]{\includegraphics[angle=0, width=0.24\textwidth]{figure/schedule_rq2_ssm_div.pdf}}
\caption{Number of Total and Unique SSHOMs Generated by Heuristics}
\label{fig:rq2total}
\end{figure}
\begin{table}[ht]
\centering
\caption{Mutant Diversity of Heuristics}
\label{tbl:rq2}
\scalebox{0.8}{
\begin{tabular}{lrrr|lrrr}
\toprule
Heuristic & \multicolumn{3}{c}{Bill's Car} & \multicolumn{3}{c}{Schedule} \\
& dScore & SSHOM & Uniq. SSHOM & dScore & SSHOM & Uniq. SSHOM \\ \midrule
Random & 0.282 & 3.0 & 2.4 & 0.457 & 3.2 & 3.2 \\
Prop & 0.226 & 10.8 & 4.4 & 0.471 & 4.8 & 4.4 \\
Dsort & 0.096 & 65.6 & 2.8 & 0.135 & 54.0 & 11.0 \\
MWM & 0.118 & 38.2 & 5.6 & 0.282 & 41.8 & 6.6 \\
\bottomrule
\end{tabular}}
\end{table}
\subsection{RQ2: Performance of each Heuristic}
Figure~\ref{fig:rq2total} shows for each benchmark programs the number of
evaluated SSHOM, dScore, and the number of distinct kill vectors for the
generated SSHOM. Table~\ref{tbl:rq2} shows the average number of calculated
metrics. Column dScore contains the diversity score of all generated HOMs (not
necessarily strongly subsuming); column SSHOM contains the average number of
generated SSHOMs, and column
Uniq. SSHOM contains the average number of generated SSHOMs with unique kill
vectors. Dsort generates the largest number of SSHOMs, which is $21.87$,
and $16.88$ times more than Random for Bill's Car and Schedule, respectively.
Although MWM produced fewer SSHOMs, it still generates $12.73$, and $13.06$
times more SSHOM than Random, respectively. This reflects the fact that Dsort
only prioritises the selection of pairs based causal effects.
Prop is less successful in generating many SSHOMs: by definition
it ends up choosing more program pairs than Dsort or MWM, resulting in
generation of fewer mutants per chosen pair. For Schedule, Prop selects 818.8
pairs on average, while Random selects 897.8, and Dsort and MWM only 21.
However, the higher diversity of chosen pairs results in higher dScore.
\begin{figure}[t]
\centering
\subfigure[MWM\label{fig:chosen_mwm}]{\includegraphics[angle=0, width=4cm]{figure/schedule_graph_mwm.pdf}}
\subfigure[Dsort\label{fig:chosen_dsort}]{\includegraphics[angle=0, width=4cm]{figure/schedule_graph_dsort.pdf}}
\caption{Pairs of Program Elements in Schedule Chosen by MWM and Dsort\label{fig:chosen_pairs}}
\end{figure}
Figure~\ref{fig:billssmdiv} and \ref{fig:schedulessmdiv} shows boxplots of the
unique number of SSHOMs generated by different heuristics. MWM produces more unique SSHOMs for Bill's Car than Dsort, but the
trend is the opposite in Schedule. We suspect that diversity of HOMs in
general (captured by dScore in Table~\ref{tbl:rq2}), and the diversity of
SSHOMs, may not align perfectly.
\begin{figure}[ht]
\centering
\subfigure[Bill's Car - \# of Survived Mutants\label{fig: Bill's Car Comparison with FOM}]{\includegraphics[angle=0, width=0.24\textwidth]{figure/billcar_rq3_survive.pdf}}
\subfigure[Schedule - \# of Survived Mutants\label{fig: Schedule Comparison with FOM}]{\includegraphics[angle=0, width=0.24\textwidth]{figure/schedule_rq3_survive.pdf}}
\caption{Survived Mutants}\label{fig:survived}
\end{figure}
MWM heuristic successfully diversified the generated mutants. Figure~\ref{fig:chosen_mwm} and \ref{fig:chosen_dsort} visualises the program element
pairs chosen by MWM and Dsort. The causal structure is shown in gray edges,
while chosen pairs are shown with blue arrowed edges. While chosen pairs from
MWM tend to be spread out in various places by not sharing common vertices,
pairs from Dsort does overlap and tend to cover only some specific regions of
the program, as expected.
\begin{table}[ht]
\centering
\caption{Average number of Survived Mutants\label{tbl:avgsurvive}}
\scalebox{0.8}{
\begin{tabular}{lrr}
\toprule
& Bill's Car & Schedule \\ \midrule
Random & 90.4 & 18.8 \\
Prop & 33.2 & 14.8 \\
Dsort & 12.6 & 14.6 \\
MWM & 94.0 & 18.2 \\
FOM & 194.0 & 104.8 \\
\bottomrule
\end{tabular}}
\end{table}
\subsection{RQ3: Survival Rate of each Heuristic}
Figure~\ref{fig:survived} shows the number of surviving mutants generated by
each heuristic, along with the number of surviving FOMs. The average values
are reported in Table~\ref{tbl:avgsurvive}. It shows that HOMs are easier to kill, possibly due to the larger semantic differences.
Among the proposed heuristics, MWM showed highest number of surviving mutants,
followed by Random. Dsort shows the lowest survival rate. The survival rate
differs a lot for Bill's Car while there are not so much variance in Schedule.
We observe that, in Bill's Car, there are specific program locations that
produce more surviving mutants than others. For example, mutants generated in
the flow of \texttt{PrintFee} function tend to survive more. Since
the main objective of the function is to print status, the return value is not
used anywhere. Consequently, it is harder to kill.
We also observe that pairs with high causal effect values tend to exist in the
part of program with main functionalities. For example, pairs chosen by Dsort
from Bill's Car are mostly from functions calculating the fee, while MWM also
chooses from the \texttt{PrintFee} function, resulting in a higher survival
rate. Random and Prop all showed high survival rate due to a similar reason.
In Schedule, survival rates of mutants are not significantly affected by the
location of mutation.
\section{Causal Program Dependence Analysis}
\label{sec:cpda}
\emph{Causal Program Dependence Analysis} (CPDA), a recently introduced dynamic program dependence analysis technique, can measure the degree of dependence between two program elements~\cite{Lee:2021aa}. Applying the causal inference~\cite{Pearl:2009aa} on the program execution trajectory, CPDA estimates how often a change of the value of a program element causes a change of the value of another program element.
CPDA calculates the dependence by a given test suite. CPDA first generates data on which program element's values are associated. It gets association data by running tests on programs that have modified part of the code in various ways, observing which program elements have simultaneously changed compared to the values they had in the original program. Given the association data, CPDA discovers the \emph{causal structure} of the program. The causal structure is a directed graph that represents the direct dependence between program elements; for each child node in the causal structure, the set of parent nodes (immediate predecessor nodes) comprises a minimal Markov blanket of the behavior of the child node~\cite{Pearl:2009aa}. Using the association data and discovered causal structure, CPDA estimates two metrics representing the degree of dependence. A \emph{causal effect} is an aggregate of the effect of each program element's change causing a change in other program elements. A \emph{direct effect} is the effect of one program element on another, excluding all the indirect effects through other program elements.
Given association data $O$, the causal effect from a program element $S_i$ to the other program element $S_j$, denoted as $\mathit{CE}_O(S_1, S_j)$, is calculated as follows:
\begin{align*}
\label{def:ce}
\mathit{CE}_O(S_i, S_j) &= P_O(S_j = 1 \mid do(S_i = 1)) \\ &\times (1 - P_O(S_j = 1 \mid do(S_i = 0))).
\end{align*}
. In the equation, $S_j = 1$ implies the value of the program element $S_j$ is changed compared to the original program, or otherwise, $S_j = 0$. $P_O(x \mid do(y))$ calculates the probability of an event $x$ \emph{caused} by the event $y$~\cite{Pearl:2009aa}. The causal effect aims to measure the difference of the effect that $S_j$ gets when $S_i$ moves its state from \emph{unchanged} (0) to \emph{changed} (1). Instead of subtracting the probability, we multiply the probability when $S_i = 1$ and the complementary probability when $S_i = 0$, keeps the causal effect having a positive value.
From the example in Sec.~\ref{sec:motivating_example}, SSHOMs can be created more easily from pairs of program locations that affect more frequently. Thus, we claim that we could efficiently search the higher-order mutant space to sample SSHOMs by utilizing the CPDA. Our main hypothesis is that the causal effect could prioritize the second-order mutation position for SSHOMs. Our empirical evaluation investigates whether there is a positive correlation between the high causal effect and the strongly subsuming rate. Assuming the hypothesis is plausible, setting up an actual guideline for searching on the higher-order mutant space is needed. In the next section, we introduce several heuristics that can efficiently sample the higher-order mutant using the causal effect.
\section{Related Work}
\label{sec:related_work}
The concept of Subsuming Higher Order Mutants was proposed by Jia et
al.~\cite{Jia2009id}, as a way to avoid equivalent mutants and to reduce the
number of mutants to examine. Jia et al. present more detailed classification
of HOMs, but we focus only on SSHOMs in this preliminary study.
One of the most widely studied topic in Higher Order Mutation Testing is how
to efficiently generate SSHOMs. Harman et al. generates SSHOMs using genetic
algorithm~\cite{Harman14ase}. Since the fitness evaluation involves executing
all candidate SSHOMs, the cost of the search-based approach can be high. Wong
et al.~\cite{Wong2020dm} uses variational execution and SAT solver to
efficiently find SSHOMs. Our approach depends on CPDA, which in turn uses
mutation analysis to compute concrete causal effect values~\cite{Lee:2021aa}.
However, compared to fitness guided search, the cost of CPDA can be controlled
by the parameters (i.e., how many mutants to consider for CPDA).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,664 |
Martin Ney ist der Name folgender Personen:
* Martin Ney (Diplomat) (Martin C. Ney; * 1956), deutscher Diplomat
Martin Ney (Serienmörder) (* 1970), deutscher Serienmörder und Pädokrimineller | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,233 |
'use strict';
var _ = require('lodash'),
nowdoc = require('nowdoc'),
tools = require('./tools');
describe('ESFive accessor properties', function () {
_.each({
'simple object accessor property with getter': {
code: nowdoc(function () {/*<<<EOS
var object = {},
title = 'This is the title';
Object.defineProperty(object, 'title', {
get: function () {
return title;
}
});
return object.title;
EOS
*/;}), // jshint ignore:line
expectedResult: 'This is the title'
},
'simple object accessor property with setter': {
code: nowdoc(function () {/*<<<EOS
var object = {},
result;
Object.defineProperty(object, 'title', {
set: function (value) {
result = value;
}
});
object.title = 'This is another title';
return result;
EOS
*/;}), // jshint ignore:line
expectedResult: 'This is another title'
},
'object with accessor property inherited via prototype chain': {
code: nowdoc(function () {/*<<<EOS
var create = function (from) {
function F() {}
F.prototype = from;
return new F();
},
object,
funkiness;
function FunkyBase() {}
function FunkySub() {}
FunkySub.prototype = create(FunkyBase.prototype);
object = new FunkySub();
Object.defineProperty(FunkyBase.prototype, 'funky', {
get: function () {
return 'and ' + funkiness;
},
set: function (value) {
funkiness = 'then some ' + value;
}
});
object.funky = 'more';
return object.funky;
EOS
*/;}), // jshint ignore:line
expectedResult: 'and then some more'
},
// Make sure we don't pass the .prototype object as the 'this' object
'object with accessor property inherited via prototype chain should set the "this" object correctly in getter and setter': {
code: nowdoc(function () {/*<<<EOS
var create = function (from) {
function F() {}
F.prototype = from;
return new F();
},
getterThisObject,
object,
setterThisObject;
function FunkyBase() {}
function FunkySub() {}
FunkySub.prototype = create(FunkyBase.prototype);
object = new FunkySub();
Object.defineProperty(FunkyBase.prototype, 'funky', {
get: function () {
getterThisObject = this;
},
set: function () {
setterThisObject = this;
}
});
object.identifier = 'This one';
object.funky;
object.funky = 6;
return {
getterThisObjectIdentifier: getterThisObject.identifier,
setterThisObjectIdentifier: setterThisObject.identifier
};
EOS
*/;}), // jshint ignore:line
expectedResultDeep: true,
expectedResult: {
getterThisObjectIdentifier: 'This one',
setterThisObjectIdentifier: 'This one'
}
},
'property lookup chain including getter that returns a function': {
code: nowdoc(function () {/*<<<EOS
var firstMember = {},
firstPerson = {},
members = {},
name = 'Dan';
Object.defineProperty(members, 'first', {
get: function () {
return firstMember;
}
});
Object.defineProperty(firstMember, 'getPerson', {
get: function () {
return function () {
return firstPerson;
};
}
});
Object.defineProperty(firstPerson, 'name', {
get: function () {
return name;
}
});
return members.first.getPerson().name;
EOS
*/;}), // jshint ignore:line
expectedResult: 'Dan'
},
'lookup of accessor property in object literal property initializer': {
code: nowdoc(function () {/*<<<EOS
var object = {},
otherObject;
Object.defineProperty(object, 'thing', {
get: function () {
return 'A thing';
}
});
otherObject = {
value: object.thing
};
return otherObject.value;
EOS
*/;}), // jshint ignore:line
expectedResult: 'A thing'
},
'lookup of accessor property on String.prototype from primitive string value': {
code: nowdoc(function () {/*<<<EOS
Object.defineProperty(String.prototype, 'doubleLength', {
get: function () {
return this.length * 2;
}
});
return 'abc'.doubleLength;
EOS
*/;}), // jshint ignore:line
expectedResult: 6
},
'lookup of two accessor properties defined with Object.defineProperties(...)': {
code: nowdoc(function () {/*<<<EOS
var me = {};
Object.defineProperties(me, {
name: {
get: function () {
return 'Dan';
}
},
occupation: {
get: function () {
return 'Engineer';
}
}
});
return me.name + ', ' + me.occupation;
EOS
*/;}), // jshint ignore:line
expectedResult: 'Dan, Engineer'
},
'lookup of accessor property on right-side of assignment to variable': {
// ...
}
}, tools.check);
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,219 |
\section{Introduction}
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth, height=0.75\linewidth]{figures/adv_intro_img.png}
\label{fig:intro_img}
\caption{\textbf{The big picture of DWP.} Based on the over-parameterized property of neural networks, we leverage weight pruning to produce additional diversified newborn models from existing white-box networks at each iteration. Since weight pruning only removes redundancy in each network, the quality of models is well-preserved. These additional diversified models benefit current ensemble-based approaches for transferable targeted attacks.}
\vspace{-15pt}
\end{figure}
While deep learning continues to achieve breakthroughs in various domains, recent studies have shown vulnerabilities of deep neural networks to adversarial attacks, causing severe threats in safety-critical applications. For example, in image classification, an attacker can create adversarial examples by adding human-imperceptible perturbations to benign images at test. These adversarial images can mislead a well-trained convolution neural network (CNN) to yield arbitrary classification results.
Several adversarial attacks have been proposed to improve and evaluate \cite{IEEESP_CW, ICLR_ens_AdvTraining, NIPS_Cross} the robustness of CNNs.
In the white-box settings, with full information of the victim model, adversarial examples can be generated effectively and efficiently. In the black-box settings, where the attacker only has limited information of the
victim model, it is still possible to create cross-model attacks by using a substitute model with white-box adversarial attack methods. This kind of black-box attacks depend on the property of transferability among different CNN models.
Many methods have been proposed to increase the transferability for untargeted attacks, where the goal is to decrease the accuracy of the victim model. However, there is still room for improvement in creating transferable targeted attacks, where the attacker need to mislead the victim model to produce a predefined specific outcome.
Recent work use an ensemble-based approach to generate transferable targeted adversarial examples with multiple CNNs models~\cite{DelvingLiu2017,NIPS_Simple}.
Inspired by the benefit of utilizing multiple CNNs simultaneously, we further enhance the ensemble-based approach by creating more CNN models from a small set of white-box models.
We focus on generating transferable targeted adversarial examples on the classical image classification task, proposing a novel approach named Diversified Weight Pruning (DWP). Weight pruning \cite{NIPS_L1Pruning} is a technique commonly used in model compression. Prior work has shown that neural networks are often over-parameterized \cite{NIPS_overparameterized}, and proposed methods to remove redundancy \cite{NIPS_OptimalBrain, NIPS_L1Pruning, ICLR_Lottery, ICLR_RethinkingPruning}.
We apply random weight pruning to each single CNN network accessible to form additional ones. These additional pruned networks with certain diversity can still preserve high accuracies comparable to that of the original model. We thus improve the ensemble-based approach with these extra diverse models.
To evaluate DWP, we experiment with an ImageNet-compatible dataset used in the NIPS 2017 adversarial competition \cite{NIPS2017_competition}. The average targeted success rate of our proposed DWP reaches 95.22\% on the commonly adopted networks. Furthermore, we follow \cite{NIPS_Simple} and test DWP in the more challenging scenarios of transferring to adversarially trained models and to distinct architectures. The results show that DWP can improve the targeted success rate with up to 8.0\% and 4.1\% on average in these two setting, respectively.
In summary, our primary contributions are as follows:
\begin{itemize}
\item We propose a novel approach DWP leveraging weight pruning to enhance the ensemble-based method for transferable targeted attacks, which is compatible with most of the current techniques.
\item Our proposed DWP is lightweight and portable without extra data or network training.
\item The experiment results show that DWP remains effective in the more challenging settings of transferring to adversarially trained models or to distinct model architectures
\item We analyze cosine similarity between perturbations acquired by attacking different pruned networks. It supports the claim that weight pruning increases the diversity of networks for generating adversarial perturbations.
\end{itemize}
\section{Related Work}
\subsection{Transferable Attack}
In this paper, we focus on simple transferable attacks \cite{NIPS_Simple}, which require neither additional data nor model training for attacking compared to resource-intensive attacks. Recent works aiming for simple transferable attacks mainly include four categories: gradient calibration, input transformation, advanced loss function, ensemble, and network augmentation.
\subsubsection{Gradient Calibration.}
Optimization-based methods are widely adopted \cite{IntriguingPropertiesAdv, HarnessAdv, IEEESP_CW, ICLR_BIM} in generating adversarial examples. With iterative methods \cite{ICLR_BIM, IEEESP_CW}, one can get better solutions to an objective function for attacking through multiple times of optimization on adversarial examples and get stronger attacking results. Adjusting gradients used to update adversarial examples at each iteration appropriately has been shown beneficial for overcoming sub-optimal results in optimization.
\cite{BoostingMomentum} combines momentum techniques with iterative attacks, accumulating gradients at each iteration to escape local optimum and stable the direction of updating.
\cite{ICLR_Nesterov} applies Nesterov accelerated gradient for optimization, giving adversarial examples an anticipatory updating to yield faster convergence.
\subsubsection{Input Transformation.}
Motivated by Data Augmentation \cite{BigData_DataAugmentation}, several works suggest attacking transformed input to prevent adversarial examples from overfitting white-box models and failing to transfer to black-box ones.
\cite{CVPRDI} uses random resizing and padding throughout the iterative attack. \cite{CVPRTI} enumerates several translated versions for each input image and fuses the gradients acquired on all of them. \cite{ICLR_Nesterov} leverages the scale-invariant property of CNNs and employs multiple scale copies from each input image. \cite{ICCV_Admix} extends the concept of mixup \cite{ICLR_Mixup}, attacking the mixup version of each input image.
\subsubsection{Modern Loss Function.}
Cross entropy loss is widely used in image classification, also serving as the objective function for adversarial attacks. However, for targeted attacks, cross entropy is pointed out the saturation problem \cite{CVPR_Poincare} as the output confidence of target class approaches to one. To this end, alternative loss functions attempt to provide more suitable gradients for optimization. \cite{CVPR_Poincare} leverages Poincar\'e distance as the loss function, which amplifies the gradient magnitude as the confidence of the target class grows. \cite{NIPS_Simple} proposes a simple logit loss, which has constant gradient magnitude regardless of the output probability.
\subsubsection{Ensemble and Network Augmentation.}
Adversarial examples generated by ensembling multiple white-box networks are more likely to transfer to black-box networks \cite{DelvingLiu2017}. Instead of simply fusing the output confidence of each white-box network, \cite{CVPR_red_var_ensemble} suggests reducing the gradient variance of white-box models during attacking.
To further improve ensemble-based approaches, Network Augmentation produces additional diverse models from the existing white-box networks. \cite{AutoMA} uses reinforcement learning to automatically find transformations suitable with white-box networks to yield more diversity. \cite{AAAI_Ghost} acquires ghost networks for ensemble through perturbing dropout and skip connections of existing ones. \cite{dual_erosion} further improves the diversified ensemble via dual-stage erosion.
\subsection{Network Pruning}
The intensive cost of computation and storage hinders applications of neural networks, especially on embedding systems. Network Compression aims to reduce the scale of networks, making them more feasible for deployment. With the over-parameterized property \cite{NIPS_overparameterized}, several works about removing redundancy in networks, known as Network Pruning, are proposed and become a branch of Network Compression. \cite{NIPS_OptimalBrain} uses the second-derivative information to find redundant weights in networks. \cite{NIPS_L1Pruning} shows that neural networks can highly preserve performance even if trimming more than half of their connections. Retraining after pruning for better preservation of accuracy is also investigated \cite{ICLR_Lottery, ICLR_RethinkingPruning}.
\section{Methodology}
To generate a targeted adversarial example $x^{\textrm{adv}}$ with target class $y^{\textrm{target}}$ for a network $\theta$ from an benign image with its label $(x,y)$, we aim to solve the following constrained optimization problem:
$$\mathop{\arg\min}_{x^{\textrm{adv}}} J(x^{\textrm{adv}}, y^{\textrm{target}};\theta) \quad \textrm{s.t.} \quad \left\|x^{\textrm{adv}}-x\right\|_\infty \leq \epsilon,$$
where $J$ is the loss function for multiclass classification, and $\epsilon$ is the perturbation budget ensuring each adversarial example keeps human-imperceptible. To circumvent the gradient decreasing problem of cross-entropy, we adopt logit loss \cite{NIPS_Simple} as our loss function $J$.
\subsection{Preliminary \& Motivation}
We start by establishing the roles of current state-of-the-art techniques in our iterative attack. After that, we demonstrate how we apply Weight Pruning to improve targeted transferability.
\subsubsection{Momentum and Nesterov Iterative Method (NI) \cite{BoostingMomentum, ICLR_Nesterov}}
Considering that the optimization of the popular Iterative-FGSM method \cite{ICLR_BIM} may fall into local optimum, Momentum Iterative-FGSM integrates the momentum technique to accumulate historical gradients and stable the update direction. Starting from $x_1=x$ and $g_0=0$, we have the iterative procedure:
$$g_n=\mu\cdot g_{n-1}+\frac{\nabla_xJ(x_n, y^{\text{target}};\theta)}{ \left\|\nabla_xJ(x_n, y^{\text{target}};\theta)\right\|_1 }$$
$$x_{n+1} = \textrm{Clip}_x^{\epsilon}(x_n - \alpha\cdot\textrm{sign}(g_n)).$$
Here $\mu$ is the decay factor of the historical gradients. The gradient computed encourages adversarial examples to increase confidence logit output by the white-box network model $\theta$ on the $\text{target}$ class through gradient ascent with learning rate $\alpha$. A clipping operation onto the $\epsilon$-ball centered at the original input image $x$ is at the end of each iteration.
Inspired by Nesterov Accelerated Gradient \cite{1983nesterov}, Nesterov Iterative Method (NI) adds the historical gradients to current adversarial examples $x_n$ and gets $x^{\textrm{nes}}_n$ in advance. Gradients at the ahead $x^{\textrm{nes}}_n$ instead of the current $x_n$ will be used for updating. The scheme helps accelerate convergence by avoiding the local optimum earlier:
$$x^{\textrm{nes}}_n=x_n+\alpha \cdot \mu \cdot g_{n-1}$$
$$g_n=\mu\cdot g_{n-1}+\nabla_xJ(x^{\textrm{nes}}_n, y^{\text{target}};\theta)$$
To preserve more information about the gradient for attacking \cite{AAAI_Indistinguishable}, we don't include the L1 normalization.
\subsubsection{Scale Invariant Method (SI) \cite{ICLR_Nesterov}}
Neural networks can preserve output even though the input image $x$ goes through scale operations such as $S_m(x)=x/2^m$. With the scale-invariant property, each composite of white-box networks and scale operations can serve as an augmentation of models. Adversarial examples can enjoy more diversity with these augmented networks:
$$g_n=\mu\cdot g_{n-1}+\dfrac{1}{M}\sum_{m=0}^{M-1}\nabla_xJ(S_m(x^{\textrm{nes}}_n), y^{\text{target}};\theta).$$
$M$ is the number of scaled versions feeding into the network for each image.
\subsubsection{Diverse Input Patterns (DI) \cite{CVPRDI}}
Inspired by data augmentation techniques \cite{BigData_DataAugmentation} used in network training, DI imposes random resizing and padding on each image before it feeds into network models to avoid overfitting. Straightforward cooperation with NI and SI is as follows:
$$g_n=\mu\cdot g_{n-1}+\dfrac{1}{M}\sum_{m=0}^{M-1}\nabla_xJ(S_m(T(x^{\textrm{nes}}_n, p_{\textrm{DI}})), y^{\text{target}};\theta).$$
The introduced $T$ decides whether to apply DI at each iteration with probability $p_{\textrm{DI}}$, which will degenerate with $p_{\textrm{DI}}=0$.
\subsubsection{Translation Invariant Method (TI) \cite{CVPRTI}}
To deal with different discriminative regions \cite{CVPRTI} of various defense neural networks, TI produces several translated versions for the current image in advance and computes the gradient for each separately. These gradients will then be fused and used to attack the current image. \cite{CVPRTI} also shows that one can approximate the gradient fusion using convolution. The approximation prevents TI from enduring the costly computation on excessive translated versions for every single image, also yielding the further revised updating procedure:
$$g_n=\mu\cdot g_{n-1}+ \textit{\textbf{W}}* \dfrac{1}{M}\sum_{m=0}^{M-1}\nabla_xJ(S_m(T(x^{\textrm{nes}}_n, p_{\textrm{DI}})), y^{\text{target}};\theta).$$
$\textit{\textbf{W}}$ is the convolution kernel matrix applied. Some typical options are linear, uniform, or Gaussian kernel.
\subsubsection{Ensemble-Based Approach \cite{DelvingLiu2017}}
\cite{DelvingLiu2017} suggests that if an adversarial example takes effect on multiple accessible white-box networks, it gets more chance to transfer to other black-box models. With the hypothesis, the ensemble-based approach is enrolled and enriches the procedure with multiple white-box models ($\theta_1, \theta_2, ..., \theta_K$) utilized:
\[
\begin{aligned}
&g_n=\mu\cdot g_{n-1}+ \\
&\textit{\textbf{W}}* \dfrac{1}{M}\sum_{m=0}^{M-1}\sum_{k=1}^{K}\beta_k\nabla_xJ(S_m(T(x^{\textrm{nes}}_n, p_{\textrm{DI}})), y^{\text{target}};\theta_k),
\end{aligned}
\]
where $\beta_k$ are the ensemble weights, $\sum_{k=1}^K\beta_k = 1$. We abbreviate the attacking procedure so far to NI-SI-TI-DI-FGSM with the combination of these techniques.
Motivated by the current methods emphasizing diversity for generating adversarial examples, we aim to acquire more diversified white-box networks from the given ones $\theta_1, \theta_2, ..., \theta_k$ to boost the ensemble-based approach. Research about Network Compression has shown that Weight Pruning can preserve accuracy even with removing up to 50\% of connections in the original network \cite{NIPS_L1Pruning}. In other words, each $\theta_k$ can produce numerous newborn networks $\theta_{k1}, \theta_{k2}, ..., \theta_{kL}$ with comparable accuracy to itself through selecting $L$ different groups of its minor connections and pruning them away. We extend the ensemble-based approach in the above updating procedure:
\[
\begin{aligned}
&g_n=\mu\cdot g_{n-1}+ \\
&\dfrac{\textit{\textbf{W}}}{M}* \sum_{m=0}^{M-1}\sum_{k=1}^{K}\sum_{l=1}^{L}\beta_{kl}\nabla_xJ(S_m(T(x^{\textrm{nes}}_n, p_{\textrm{DI}})), y^{\text{target}};\theta_{kl}),
\end{aligned}
\]
where $\beta_{kl}$ are the ensemble weights, $\sum_{k=1}^{K}\sum_{l=1}^{L}\beta_{kl} = 1$.
\subsection{Diversified Weight Pruning}
Instead of producing all the newborn models beforehand, we integrate our weight pruning method into the iterative attacking. We acquire newborn models at each iteration right before gradient computing. Without keeping all of them simultaneously, storage and memory overhead are almost identical to the original ensemble-based approach.
We name the proposed approach Diversified Weight Pruning (DWP) due to the increased diversity of white-box models for ensemble via Weight Pruning.
To guarantee the newborn model has acceptable performance as the original one, we sort the connections of each white-box network by the L1 norm of their weight values. With a predefined rate $r$, we only consider the lowest ($100\cdot r$)\% ``prunable'' since weights with small values are shown unnecessary \cite{NIPS_L1Pruning}. Networks can preserve accuracy after these connections are pruned away even without retraining \cite{NIPS_L1Pruning}.
For our pruning operation, we first identify the set of prunable weights. Let $\gamma$ be the ($100\cdot(1-r)$)-th percentile of weights in $\theta$. We formulate the prunable set:
$$\Gamma(\theta, r)=\{w\in\theta|w<\gamma\} \subseteq \theta.$$
With $\Gamma(\theta, r)$ collecting all the prunable weights of $\theta$, we introduce an indicator vector for it:
$$\Pi_{\Gamma(\theta, r)}=(\lambda_1, \lambda_2, ..., \lambda_{\kappa}),$$
where $\kappa$ is the total number of weights in $\theta=\{w_1, w_2, ..., w_\kappa\}$ including non-prunable ones. $\lambda_i$ is determined by whether its corresponding $w_i\in\theta$ is in the prunable subset $\Gamma(\theta, r)$:
\[
\lambda_i=
\begin{cases}
1, & \text{if } w_i\in \Gamma(\theta, r)\\
0, & \text{otherwise}
\end{cases}.
\]
Supported by the indicator vector $\Pi_{\Gamma(\theta, r)}$, our pruning operation $P(\cdot)$ can protect the non-prunable weights by masking:
$$P(\theta, r)=(\textbf{1}_{\kappa}-\Pi_{\Gamma(\theta, r)}\odot \textbf{b})\odot \theta,$$
where $\odot$ denotes the element-wise multiplication and $\textbf{1}_{\kappa}=(1,1, ..., 1)\in R^\kappa$ denotes an all-one vector. $\textbf{b}=(b_1, b_2, ..., b_\kappa)$ is a vector with $b_i\overset{\textrm{i.i.d}}{\sim}\textrm{Bernoulli}(p_{\textrm{bern}})$.
To be specific, $\Pi_{\Gamma(\theta, r)}$ and $\textbf{b}$ both are binary masks with identical layout as $\theta$. $\Pi_{\Gamma(\theta, r)}$ is responsible for protecting non-prunable weights, while $\textbf{b}$ is for random pruning. Each binary element in $\Pi_{\Gamma(\theta, r)}\odot \textbf{b}$ indicates whether to prune the corresponding weight value in $\theta$. The main difference from a similar technique, Dropout \cite{JMLR_Dropout, AAAI_Ghost}, is that DWP only considers prunable weights. We will show the significance of protecting non-prunable weights in Ablation Study. Once we obtain the gradients, we will revoke the pruning operation to keep the white-box networks intact for successive iterations.
With the participation of DWP, we extend the abbreviation as DWP-NI-SI-TI-DI-FGSM, further modifying our updating scheme:
\[
\begin{aligned}
&g_n=\mu\cdot g_{n-1}+\\
&\dfrac{\textit{\textbf{W}}}{M}* \sum_{m=0}^{M-1}\sum_{k=1}^{K}\beta_{k}\nabla_xJ(S_m(T(x^{\textrm{nes}}_n, p_{\textrm{DI}})), y^{\text{target}};P(\theta_{k}, r)).
\end{aligned}
\]
Since we're not aiming for actually compressing networks, the pruning operations only mask unwanted connections rather than truly delete them. Thus, the layout of each network is hardly affected, allowing DWP to be more portable and usable. Also, benefiting from no dependency on network retraining and extra data, our proposed DWP is simple and lightweight. As there is no further retraining, we select the L1 norm for pruning since it is better than L2 on preserving accuracy \cite{NIPS_L1Pruning}.
\section{Experiments}
\subsection{Experimental Setup}
\subsubsection{Dataset.}
To evaluate targeted attack, we use an ImageNet-compatible dataset{\color{red}\footnote{\url{https://github.com/cleverhans-lab/cleverhans/blob/11ea10/examples/nips17_adversarial_competition/dataset/dev_dataset.csv}}} provided by NIPS 2017 adversarial competition \cite{NIPS2017_competition} containing 1,000 images. Each image in the dataset has an officially assigned target class for consistency.
\subsubsection{Networks.}
We perform experiments on eight naturally trained models: Inception-v3 (Inc-v3) \cite{CVPR_InceptionV3}, Inception-v4 (Inc-v4), Inception-Resnet-v2 (IncResv2) \cite{AAAI_IncV4_IncResV2}, ResNet-50 (Res-50), ResNet-101 (Res-101), ResNet-152 (Res-152) \cite{CVPR_ResNet}, VGGNet-16 (VGG-16) \cite{ICLR_VGG} and DenseNet-121 (Den-121) \cite{CVPR_DenseNet}, and two adversarially trained models: ens3-adv-Inception-v3 (Inc-v3ens3) and ens-adv-inception-resnet-v2 (IncRes-v2ens) \cite{ICLR_ens_AdvTraining}.
All the ten networks are publicly accessible.
\subsubsection{Hyper-parameters.}
Our method includes three input transformations TI, DI, and SI. We follow \cite{CVPR_Poincare}, setting the probability $p_{\textrm{DI}}$ of DI to be $0.7$ and selecting a Gaussian kernel with the kernel length equaling $5$ for $\textit{\textbf{W}}$ in TI. For SI, due to the limitation in computing resources, we set the number of scale copies $M=3$.
Following \cite{BoostingMomentum, ICLR_Nesterov, CVPR_Poincare, NIPS_Simple}, the momentum decay factor $\mu$ is assigned $1$.
As for the total iteration for iterative attacks, we conduct experiments under $20$ and $100$ respectively with learning rate $\alpha=2/255$ following \cite{NIPS_Simple} to observe results under the different extent of converging.
All the experiments comply with perturbations less than the budget $\epsilon=16$ under $L_{\infty}$ norm to keep adversarial examples imperceptible to humans.
Last but not least, for our proposed DWP, the probability $p_{\textrm{bern}}$ is $0.5$ and the prunable rate $r$ is $0.7$, which means at each iteration, we prune $35\%$ of connections of each network in expectation.
\subsubsection{Baseline Methods.}
To inspect the compatibility of DWP, we consider the entire NI-SI-TI-DI-FGSM and each of its components as our baselines. We expect DWP to improve each technique alone. The middle column of Table~\ref{tab:Baseline-Methods} summarizes abbreviations of our baseline methods and their relations to the hyper-parameters, which are NI-FGSM, SI-FGSM, TI-FGSM, DI-FGSM, and NI-SI-TI-DI-FGSM.
\begin{table}[h]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{|l| l | l |}
\hline
Hyper-parameters & NI-SI-TI-DI-FGSM & DWP-NI-SI-TI-DI-FGSM\\
\hline
{\begin{tabular}[c]{@{}l@{}}$M=0, p_{\textrm{DI}}=0,$\\Identity kernel $\textit{\textbf{W}}$\cite{BOOK_DIP}\end{tabular}} & NI-FGSM & DWP-NI-FGSM\\
\hline
{\begin{tabular}[c]{@{}l@{}}$\mu=0, M=0, p_{\textrm{DI}}=0$\end{tabular}} & TI-FGSM & DWP-TI-FGSM\\
\hline
{\begin{tabular}[c]{@{}l@{}}$\mu=0, M=0,$\\Identity kernel $\textit{\textbf{W}}$\end{tabular}} & DI-FGSM & DWP-DI-FGSM\\
\hline
{\begin{tabular}[c]{@{}l@{}}$\mu=0, p_{\textrm{DI}}=0,$\\Identity kernel $\textit{\textbf{W}}$\end{tabular}} & SI-FGSM & DWP-SI-FGSM\\
\hline
\end{tabular}
}
\caption{
The abbreviations of baseline methods. For example, TI-FGSM is the special case of NI-SI-TI-DI-FGSM with
$\mu=0$, $M=0$ and $p_{\textrm{DI}}=0$. DI-FGSM, NI-FGSM and SI-FGSM are also special cases under certain hyper-parameters. DWP
will be combined with each baseline method for evaluation.
}
\label{tab:Baseline-Methods}
\end{table}
\subsection{Transferable Targeted Attack in various scenarios}
We consider targeted transferability under three scenarios: common transferring, transferring to distinct architectures and transferring to adversarially trained models. We will prepare specified networks for each case.
Each time we select a network as the black-box model and take the others as white-box models. We generate adversarial examples on the ensemble of the white-box models and evaluate targeted success rates on the black-box model left alone. No access to the black-box model is allowed during attacking. Note that for ensembling, we use equal ensemble weights $\beta_k=1/K$ for each of the $K$ white-box models.
\subsubsection{Common Transferring}
In common transferring, we select Inc-v3, Inc-v4, IncResv2, Res-50, Res-101, and Res-152. The six normally trained networks are commonly adopted to provide a preliminary estimation of the transferability of attacks \cite{BoostingMomentum, CVPRDI, CVPRTI, CVPR_Poincare, NIPS_Simple} for evaluation.
Table~\ref{tab:common-transfer-targeted-success-rate} shows results of transferable targeted attack under the common transferring scenario. DWP boosts almost all the attack methods, especially the standalone ones. When all the state-of-the-art techniques collaborate with DWP, the average targeted success rate can even achieve 95.22\%.
\begin{table*}[t]
\centering
\begin{tabular}{l| l l l l l l || l}
\hline
Attack Method & -Inc-v3 & -Inc-v4 & -IncRes-v2& -Res-50 & -Res-101 & -Res-152 & Average\\
\hline
TI-FGSM & 21.5/24.2 & 21.1/25.1 & 23.8/29.1 & 52.2/63.6 & 56.4/71.4 & 53.9/67.5 & 38.15/46.82\\
DWP-TI-FGSM & 24.1/43.9 & 24.8/46.6 & 25.3/53.1 & 55.6/78.7 & 63.0/84.0 & 57.8/83.0 & \textbf{41.77}/\textbf{64.88}\\
\hline
DI-FGSM & 55.7/78.4 & 57.7/81.8 & 59.0/84.4 & 67.7/91.2 & 71.1/91.9 & 73.0/93.4 & 64.03/86.85 \\
DWP-DI-FGSM & 59.8/87.4 & 60.7/89.5 & 62.3/89.1 & 70.1/93.7 & 73.5/93.9 & 74.4/94.7 & \textbf{66.80}/\textbf{91.38} \\
\hline
NI-FGSM & 18.8/35.9 & 16.7/35.1 & 17.8/36.9 & 45.3/63.1 & 50.0/66.7 & 49.2/67.9 & 32.97/50.93\\
DWP-NI-FGSM & 31.7/50.9 & 26.3/49.1 & 28.6/53.4 & 58.7/74.6 & 64.7/80.3 & 61.6/81.3 & \textbf{45.27}/\textbf{64.93}\\
\hline
SI-FGSM & 39.5/41.5 & 35.8/38.4 & 38.0/43.1 & 66.5/75.7 & 71.2/79.8 & 71.0/79.5 & 53.67/59.67\\
DWP-SI-FGSM & 40.9/64.7 & 42.0/66.4 & 47.1/72.0 & 72.6/89.7 & 76.8/91.6 & 77.5/91.9 & \textbf{59.48}/\textbf{79.38} \\
\hline
\hline
NI-SI-TI-DI-FGSM & 71.1/91.3 & 75.1/92.7 & 76.3/92.7 & 81.7/94.4 & 84.0/95.5 & 85.5/95.6 & 78.95/93.70\\
DWP-NI-SI-TI-DI-FGSM & \textbf{77.2}/\textbf{94.1} & \textbf{79.3}/\textbf{94.1} & \textbf{78.9}/\textbf{94.3} & \textbf{82.3}/\textbf{95.8} & \textbf{85.1}/\textbf{96.3} & \textbf{87.2}/\textbf{96.7} & \textbf{81.67}/\textbf{95.22}\\
\hline
\end{tabular}
\caption{The targeted success rates (\%) of common transferring. ``-'' in each column stands for the black-box network while the other five serve as white-box ones for ensemble. Each cell is of results under 20/100 iterations. DWP improves all the methods, especially for the standalone ones.}
\label{tab:common-transfer-targeted-success-rate}
\end{table*}
\subsubsection{Transferring to Distinct Architectures}
In general, information about the networks used by defenders remains unknown to attackers. A targeted attack method will be more practical if adversarial examples generated can transfer to black-box architectures less similar to the white-box networks utilized by attackers. Therefore, we follow \cite{NIPS_Simple} and evaluate the attack methods on the four networks with distinct architectures: Res-50, VGG-16, Den-121, and Inc-v3.
Results of transferring to distinct architectures are shown in Table~\ref{tab:Transfer-to-Dissimilar-architectures-Targeted-Success-Rate}. Due to the less similarity between networks' architectures, there is more room for improvement on targeted transferability under the scenario than the common transferring. DWP enhances NI-SI-TI-DI-FGSM with 4.1\% on average after 100 iterations, which is more than 1.5\% under the common transferring. These results demonstrate the benefits of diversified ensemble in attacking distinct black-box architectures.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{l| l l l l || l}
\hline
Attack Method & -Res-50 & -Den-121 & -VGG16 & -Inc-v3 & Average\\
\hline
TI-FGSM & 17.7/25.2 & 20.3/28.1 & 22.3/30.3 & 6.3/8.8 & 16.65/23.10\\
DWP-TI-FGSM & 21.3/42.0 & 25.5/41.5 & 27.5/55.4 & 7.7/15.7 & \textbf{20.50}/\textbf{38.65}\\
\hline
DI-FGSM & 39.1/67.3 & 44.8/76.3 & 47.9/75.5 & 24.7/49.5 & 39.13/67.15\\
DWP-DI-FGSM & 40.9/73.0 & 49.0/80.1 & 51.3/82.6 & 25.1/54.8 & \textbf{41.58}/\textbf{72.63}\\
\hline
NI-FGSM & 13.2/27.1 & 17.0/33.4 & 15.9/30.6 & 6.8/17.4 & 13.23/27.13\\
DWP-NI-FGSM & 19.3/36.2 & 22.6/43.9 & 28.4/46.6 & 8.9/20.5 & \textbf{19.80}/\textbf{36.80}\\
\hline
SI-FGSM & 28.2/34.5 & 35.1/45.5 & 29.1/38.0 & 13.6/16.8 & 26.50/33.70\\
DWP-SI-FGSM & 33.4/54.1 & 40.4/61.7 & 42.7/64.7 & 16.2/27.4 & \textbf{33.18}/\textbf{51.98}\\
\hline
\hline
NI-SI-TI-DI-FGSM & 57.6/80.6 & 70.3/88.1 & 60.4/80.2 & 48.1/72.0 & 59.10/80.23 \\
DWP-NI-SI-TI-DI-FGSM & \textbf{59.7}/\textbf{83.1} & \textbf{72.9}/\textbf{90.4} & \textbf{63.8}/\textbf{86.8} &
\textbf{49.5}/\textbf{77.0} & \textbf{61.48}/\textbf{84.33} \\
\hline
\end{tabular}
}
\caption{The targeted success rates of transferring to distinct architectures. ``-'' stands for the black-box network with the other three serving as the white-box ones for ensemble. Each cell is of results under 20/100 iterations. DWP appears to be compatible with each of these state-of-the-art techniques, yielding improvement with them on average.}
\label{tab:Transfer-to-Dissimilar-architectures-Targeted-Success-Rate}
\end{table*}
\subsubsection{Transferring to Adversarially Trained Models}
Adversarial training \cite{ICLR_ens_AdvTraining, ICLR_MadryPGDAdvTraining} is one of the primary techniques for defending against malicious attacking. It brings robustness to models by training them with adversarial examples. Under the scenario of transferring to adversarially trained models, we ensemble only the six naturally trained networks (Inc-v3, Inc-v4, In-cResv2, Res-50, Res-101, and Res-152) as white-box models to simulate the situation where attackers have few details about defense. The two adversarially trained networks (Inc-v3ens3 and IncRes-v2ens) will act as our black-box model separately.
Table~\ref{tab:Transfer-to-Robust-Models-Targeted-Success-Rate} summarizes the results of transferring to adversarially trained networks. Targeted success rates under this scenario are lower than the other two due to the robustness of adversarially trained networks. Under such a challenging scenario, DWP still helps alleviate the discrepancy between white-box naturally trained and black-box adversarially trained networks, bringing about up to 8.0\% and 6.75\% improvement on average under 20 and 100 iterations. The necessity of the diversified ensemble is highlighted again, especially for black-box networks with significant differences from white-box ones.
\begin{table}[t]
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{l| c c || l }
\hline
Attack Method & Inc-v3ens3 & IncRes-v2ens & Average\\
\hline
TI-FGSM & 6.0/3.7 & 1.1/0.2 & 3.55/1.95\\
DWP-TI-FGSM & 6.8/10.3 & 1.5/0.7 & \textbf{4.15}/\textbf{5.5}\\
\hline
DI-FGSM & 23.0/27.5 & 3.6/2.0 & 13.3/14.75\\
DWP-DI-FGSM & 25.3/42.1 & 5.3/4.9 & \textbf{15.3}/\textbf{23.50}\\
\hline
NI-FGSM & 3.7/9.9 & 0.3/0.6 & 2.00/5.25\\
DWP-NI-FGSM & 8.1/11.9 & 1.0/1.1 & \textbf{4.55}/\textbf{6.50}\\
\hline
SI-FGSM & 17.2/15.3 & 4.1/1.0 & 10.65/8.15\\
DWP-SI-FGSM & 17.7/30.4 & 5.0/3.8 & \textbf{11.35}/\textbf{17.10}\\
\hline
\hline
NI-SI-TI-DI-FGSM & 55.2/77.7 & 43.6/54.8 & 49.40/66.25\\
DWP-NI-SI-TI-DI-FGSM & \textbf{64.1}/\textbf{83.1} & \textbf{50.7}/\textbf{62.7} & \textbf{57.40}/\textbf{72.90}\\
\hline
\end{tabular}
}
\caption{The targeted success rates of transferring to adversarially trained networks. Adversarial examples are generated by ensembling only the six naturally trained networks in Table~\ref{tab:common-transfer-targeted-success-rate}. Each cell is of success rates under 20/100 iterations. The results show that DWP can alleviate the differences between naturally and adversarially trained networks, improving attacks under the challenging scenario.}
\label{tab:Transfer-to-Robust-Models-Targeted-Success-Rate}
\end{table}
\subsection{Perturbations from Different Newborn Models}
\cite{DelvingLiu2017} observes that adversarial perturbations from white-box neural networks with different architectures will be orthogonal. The orthogonality implies zero cosine similarity, implying diversity to a certain degree. Since DWP produces additional newborn models, the relationship between perturbations generated from these models is noteworthy. In this section, we analyze perturbation vectors from newborn models belonging to identical networks and different networks using cosine similarity following \cite{DelvingLiu2017}.
We study the four networks same as transferring to distinct architectures. Firstly, we acquire five newborn models from each of the four networks via pruning different connections. For calculating cosine similarity between perturbations, we take the average of the first ten images from the ImageNet-compatible dataset to avoid cherry-picking. To prevent the influence of factors other than newborn models, we use NI-FGSM only to generate adversarial perturbations excluding TI, DI, and SI. Note that all the other hyper-parameters are the same as attacking. Unlike DWP, we don't revoke pruning in this experiment to ensure each perturbation is from an identical newborn model throughout attacking.
For perturbations from newborn models belonging to the same architecture, we take ResNet-50 as an example, summarizing the cosine similarity in Table~\ref{tab:ortho-between-same-arch}. The five newborn models of ResNet-50 yield ten ($C^5_2$) non-diagonal values in pair and five diagonal values with themselves. The diagonal ones are all 1.0 since they are from two identical perturbation vectors. As for the non-diagonal, all the values are close to 0, implying orthogonality.
The results show that even though these newborn models are from the same architecture (ResNet-50) with merely different connections pruned, perturbations generated with them will still be orthogonal as observed in different architectures \cite{DelvingLiu2017}.
For perturbations from newborn models belonging to two different architectures and also the cases of single architectures besides ResNet-50, we summarize the average cosine similarity in Table~\ref{tab:ortho-between-different-arch}. Each diagonal cell ($i,i$) averages cosine similarity from the 10 pairs ($C^5_2$) of five newborn models of the architecture $i$ (column). The non-diagonal cells ($i,j$) averages the one from 25 pairs of newborn models of architecture $i$ (row) and $j$ (column) respectively. All the values in Table~\ref{tab:ortho-between-different-arch} are close to zero, suggesting that whether two newborn models belong to the same or different architectures, perturbations generated from them will always be nearly orthogonal.
From the viewpoint of geometry, orthogonal perturbations from various networks participating in the ensemble-based approach enrich the overall direction for updating adversarial examples. These observations on orthogonality support our claim that newborn models obtained via Weight Pruning provide more diversity for attacking.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{l| l l l l l}
\hline
Res-50 & newborn1 & newborn2 & newborn3 & newborn4 & newborn5\\
\hline
newborn1 & 1.0000 & 0.0009 & 0.0005& 0.0000 & -0.0004\\
newborn2 & - & 1.0000 & 0.0006& -0.0010 & -0.0004\\
newborn3 & - & - & 1.0000 & 0.0002 & 0.0005\\
newborn4 & - & - & - & 1.0000 & 0.0002\\
newborn5 & - & - & - & - & 1.0000\\
\hline
\end{tabular}
}
\caption{The perturbation cosine similarity between newborn models belonging to Res-50. Although the models are all acquired from Res-50, their perturbations are still orthogonal. Note that the table is symmetric with the values rounded to four decimal places.}
\label{tab:ortho-between-same-arch}
\end{table}
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{l| l l l l }
\hline
& Res-50 & VGG-16 & Den-121 & Inc-v3\\
\hline
Res-50 & 0.0001 & -0.0001 & 0.0003 & -0.0001\\
VGG-16 & - & 0.0004 & 0.0000 & -0.0002 \\
Den-121 & - & - &-0.0001 & 0.0001\\
Inc-v3 & - & - & - & 0.0005 \\
\hline
\end{tabular}
}
\caption{The average perturbation cosine similarity between each pair of architectures. The non-diagonal cells take an average on 25 pairs of newborn models from the two corresponding architectures, while the diagonal ones average ten. The table shows newborn models of whether distinct or identical architectures yield orthogonal perturbations.}
\label{tab:ortho-between-different-arch}
\end{table}
\subsection{Ablation Study}
In this section, we conduct ablation experiments to study the two key points of DWP: the quality and the quantity of pruning.
\subsubsection{Quality of Pruning: In Comparison with Dropout.}
While Weight Pruning considers the importance of each connection and only prunes the redundant ones away, Dropout \cite{JMLR_Dropout, AAAI_Ghost} deactivates activation functions without any examination of their values. The reason is that instead of compressing networks, Dropout is mainly used to prevent networks from overfitting \cite{JMLR_Dropout} during training. In this section, we produce newborn models using Dropout, comparing the results of targeted attacks with DWP. In the following experiments, we will show considering connection values to guarantee the quality of newborn models is necessary.
Dropout can be viewed as a special case of our Diversified Weight Pruning with all the connections in a network treated as prunable regardless of their weight values. Therefore, we simulate Dropout with the prunable rate $r$ set to $1.0$, giving DWP-NI-SI-TI-DI-FGSM with $r=1.0$ an alias, Dropout-NI-SI-TI-DI-FGSM. Both DWP-NI-SI-TI-DI-FGSM with our original settings ($r=0.7$) and the newly introduced Dropout-NI-SI-TI-DI-FGSM will serve for generating adversarial examples under the scenario of transferring to distinct architectures. We evaluate their targeted success rates on the black-box model every five iterations for more precise observation.
Results of the comparison between DWP-NI-SI-TI-DI-FGSM and Dropout-NI-SI-TI-DI-FGSM are shown in Figure~\ref{fig:DWP-vs-Dropout}. NI-SI-TI-DI-FGSM is also included for reference. Roughly, targeted attack success rates of all the methods on the four black-box networks increase with more iterations \cite{NIPS_Simple}.
With a closer look, DWP-NI-SI-TI-DI-FGSM in orange curves is much better than all the other methods. The green curves standing for Dropout-NI-SI-TI-DI-FGSM are found even worse than the blue ones of our baseline NI-SI-TI-DI-FGSM.
The relatively poor results of Dropout-NI-SI-TI-DI-FGSM are due to the ignorance of weight values when pruning, which may yield newborn models with unstable performance and mislead the directions of targeted adversarial perturbations.
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/ResNet50_dwp_vs_dropout.png}
\caption{ResNet-50}
\label{fig:res50}
\end{subfigure}
\begin{subfigure}[t]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/DenseNet121_dwp_vs_dropout.png}
\caption{DenseNet-121}
\label{fig:DenseNet121}
\end{subfigure}
\begin{subfigure}[t]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/VGG16_dwp_vs_dropout.png}
\caption{VGGNet-16}
\label{fig:VGG16}
\end{subfigure}
\begin{subfigure}[t]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/InceptionV3_dwp_vs_dropout.png}
\caption{Inception-v3}
\label{fig:InceptionV3}
\end{subfigure}
\caption{\textbf{Comparison between DWP-NI-SI-TI-DI-FGSM, Dropout-NI-SI-TI-DI-FGSM, and our baseline NI-SI-TI-DI-FGSM on targeted transferability with different numbers of iterations.} Each curve shows targeted success rates on the black-box network specified in the caption below. Our DWP-NI-SI-TI-DI-FGSM (orange) is much better than the baseline (blue) and Dropout-NI-SI-TI-DI-FGSM (green). The results demonstrate the significance of protecting necessary weights and ensuring the quality of newborn models for ensembling.}
\label{fig:DWP-vs-Dropout}
\end{figure*}
\subsubsection{Quantity of Pruning: Influence of Prunable Rates.}
In this section, we focus on targeted attack success rates under different prunable rates. As the prunable rate determines the number of connections possible to be pruned during attacking, white-box models can produce more diverse newborn models using higher prunable rates. However, with an excessive number of connections pruned away, the quality of newborn networks will be unstable.
To find the sweet spot to the trade-off, we enumerate different prunable rates, conducting the attack experiments with all the other hyper-parameters as default. We consider the four models of transferring to distinct architectures since the scenario is closer to the real situation. Moreover, we will provide observations on accuracy decay regarding pruning for reference using the 1,000 clean images of the ImageNet-compatible dataset.
As shown in Figure~\ref{fig:suc_prune}, the curves of targeted attack success rates excluding each black-box network are roughly mountain-like due to the trade-off. We select $r=0.7$ throughout our experiments as the curves reach maximum success rates with the prunable rate. Figure~\ref{fig:acc_prune} shows the accuracy decay in respect of the ratio of connections pruned in each network. With our designated prunable rate $r=0.7$, DWP will prune $35\%$ of connections approximately. According to Figure~\ref{fig:acc_prune}, there is no apparent damage to the accuracy of each network with the quantity of pruning. Therefore, the quality of newborn models remains stable.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.495\linewidth}
\centering
\includegraphics[width=1.02\linewidth]{figures/suc_prune100.png}
\caption{}
\label{fig:suc_prune}
\end{subfigure}
\begin{subfigure}[t]{0.495\linewidth}
\centering
\includegraphics[width=1.02\linewidth]{figures/acc_prune.png}
\caption{}
\label{fig:acc_prune}
\end{subfigure}
\caption{(a): \textbf{The targeted success rates under different prunable rates $r$ on each black-box model.} Each mountain-like curve shows the trade-off between diversity and stability of newborn models. All four curves reach the sweet spot at $r=0.7$. (b): \textbf{The decay on the accuracy of each network with respect to the rate of parameters pruned.} With the prunable rate $r=0.7$, there are 35\% of weights pruned away in expectation at each iteration. The quantity is not harmful to the accuracy of the networks according to the curves.}
\end{figure}
\section{Conclusion}
In this paper, we propose Diversified Weight Pruning (DWP) leveraging network compression to improve the targeted transferability of adversarial attacks. DWP aims to produce more white-box models from the existing ones via network pruning for ensemble. Due to the over-parameterized property of neural networks, the performance of models newly produced by DWP is well-preserved. Experiments show that assuring the quality of networks participating in ensemble significantly influences targeted transferability. By evaluating DWP on ImageNet, we show that DWP improves the state-of-the-art simple transferable attacks, especially for challenging scenarios such as adversarially trained models and distinct architectures. With the longitudinal ensemble, DWP benefits most of the existing attacks without imposing extra costs. We hope that our work can serve as a bridge between network compression and transferable attack, inspiring more collaboration.
\section{Using \LaTeX{} to Format Your Paper}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,476 |
Kapil Sharma's poor health reportedly led to the cancellation of yet another episode of his show. On Wednesday, Team Mubarakan arrived on the sets of The Kapil Sharma Show but had to leave without filming because the comedian failed to make it to the shoot, reports Indian Express. "As Kapil's health was involved, the team decided to wait for him for some more time and finish the shoot," the source said. Actors Anil Kapoor, Arjun Kapoor, Athiya Shetty and Ileana D'Cruz arrived as per schedule at 5.30 pm and reportedly left after 10 pm. "The shoot was scheduled to begin around 5.30 pm. But as Kapil had not come till then, we made some calls and it was then we were told that he is not keeping well but will try to come in for the shoot as soon as possible. But till 10 pm when he did not turn up, everyone decided that they shall leave after waiting for over four hours," the source told Indian Express.
Last week, Shah Rukh Khan and Anushka Sharma also left the sets of The Kapil Sharma Show without shooting for the episode after the host collapsed, reported Hindustan Times. Kapil's team even cancelled a shoot with the stars of Guest Iin London mid-way because of Kapil's deteriorating health. A source told Hindustan Times that Kapil is stressed due to the impending arrival of his rival Krushna Abhishek's The Drama Company.
Krushna's showrunners have hired The Kapil Sharma Show's creative director Preeti Simoes along with his former co-star actor Ali Asgar. The show also featured actor Mithun Chakraborty. Meanwhile, comedienne Bharti Singh has joined Kapil's show. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,808 |
Valentin Vornicu (* 13. Juni 1983 in Bukarest) ist ein professioneller rumänisch-amerikanischer Pokerspieler und Mathematiker. Mit zwölf Ringen ist er einer der erfolgreichsten Spieler bei den Circuitturnieren der World Series of Poker (WSOP).
Persönliches
Vornicu wuchs in Bukarest auf. Er nahm in den Jahren 2001 und 2002 für Rumänien an der Internationalen Mathematik-Olympiade teil und gewann 2002 eine Bronze-Medaille. Anschließend gründete er das Bildungsunternehmen MathLinks, das zunächst u. a. Schüler auf die Mathematik-Olympiade vorbereiten sollte. Das Forum der Website wurde schnell populär und im Jahr 2004 mit der Website Art of Problem Solving zusammengeführt. Vornicu studierte an der Universität Bukarest und schloss sein Studium 2008 mit einem Master in Algebra und Zahlentheorie ab. Er arbeitet als Mathelehrer und bereitet seine Schüler dabei auf eine Teilnahme an der United States of America Mathematical Olympiad vor. Vornicu lebt in San Diego.
Pokerkarriere
Werdegang
Vornicu spielte online unter den Nicknames busto23 (partypoker), mBlasterX (PokerStars sowie Lock Poker) und Pupp3tMast3r (Full Tilt Poker). Seit 2011 nimmt er an renommierten Live-Turnieren teil.
Vornicu gewann Mitte Oktober 2011 ein Turnier beim WSOP-Circuit in Hammond und sicherte sich damit seinen ersten Ring sowie eine Siegprämie von mehr als 50.000 US-Dollar. Im Juni 2013 war er erstmals bei der Hauptturnierserie der World Series of Poker im Rio All-Suite Hotel and Casino am Las Vegas Strip erfolgreich und kam bei einem Turnier der Variante No Limit Hold'em in die Geldränge. Mitte Dezember 2015 gewann Vornicu das Main Event des WSOP-Circuits in Los Angeles mit einem Hauptpreis von knapp 200.000 US-Dollar. Bei der WSOP 2016 erreichte er den siebten Turniertag im Main Event und belegte dort den 23. Platz, der mit rund 270.000 US-Dollar bezahlt wurde. Mitte August 2017 wurde Vornicu bei einem Event der Seminole Hard Rock Poker Open in Hollywood Vierter und erhielt ein Preisgeld von rund 80.000 US-Dollar.
Insgesamt hat sich Vornicu mit Poker bei Live-Turnieren mehr als eine Million US-Dollar erspielt.
Ringübersicht
Vornicu kam bei den WSOP-Circuitturnieren 85-mal ins Geld und gewann zwölf Ringe:
Weblinks
Einzelnachweise
Pokerspieler (Rumänien)
Pokerspieler (Vereinigte Staaten)
Mathematiker (21. Jahrhundert)
Sportler (Bukarest)
Rumäne
US-Amerikaner
Geboren 1983
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,375 |
package co.cask.tigon.api.flow.flowlet;
import co.cask.tigon.api.ResourceSpecification;
import co.cask.tigon.api.common.PropertyProvider;
import co.cask.tigon.internal.flowlet.DefaultFlowletSpecification;
import com.google.common.base.Preconditions;
import com.google.common.collect.ImmutableMap;
import com.google.common.collect.ImmutableSet;
import java.util.Map;
/**
* This class provides specification of a Flowlet. Instance of this class should be created through
* the {@link Builder} class by invoking the {@link Builder#with()} method.
*
* <pre>
* {@code
* FlowletSpecification flowletSpecification =
* FlowletSpecification flowletSpecification =
* FlowletSpecification.Builder.with()
* .setName("tokenCount")
* .setDescription("Token counting flow")
* .setFailurePolicy(FailurePolicy.RETRY)
* .build();
* }
* </pre>
*/
public interface FlowletSpecification extends PropertyProvider {
/**
* @return Class name of the {@link co.cask.tigon.api.flow.flowlet.Flowlet} class.
*/
String getClassName();
/**
* @return Name of the flowlet.
*/
String getName();
/**
* @return Description of the flowlet.
*/
String getDescription();
/**
* @return The failure policy of the flowlet.
*/
FailurePolicy getFailurePolicy();
/**
* @return The {@link co.cask.tigon.api.ResourceSpecification} for the flowlet.
*/
ResourceSpecification getResources();
/**
* @return The maximum instances allowed for this flowlet.
*/
int getMaxInstances();
/**
* Builder for creating instance of {@link FlowletSpecification}. The builder instance is
* not reusable, meaning each instance of this class can only be used to create one instance
* of {@link FlowletSpecification}.
*/
static final class Builder {
private String name;
private String description;
private FailurePolicy failurePolicy = FailurePolicy.RETRY;
private Map<String, String> arguments;
private ResourceSpecification resources = ResourceSpecification.BASIC;
private int maxInstances = Integer.MAX_VALUE;
/**
* Creates a {@link Builder} for building instance of this class.
*
* @return A new builder instance.
*/
public static NameSetter with() {
return new Builder().new NameSetter();
}
public final class NameSetter {
/**
* Sets the name of a flowlet.
* @param name Name of the flowlet.
* @return An instance of {@link DescriptionSetter}
*/
public DescriptionSetter setName(String name) {
Preconditions.checkArgument(name != null, "Name cannot be null.");
Builder.this.name = name;
return new DescriptionSetter();
}
}
/**
* Class defining the description setter that is used as part of the builder.
*/
public final class DescriptionSetter {
/**
* Sets the description of the flowlet.
* @param description Descripition to be associated with flowlet.
* @return An instance of what needs to be done after description {@link AfterDescription}
*/
public AfterDescription setDescription(String description) {
Preconditions.checkArgument(description != null, "Description cannot be null.");
Builder.this.description = description;
return new AfterDescription();
}
}
/**
* Class defining the action after defining the description for a flowlet.
*/
public final class AfterDescription {
/**
* Sets the failure policy of a flowlet.
* @param policy Policy to be associated with a flowlet for handling processing failures.
* @return An instance of {@link AfterDescription}
*/
public AfterDescription setFailurePolicy(FailurePolicy policy) {
Preconditions.checkArgument(policy != null, "FailurePolicy cannot be null");
failurePolicy = policy;
return this;
}
/**
* Adds a map of arguments that would be available to the flowlet through
* the {@link co.cask.tigon.api.flow.flowlet.FlowletContext} at runtime.
*
* @param args The map of arguments.
* @return An instance of {@link AfterDescription}.
*/
public AfterDescription withArguments(Map<String, String> args) {
arguments = ImmutableMap.copyOf(args);
return this;
}
public AfterDescription withResources(ResourceSpecification resourceSpec) {
Preconditions.checkArgument(resourceSpec != null, "Resources cannot be null.");
resources = resourceSpec;
return this;
}
/**
* Set the maximum instances allowed for this flowlet.
* @param maxInstances The maximum number of instances allowed for this flowlet.
* @return An instance of {@link AfterDescription}.
*/
public AfterDescription setMaxInstances(int maxInstances) {
Builder.this.maxInstances = maxInstances;
return this;
}
/**
* Creates an instance of {@link FlowletSpecification}.
* @return An instance of {@link FlowletSpecification}.
*/
public FlowletSpecification build() {
return new DefaultFlowletSpecification(name, description, failurePolicy, arguments, resources, maxInstances);
}
}
/**
* Private builder to maintain builder contract.
*/
private Builder() {
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,420 |
ChiefsPlanet > The Lounge > Other Sports Mike Piazza decides to retire after 16 seasons
View Full Version : Other Sports Mike Piazza decides to retire after 16 seasons
Deberg_1990
So is he a Hall of Famer?? You have to consider he played in the steriod era with power and came out clean.
http://news.yahoo.com/s/ap/20080520/ap_on_sp_ba_ne/bbo_piazza_retires
BEVERLY HILLS, Calif. - Mike Piazza is retiring from baseball following a 16-season career in which he became one of the top-hitting catchers in history.
"After discussing my options with my wife, family and agent, I felt it was time to start a new chapter in my life," he said in a statement released Tuesday by his agent, Dan Lozano. "It has been an amazing journey ... So today, I walk away with no regrets.
"I knew this day was coming and over the last two years. I started to make my peace with it. I gave it my all and left everything on the field."
The 39-year-old Piazza batted .275 with eight homers and 44 RBIs as a designated hitter for Oakland last season, became a free agent and did not re-sign. He was not available to discuss his decision, according to Josh Goldberg, a spokesman for Lozano.
Taken by the Los Angeles Dodgers on the 62nd round of the 1988 amateur draft, Piazza became a 12-time All-Star, making the NL team 10 consecutive times starting in 1993.
"He was one of those hitters who could change the game with one swing. He was certainly the greatest-hitting catcher of our time, and arguably of all time," said Atlanta pitcher Tom Glavine, Piazza's former teammate on the New York Mets.
Piazza finished with a .308 career average, 427 home runs and 1,335 RBIs for the Dodgers (1992-98), Florida (1998), Mets (1998-05), San Diego (2006) and Oakland (2007).
"It's the end of a Hall of Fame career," Mets manager Willie Randolph said. "It was a privilege to manage him for the short time that I did."
Los Angeles Angels manager Mike Scioscia was a teammate of Piazza's on the 1992 Los Angeles Dodgers and remembered back to Piazza's first season in the majors and what he accomplished.
"To put yourself in the same ballpark with what a guy like Roy Campanella did is saying something and Mike is definitely up there with what Roy did," Scioscia said.
Piazza's 396 homers are easily the most as a catcher, according to the Elias Sports Bureau. Carlton Fisk is second with 351, followed by Johnny Bench (327) and Yogi Berra (306).
"If I'm half the hitter he was, I'll have a pretty successful career," said Atlanta's Brian McCann, one of the top-hitting catchers currently in the majors. "He did a lot of great things for the catching position."
Piazza never had a great throwing arm but was praised by pitchers for his game-calling.
"You'd have to really go back and see Mike from the early days of trying to catch to where he ended up, the hard work he put in, the dedication he had to get good enough on the defensive end to where he could get his at-bats," Scioscia said. "He made himself into a guy who could go out there and catch and do the job he needed behind the plate."
Piazza thanked his family, teams and managers, some of his teammates — and even owners, general managers, minor league staffs and reporters.
"Within the eight years I spent in New York, I was able to take a different look at the game of baseball," Piazza said. "I wasn't just a young kid that was wet behind the ears anymore — I was learning from other veteran guys like Johnny Franco, who taught me how to deal with the pressures of playing in New York, and Al Leiter, who knew what it took to win a world championship."
He did not bring up two of the more memorable moments in his career: When the Yankees' Roger Clemens beaned him on July 8, 2000, and when Clemens threw the broken barrel of Piazza's bat in his direction in Game 2 of the World Series that October. Clemens denied intent both times.
"Last but certainly not least, I can't say goodbye without thanking the fans," Piazza said. "I can't recall a time in my career where I didn't feel embraced by all of you. Los Angeles, San Diego, Oakland and Miami — whether it was at home or on the road, you were all so supportive over the years.
"But I have to say that my time with the Mets wouldn't have been the same without the greatest fans in the world. One of the hardest moments of my career, was walking off the field at Shea Stadium and saying goodbye. My relationship with you made my time in New York the happiest of my career and for that, I will always be grateful."
H O F
Definitely. He'll be ranked with the likes of Bench as one of the greatest hitting catchers of all time.
He was my favorite player when I was a kid.
He put up epic numbers for a catcher. Easily first ballot HOF.
Though it's interesting because he was so poor defensively.
Last time I hear Piazza discussed
it was if he was a pitcher or catcher
I always though he was at least serviceable even though he threw out runners at such a poor rate. However later in his career he was so terrible that guys could run on him pretty freely.
Ultra Peanut
God, I remember when he was the Rookie of the Year.
This makes me feel old. I'm sure it makes him feel even older, since I was only like eight at the time, but still.
One of the best hitting catchers of all time. Of course he's in.
Having that said he was absolutely AWFUL defensively.
Frazod
I guess if Clemens wants to throw shit at him now, he'll have to go to his house. :D
I remember Mike Piazza endorsed a Nintendo 64 game "Mike Piazza's Baseball", and it was the most unrealistic game ever. This game wasn't like Blitz or the Sluggfest, this game was supposed to be like Triple Play or All Star Baseball it was supposed to be realistic.
Then i hit my first homerun in the 2nd game I played.
950ft shot.
I then returned the game after that.
Sure-Oz
Wasn't he like a 67th rounder | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 510 |
package nl.knaw.huygens.timbuctoo.v5.jsonfilebackeddata;
import com.fasterxml.jackson.core.type.TypeReference;
import com.google.common.base.Charsets;
import org.junit.Test;
import java.io.File;
import java.nio.file.Files;
import java.util.HashMap;
import java.util.Map;
import static org.hamcrest.MatcherAssert.assertThat;
import static org.hamcrest.Matchers.nullValue;
import static org.hamcrest.core.Is.is;
public class JsonFileBackedDataTest {
@Test
public void itStoresData() throws Exception {
File tmpFile = Files.createTempFile("prefix", "asd").toFile();
try {
tmpFile.delete(); //the file must either not exist, or contain valid data
JsonFileBackedData<Map<String, String>> instance = JsonFileBackedData.getOrCreate(
tmpFile,
() -> new HashMap<>(),
new TypeReference<Map<String, String>>() {
}
);
assertThat(instance.getData().isEmpty(), is(true));
assertThat(new String(Files.readAllBytes(tmpFile.toPath()), Charsets.UTF_8), is("{ }"));
instance.updateData(data -> {
data.put("foo", "utf-8 ☃");
return data;
});
assertThat(new String(Files.readAllBytes(tmpFile.toPath()), Charsets.UTF_8), is("{\n \"foo\" : \"utf-8 ☃\"\n}"));
JsonFileBackedData.regenerateSoWeCanTestHowWellLoadingWorks(tmpFile);
JsonFileBackedData<Map<String, String>> reloadedInstance = JsonFileBackedData.getOrCreate(
tmpFile,
() -> new HashMap<>(),
new TypeReference<Map<String, String>>() {
}
);
assertThat(reloadedInstance.getData().get("foo"), is("utf-8 ☃"));
} finally {
tmpFile.delete();
}
}
@Test
public void itCanBeInitializedToNull() throws Exception {
File tmpFile = Files.createTempFile("prefix", "asd").toFile();
try {
tmpFile.delete(); //the file must either not exist, or contain valid data
JsonFileBackedData<Map<String, String>> instance = JsonFileBackedData.getOrCreate(
tmpFile,
() -> null,
new TypeReference<Map<String, String>>() {
}
);
assertThat(instance.getData(), is(nullValue()));
assertThat(new String(Files.readAllBytes(tmpFile.toPath()), Charsets.UTF_8), is("null"));
} finally {
tmpFile.delete(); //the file must either not exist, or contain valid data
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,012 |
Q: Neo4j bidirectional traversal api I am playing around with Neo4j and so far I have a geographical graph where an AIRPORT is connect to a CITY, the CITY to a COUNTRY and the COUNTRY to a CONTINENT, as depicted in the picture
Labels on the arrows translate to org.neo4j.graphdb.RelationshipType into my code. So far, I can build the path between the start node MXP to the end node LTN using the following mono-directional traversal.
Traverser traverse = database.traversalDescription().depthFirst()
.relationships(CITY, BOTH)
.relationships(CONTINENT, BOTH)
.relationships(COUNTRY, BOTH)
.relationships(REGION, BOTH)
.evaluator(Evaluators.includeWhereEndNodeIs(endNode)).traverse(startNode);
With this, I get a single path MXP -> Milan -> Italy -> Europe <- England <- London <- LTN, which is correct given the graph description, the traversal description and of course my understanding my understanding of such description.
I am trying to change this code to perform a bidirectional traversal, meaning I want to start from both MXP and LTN and stop at the collision point. I tried with the following snippet, where comments mean my understanding so it may easier to point out the problem.
TraversalDescription startSide = database.traversalDescription().depthFirst() //Depth first algorithm
.relationships(CITY, OUTGOING) //consider CITY relationship, only outgoing
.relationships(REGION, OUTGOING) //consider REGION relationship, only outgoing
.relationships(COUNTRY, OUTGOING) //consider COUNTRY relationship, only outgoing
.relationships(CONTINENT, OUTGOING) //consider CONTINENT relationship, only outgoing
.evaluator(Evaluators.excludeStartPosition()); //do not consider the starting point.
//Here I tried also with all, with the same result
//with includeWhereEndNodeIs(endNode), again with same result
//and combining includeWhereEndNodeIs and excludeStartPosition, once more with same result.
//All tries I mirrored for the endSide description, changing endNode to startNode where I feel it was needed
TraversalDescription endSide = database.traversalDescription().depthFirst()
.relationships(CITY, OUTGOING)
.relationships(REGION, OUTGOING)
.relationships(COUNTRY, OUTGOING)
.relationships(CONTINENT, OUTGOING)
.evaluator(Evaluators.excludeStartPosition());
List<Node> asList = Arrays.asList(startNode, endNode);
Traverser traverse = database.bidirectionalTraversalDescription().endSide(endSide).startSide(startSide).traverse(asList, asList);
Here, instead of the path I am getting with the monodirectional traversal try, I get two paths, one with only MXP and one with only LTN.
At this point I seriously believe I am completely misunderstanding the bidirectional traversal and maybe even its purpose. Where is my mistake? Why I do not get the same output?
A: I finally got a working solution. The problem in my code was related to the concept of uniqueness. Interesting points for my problem are
Sets the rules for how positions can be revisited during a traversal as stated in Uniqueness. Default if not set is NODE_GLOBAL.
NODE_GLOBAL uniqueness: No node in the entire graph may be visited more than once. This could potentially consume a lot of memory since it requires keeping an in-memory data structure remembering all the visited nodes.
NODE_PATH uniqueness: A node may not occur previously in the path reaching up to it.
These descriptions are somehow different from the official API so I played around trying different combination and ended up with the following code:
TraversalDescription bothSide = database.traversalDescription().depthFirst()
.relationships(CITY, OUTGOING)
.relationships(REGION, OUTGOING)
.relationships(COUNTRY, OUTGOING)
.relationships(CONTINENT, OUTGOING)
.uniqueness(NODE_PATH);
Traverser traverser = database
.bidirectionalTraversalDescription()
.startSide(bothSide)
.endSide(bothSide)
.traverse(node, endNode);
Basically, I defined a common TraversalDescription for both the end and the start side, where I want to follow only OUTGOING relationships and I want to consider paths only where nodes are unique inside the path itself.
Then, I defined a bidirectional traverser which simply sets up the end and the start side and traverses the graph from the starting node node to the end node endNode (well, actually it traverses from start to end AND from end to start at the same time and stops when the two traversal collide, merging the resulting paths into a single path leading from start to end).
NOTE: I am not completely sure about the meaning of NODE_GLOBAL, since in my database each node represents a geographic entity, so each node in the path MXP -> Milan -> Italy -> Europe <- England <- London <- LTN should be visited only once and thus there should be no difference between NODE_GLOBAL and NODE_PATH in this context.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,031 |
Pianist Jesse Stacken Releases Bagatelles for Trio Out June 12, 2012 on Fresh Sound New Talent
Posted by Rob Young in Creative Music, Improvised Music, Modern Jazz, New Music, Press Release, What's New?
≈ Comments Off on Pianist Jesse Stacken Releases Bagatelles for Trio Out June 12, 2012 on Fresh Sound New Talent
Eivind Opsvik, JEFF DAVIS, Jesse Stacken
With EIVIND OPSVIK (bass) & JEFF DAVIS (drums)
Jesse Stacken, Bagatelles for Trio
Béla Bartók's "Fourteen Bagatelles" (1908) were among the Hungarian composer's earliest masterpieces. The collection of piano miniatures marked a formative moment in the young composer's life, harkening back to the Romanticism of the century just ended, showing nascent traces of his exploration of his native country's folk music, and taking bold new steps in the direction of the avant-garde.
At a similarly early stage in his own already-impressive career, pianist/composer Jesse Stacken has composed his own set of Bagatelles – one less in number than Bartók's, but expanded for a piano trio and open to jazz improvisation.
Jesse Stacken
"I really liked Bartók's 'Bagatelles' and studied them," Stacken says. "And as I was learning them, I thought that a lot of them were almost perfect for inspiring some sort of improvisation. Each one is short, concise, and portrays one or two very direct ideas. So I thought about composing something with that in mind."
Stacken's "Bagatelles" were conceived for his working trio with bassist Eivind Opsvik and drummer Jeff Davis. The trio formed in 2005 and has recorded two previous albums for Fresh Sound New Talent Records: That That (2007), which also focused largely on shorter works, and Magnolia(2009), which explored more extended forms. Stacken calls Bagatelles for Trioboth an extension of and departure from those previous releases.
Recorded shortly after achieving his master's degree from the Manhattan School of Music, "the first record was me trying to get away from school," Stacken says. "I was just trying to respectfully reject some of that college-sounding jazz, get out of my comfort zone and get into some different sounds. Magnolia was more about rediscovering the sound of the piano."
This third release, Stacken continues, became driven by "a return to melody." That goal is clear throughout, in each of the Bagatelles' starkly bare-bones presentation of its core elements. Unadorned melody stands in skeletal relief on pieces like the ethereally introspective "No. 3," the slyly darting "No. 4," or the stutter-stop "No. 8." But elements that inflected those previous discs recur here: the resonant intonations of the piano, influenced by the work of composer Morton Feldman, motivates a number of pieces, "No. 1" in particular. At the same time, even in their abbreviated form, several works are constructed with intricate frameworks; "No. 8" and "No. 11" draw inspiration from Schoenberg's 12-tone serial technique and "No. 12" from the immense chords of Messiaen's compositions for organ.
More than any individual piece, however, the entire set of Bagatelles for Trio can be viewed as a single extended work, and Stacken labored to craft an overarching unity for the album as a whole. The trio performs the piece only in its entirety, in the order represented on the CD, never as individual selections out of context.
Stacken insists that the composition of the Bagatelles was heavily influenced by the individual voices of his longtime triomates. He initially met Davis while both were playing with saxophonist Peter Van Huffel, though when it came time to form his own group, "Jeff was not the most comfortable choice," Stacken says. "He's got a very warm feeling to his playing, but he's not a guy who lays down the time. He plays around the kit in a very melodic way. I decided to go with Jeff to get me going in a different direction."
Stacken calls Opsvik, "an amazing player, but he also has a really strong personality and he a lot of artistic savvy. He has a really good ear." It was Opsvik, in fact, who finally pointed out that a fourteenth Bagatelle seemed not to fit; Stacken agreed and removed the piece.
"The chemistry between us seems to work and develop really well," Stacken says of the trio. "We're friends and have a really good time together on and off the bandstand. It seems to be getting better and better as we go."
Originally from Hopkins, Minnesota, Stacken studied music education at the University of Wisconsin-Eau Claire before earning his master's at the Manhattan School of Music. He has since become an active participant in New York's creative music scene. Besides his trio, he works regularly with trumpeter/cornetist Kirk Knuffke; the two have recorded a pair of albums as a duo and a third, as a trio with drummer Kenny Wolleson, was recently released. Stacken recently formed a quartet called For the Mill with some of New York's most inventive musicians: saxophonists Andrew D'Angelo and Josh Sinton and drummer Mike Pride.
Jesse Stacken: Upcoming Performances
Thursday May 3: Jesse Stacken Solo Piano, I-Beam, 8:30 PM
Sat May 12: Stacken/Knuffke Duo, I-Beam, 8:30 PM
Thurs June 7: Liam Sillery Quintet, Somethin' Jazz Club 7pm
Fri June 29: Jesse Stacken Trio, Tenri Cultural Institute, 8pm
Bagatelles for Trio CD Release
For more information, please visit jessestacken.com and freshsoundrecords.com
..:: SOURCE: Fully Altered Media ::.. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 39 |
Gum disease is a common condition that leads to tooth loss. Bacteria contained in plaque and tartar causes inflammation and infection both superficially and beneath the gum line. Without proper treatment, the condition progresses until teeth become loose and fall out. Symptoms of early gum disease include red, tender, or puffy gums, persistent bad breath, bleeding during brushing or flossing, receding gums, changes in bite, and the formation of deep, infected pockets between teeth and gums.
Anyone working in family dentistry will tell you that gum disease is largely preventable with regular dental care. Start by maintaining good oral hygiene. Brush carefully twice a day or more, and floss daily. Use an antibacterial mouth rinse to kill oral bacteria. Other ways to improve your oral health include avoiding tobacco, reducing stress, eating a healthy diet, and not grinding or clenching your teeth. For the best plaque control, have your teeth professionally cleaned twice a year. If you don't have dental insurance, find an affordable dentist in Englewood who offers discounted dental care or a payment plan. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,623 |
import logging
# Third-Party modules
import flask
# Project modules
logging.basicConfig(level=logging.INFO)
app = flask.Flask(__name__)
@app.route('/old')
def index():
return flask.render_template('/old/index.html')
@app.route('/')
def home():
return flask.redirect(flask.url_for('index'))
@app.route('/google8b87abaa24c74d5d.html')
def google_verification():
return flask.render_template('google8b87abaa24c74d5d.html')
if __name__ == '__main__':
app.run()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,382 |
Черна вдовица (Latrodectus mactans) е един от 31-те вида отровни паяци от рода на черните вдовици (Latrodectus). Отровата при ухапване от женския паяк е много опасна за човека, понякога може да е смъртноносна. Мъжките индивиди почти никога не хапят.
Описание
Среща се основно в Америка, сравнително малък е, тялото достига най-често 15 mm при женските и 9 mm при мъжките. Черен е на цвят с много характерно червено оцветяване отдолу на корема във формата на пясъчен часовник. Паяжината на черната вдовица е много здрава. Черната вдовица се храни с насекоми. Нарича се вдовица заради това, че когато се размножава тя често убива мъжкия след контакта, както е и при богомолките.
Външни препратки
Черна вдовица
Черни вдовици
Отровни животни | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,344 |
Richard Francis (1827 - November 30, 1888) was a famous Black bartender in pre-Prohibition Washington, D.C. Francis was born in 1827 to free Black parents in Surry County, Virginia. By 1848 he was in Washington, D.C., where he worked at Hancock's bar on 12th and Pennsylvania Avenue for almost four decades. He was a friend and confidant to a wide range of Washington politicians, including reportedly Clay, Calhoun, and Webster. In 1884, his friend, Senator George F. Edmunds, who was at that time President pro tempore of the Senate, gave him the patronage role of managing the private restaurant and bar that then existed in the U.S. Senate. Cocktail historian Dave Wondrich reports that, while the record is fragmentary, the first Black bartender for Congress was an individual by the name of Carter in the 1830s to 1850s, and Francis is believed to be the second Black bar manager for Congress.
He died in 1888 a wealthy man due to his investments in DC real estate; his son, Dr. John R. Francis, later purchased Hancock's bar.
He died of a paralytic stroke at home the morning of November 30, 1888 at the age of 62.
References
1827 births
1888 deaths
People from Surry County, Virginia
People from Washington, D.C.
American bartenders | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 970 |
<?php
require __DIR__ . '/product_simple_rollback.php';
require __DIR__ . '/../../../Magento/Checkout/_files/active_quote_rollback.php';
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,640 |
Q: iPhone: how to stop Record Automated tests in Instruments I wrote some scripts for automation testing, then I put them in .js file. And using Instruments, I start them by taping on Record button. What I want to know, is there any way to stop recording in script ? Because, if my Test1 fail, I don't want Test2 to be run. Thanks...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,084 |
Q: Inverting digits of a hex value I need to take all the digits in a hex number and "invert" them: all zeroes become non-zeroes (F) and all non-zeroes become zeroes.
I tried:
void someFunction(DWORD hexVal)
{
//...
hexVal = ~hexVal;
//...
}
and this changed 0xE0000000 to 0x1FFFFFFF instead of 0x0FFFFFFF.
How can I produce the desired result?
A: This should give you the desired result for 2 bytes. You get the idea for 4 bytes.
hexval = ((hexval & 0xf000) ? 0 : 0xf000) |
((hexval & 0xf00) ? 0 : 0xf00) |
((hexval & 0xf0) ? 0 : 0xf0) |
((hexval & 0xf) ? 0 : 0xf);
A: Assuming you really meant that you want zero->non-zero and vice-versa, on a digit-by-digit basis:
DWORD invertDigits(DWORD in) {
return (
((in & (0xF << 28)) ? 0x0 : (0xF << 28)) |
((in & (0xF << 24)) ? 0x0 : (0xF << 24)) |
((in & (0xF << 20)) ? 0x0 : (0xF << 20)) |
((in & (0xF << 16)) ? 0x0 : (0xF << 16)) |
((in & (0xF << 12)) ? 0x0 : (0xF << 12)) |
((in & (0xF << 8)) ? 0x0 : (0xF << 8)) |
((in & (0xF << 4)) ? 0x0 : (0xF << 4)) |
((in & (0xF << 0)) ? 0x0 : (0xF << 0))
);
}
A: That is the desired result for the bitwise NOT operation. 0xE0000000 + 0x1FFFFFFF = 0xFFFFFFFF
The absolute fastest way to do what you want would be to split it into bytes and use a lookup table.
This solution takes the processor equivalent of about: 24 adds, 4 multiplies, and 4 memory lookups. The multiplies are part of the array indexing. All simple mathematical operations run at about the same speed, except multiplies and memory lookups which are slightly longer. Your mileage may vary depending on your processor architecture and the compiler optimizations performed.
unsigned int transform1(unsigned int value)
{
// static const unsigned char ZZ = 0x0, ZF = 0xF, FZ = 0xF0, FF = 0xFF; // for C++
#define ZZ (unsigned char) 0x00
#define FZ (unsigned char) 0xF0
#define ZF (unsigned char) 0x0F
#define FF (unsigned char) 0xFF
static const unsigned char lookup[256] =
{
FF, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ, FZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
ZF, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ, ZZ,
}; // array takes up 1KB of RAM
unsigned int result = 0;
result |= lookup[(unsigned int)((value & (FF << 0 )) >> 0) ] << 0;
result |= lookup[(unsigned int)((value & (FF << 8 )) >> 8) ] << 8;
result |= lookup[(unsigned int)((value & (FF << 16)) >> 16)] << 16;
result |= lookup[(unsigned int)((value & (FF << 24)) >> 24)] << 24;
return result;
}
A: You may have to go byte by byte, starting with the MSB. Check if the value is between 16^6 and 16^7 (assuming this is unsigned). If it is, add to the new number 0. If it's not, add to the new number 2^31+2^30+2^29+2^28.
See what I'm getting at?
A: So inversion and negation are two different things.
Inversion takes each bit and produces its complement like so:
0xE0000000 = 1110 0000 0000 0000 0000 0000 0000 0000
~0xE0000000 = 0001 1111 1111 1111 1111 1111 1111 1111 = 0x1FFFFFFF
If you want "Anything other than zero needs to become zero" you want boolean negation i.e.
hexVal = !hexVal;
EDIT: Okay, so I finally got what the asker was asking after reading some of the other answers, here's my personal version using one giant bit math expression
n = ~(n | ((n & 0x77777777) << 1) | ((n & 0x88888888) >> 3)
| ((n & 0x33333333) << 2) | ((n & 0xCCCCCCCC) >> 2)
| ((n & 0x11111111) << 3) | ((n & 0xEEEEEEEE) >> 1));
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,789 |
Дар'ївка, Дарів — гірська річка в Україні, у Рожнятівському районі Івано-Франківської області. Ліва притока Лімниці, (басейн Дністра).
Опис
Довжина річки 10 км, похил річки 26 м/км, площа басейну водозбору 29,8 км², найкоротша відстань між витоком і гирлом — 8,94 км, коефіцієнт звивистості річки — 1,12. Формується багатьма безіменними гірськими потоками. Річка розташована в Українських Карпатах.
Розташування
Бере початок на південних схилах хребта Дарів (1187,8 м). Тече переважно на північний схід понад хребтом Верхній Дарів (1236,0 м) і у селі Осмоло́да впадає у річку Лімницю, праву притоку Дністра.
Цікаві факти
На лівому березі річки розташована гора Студенен (1600 м), а на правому березі — гора Овал (1609,6 м).
У пригирловій частині на правому березі річки розташована станція вузькоколійної залізниці Дар'їв. Маршрутна схема (Брошнів — Рожнятів — Перегінське — Закерничне (гілка на Гуту) — Ангелове — Осмолода — Дар'їв).
У селі Осмолода річку перетинає тупиковий автошлях .
Примітки
Джерела
«Каталог річок України» . — К. : Видавництво АН УРСР, 1957. — С. 37. — (№ 483).
Словник гідронімів України — К.: Наукова думка, 1979. — С. 165 (Дарів)
Малі річки України
Річки Івано-Франківської області
Річки Рожнятівського району
Статті про річки Івано-Франківської області без зображення в картці | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,505 |
the link below leads to the rest of what you can start with pasted herein. it shows more fully how the empire works its nefarious ways;
On the world's Grand Chessboard, the US is fighting for control and influence. And there are countries where its ambassadors are perceived more as imperial governors than simple channels of communication.
At the height of the Maidan protests in Kiev in early 2014, a conversation was leaked between the US ambassador to Ukraine, Geoffrey Pyatt, and the then-Assistant Secretary of State in the Obama administration, Victoria Nuland. The conversation gained notoriety because Nuland said to Pyatt, "F**k the EU" and the recording was almost instantly available on Youtube.
More shocking than Nuland's bad language, however, was what the conversation was about. The US government officials were discussing how to put their men into power in Ukraine – which of the three then opposition factions would dominate, who would take the lead (Arseniy Yatsenyuk) and who would be excluded (Vladimir Klitschko). At the time of this conversation, early February 2014, their enemy Viktor Yanukovych was still president. The leaked recording proved that the US and its Kiev embassy were actively involved in a regime change operation. The composition of the post-Maidan government corresponded exactly with US plans.
What few people knew at the time was that such levels of control over the composition of foreign governments had become standard practice for US embassies all over the world. As I could see on my very numerous travels around the Balkans in the late 1990s and early 2000s, the US ambassador was treated by the political class and the media in those countries not as the officially accredited representative of a foreign government but instead as an imperial governor whose pronunciamentos were more important than those of the national government.
https://www.globalresearch.ca/us-diplomats-act-like-imperial-governors-riding-roughshod-over-sovereignty-of-national-governments/5648227 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,578 |
{"url":"http:\/\/cnx.org\/content\/m11051\/latest\/","text":"# Connexions\n\nYou are here: Home \u00bb Content \u00bb C62x Assembly Primer II\n\n### Lenses\n\nWhat is a lens?\n\n#### Definition of a lens\n\n##### Lenses\n\nA lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.\n\n##### What is in a lens?\n\nLens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.\n\n##### Who can create a lens?\n\nAny individual member, a community, or a respected organization.\n\n##### What are tags?\n\nTags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.\n\n#### Affiliated with (What does \"Affiliated with\" mean?)\n\nThis content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.\n\u2022 Rice Digital Scholarship\n\nThis module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: \"Finite Impulse Response\"\n\nClick the \"Rice Digital Scholarship\" link to see all content affiliated with them.\n\n### Recently Viewed\n\nThis feature requires Javascript to be enabled.\n\n# C62x Assembly Primer II\n\nModule by: Hyeokho Choi. E-mail the author\n\n## Typical Assembly Operations\n\nQuite often you need to load a register with a constant. The C62x instructions you can use for this task are MVK, MVKL, and MVKH. Each of these instructions can load a 16-bit constant to a register. Read and understand the description of these instructions in the manual.\n\n#### Exercise 1\n\n1. Load the 16-bit constant 0xff12 to A1.\n2. Load the 32-bit constant 0xabcd45ef to B0.\n\n##### Solution\n\nIntentionally left blank.\n\n### Register moves, zeroing\n\nContents of one register can be copied to another register by using the MV instruction. There is also the ZERO instruction to set a register to zero. Learn how to use these instructions by reading the appropriate TI manual pages.\n\nBecause the C62x processor has the so-called load\/store architecture, you must first load up the content of memory to a register to be able to manipulate it. The basic assembly instructions you use for loading are LDB, LDH, and LDW for loading up 8-, 16-, and 32-bit data from memory. (There are some variations to these instructions for different handling of the signs of the loaded values.) Read and understand how these instructions work.\n\nHowever, to specify the address of the memory location to load from, you need to load up another register (used as an address index) and you can use various addressing modes to specify the memory locations in many different ways. The addressing modes is the method by which an instruction calculates the location of an object in memory. The table below lists all the possible different ways to handle the address pointers in C62x CPU. Note the similarity with the C pointer manipulation.\n\nTable 1\nSyntax Memory address accessed Pointer modification\n*R R None\n*++R R Preincrement\n*--R R Predecrement\n*R++ R Postincrement\n*R-- R Postdecrement\n*+R[disp] R+disp None\n*-R[disp] R+disp None\n*++R[disp] R+disp Preincrement\n*--R[disp] R+disp Predecrement\n*R++[disp] R+disp Postincrement\n*R--[disp] R+disp Postdecrement\n\nThe [disp] specifies the number of elements in word, halfword, or byte, depending on the instruction type and it can be either 5-bit constant or a register. The increment\/decrement of the index registers are also in terms of the number of bytes in word, halfword or byte. The addressing modes with displacements are useful when a block of memory locations is accessed. Those with automatic increment\/decrement are useful when a block is accessed consecutively to implement a buffer, for example, to store signal samples to implement a digital filter.\n\n#### Exercise 2\n\n(Load from memory): Assume the following values are stored in memory addresses:\n\n\n100h fe54 7834h\n104h 3459 f34dh\n108h 2ef5 7ee4h\n10ch 2345 6789h\n110h ffff eeddh\n114h 3456 787eh\n118h 3f4d 7ab3h\n\n\nSuppose A10 = 0000 0108h. Find the contents of A1 and A10 after executing the each of the following instructions.\n\n1. LDW .D1 *A10, A1\n2. LDH .D1 *A10, A1\n3. LDB .D1 *A10, A1\n4. LDW .D1 *-A10[1], A1\n5. LDW .D1 *+A10[1], A1\n6. LDW .D1 *+A10[2], A1\n7. LDB .D1 *+A10[2], A1\n8. LDW .D1 *++A10[1], A1\n9. LDW .D1 *--A10[1], A1\n10. LDB .D1 *++A10[1], A1\n11. LDB .D1 *--A10[1], A1\n12. LDW .D1 *A10++[1], A1\n13. LDW .D1 *A10--[1], A1\n\n##### Solution\n\nIntentionally left blank.\n\n### Storing data to memory\n\nStoring the register contents uses the same addressing modes. The assembly instructions used for storing are STB, STH, and STW. Read and understand these instructions in the TI manual.\n\n#### Exercise 3\n\n(Storing to memory): Write assembly instructions to store 32-bit constant 53fe 23e4h to memory address 0000 0123h.\n\n##### Solution\n\nIntentionally left blank.\n\nSometimes, it becomes necessary to access part of the data stored in memory. For example, if you store the 32-bit word 0x11223344 at memory location 0x8000, the four bytes having addresses location 0x8000, location 0x8001, location 0x8002, and location 0x8003 contain the value 0x11223344. Then, if I read the byte data at memory location 0x8000, what would be the byte value to be read?\n\nThe answer depends on the endian mode of the memory system. In the little endian mode, the lower memory addresses contain the LSB part of the data. Thus, the bytes stored in the four byte addresses will be as shown in Table 2.\n\n 0x8000 0x44 0x8001 0x33 0x8002 0x22 0x8003 0x11\n\nIn the big endian mode, the lower memory addresses contain the MSB part of the data. Thus, we have\n\n 0x8000 0x11 0x8001 0x22 0x8002 0x33 0x8003 0x44\n\nIn this course, we use the little endian mode by default and all the lab programming must assume the little endian mode.\n\n#### Exercise 4\n\n(Little endian mode): What will be the value in A0 after executing the following assembly instructions? (functional unit specifications were omitted.)\n\n1. MVKL 0x80000000, A10\n2. MVKH 0x80000000, A10\n3. MVKL 0x12345678, A9\n4. MVKH 0x12345678, A9\n5. STW A9, *A10\n6. LDB *+A10[2],A0\nWhat will be the value in A0 if the system uses the big endian mode?\n\n##### Solution\n\nIntentionally left blank.\n\nIn fact, the above addressing method describes the so-called linear addressing mode (default upon reset), where the offset or increment\/decrement of pointers occur without bound. There is a circular addressing modes that can handle a finite size buffer efficiently. You will implement circular buffers for the FIR filtering algorithm in the FIR filtering experiments later.\n\nIn the C62x CPU, it takes exactly one CPU clock cycle to execute each instruction. However, the instructions such as LDW need to access the slow external memory and the results of the load are not available immediately at the end of the execution. This delay of the execution results is called delay slots.\n\n#### Example 1\n\nFor example, let's consider loading up the content of memory content at address pointed by A10 to A1 and then moving the loaded data to A2. You might be tempted to write simple 2 line assembly code as follows:\n\n\n1 LDW .D1 *A10, A1\n2 MV .D1 A1,A2\n\n\nWhat is wrong with the above code? The result of the LDW instruction is not available immediately after LDW is executed. As a consequence, the MV instruction does not copy the desired value of A1 to A2. To prevent this undesirable execution, we need to make the CPU wait until the result of the LDW instruction is correctly loaded to A1 before executing the MV instruction. For load instructions, we need extra 4 clock cycles until the load results are valid. To make the CPU wait for 4 clock cycles, we need to insert 4 NOP (no operations) instructions between LDW and MV. Each NOP instruction makes the CPU idle for one clock cycle. The resulting code will be like this:\n\n\n1 LDW .D1 *A10, A1\n2 NOP\n3 NOP\n4 NOP\n5 NOP\n6 MV .D1 A1,A2\n\n\nor simply you can write\n\n\n1 LDW .D1 *A10, A1\n2 NOP 4\n3 MV .D1 A1,A2\n\n\nThen, why didn't the designer of the CPU make such that LDW instruction takes 5 clock cycles to begin with, rather than let the programmer insert 4 NOPs? The answer is that you can insert other instructions other than NOPs as far as those instructions do not use the result of the LDW instruction above. By doing this, the CPU can execute additional instructions while waiting for the result of the LDW instruction to be valid, greatly reducing the total execution time of the entire program.\n\n### More on instructions with delay slots\n\nThe Table 3-5 in TI's instruction set description shows the execution of the instructions with delay slots in more detail. The instructions with delay slots are multiply (MPY, 1 delay slot), the load (LDB, LDW etc. 4 delay slots) instructions, and the branch (B, 5 delay slots) instruction.\n\nThe functional unit latency indicates for how many clock cycles each instructions actually use a functional unit. All C62x instructions have 1 functional unit latency, meaning that each functional unit is ready to execute the next instruction after 1 clock cycle regardless of the delay slots of the instructions. Therefore, the following instructions are valid:\n\n\n1 LDW .D1 *A10, A4\n\n\nAlthough the first LDW instruction do not load the A4 register correctly while the ADD is executed, the D1 functional unit becomes available in the clock cycle right after the one in which LDW is executed.\n\nTo clarify the execution of instructions with delay slots, let's think of the following example of LDW instruction. Let's assume A10 = 0x0100 A2=1, and your intent is loading A9 with the 32-bit word at the address 0x0104. The 3 MV instructions are not related to the LDW instruction. They do something else.\n\n\n1 LDW .D1 *A10++[A2], A9\n2 MV .L1 A10, A8\n3 MV .L1 A1, A10\n4 MV .L1 A1, A2\n5 ...\n\n\nWe can ask several interesting questions at this point:\n\n1. What is the value loaded to A8? That is, in which clock cycle, the address pointer is updated?\n2. Can we load the address offset register A2 before the LDW instruction finishes the actual loading?\n3. Is it legal to load to A10 before the first LDW finishes loading the memory content to A9? That is, can we change the address pointer before the 4 delay slots elapse?\n1. Although it takes extra 4 clock cycles for the LDW instruction to load the memory content to A9, the address pointer and offset registers (A10 and A2) are read and updated in the clock cycle the LDW instruction is issued. Therefore, in line 2, A8 is loaded with the updated A10, that is A10 = A8 = 0x104.\n2. Because the LDW reads the A10 and A2 registers in the first clock cycle, you are free to change these registers and do not affect the operation of the first LDW.\n\nSimilar theory holds for MPY and B (when using a register as a branch address) instructions. The MPY reads in the source values in the first clock cycle and loads the multiplication result after the 2nd clock cycle. For B, the address pointer is read in the first clock cycle, and the actual branching occurs after the 5th clock cycle. Thus, after the first clock cycle, you are free to modify the source or the address pointer registers. For more details, refer Table 3-5 in the instruction set description or read the description of the individual instruction.\n\nThere are several instructions for addition, subtraction and multiplication on C62x CPU. The basic instructions are ADD, SUB, and MPY. Learn about these instructions in the TI manual. ADD and SUB have 0 delay slots (meaning the results of operation are immediately available), but the MPY has 1 delay slot (the result of multiplication is valid after additional 1 clock cycle).\n\n#### Exercise 5\n\n(Add, subtract, and multiply): Write an assembly program to compute ( 0000 ef35h + 0000 33dch - 0000 1234h ) * 0000 0007h\n\n##### Solution\n\nIntentionally left blank.\n\n### Branching and conditional operations\n\nOften you need to control the flow of the program execution by branching to another block of code. The B instruction does the job in the C62x CPU. The address of the branch can be specified either by displacement or stored in a register to be used by the B instruction. Read and understand the B instruction in the manual. The B instruction has 5 delay slots, meaning that the actual branch occurs in the 5th clock cycle after the instruction is executed.\n\nIn many cases, depending on the result of previous operations, you execute the branch instruction conditionally. For example, to implement a loop, you decrement the loop counter by 1 each time you run a set of instructions and whenever the loop counter is not zero, you need to branch to the beginning of the code block to iterate the loop operations. In C62x CPU, this conditional branching is implemented using the conditional operations. Although B may be the instruction implemented using conditional operations most often, all instructions in C62x can be conditional.\n\nConditional instructions are represented in code by using square brackets, [ ], surrounding the condition register name. For example, the following B instruction is executed only if B0 is nonzero:\n\n\n1 [B0] B .L1 A0\n\n\nTo execute an instruction conditionally when the condition register is zero, we use ! in front of the register. For example, the B instruction is executed when B0 is zero.\n\n\n1 [!B0] B .L1 A0\n\n\nNot all registers can be used as the condition registers. In C62x CPU, the registers that can be tested in conditional operations are B0, B1, B2, A1, A2.\n\n#### Exercise 6\n\n(Simple loop): Write an assembly program computing the summation n=1100n n 1 100 n by implementing a simple loop.\n\n##### Solution\n\nIntentionally left blank.\n\n### Logical operations and bit manipulation\n\nThe logical operations and bit manipulations are accomplished by the AND, OR, XOR, CLR, SET, SHL, and SHR instructions. Read and understand the operations of these instructions.\n\n### Other assembly instructions\n\nOther useful instructions include IDLE and compare instructions such as CMPEQ etc. Read and understand the operations of these instructions.\n\n### C62x instruction set summary\n\nThe set of instructions that can be performed in each functional unit is as follows (See Table 4, Table 5, Table 6 and Table 7). Please refer to TMS320C62x\/C67x CPU and Instruction Set Reference Guide for detailed description of each instruction.\n\nTable 4: .S Unit\nInstruction Description\nADD(U) signed or unsigned integer addition without saturation\nADDK integer addition using signed 16-bit constant\nADD2 two 16-bit integer adds on upper and lower register halves\nB branch using a register\nCLR clear a bit field\nEXT extract and sign-extend a bit field\nMV move from register to register\nMVC move between the control file and the register file\nMVK move a 16-bit constant into a register and sign extend\nMVKH move 16-bit constant into the upper bits of a register\nNEG negate (pseudo-operation)\nNOT bitwise NOT\nOR bitwise OR\nSET set a bit field\nSHL arithmetic shift left\nSHR arithmetic shift right\nSSHL shift left with saturation\nSUB(U) signed or unsigned integer subtraction without saturation\nSUB2 two 16-bit integer integer subs on upper and lower register halves\nXOR exclusive OR\nZERO zero a register (pseudo-operation)\nTable 5: .L Unit\nInstruction Description\nABS integer absolute value with saturation\nADD(U) signed or unsigned integer addition without saturation\nAND bitwise AND\nCMPEQ integer compare for equality\nCMPGT(U) signed or unsigned integer compare for greater than\nCMPLT(U) signed or unsigned integer compare for less than\nLMBD leftmost bit detection\nMV move from register to register\nNEG negate (pseudo-operation)\nNORM normalize integer\nNOT bitwise NOT\n+OR bitwise OR\nSADD integer addition with saturation to result size\nSAT saturate a 40-bit integer to a 32-bit integer\nSSUB integer subtraction with saturation to result size\nSUBC conditional integer subtraction and shift - used for division\nXOR exclusive OR\nZERO zero a register (pseudo-operation)\nTable 6: .D Unit\nInstruction Description\nADD(U) signed or unsigned integer addition without saturation\nADDAB (B\/H\/W) integer addition using addressing mode\nLDB (B\/H\/W) load from memory with a 15-bit constant offset\nMV move from register to register\nSTB (B\/H\/W) store to memory with a register offset or 5-bit unsigned constant offset\nSUB(U) signed or unsigned integer subtraction without saturation\nSUBAB (B\/H\/W) integer subtraction using addressing mode\nZERO zero a register (pseudo-operation)\nTable 7: .M Unit\nInstruction Description\nMPY (U\/US\/SU) signed or unsigned integer multiply 16lsb*16lsb\nMPYH (U\/US\/SU) signed or unsigned integer multiply 16msb*16msb\nMPYLH signed or unsigned integer multiply 16lsb*16msb\nMPYHL signed or unsigned integer multiply 16msb*16lsb\nSMPY (HL\/LH\/H) integer multiply with left shift and saturation\n\n## Useful assembler directives\n\nOther than the CPU instruction set, there are special commands to the assembler that direct the assembler to do various jobs when assembling the code. You should learn about some of these assembler directives to be able to write an assembly program. There are useful assembler directives you can use to let the assembler know various settings, such as .set, .macro, .endm, .ref, .align, .word, .byte .include.,\n\nThe .set directive defines a symbolic name. For example, you can have\n\n\n1 count .set 40\n\n\nThen, the assembler replaces each occurrence of count with 40.\n\nYou have already seen how the .ref directive is used to declare symbolic names defined in another file. It is similar to the extern declaration in C.\n\nThe .space directive reserves a memory space with specified number of bytes. For example, you can have\n\n\n1 buffer .space 128\n\n\nto define a buffer of size 128 bytes. The symbol buffer has the address of the first byte reserved by .space. The .bes directive is similar to .space, but the label has the address of the last byte reserved.\n\nTo put a constant value in the memory, you can use .byte, .word, etc. If you have\n\n\n1 const1 .word 0x1234\n\n\nthe assembler places the word constant 0x1234 at a memory location and const1 has the address of the memory location. .byte etc. works similarly.\n\nSometimes you need to place your data or code at a specific memory address boundaries such as word, halfword, etc. You can use the .align directive to do this. For example, if you have\n\n\n1 .align 4\n2 buffer .space 128\n3 ...\n\n\nThen, the first address of the reserved 128 bytes is at the word boundary in memory, that is the 2 LSBs of the address (in binary) are 0. Similarly, for half-word alignment, you should have .align directive to do this. For example, if you have\n\n\n1 .align 2\n2 buffer .space 128\n3 ...\n\n\nThe .include directive is used to read the source lines from another file. If you have\n\n\n1 .include other.asm''\n\n\nwill input the lines in other.asm at this location. This is useful when working with multiple files. Instead of making a project having multiple files, you can simply include these different files in one file.\n\nOther assembler directives include .end, etc. You will learn about the macro directives .macro, .endm later .\n\nHow do you write comments in your assembly program? Anything that follows ; is considered as a comment and ignored by the assembler. For example,\n\n\n1 ; this is a comment\n\n\n## Assigning functional units\n\nEach instruction has particular functional units that can execute it. For a complete list of the instructions that can be executed in each functional unit, see Table 3-2 in the instruction set manual. Note that some instructions can be executed by several different functional units.\n\n(Reference) shows how data and addresses can be transfered between the registers, functional units and the external memory. If you observe carefully, the destination path (marked as dst) going out of the .L1, .S1, .M1 and D1 units are connected to the register file A.\n\n### note:\n\nThis means that any instruction with one of the A registers as destination (the result of operation is stored in one of A registers) should be executed in one of these 4 functional units.\nFor the same reason, if the instructions have B registers as destination, the .L2, .S2, .M2 and D2 units should be used.\n\nTherefore if you know the instruction and the destination register, you should be able to assign the functional unit to it.\n\n### Exercise 7\n\n(Functional units): List all the functional units you can assign to each of these instructions:\n\n1. ADD .?? A0,A1,A2\n2. B .?? A1\n3. MVKL .?? 000023feh, B0\n4. LDW .?? *A10, A3\n\n#### Solution\n\nIntentionally left blank.\n\nIf you look at (Reference) again, each functional unit must receive one of the source data from the corresponding register file. For example, look at the following assembly instruction:\n\n\n\n\nThe .L1 unit gets data from A0 (this is natural) and B0 (this is not) and stores the result in A1 (this is a must). The data path through which the content of B0 is conveyed to the .L1 unit is called 1X cross path. When this happens, we add x to the functional unit to designate the cross path:\n\n\n\n\nSimilarly the data path from register file B to the .M2, .S2 and .L2 units are called 2X cross path.\n\n### Exercise 8\n\n(Cross path): List all the functional units that can be assigned to each of the instruction:\n\n1. ADD .??? B0,A1,B2\n2. MPY .??? A1,B2,A4\n\n#### Solution\n\nIntentionally left blank.\n\nIn fact, when you write an assembly program, you can omit the functional unit assignment altogether. The assembler figures out the available functional units and properly assigns them. However, manually assigned functional units help you to figure out where the actual execution takes place and how the data move around between register files and functional units. This is particularly useful when you put multiple instructions in parallel. We will learn about the parallel instructions later on.\n\n## Writing the inner product program\n\nNow you should know enough about C62x assembly to implement the inner product algorithm to compute y=n=110 a n \u00d7 x n y n 1 10 a n x n\n\n### Exercise 9\n\n(Inner product): Write the complete inner product assembly program to compute y=n=110 a n \u00d7 x n y n 1 10 a n x n where a n a n and x n x n take the following values:\n\n\na[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, a }\nx[] = { f, e, d, c, b, a, 9, 8, 7, 6 }\n\n\nThe a n a n and x n x n values must be stored in memory and the inner product is computed by reading the memory contents.\n\n#### Solution\n\nIntentionally left blank.\n\n## Pipeline, Delay slots and Parallel instructions\n\nWhen an instruction is executed, it takes several steps, which are fetching, decoding, and execution. If these steps are done one at a time for each instruction, the CPU resources are not fully utilized. To increase the throughput, CPUs are designed to be pipelined, meaning that the foregoing steps are carried out at the same time.\n\nOn the C6x processor, the instruction fetch consists of 4 phases; generate fetch address (F1), send address to memory (F2), wait for data (F3), and read opcode from memory (F4). Decoding consists of 2 phases; dispatching to functional units (D1) and decoding (D2). The execution step may consist of up to 6 phases (E1 to E6) depending on the instructions. For example, the multiply (MPY) instructions has 1 delay resulting in 2 execution phases. Similarly, load (LDx) and branch (B) instructions have 4 and 5 delays respectively.\n\nWhen the outcome of an instruction is used by the next instruction, an appropriate number of NOPs (no operation or delay) must be added after multiply (one NOP), load (four NOPs, or NOP 4), and branch (five NOPs, or NOP 5) instructions in order to allow the pipeline to operate properly. Otherwise, before the outcome of the current instruction is available (which is to be used by the next instruction), the next instructions are executed by the pipeline, generating undesired results. The following code is an example of pipelined code with NOPs inserted:\n\n\n1 MVK 40,A2\n2 loop: LDH *A5++,A0\n3 LDH *A6++,A1\n4 NOP 4\n5 MPY A0,A1,A3\n6 NOP\n8 SUB A2,1,A2\n9 [A2] B loop\n10 NOP 5\n11 STH A4,*A7\n\n\nIn line 4, we need 4 NOPs because the A1 is loaded by the LDH instruction in line 3 with 4 delays. After 4 delays, the value of A1 is available to be used in the MPY A0,A1,A3 in line 5. Similarly, we need 5 delays after the [A2] B loop instruction in line 9 to prevent the execution of STH A4,*A7 before branching occurs.\n\nThe C6x Very Large Instruction Word (VLIW) architecture, several instructions are captured and processed simultaneously. This is referred to as a Fetch Packet (FP). This Fetch Packet allows C6x to fetch eight instructions simultaneously from on-chip memory. Among the 8 instructions fetched at the same time, multiple of them can be executed at the same time if they do not use same CPU resources at the same time. Because the CPU has 8 separate functional units, maximum 8 instructions can be executed in parallel, although the type of parallel instructions are limited because they must not conflict each other in using CPU resources. In assembly listing, parallel instructions are indicated by double pipe symbols (||). When writing assembly code, by designing code to maximize parallel execution of instructions (through proper functional unit assignments, etc.) the execution cycle of the code can be reduced.\n\n## Parallel instructions and constraints\n\nWe have seen that C62x CPU has 8 functional units. Each assembly instruction is executed in one of these 8 functional units, and it takes exactly one clock cycle for the execution. Then, while one instruction is being executed in one of the functional units, what are other 7 functional units doing? Can other functional units execute other instructions at the same time?\n\nThe answer is YES. Thus, the CPU can execute maximum 8 instructions in each clock cycle. The instructions executed in the same clock cycle are called parallel instructions. Then, what instructions can be executed in parallel? A short answer is: as far as the parallel instructions do not use the same resource of the CPU, they can be put in parallel. For example, the following two instructions do not use the same CPU resource and they can be executed in parallel.\n\n\n\n\n### Resource constraints\n\nThen, what are the constraints on the parallel instructions? Let's look at the resource constraints in more detail.\n\n#### Functional unit constraints\n\nThis is simple. Each functional unit can execute only one instruction per each clock cycle. In other words, instructions using the same functional unit cannot be put in parallel.\n\n#### Cross paths constraints\n\nIf you look at the data path diagram of the C62x CPU, there exists only one cross path from B register file to the L1, M1 and S1 functional units. This means the cross path can be used only once per each clock cycle. Thus, the following parallel instructions are invalid because the 1x cross path is used for both instructions.\n\n\n2 || MPY .M1x A5,B0,A3\n\n\nThe same rule holds for the 2x cross path from the A register file to the L2, M2 and S2 functional units.\n\nThe D units are used for load and store instructions. If you examine the C62x data path diagram, the addresses for load\/store can be obtained from either A or B side using the multiplexers connecting crisscross to generate the addresses DA1 and DA2. Thus, the instructions such as\n\n\n1 LDW .D2 *B0, A1\n\n\nis valid. The functional unit must be on the same side as the address source register (address index in B0 and therefore D2 above), because D1 and D2 units must receive the addresses from A and B sides, respectively.\n\nAnother constraint is that while loading a register in one register file from memory, you cannot simultaneously store a register in the same register file to memory. For example, the following parallel instructions are invalid:\n\n\n1 LDW .D1 *A0, A1\n2 || STW .D2 A2, *B0\n\n\nYou cannot have more than four reads from the same register in each clock cycle. Thus, the following is invalid:\n\n\n1 ADD .L1 A1, A1, A2\n2 || MPY .M1 A1, A1, A3\n3 || SUB .D1 A1, A4, A5\n\n\n#### Constraints on register writes\n\nA register cannot be written to more than once in a single clock cycle. However, note that the actual writing to registers may not occur in the same clock cycle during which the instruction is executed. For example, the MPY instruction writes to the destination register in the next clock cycle. Thus, the following is valid:\n\n\n1\t ADD .L1 A1, A1, A2\n2 || MPY .M1 A1, A1, A2\n\n\nThe following two instructions (not parallel) are invalid (why?):\n\n\n1 MPY .M1 A1, A1, A2\n2 ADD .L1 A3, A4, A2\n\n\nSome of these write conflicts are very hard to detect and not detected by the assembler. Extra caution should be exercised with the instructions having nonzero delay slots.\n\nAt this point, you might have wondered why the C62x CPU allows parallel instructions and generate so much headache with the resource constraints, especially with the instructions with delay slots. And, why not just make the MPY instruction take 2 clock cycles to execute so that we can always use the multiplied result after issuing it?\n\nThe reason is that by executing instructions in parallel, we can reduce the total execution time of the program. A well-written assembly program executes as many instructions as possible in each clock cycle to implement the desired algorithm.\n\nThe reason for allowing delay slots is that although it takes 2 clock cycles for an MPY instruction generate the result, we can execute another instruction while waiting for the result. This way, you can reduce the clock cycles wasted while waiting for the result from slow instructions, thus increasing the overall execution speed.\n\nHowever, how can we put instructions in parallel? Although there's a systematic way of doing it (we will learn a bit later), at this point you can try to restructure your assembly code to execute as many instructions as possible in parallel. And, you should try to execute other instructions in the delay slots of those instructions such as MPY, LDW, etc., instead of inserting NOPs to wait the instructions produce the results.\n\n### Exercise 10\n\n(parallel instructions): Modify your assembly program for the inner product computation in the previous exercise to use parallel instructions as much as possible. Also, try to fill the delay slots as much as possible. Using the code composer's profiling, compare the clock cycles necessary for executing the modified program. How many clock cycles could you save?\n\n#### Solution\n\nIntentionally left blank.\n\n## Content actions\n\nPDF | EPUB (?)\n\n### What is an EPUB file?\n\nEPUB is an electronic book format that can be read on a variety of mobile devices.\n\nMy Favorites (?)\n\n'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.\n\n| A lens I own (?)\n\n#### Definition of a lens\n\n##### Lenses\n\nA lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.\n\n##### What is in a lens?\n\nLens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.\n\n##### Who can create a lens?\n\nAny individual member, a community, or a respected organization.\n\n##### What are tags?\n\nTags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.\n\n| External bookmarks","date":"2014-03-11 12:07:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.205831840634346, \"perplexity\": 4030.7674354759165}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1394011188282\/warc\/CC-MAIN-20140305091948-00019-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Traffic congestion has become a major problem affecting many human activities on a daily basis and resulting in both serious transportation delays and environmental damages. Continuously collecting information about the state of the road network (e.g. occupancy rates of the road segments) can be valuable for a better understanding of the traffic flow and a wiser planning and restructuring of the road network. Due to the high deployment and maintenance costs of dedicated traffic sensors, a more attractive approach to achieve this aim is to collect traces directly from GPS-equipped vehicles. The collected data can be map matched to corresponding road segments and can be used to deduce the state of the network in real time. It can also be stored and used later to conduct further, more complex data analysis tasks.
.
The problem addressed throughout this article is how to discover clusters of network-constrained trajectories, i.e. how to group together trajectories that moved along the same parts of the road network. This post-analysis step is conducted on a considerable amount of collected data. It can lead to a better understanding of global movement patterns and tendencies that go unnoticed on the individual level as well as a better grasp of the reasons that lead to congestion situations.
Section \ref{sec:ProblemStatement} presents our formulation of the network-constrained moving object trajectories clustering problem. Our approach to solve this problem is presented in Section \ref{sec:ClusteringApproach}. Experimental results are exposed in Section \ref{sec:Results} whereas related work is briefly discussed in Section \ref{sec:RelatedWork}. Finally, Section \ref{sec:Conclusion} concludes this paper.
\section{Data Model and Problem Statement}
\label{sec:ProblemStatement}
A road network can be represented as a directed graph $G = (V, E)$. $V$ is the set of nodes (or vertices) representing the road intersections in the network whereas $E$ is the set of edges denoting road segments that interconnect these intersections. The direction of a given edge $e=(v_1,v_2)$ linking two vertices $v_1$ and $v_2$ indicates that the corresponding road segment can be travelled from $v_1$ to $v_2$ and not the other way around. A trajectory $T$ traveling along this road network can be modeled as the ordered sequence of visited segments. If travel times are to be taken into account, each segment $e_i$ can be timestamped with the date $t_i$ the trajectory $T$ visited it: $T = \left\{ (t_1, e_1), ... , (t_i, e_i), ... , (t_n, e_n) \right\}$ ($n$ being the number of segments contained in $T$).
Given a dataset of trajectories $\mathcal{T}$ that travelled along a road network $G$, the network-constrained trajectory clustering problem consists in discovering sub-groups (or clusters) $\mathcal{C} = \{C_1, C_2, ... , C_m\}$ of trajectories exhibiting similar behavior. Resemblance between trajectories of the same cluster $C_i$ should be as high as possible and trajectories across two different clusters $C_i$ and $C_j$ should be as different as possible.
\section{Clustering Approach}
\label{sec:ClusteringApproach}
Our clustering approach proceeds in two steps. First, it calculates a similarity graph from $\mathcal{T}$. Then, in the second step, the graph is used to conduct modularity-based graph clustering and regroup similar trajectories together.
\subsection{Similarity between trajectories}
\label{sec:Similarity}
We consider trajectories as bags-of-segments: comparisons between trajectories are done on a segment-basis (i.e. each segment is checked individually, without taking account of the order or the presence of other segments). This choice comes from the fact that: i. in a context of traffic analysis, congestion situations appear first on the level of individual, isolated segments then spread naturally among adjacent road segments. Inspecting each segment apart is, therefore, sufficient to detect these situations; and ii. even if the approach bag-of does not directly take into account the order of the segments, thanks to the fact that the underlying network is a directed graph the order of travel through the segments is implicitly respected.
To assess their relevance to each trajectory, we assign weights to road segments based on their frequency in the data set in a TF-IDF (\textit{Term Frequency - Inverse Document Frequency}) fashion: the spatial weight of a segment $e$ in a trajectory $T$ is defined, analogously to the TF-IDF weight, as follows:
$$
\omega_{e,T} = \frac{\mbox{length}(e)}{\sum_{e' \in T}\mbox{length}(e')} \cdot \log\frac{| \mathcal{T} |}{| \{T' : e \in T' \} |}
$$
$\mbox{length}(e)$ is the spatial length of the segment $e$, $|\mathcal{T}|$ is the cardinality of the dataset $\mathcal{T}$ and $| \{T' : e \in T' \} |$ the cardinality of the subset of trajectories containing the segment $e$. The first term of the multiplication is the equivalent to the term frequency whereas the second is the equivalent to the inverse document frequency.
To compare two trajectories $T_i$ and $T_j$ we calculate their cosine similarity:
$$
\mbox{Similarity}(T_i,T_j) = \frac{\sum_{e \in E} \omega_{e,T_i} \cdot \omega_{e,T_j}}{\sqrt{\sum_{e \in E} \omega_{e,T_i}^2} \cdot \sqrt{\sum_{e \in E} \omega_{e,T_j}^2}}
$$
This weighting and similarity calculation approach takes only the spatial dimension into consideration. The reason is that, for traffic monitoring and optimization purposes, decision making might involve rerouting traffic from a portion of the network to another. Thus affecting all trajectories that passed by the concerned zone whether they travelled together or at different dates.
\subsection{Clustering Algorithm}
As mentioned before, the first step of our clustering algorithm consists in constructing a weighted, undirected similarity graph $G_{\mbox{Sim}} = (\mathcal{T}, E', W)$. Each trajectory from the dataset $\mathcal{T}$ corresponds to a node in $G_{\mbox{Sim}}$. An edge $e \in E'$ links two trajectories $T_i$ and $T_j$ if and only if $\mbox{Similarity}(T_i,T_j) > 0$ in which case the similarity is assigned as a weight ($\omega_e \in W$) to the edge. The choice of graph representation is not only natural but it also puts extra emphasis on the fact that two trajectories that have nothing in common should never be put directly into the same cluster (since no edge provides a direct link between the two).
Since we are interested in analyzing an important number of trajectories and since an edge links two trajectories together if they share at least one road segment in common, $G_{\mbox{Sim}}$ tends to be very large and its vertices have generally high degrees. For these reasons, modularity-based community detection algorithms are efficient for clustering such graphs \cite{Fortunato_2010}. Modularity measures the classification quality by inspecting the arrangement of edges within clusters (commonly called communities) of vertices. A high modularity indicates that the edges within communities are more numbered (or have more important weights) than in the case of randomly distributed edges.
We opted for an implementation of the modularity-based hierarchical graph clustering algorithm proposed in \cite{Noack_2009} for our clustering step. The algorithm performs modularity optimization on the nodes of the similarity graph and the structure of the discovered communities is validated by means of comparison against random graphs generated with the same set of nodes. If validated, the communities are retained and the algorithm proceeds recursively on each isolated community (i.e. considering only the sub-graph including the nodes of the community). The final result is a hierarchy of nested clusters that can either be explored level by level or using a greedy approach that expands, at each given step, the cluster that induces the minimal loss of modularity.
\section{Experimental Results}
\label{sec:Results}
For our experimental study, we used a synthetic dataset\footnote{The dataset is available at: \url{http://perso.telecom-paristech.fr/~mahrsi/ESANN2012/}} of 10000 trajectories generated with the Brinkhoff generator \cite{Brinkhoff_2002} using the Oldenburg map which contains 6105 nodes and 7035 undirected edges that can be travelled in both directions.
We started by comparing different similarity calculation approaches: we compared our cosine similarity using spatial weighting (cf. Section \ref{sec:Similarity}) with the Jaccard index and cosine similarity with classic TD-IDF weighting. The clustering step achieved the highest modularity optimization with spatial weighting (0.5635 vs 0.5251 for Jaccard index and only 0.4861 for classic TF-IDF). The algorithm produced a hierarchy of clusters that spans on 6 levels with 9 clusters on the top level and 648 clusters in the lowest level. Figure \ref{fig:clusters} shows examples of clusters produced by our algorithm on the third level of the clustering hierarchy (we chose to expand level by level since this approach seemed to give the most balanced clusters for this dataset). Each sub-figure shows the distribution of departure and arrival points of the trajectories, in a given cluster, that moved along the most visited road segment.
\begin{figure}[h]
\centering
\subfigure{
\includegraphics[scale=0.37]{cluster_82.eps}
}
\subfigure{
\includegraphics[scale=0.37]{cluster_202.eps}
}
\subfigure{
\includegraphics[scale=0.37]{cluster_131.eps}
}
\caption{Departures points (crosses) and arrival points (empty squares) of some clusters that travelled along the most occupied road segment.}
\label{fig:clusters}
\end{figure}
Visualization of the generated clusters shows promising results as of the capability of the clustering approach to divide trajectories into well separated groups. Moreover, the fact that the algorithm produces a hierarchy of nested clusters can be very useful for the understanding and visualization of traffic: one can start with a given level and locate interesting clusters (e.g. clusters that crossed a given area of interest) and see how these clusters are expanded into sub-clusters on lower levels.
We also compared our clustering approach to classic hierarchical agglomerative clustering. To this end, we used a distance that is complementary to our similarity measure: $ \mbox{distance}(T_1,T_2) = 1-\mbox{Similarity}(T_1,T_2)$. We calculated an adjacency matrix based on this distance and we used it for the agglomerative clustering (with single, average and full linkage). Comparisons were conducted for the same number of cluster: for each of the 6 hierarchy levels produced in our approach, we cut the hierarchical clustering at the same number of clusters and we compared the two. Quality measures that we used are: i. interclass and intraclass inertia of the start points of the trajectories; ii. interclass and intraclass inertia of the end points of the trajectories; and 3. interclass and intraclass overlap of the trajectories. The first two are used to assess the compactness of the regrouped trajectories start/end points . Overlap measures give an appreciation of how much road segments trajectories share among and across the resulting clusters ($\mathcal{C}$ being the set of resulting clusters, and $|C|$ the number of trajectories in the cluster $C$):
$$
\mbox{intraclass ovelap} = \sum_{C \in \mathcal{C}} \frac{1}{|C|} \sum_{T_i,T_j \in C} \frac{\sum_{e \in T_i, e \in T_j}\mbox{length}(e)}{\sum_{e \in T_i}\mbox{length}(e)}
$$
$$
\mbox{interclass overlap} = \sum_{C_i \in \mathcal{C}} \frac{1}{|\mathcal{T}| - |C_i|} \sum_{C_j \in \mathcal{C}, j \neq i} \sum_{T \in C_i, T' \in C_j}\frac{\sum_{e \in T, e \in T'}\mbox{length}(e)}{\sum_{e \in T}\mbox{length}(e)}
$$
Due to lack of space, we only show the results of our interclass and intraclass overlap comparison (Table \ref{tab:overlap}). Modularity-optimization clustering achieves the best intraclass overlap among the tested approaches. This indicates that trajectories within a same cluster are more similar and share more road segments than in the case of classic hierarchical clustering. The lower (at first glance better) interclass overlap achieved by single linkage hierarchical clustering comes from the fact that this approach produces very unbalanced clusters (it tends to produce a huge cluster regrouping most of the trajectories in the dataset while the other clusters are very small). This problem is also visible with average and full linkage hierarchical clustering but only when the number of clusters is small. When the number of clusters gets bigger, average and full linkage as well as modularity-optimization clustering all behave in a similar manner w.r.t. interclass overlap.
\begin{table}[htdp]
\begin{center}
\small
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Nbr. of & \multicolumn{4}{c|}{Intraclass overlap} & \multicolumn{4}{c|}{Interclass overlap}\\
\cline{2-9}
clusters & HC(S) & HC(A) & HC(F) & Mod. & HC(S) & HC(A) & HC(F) & Mod.\\
\hline
9 & 111 & 111 & 123 & 608 & 0.5 & 0.2 & 38.9 & 42.8\\
\hline
45 & 149 & 155 & 349 & 1768 & 5.1 & 2.9 & 78.4 & 62.2\\
\hline
159 & 270 & 1594 & 1877 & 3121 & 8.8 & 59.1 & 97.9 &79.4\\
\hline
419 & 569 & 3073 & 3682 & 4264 & 22.4 & 69.5 & 87.1 & 87.9\\
\hline
621 & 822 & 3999 & 4419 & 4780 & 30.3 & 76.2 & 85.8 & 89.7\\
\hline
648 & 881 & 4106 & 4491 & 4823 & 35.4 & 76.5 & 85.8 & 89.8\\
\hline
\end{tabular}
\end{center}
\caption{Interclass and intraclass overlaps achieved by modularity-optimization trajectory clustering and hierarchical agglomerative clustering (with S: single linkage, A: average linkage and F: full linkage).}
\label{tab:overlap}
\end{table}
\section{Related Work}
\label{sec:RelatedWork}
Clustering trajectory data attracted many research in the last few years. Existing proposals include TraClus \cite{Lee_2007}, convoy \cite{Jeung_2008b} and flock \cite{Benkert_2008} patterns and many others. These are mainly density-based approaches that suppose that the moving objects can move freely on an euclidean space. Therefore, these approaches are substantially different from the problem at hand where the movement is constrained. The case of constrained trajectories started to attract attention only recently. In \cite{Kharrat_2008}, the authors propose a density-based approach to discover dense paths in such trajectories. Like all density-based method, the approach is very sensitive to the configuration of the $minPts$ and $\epsilon$ parameters. Furthermore, the approach clusters sub-trajectories together and does not conserve trajectory participation on the whole dense path (i.e. a trajectory might participate only partially in the path). Our approach, on the contrary, regroups whole trajectories. To our knowledge, our work is the first to apply modularity-optimization graph clustering in the context of trajectory data.
\section{Conclusion}
\label{sec:Conclusion}
In this article, we presented a novel approach to cluster trajectories constrained by an underlying road network. The approach starts by computing a similarity graph between trajectories based on the cosine spatial similarity that we defined. The graph is then used to conduct hierarchical modularity-optimization clustering to discover communities of trajectories that exhibited similar behavior. Results on synthetic trajectory data are promising and showed that the proposed approach yields better, more relevant clusters than the classic hierarchical clustering. For future research directions, we are mainly interested in the visual exploitation of the clustering results as well as the study of their relevance in decision making in a context of traffic rerouting and optimization.
\begin{footnotesize}
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
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Inge Sørensen (18 July 1924 – 9 March 2011), later Inge Tabur, sometimes known as "Lille henrivende Inge" ("Little Lovely Inge") was a Danish swimmer, who at age 12 won a bronze medal in 200 meter breaststroke at the 1936 Summer Olympics in Berlin. This makes her the youngest Olympic medal winner in an individual competition.
During 1936-1944 she won nine Danish championships, two Nordic championships and one European championship. She set 14 Danish records in breaststroke. She also broke the world record on 400 m and 500 m breaststroke and became the first Danish female swimmer under 3 minutes on 200 m breaststroke.
Her swimming career was cut short by World War II. After the war she married a Danish engineer and moved abroad eventually residing in USA.
Early life
Inge Sørensen was born in Skovshoved, Denmark, the daughter of a fishmonger in 1924. She began swimming at age 3 in the harbor of Skovshoved north of Copenhagen and won her first Danish championship age 11 in 1936 on 200 m breaststroke.
Because of this she was selected for the Olympic Games in Berlin the same year.
Olympic Games in Berlin 1936
At the Olympic Games at age 12 she won a bronze medal on 200 m breaststroke. She is the youngest medal-winner at the Olympic Games in an individual competition. Her final on August 11 was the first event to be broadcast live on air by Danish radio, which itself was a sensation. During the games she got the nickname Lille henrivende Inge (Little Lovely Inge).
On return from Berlin she and the other Danish female medal-winner, Ragnhild Hveger, received a chaotic welcome in Copenhagen.
European Championships in London 1938
Between 1936 and 1938 Inge Sørensen improved her times in swimming though it was feared that puberty would slow her down. At the European championships in swimming in London 1938 she won a gold medal in 200 m breaststroke.
She was not the only Danish woman with success because Denmark became best nation for female swimmers.
World War II
The period 1939-1941 was the height of her career. In 1939 she broke the world record on 400 m and 500 m breaststroke and in 1941 she became the first Danish female swimmer under 3 minutes on 200 m breaststroke.
Nevertheless, World War II meant that the Summer Olympics in 1940 and 1944 were cancelled together with the European Championships in 1942. This made her end her swimming career in 1944.
Unlike other Danish swimmers, such as Ragnhild Hveger, she did not go to Germany for international competition during the war.
Later life
Inge Sørensen took an education in instructing and coaching in 1946 and afterwards taught gymnastics and swimming. After being married in 1948 to Danish engineer Janus Tabur, she followed him to South Africa and later Canada and from 1951 United States. She lived in New Jersey until her death in 2011.
Notes
References
Bibliography
1924 births
2011 deaths
Danish female swimmers
Female breaststroke swimmers
Olympic bronze medalists for Denmark
Olympic swimmers of Denmark
Olympic bronze medalists in swimming
People from Mount Laurel, New Jersey
People from Gentofte Municipality
European Aquatics Championships medalists in swimming
Medalists at the 1936 Summer Olympics
Swimmers at the 1936 Summer Olympics
Sportspeople from the Capital Region of Denmark | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,383 |
Q: space of projective plane curves of degree d This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $\mathbb P^N$ where $N=\binom{d+2}{2}-1$. All seems reasonable and I can easily produce this canonical correspondence.
However he writes that this is not 1-1 if we look at the image of a curve of degree $d$ that is reducible. I seem to be too stupid to find two points in the preimage of a point in $\mathbb P^N$. Perhaps I misunderstood something?
A: $\newcommand\P{\mathbb P}$I think I know why I was not getting the example:
*
*I was thinking of a function from the set of projective plane curves of degree $d$ to $\P^N$
*I thought I could get an example if $d=2$ (maybe this is not posssible).
So Hartshorne meant a map from points in $\P^N$ to the set of curves that can be defined by forms of degree $d$. So not only reverse of what I thought, but also we are not talking about the degree of the curve here which I misunderstood as well. One way to get a fiber with more than one point using this map is to consider $d=4$ and now the point in $\P^N$ corresponding to $x^2y^2$ which is different from the point in $\P^N$ corresponding to $x^3y$ map to the same curve.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,443 |
QtConfiguration Library
=======================
Settings API with change notifications.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 76 |
{"url":"https:\/\/www.physicsforums.com\/threads\/showing-that-a-set-is-an-ideal.913655\/","text":"# Showing that a set is an ideal\n\n1. May 5, 2017\n\n### Mr Davis 97\n\n1. The problem statement, all variables and given\/known data\nShow that the collection of all nilpotent elements of a commutative ring $R$ is an ideal.\n\n2. Relevant equations\n\n3. The attempt at a solution\nShowing that something is an ideal is somewhat straightforward, but I am a little confused as to what explicitly I have to show. If we denote $N$ as the set in question, then I know that we have to show that $aN \\subseteq N$ and $Nb \\subseteq$ for all $a,b \\in R$. But what else do I have to show? Do I have to show that N is an additive subgroup?\n\n2. May 5, 2017\n\nYes\n\n3. May 5, 2017\n\n### Mr Davis 97\n\nConsider $(a + b)^{m+n}$, where $a,b \\in N$. In the binomial expansion, each summand contains a term $a^{i}b^{m+n\u2212i}$. Now\neither $i \\ge m$ so that $a^i$ = 0 or $m+ n \u2212 i \\ge n$ so that $b^{m+n\u2212i} = 0$. Thus each summand of $(a + b)^{m+n}$\nis zero, so $(a + b)^{m+n}$= 0 and $N$ is closed under addition. Also, since $0^1 = 0$, $0 \\in N$.\nAlso $(\u2212a)^m$ is either $a^m$ or $\u2212a^m$, so $(\u2212a)^m = 0$ and $-a \\in N$.\n\nDoes this show that $N$ is a additive subgroup of $R$?\n\n4. May 5, 2017\n\n### Staff: Mentor\n\nI think you need the power $n+m+1$. But basically, yes. For an ideal you also need $r \\cdot a \\in N$.\n\n5. May 5, 2017\n\n### Mr Davis 97\n\nIn the solution to this problem in the book, it also shows that $N$ is closed under multiplication of elements in $N$. It shows that $(ab)^{mn} = (a^m)^n(b^n)^m = (0)(0) = 0$, so $ab \\in N$. Isn't this unnecessary? Don't I only need to show that $N$ is an additive subgroup and that is satisfies the absorption property for ideals? Where would being internally closed under multiplication fit in?\n\n6. May 5, 2017\n\n### Staff: Mentor\n\nWell, it does no harm and is almost obvious in a commutative ring. It shows, that $N$ carries also a ring structure. But if $r \\cdot a \\in N$ for all $r \\in R\\, , \\,a \\in N$, doesn't this imply $a\\cdot b \\in N$ for all $a,b \\in N$?\n\n7. May 5, 2017\n\n### Mr Davis 97\n\nThat's what I mean. It seems superfluous. I just want to make sure I know what is strictly necessary to show that some set is an ideal of a ring.\n\n8. May 5, 2017\n\n### Staff: Mentor\n\nThen (minimal) it's (for left ideals)\n1. $N \\neq \\emptyset$\n2. $a - b \\in N$ for all $a,b \\in N$\n3. $r \\cdot a \\in N$ for all $r\\in R \\, , \\,a \\in N$\nOf course left and right ideals don't have to be distinguished in a commutative ring, but in general they have to be. And the first condition cannot be omitted by the second ($a-a \\in N)$, as in case $N$ is empty, the second condition is still true, whereas $0 \\notin N$, which is needed. So we can also write $0 \\in N$ as first condition.","date":"2018-03-22 06:30:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8088934421539307, \"perplexity\": 266.4351133538612}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257647777.59\/warc\/CC-MAIN-20180322053608-20180322073608-00025.warc.gz\"}"} | null | null |
Q: How to fix Flutter Failed to build iOS app? I updated my Xcode to Version 13.4 and android studio to Chipmunk and after this, I am getting below error while trying to build for ios. This issue is only occurring for ios builds.
Below is the log I am getting.
Xcode build done. 19.5s
Failed to build iOS app
Error output from Xcode build:
↳
2022-05-27 19:18:46.624 xcodebuild[62430:463863] Requested but did not find extension point with identifier Xcode.IDEKit.ExtensionSentinelHostApplications for extension Xcode.DebuggerFoundation.AppExtensionHosts.watchOS of plug-in com.apple.dt.IDEWatchSupportCore
2022-05-27 19:18:46.624 xcodebuild[62430:463863] Requested but did not find extension point with identifier Xcode.IDEKit.ExtensionPointIdentifierToBundleIdentifier for extension Xcode.DebuggerFoundation.AppExtensionToBundleIdentifierMap.watchOS of plug-in com.apple.dt.IDEWatchSupportCore
** BUILD FAILED **
Xcode's output:
↳
Writing result bundle at path:
/var/folders/6v/14mdnyyd4vs7gh4r1ry0hfxh0000gn/T/flutter_tools.IBcqbj/flutter_ios_build_temp_dirq4Dibw/temporary_xcresult_bundle
Failed to package /Users/bhaskarrajaryal/AndroidStudioProjects/PersonalProjects/FlutterProjects/testing.
Command PhaseScriptExecution failed with a nonzero exit code
note: Using new build system
note: Planning
note: Build preparation complete
note: Building targets in dependency order
Result bundle written to path:
/var/folders/6v/14mdnyyd4vs7gh4r1ry0hfxh0000gn/T/flutter_tools.IBcqbj/flutter_ios_build_temp_dirq4Dibw/temporary_xcresult_bundle
Could not build the application for the simulator.
Error launching application on iPhone 11.
Can anyone help me with this as I am stuck for fews day because of this?
A:
The problem here is that the Podfile that Flutter template creates by
default has no specific iOS version set unfortunately.
Do this to fix this problem:
in ios/ folder of your project, open the Podfile.
At top of Podfile, make sure this line is not commented out and change the iOS version to 12.0.
change from:
#platform :ios, '8.0'
to:
platform :ios, '12.0'
Run pod deintegrate in Terminal inside the ios/ folder of your project.
Run pod install --repo-update in your ios/ folder
This should do the trick!
If after this you are getting the following error
CocoaPods did not set the base configuration of your project because your project already has a custom config set. In order for CocoaPods integration to work at all, please either set the base configurations of the target
Then you need to open your iOS workspace in Xcode and select your root project on top left, then inside the Info tab, choose your configuration (in this case Debug) and change it to None. After that, do pod install again.
HERE PICTURE :
CLICK HERE
| {
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}\label{sec:intro}
Williams showed \cite{Wil1,Wil2} that expanding uniformly hyperbolic attractors can be represented as inverse limits on branched one-manifolds and the dynamics can be completely understood in the terms of the underlying one-dimensional map. Moreover, such attractors are locally homeomorphic to the product of a Cantor set and an (open) arc. Thus, in order to understand the properties of non-hyperbolic systems, we need to focus on those which have inhomogeneous global attractors. The simplest, yet topologically interesting such systems are
inverse limits $\underleftarrow{\lim}(I, f)$, with a continuous bonding map $f\colon I\to I$. It is well-known that every $\underleftarrow{\lim}(I, f)$ can be embedded in the plane as a global attractor of a planar homeomorphism, which is conjugate to the natural extension of $f$ on its global attractor, see \cite{BaMa}. Inverse limits of interval maps contain points which are locally not homeomorphic to the Cantor set of arcs, called {\em folding points}. The term folding point was introduced by Raines in \cite{Raines} and the
name emphasizes the occurrence of ``folds" in arbitrary small neighbourhoods of such a point, see Figure~\ref{fig:uils}. Our goal is to understand the structure and properties of the set of folding points in terms of the dynamics of the bonding map $f$.
A {\em continuum} is a nonempty compact connected metric space. It is called {\em chainable} (or {\em arc-like}), if it admits an $\eps$-mapping onto
the interval $[0,1]$ for every $\eps>0$. Every chainable continuum can be represented as an inverse limit on intervals, possibly with varying bonding maps (see \eg \cite{InMah}).
In this paper we study topological properties of a wide class of chainable continua $\underleftarrow{\lim}\{I,f\}$, where $f\colon I\to I$ is a {\em long-zigzag} map.
\begin{definition}\label{def:longzigzag}
A map $f\colon I\to I$ is called {\em long-zigzag}
if there exists $\eps>0$ such that for every $n\in\N_0$ and every interval $[a,b]\subset I$ such that $f^n([a, b])=[f^n(a), f^n(b)]$ (or $f^n([a,b])=[f^n(b), f^n(a)]$), $f^n(x)\not\in\{f^n(a),f^n(b)\}$ for every $x\in (a,b)$, and $|f^n(a)-f^n(b)|<\eps$, it holds that $f^n$ maps $[a,b]$ homeomorphically onto $f^n([a,b])$.
\end{definition}
Intuitively, long-zigzag maps are interval maps for which the lengths of possible {\em zigzags} (see Figure~\ref{fig:zigzag}) in all iterates are bounded from below.
Specifically, a map $f$ will be called {\em zigzag-free} if it has a property that whenever $f([a,b])=[f(a),f(b)]$ or $f([a,b])=[f(b),f(a)]$ and $f(x)\not\in \{f(a),f(b)\}$ for all $x\in(a,b)$, then $f|_{[a,b]}$ is a homeomorphism, see Definition~\ref{def:zigzagfree}.
Every zigzag-free map, and this includes all unimodal maps (see Lemma 2.1. in \cite{ABC1}), is long-zigzag.
On the other hand the interval maps which produce the {\em pseudo-arc} (\eg Henderson's map \cite{Henderson}) are not long-zigzag maps.
We show that the assumption of a map being long-zigzag is actually very mild - if interval map $f$ has finitely many critical points $\{c_1, \ldots, c_n\}$ such that if $f^k|_{[c_i,c_{i+1}]}$ is one-to-one for every $k\in\N$, then the length of branches $|f^k([c_i,c_{i+1}])|$ is bounded from below, then $f$ is long-zigzag. Specifically, every piecewise monotone $f$ which is locally eventually onto is long-zigzag, see Corollary~\ref{thm:leo+fpwl=lzzb}. We say that $f$ is \emph{locally eventually onto (leo)}, if every open interval in $I$ is eventually mapped onto the whole $I$. To assume leo is also not a severe restriction in our context since any interval map with finitely many critical points and without restrictive intervals,
periodic attractors or wandering intervals is conjugate to a piecewise linear leo map, or semi-conjugate otherwise
(see \cite{MT} or Theorem 8.1 from \cite{dMvS}).
It is easy to see that there also exist long-zigzag interval maps with infinitely many critical points.
Moreover, the space of long-zigzag interval maps is dense in the $C^k$-topology for every $k\in \N$ in the space of all
continuous interval maps. This follows since long-zigzag maps include long-branched interval maps and the later contain
the set of hyperbolic maps. Hyperbolic maps are dense in the set of surjective interval maps in the $C^k$-topology for every $k\in\N$,
see \cite{dMvS, KSvS} for details.
Our study is motivated by results on the structure of local inhomogeneities of unimodal inverse limit spaces, see \eg \cite{BM,Raines,Br1,AABC}, and by the lack of basic tools to study inhomogeneities of indecomposable one-dimensional continua beyond the unimodal setting.
An interval map $f$ is called {\em unimodal}
if there exists a unique critical point $c\in (0,1)$. Unimodal inverse limits present a simplified models for global attractors of H\'enon maps $H_{a,b}\colon \R^2\to\R^2$, $H_{a,b} = (1-ax^2+by, x)$ (sometimes they are indeed an actual model, see \cite{BaHo}). They have recently generated a significant research interest, both in continuum theory, culminating with the proof of the Ingram conjecture in \cite{BBS}, and in dynamical systems, by the works of Boyland, de Carvalho and Hall \cite{3G,3G1,3G3,3G2} (from both topological and measure theoretical perspective).
When studying the local structure of unimodal inverse limits, it turns out that one of the key properties is that unimodal maps are long-zigzag. Using only analytic observations for interval maps (specifically avoiding symbolic arguments used in the unimodal setting), we prove that the following results on unimodal inverse limits hold in a much wider setting.
{\bf (1) The properties of endpoints.} If $f\colon I\to I$ is unimodal, then every endpoint in $\underleftarrow{\lim}\{I,f\}$ is an endpoint of every arc which contains it (Bruin \cite{Br1}).
A point $x$ in a chainable continuum $K$ is called an {\em endpoint}, if for every two subcontinua $x\in A, B\subset K$ it holds that $A\subset B$ or $B\subset A$. A point $x$ in a continuum $K$ is called an {\em $L$-endpoint} if $x$ is an endpoint of every arc in which it is contained (this definition was given by Lelek~\cite{Le} for the study of dendroids).
\begin{definition}
Let $X=\underleftarrow{\lim}\{I, f_i\}$ and $x=(x_0, x_1, \ldots)\in X$. For $i\in\N_0$ we define the {\em $i$-basic arc} $A_i(x)$ as the maximal arc in $X$ such that $x\in A_i(x)$ and $\pi_i|_{A_i(x)}$ is one-to-one (can be degenerate).
\end{definition}
Bruin~\cite{Br1} (see also Brucks and Diamond \cite{BrDi}) introduced this notion of a basic arc and then proves that in unimodal inverse limits a point $x$ is an endpoint of $K$ if and only if it is an endpoint of its $i$-basic arc for every $i\geq 0$ (here we call that a {\em $B$-endpoint}).
It follows that every endpoint in $\underleftarrow{\lim}\{I,f\}$ is an $L$-endpoint, if $f\colon I\to I$ is unimodal.\\
The following theorem generalizes Bruin's result for the class of long-zigzag leo maps.
{\bf Theorem 4.7.}
If $f\colon I\to I$ is a long-zigzag leo map, then the sets of $B$-endpoints, $L$-endpoints and endpoints in $\underleftarrow{\lim}\{I,f\}$ are the same.
When studying $L$-endpoints, a natural question which arises is how to characterize interval inverse limits $\underleftarrow{\lim}\{I,f_n\}$ which are homeomorphic to an arc. In Appendix~\ref{appB} we provide a characterization of an arc as an inverse limit with a single bonding map which has a finite set of critical points. That is a fundamental result, interesting also on its own. The characterization in a more general setting is still outstanding.
{\bf (2) Dynamical characterization of folding points.} Let $C$ denote the {\em set of critical points} of a
continuous interval map $f$ including the points $0$ and $1$.
The {\em omega limit set $\omega(C)$} of $C$ is the set of all limit points of sequences $(f^n(c))_{n\in\N}$, for all $c\in C$. For a large class of interval maps $f$, which contains unimodal maps, a folding point of $\underleftarrow{\lim}\{I,f\}$ can be characterized as a point for which all projections are in $\omega(C)$ (Raines \cite{Raines}). The class of maps from Raines' theorem in contained in a class of long-zigzag maps, see Corollary~\ref{cor:Raines}.
We generalize Raines' characterization to long-zigzag interval maps $f$. For every $n\in\N$
we denote the {\em $n$-th natural coordinate projection} of $\underleftarrow{\lim}\{I, f\}$ to $I$ by $\pi_n$.
{\bf Theorem 5.1.}
Assume $f\colon I\to I$ is long-zigzag map. Then $x\in\underleftarrow{\lim}\{I, f\}$ is a folding point if and only if $\pi_n(x)\in\omega(C)$ for every $n\in\N_0$.
{\bf (3) Retractability and endpoints.} If $f$ is unimodal, but not conjugate to the full tent map, then the set of endpoints in $\underleftarrow{\lim}\{I,f\}$ is equal to the set of folding points if and only if $c$ is persistently recurrent, see \cite{AABC}.
The notion of persistent recurrence was first introduced in \cite{BL} in connection with the existence
of wild attractors of unimodal interval maps and was used in the construction of wild Cantor set attractors in \cite{BKNS}.
We generalize the notion of persistent recurrence and characterize the class of long-zigzag inverse limits for which the set of folding points equals the set of endpoints. We use both Theorem~\ref{thm:llzzb} and Theorem~\ref{thm:folding} in the proof of the theorem below.
\begin{definition}
The map $f$ is {\em non-retractable along $\omega(C)$} if for every
backward orbit $x=(x_0, x_1, x_2, \ldots)$ in $\omega(C)$ and every open interval $U\subset I$ such that $x_0\in U$, the {\em pull-back} of $U$ along $x$ is not monotone.
\end{definition}
Definition~\ref{def:pullback} explains precisely what we mean by a monotone pull-back along $x$.
Note that the definition does not require recurrence of critical points. For example, the critical set $C=\{0,1/2,1\}$ of the full tent map $T_2(x)=\min\{2x,2(1-x)\}$, $x\in I$, is non-retractable, yet $c=1/2$ is not recurrent. Due to the lack of recurrence in this more general setting, we avoid the use of the term ''persistent recurrence''.
{\bf Theorem 6.3.}
Let $f\colon I\to I$ be a long-zigzag leo map with the critical set $C$. Then for $X=\underleftarrow{\lim}\{I, f\}$ it holds that every folding point is an endpoint if and only if $f$ is non-retractable along $\omega(C)$.
Since it is in general hard to see from the bonding map if the map is non-retractable or not, we provide in Appendix~\ref{appA} jointly with Henk Bruin a characterization of interval maps that are retractable through their dynamical properties.
{\bf (4) Effect of critical recurrence on the existence of endpoints.} If $f\colon I\to I$ is unimodal, then $\underleftarrow{\lim}\{I,f\}$ contains an endpoint if and only if $c$ is recurrent, \ie $c\in\omega(c)$. Moreover, if the orbit of $c$ is infinite, then $\underleftarrow{\lim}\{I,f\}$ has uncountably many endpoints, see \cite{BM} and \cite{Br1}.
We show that the similar result holds true for long-zigzag leo maps $f$, assuming that at least one critical point recurs on itself (\ie $c\in\omega(c)$).
We emphasize that we avoid using the technical characterization of endpoints from Theorem 1.4 of \cite{BM}, which makes our proof more accessible, even in the unimodal setting.
{\bf Theorem 7.3.}
If $f$ is long-zigzag leo map, and there exists $c\in C$ such that $c\in\omega(c)$, then there exists an endpoint in $\underleftarrow{\lim}\{I,f\}$. If additionally $\orb(c)$ is infinite, then $\underleftarrow{\lim}\{I,f\}$ has uncountably many endpoints.
We note that the mentioned results present foundations for more general theory in the setting of graph inverse limits (and thus $1$-dimensional continua in general), once the definitions given in this paper are adjusted accordingly.
For example, the notion of a long-zigzag circle map $f\colon S^1\to S^1$ needs to be defined on a lift $F\colon\R\to\R$, and then the arguments in the proofs of four theorems stated above follow analogously. Therefore, the above four theorems hold in the setting of inverse limits with circle bonding maps as well. When underlying graphs have branching points, the definition of a folding point has to be given more precisely in order to give an applicable theory.
Let us briefly outline the structure of the paper. After fixing notation and giving preliminaries we discuss zigzags in interval maps and show that every leo map with finitely many
critical points is long-zigzag in Section~\ref{sec:zigzags}. In Section~\ref{sec:endpoints} we discuss different notions of endpoints,
and finish our discussion with the proofs of propositions which imply Theorem~\ref{thm:llzzb}.
In Sections~\ref{sec:fp},~\ref{sec:persistence} and ~\ref{sec:recurrence}, we prove Theorems~\ref{thm:folding}, ~\ref{thm:FisE} and ~\ref{thm:endpt} respectively. Finally, Appendix~\ref{appA} gives a characterization of retractability along $\omega(C)$ through properties of interval maps and Appendix~\ref{appB} provides a characterization of arc when the bonding map has finitely many critical points. Whenever possible, we include examples which show that the standing assumptions are indeed optimal.
\section{Preliminaries and notation}\label{sec:prel}
The set of {\em natural numbers} is denoted by $\N$, and $\N_0=\N\cup\{0\}$.
The {\em interior} and the {\em closure} of a set $S$ will be denoted by $\Int(S)$ and $\overline{S}$,
respectively. By $I$ we denote the {\em unit interval} $[0,1]$. A {\em map} $f\colon I\to I$ always
means a continuous surjective function, and $f^n=f\circ\ldots\circ f$ is the $n$-th iterate of $f$.
A point $c\in (0,1)$ is called a {\em critical point} of $f\colon I\to I$ if for every open $J\ni c$,
$f|_J$ is not one-to-one.
The set $C$ of all critical points of $f$ including $0$ and $1$ is called the {\em critical set}.
An interval map is called {\em piecewise monotone}, if it has finitely many critical points. Given $f\colon I\to I$, an {\em orbit of a point} $x\in I$ is the set $\orb(x)=\{f^n(x): n\geq 0\}$, and an {\em orbit of a set} $S\subset I$ is $\orb(S)=\{\orb(x): x\in S\}$. The {\em omega-limit set} of a set $S\subset I$, denoted by $\omega(S)$, is the set of all limit points of $\orb(S)$.
A {\em continuum} is a non-empty, compact, connected, metric space. In the definition of inverse limit space we can assume without loss of generality that the bonding functions are surjective.
Given a sequence of maps
$f_i\colon I\to I$, $i\in\N$, the inverse limit is given by:
$$\underleftarrow{\lim}\{I, f_i\}=\{(x_0, x_1, \ldots): f_i(x_i)=x_{i-1}, i\in\N\}\subset I^{\N_0}.$$
Equipped with the {\em product topology}, $\underleftarrow{\lim}\{I, f_i\}$ is a continuum, and it is {\em chainable},
\ie for every $\eps>0$ there exists an $\eps$-mapping onto an interval.
Moreover, every chainable continuum is an inverse limit on intervals, see \cite{InMah}. The {\em coordinate projections} are defined
by $\pi_i\colon \underleftarrow{\lim}\{I, f_i\}\to I$, $\pi_i((x_0, x_1, \ldots))=x_i$, for $i\in\N_0$ and they are all continuous.
Also, for $i\in\N_0$, an {\em $i$-box} is the set of the form $\pi_i^{-1}(U)$, where $U\subset I$ is an open set.
The set of all $i$-boxes, $i\geq 0$, forms a basis for the topology of $\underleftarrow{\lim}\{I, f_i\}$.
If $f_i=f$ for all $i\in\N$, we denote the inverse limit by $X=\underleftarrow{\lim}\{I, f\}$. In that case there exists a
homeomorphism $\hat f\colon X\to X$, given by $$\hat f((x_0, x_1, \ldots)):=(f(x_0), f(x_1), f(x_2), \ldots)=(f(x_0), x_0, x_1, \ldots).$$
It is called the {\em natural extension} of $f$ (or sometimes {\em shift homeomorphism}). If $f$ is {\em unimodal}
then $X$ is called a {\em unimodal inverse limit space}.
\section{Zigzags in interval maps}\label{sec:zigzags}
In this section we define the basic notions that we need in the rest of the paper and prove in Corollary~\ref{thm:leo+fpwl=lzzb} that every piecewise monotone leo map is long-zigzag.
\begin{definition}\label{def:zigzagfree}
A map $f\colon I\to I$ is called \em{zigzag-free} if for every interval $[a, b]\subset I$ such that $f([a,b]))=[f(a), f(b)]$ (or $f([a,b])=[f(b),f(a)]$), and $f(x)\not\in\{f(a),f(b)\}$ for every $x\in (a,b)$, it holds that $[a,b]$ is mapped homeomorphically onto $f([a,b])$ (see examples of maps which are not zigzag-free in Figure~\ref{fig:zigzag}).
\end{definition}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[scale=4]
\draw (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw [thick] (0,0)--(1/5,1)--(2/5,1/4)--(3/5,3/4)--(4/5,1/2)--(1,1);
\draw (2/5,-0.02)--(2/5,0.02);
\node at (2/5, -0.07) {\scriptsize $a$};
\node at (-0.1, 1/4) {\scriptsize $f(a)$};
\node at (-0.1, 1) {\scriptsize $f(b)$};
\draw[dashed] (2/5,0)--(2/5,1/4)--(0,1/4);
\draw (1,-0.02)--(1,0.02);
\node at (1, -0.07) {\scriptsize $b$};
\draw (2/5,1/4)--(3/5,3/4)--(4/5,1/2)--(1,1);
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[scale=4]
\draw (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw[thick] (0,0)--(0.2,0.85)--(0.4,0.5)--(0.6,0.75)--(0.8,0.25)--(1,1);
\draw (0,-0.02)--(0,0.02);
\node at (0, -0.07) {\scriptsize $a$};
\node at (-0.1,0) {\scriptsize $f(a)$};
\node at (-0.1, 1) {\scriptsize $f(b)$};
\draw (1,-0.02)--(1,0.02);
\node at (1, -0.07) {\scriptsize $b$};
\draw (0,0)--(0.2,0.85)--(0.4,0.5)--(0.6,0.75)--(0.8,0.25)--(1,1);
\end{tikzpicture}
\caption{Maps $f$ which are not zigzag-free.}
\label{fig:zigzag}
\end{figure}
Note that there is no smoothness condition regarding the function $f$ in the definition above, however for simplicity we will draw all pictures piecewise linear.
In the next two results we need the following observation.
\begin{observation}\label{obs:compzz}
For maps $f,g\colon I\to I$ write $F=f\circ g$ and assume $a<b\in I$ are such that $F([a,b])\in\{[F(a),F(b)], [F(b),F(a)]\}$ and $F(x)\not\in\{F(a),F(b)\}$ for all $x\in(a,b)$. Then also $g([a,b])\in\{[g(a), g(b)], [g(b), g(a)]\}$ and $g(x)\not\in\{g(a),g(b)\}$ for all $x\in(a,b)$. The same holds for $f$, \ie $f([g(a),g(b)])\in\{[f\circ g(a), f\circ g(b)]\}$, and $f(x)\not\in\{f\circ g(a), f\circ g(b)\}$ for all $x\in (g(a),g(b))$. Thus $f|_{[g(a),g(b)]}$ is either monotone, or $f$ has at least two critical points in $(g(a),g(b))$, in which case $|F(a)-F(b)|\geq\min\{|f(x)-f(y)|: x\neq y \text{ neighbouring critical points of $f$} \}$.
\end{observation}
The last statement in the observation above will be used later in Theorem~\ref{thm:lzzbcond}.
\begin{lemma}\label{lem:itzigzag}
If $f$ is zigzag-free, then $f^n$ is zigzag-free for every $n\in\N$.
\end{lemma}
\begin{proof}
We use Observation~\ref{obs:compzz} in the case when $g=f$.
Assume that $0\leq a<b\leq 1$ are such that $f^n([a,b])=[f^n(a), f^n(b)]$ or $f^n([a,b])=[f^n(b), f^n(a)]$ and $f^n(x)\not\in\{f^n(a),f^n(b)\}$ for every $x\in(a,b)$.
Then $f([a,b])=[f(a),f(b)]$ or $f([a,b])=[f(b),f(a)]$, and $f(x)\not\in\{f(a),f(b)\}$ also, so since $f$ is zigzag-free, $f|_{[a,b]}$ is monotone. The same holds for $f^{i+1}|_{[f^i(a),f^i(b)]}$ for every $i\in\{1,\ldots, n-2\}$. It follows that $f^n|_{[a,b]}$ is one-to-one, so $f^n$ is zigzag-free.
\end{proof}
\begin{definition}\label{def:longzigzag}
A map $f\colon I\to I$ is called {\em long-zigzag}
if there exists $\eps>0$ such that for every $n\in\N_0$ and every interval $[a,b]\subset I$ such that $f^n([a, b])=[f^n(a), f^n(b)]$ (or $f^n([a,b])=[f^n(b), f^n(a)]$), $f^n(x)\not\in\{f^n(a),f^n(b)\}$ for every $x\in (a,b)$, and $|f^n(a)-f^n(b)|<\eps$, it holds that $f^n$ maps $[a,b]$ homeomorphically onto $f^n([a,b])$.
\end{definition}
\begin{remark}
Map $f\colon I\to I$ is called {\em long-branched} if there exists $\eps>0$ such that for every $n\in\N$ and every adjacent critical points $a<b$ of $f^n$ it holds that $|f^n([a,b])|>\eps$. Note that every long-branched map is also long-zigzag. For example, Minc's map (see Figure~\ref{fig:Minc}) is long-branched, since the lengths of branches of $f^n$ are bounded below by $1/3$ for every $n\in \N$. However, there are non-longbranched maps which are long-zigzag. For example, there is a dense $G_{\delta}$ set of slope values in $(\sqrt{2},2]$ of such maps in the tent map family (see \cite{BBD}). Tent maps are zigzag-free, see Lemma 2.1 from \cite{ABC1}, and therefore tent maps are long-zigzag. Nevertheless, tent maps can have arbitrary small branches.
\end{remark}
\begin{figure}
\begin{tikzpicture}[scale=4]
\draw [thick](0,0) -- (1/3, 1) -- (5/12, 1/3) -- (7/12,2/3) -- (2/3, 0) -- (1,1);
\draw (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);
\draw[dashed] (1/3, 0) -- (1/3, 1);
\draw[dashed] (1/2, 0) -- (1/2, 1);
\draw[dashed] (2/3, 0) -- (2/3, 1);
\draw[dashed] (0, 1/3) -- (1, 1/3);
\draw[dashed] (0, 1/2) -- (1, 1/2);
\draw[dashed] (0, 2/3) -- (1, 2/3);
\node at (1/3,-0.1) {\small $\frac{1}{3}$};
\node at (1/2,-0.1) {\small $\frac{1}{2}$};
\node at (2/3,-0.1) {\small $\frac{2}{3}$};
\node at (-0.1,1/3) {\scriptsize $1/3$};
\node at (-0.1,1/2) {\scriptsize $1/2$};
\node at (-0.1,2/3) {\scriptsize $2/3$};
\draw[solid, fill] (1/2,0.5) circle (0.015);
\node at (0.9,0.9) {\small $f$};
\node at (0.53,0.45) {\scriptsize $p$};
\end{tikzpicture}
\caption{Example by Minc (2001) suggested as a candidate for a counterexample to the Nadler-Quinn problem \cite{Nadler}. The map is long-branched and thus also long-zigzag. However, it is not zigzag-free.}
\label{fig:Minc}
\end{figure}
In the following theorem we show that the set of long-zigzag maps is large. As a corollary we obtain that every locally eventually onto map $f\colon I\to I$ with a finite set of critical points is long-zigzag.
\begin{theorem}\label{thm:lzzbcond}
Let $f\colon I\to I$ be a map with finite critical set $C=\{0=c_1<c_2,\ldots, c_N=1\}$, such that for every $i\in\{1,\ldots, N-1\}$ one of the following holds:
\begin{itemize}
\item[(a)] there exists $k(i)\in\N$ such that $f^{k(i)}((c_i,c_{i+1}))\cap C\neq\emptyset$, or
\item[(b)] $f^k|_{[c_i,c_{i+1}]}$ is a homeomorphism onto its image for every $k\in\N$, and there is $\delta(i)>0$ such that $|f^k([c_i,c_{i+1}])|\geq\delta(i)$ for all $k\in\N$.
\end{itemize}
Then $f$ is a long-zigzag map.
\end{theorem}
\begin{proof}
Let $\Sigma=\{i\in\{1, \ldots, N-1\}: i \textrm{ satisfies property (a)}\}$, and $\Delta=\{i\in\{1, \ldots, N-1\}: i \textrm{ satisfies property (b)}\}$. Then $\Sigma\cap \Delta=\emptyset$.
For $i\in \Sigma$ let $k(i)\in\N$ be the minimal natural number such that there is a critical point of $f$ in $f^{k(i)}((c_i,c_{i+1}))$.
Define
$$\eps:=\min\left\{\left\{\min_{1\leq j\leq k(i)}|f^j([c_i,c_{i+1}])|:i\in \Sigma\right\}\cup\{\delta(i): i\in \Delta\}\right\}$$
and observe that $\eps>0$.
We will show that for every $n\in\N$ and every $[a,b]\subset I$ such that $f^n|_{[a,b]}$ is not monotone, if $f^n([a,b])=[f^n(a),f^n(b)]$ (or $[f^n(b),f^n(a)]$) and $f^n(x)\not\in\{f(a),f(b)\}$ for all $x\in(a,b)$, then it holds that $|f^n(a)-f^n(b)|\geq \eps$.
Fix $n\in\N$ and $[a,b]\subset I$. By Observation~\ref{obs:compzz}, either $f|_{f^{n-1}([a,b])}$ is monotone, or $f$ contains at least two critical points in $f^{n-1}([a,b])$. In the later case $|f^n(a)-f^n(b)|\geq \min\{|f(c_i)-f(c_{i+1})|: i\in\{1,\ldots, N-1\}\}\geq \eps.$
Assume $f|_{f^{n-1}([a,b])}$ is monotone. Again, either $f|_{f^{n-2}([a,b])}$ is monotone, or $f$ contains at least two critical points in $f^{n-2}([a,b])$. In the later case $|f^{n-1}(a)-f^{n-1}(b)|\geq \min\{|f(c_i)-f(c_{i+1})|: i\in\{1,\ldots, N-1\}\}.$ Since $f|_{f^{n-1}([a,b])}$ is monotone, it follows that $f^2|_{[c_i,c_{i+1}]}$ is monotone, and we conclude that $|f^n(a)-f^n(b)|\geq |f^2([c_i,c_{i+1}])|\geq \eps$.
We continue inductively and conclude that $|f^n(a)-f^n(b)|\geq\eps$.
\end{proof}
\begin{definition}
We say that a map $f\colon I\to I$ is {\em locally eventually onto (leo)} if for every $[a, b]\subset I$ there exists $n\in\N$ such that $f^n([a,b])=I$.
\end{definition}
\begin{corollary}\label{thm:leo+fpwl=lzzb}
Every piecewise monotone leo map $f\colon I\to I$ is long-zigzag.
\end{corollary}
\begin{proof}
Since for every $[c_i,c_{i+1}]$ there is $k\in\N$ such that $f^k([c_i, c_{i+1}])=I$, the proof follows directly from the previous theorem.
\end{proof}
\begin{remark}
It is obviously not enough to assume only piecewise monotonicity of $f$ in Corollary~\ref{thm:leo+fpwl=lzzb}, see \eg map on Figure~\ref{fig:doublespiral1} and its iterations.
\end{remark}
\begin{corollary}\label{cor:Raines}
If $f\colon I\to I$ is piecewise monotone map with the property that for every interval $J\subset I$, and every connected component $A$ of $f^{-1}(J)$ it holds that $|A|\leq |J|$, then $f$ is long-zigzag.
\end{corollary}
\begin{proof}
Let $C=\{0=c_1< c_2 < \ldots < c_N=1\}$ be the critical set of $f$. Note that for every $i\in\{1,\ldots, N-1\}$ for which $f^k|_{[c_i,c_{i+1}]}$ is one-to-one, it holds that $|f^k([c_i,c_{i+1}])|\geq |c_{i+1}-c_i|$. We define $\delta_i:=|c_{i+1}-c_i|$ and apply Theorem~\ref{thm:lzzbcond}.
\end{proof}
\section{Types of endpoints}\label{sec:endpoints}
In this section we study different notions of an endpoint in interval inverse limits and how they emerge. We study $\underleftarrow{\lim}\{I, f_i\}$, where $f_i\colon I\to I$ is a continuous surjection for every $i\in\N$.
\begin{definition}
Let $K$ be a chainable continuum.
We say that $x\in K$ is an {\em endpoint} if for every subcontinua $A, B\subset K$ such that $x\in A\cap B$, it holds that $A\subset B$ or $B\subset A$. We say that $x\in K$ is an {\em $L$-endpoint} (definition is due to Lelek \cite{Le}) if it is an endpoint of every arc in $K$ which contains it.
\end{definition}
Note that if $x$ is in no arc of $K$, $x$ is automatically an $L$-endpoint.
Lelek's definition of an endpoint is usually used in the case when $K$ is a dendroid, and the first definition is used when $K$ is chainable. However, we will see that $L$-endpoints are of significant importance also in chainable continua. Note that every endpoint is also $L$-endpoint, but the converse is not true. For instance, every vertex of degree one in a simple triod is an $L$-endpoint but not an endpoint. Counterexamples can also be found among chainable continua, for example if we attach an arc to any point of the pseudo-arc; then the point of the intersection is an $L$-endpoint but not an endpoint.
In this section we partially answer the following problem, and use the obtained results in the subsequent sections.
\begin{problem}
For which chainable continua $\underleftarrow{\lim}\{I, f_i\}$ are the two definitions of endpoints equivalent?
\end{problem}
Bruin shows in \cite{Br1} that the definitions are equivalent for inverse limits on intervals generated by a single {\em unimodal} bonding map. There he introduces a notion of a basic arc and then proves that in unimodal inverse limits a point $x$ is an endpoint if and only if it is an endpoint of its basic arc and is not ``glued" to any other basic arc in $x$.
We generalize the notion of a basic arc here and then reformulate Bruin's result.
\begin{definition}
Let $X=\underleftarrow{\lim}\{I, f_i\}$ and $x=(x_0, x_1, \ldots)\in X$. For $i\in\N_0$ we define
the {\em $i$-basic arc} $A_i(x)$ as the maximal arc in $X$ such that $x\in A_i(x)$ and $\pi_i|_{A_i(x)}$ is one-to-one (can be degenerate).
\end{definition}
Observe that if $J_{i+k}\subset I$ are closed intervals such that $x_{i+k}\in J_{i+k}$ and $f_{i}\circ\ldots\circ f_{i+k-1}|_{J_{i+k}}$ is one-to-one and onto $J_i$ for every $k>0$, then $\{x\in X: x_{i+k}\in J_{i+k}, k> 0\}\subset A_i(x)$.
\begin{proposition}[Proposition 2 in \cite{Br1}]\label{prop:Bruin}
Let $T_s$ be a tent map with slope $s\in (1, 2]$. Then $x\in \underleftarrow{\lim}\{I, T_s\}$ is an endpoint of $\underleftarrow{\lim}\{I, T_s\}$ if and only if it is an endpoint of $A_i(x)$ for every $i\in\N_0$ (see Figure~\ref{fig:uils}).
\end{proposition}
\begin{figure}
\begin{tikzpicture}[scale=4]
\draw (0,1.6)--(1,1.6);
\draw (0,1.5)--(1,1.5);
\draw (0,1.58)--(1,1.58);
\draw (0,1.52)--(1,1.52);
\draw (0,1.38)--(1,1.38);
\draw (0,1.3)--(1,1.3);
\draw (0,1.365)--(1,1.365);
\draw (0,1.315)--(1,1.315);
\draw (0,1.16)--(1,1.16);
\draw (0,1.1)--(1,1.1);
\draw (0,1.15)--(1,1.15);
\draw (0,1.11)--(1,1.11);
\draw[domain=90:270] plot ({0.05*cos(\x)}, {1.55+0.05*sin(\x)});
\draw[domain=90:270] plot ({0.03*cos(\x)}, {1.55+0.03*sin(\x)});
\draw[domain=90:270] plot ({0.04*cos(\x)}, {1.34+0.04*sin(\x)});
\draw[domain=90:270] plot ({0.025*cos(\x)}, {1.34+0.025*sin(\x)});
\draw[domain=90:270] plot ({0.03*cos(\x)}, {1.13+0.03*sin(\x)});
\draw[domain=90:270] plot ({0.02*cos(\x)}, {1.13+0.02*sin(\x)});
\draw[thick] (0, 1)--(1,1);
\node[circle,fill, inner sep=1] at (0,1){};
\node at (-0.07,0.98) {$x$};
\node at (0.5,0.85) {\small flat endpoint ($\E_F$)};
\end{tikzpicture}
\begin{tikzpicture}[scale=1]
\draw[dashed] (1,1) circle (1);
\node[circle,fill, inner sep=1] at (1,1){};
\node at (1,0.8) {\small $x$};
\node at (0.5,1) {?};
\node at (1.5,1) {?};
\node at (1,1.5) {?};
\node at (1,0.3) {?};
\node at (0.9,-0.5) {\small nasty endpoint ($\E_N$)};
\end{tikzpicture}
\vspace{10pt}
\begin{tikzpicture}[scale=5]
\draw (0.1,0.75)--(0.8,0.75);
\draw (0.3,0.62)--(0.8,0.62);
\draw (0.3,0.56)--(0.65,0.56);
\draw (0.4,0.53)--(0.65,0.53);
\draw (0.4,0.51)--(0.55,0.51);
\draw[domain=270:450] plot ({0.8+0.065*cos(\x)}, {0.685+0.065*sin(\x)});
\draw[domain=90:270] plot ({0.3+0.03*cos(\x)}, {0.59+0.03*sin(\x)});
\draw[domain=270:450] plot ({0.65+0.015*cos(\x)}, {0.545+0.015*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.01*cos(\x)}, {0.52+0.01*sin(\x)});
\node[circle,fill, inner sep=1] at (0.5,0.47){};
\node at (0.5,0.42) {$x$};
\node at (0.5,0.3) {\small spiral endpoint ($\E_S$)};
\end{tikzpicture}
\caption{Different types of endpoints in unimodal inverse limits. For further properties of endpoints in the unimodal setting see \cite{AABC}.}
\label{fig:uils}
\end{figure}
Proposition~\ref{prop:Bruin} motivates the third definition of an endpoint in $X=\underleftarrow{\lim}\{I, f_i\}$.
\begin{definition}
We say that $x\in X=\underleftarrow{\lim}\{I, f_i\}$ is a $B$-endpoint if it is an endpoint of $A_i(x)$ for every $i\in\N_0$.
\end{definition}
Note that the definition above depends on the inverse limit representation of $X$. For example, an arc can be represented as an inverse limit on $I$ generated by the identity, or as a double spiral generated by a bonding map in Figure~\ref{fig:doublespiral1}. In the first representation the whole inverse limit space is a single basic arc and a point is an endpoint if and only if it is a $B$-endpoint. In the latter representation, for spiral point $x=(1/2, 1/2, \ldots)$ every $A_i(x)$ is degenerate and thus $x$ is a $B$-endpoint. Obviously, $x$ is not an endpoint.
\begin{figure}
\centering
\begin{tikzpicture}[scale=5]
\draw (0.1,0.75)--(0.8,0.75);
\draw (0.3,0.62)--(0.8,0.62);
\draw (0.3,0.56)--(0.65,0.56);
\draw (0.4,0.53)--(0.65,0.53);
\draw (0.4,0.51)--(0.55,0.51);
\draw[domain=270:450] plot ({0.8+0.065*cos(\x)}, {0.685+0.065*sin(\x)});
\draw[domain=90:270] plot ({0.3+0.03*cos(\x)}, {0.59+0.03*sin(\x)});
\draw[domain=270:450] plot ({0.65+0.015*cos(\x)}, {0.545+0.015*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.01*cos(\x)}, {0.52+0.01*sin(\x)});
\node[circle,fill, inner sep=1] at (0.5,0.47){};
\begin{scope}[yscale=-1,xscale=-1,yshift=-0.94cm,xshift=-1cm]
\draw (0.1,0.75)--(0.8,0.75);
\draw (0.3,0.62)--(0.8,0.62);
\draw (0.3,0.56)--(0.65,0.56);
\draw (0.4,0.53)--(0.65,0.53);
\draw (0.4,0.51)--(0.55,0.51);
\draw[domain=270:450] plot ({0.8+0.065*cos(\x)}, {0.685+0.065*sin(\x)});
\draw[domain=90:270] plot ({0.3+0.03*cos(\x)}, {0.59+0.03*sin(\x)});
\draw[domain=270:450] plot ({0.65+0.015*cos(\x)}, {0.545+0.015*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.01*cos(\x)}, {0.52+0.01*sin(\x)});
\end{scope}
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=3]
\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);
\draw[thick] (0,0)--(1/3,5/9)--(2/3,4/9)--(1,1);
\draw[dashed] (0,5/9)--(5/9,5/9)--(5/9,0);
\draw[dashed] (1,4/9)--(4/9,4/9)--(4/9,1);
\node[circle,fill, inner sep=1] at (0.5,0.5){};
\draw[dashed] (0,0)--(1,1);
\end{tikzpicture}
\caption{Representation of an arc in which point in the interior is a $B$-endpoint.}
\label{fig:doublespiral1}
\end{figure}
It follows directly from the definitions that every $L$-endpoint is a $B$-endpoint. For further examples of $B$-endpoints which are not $L$-endpoints see Figure~\ref{fig:one} and Figure~\ref{fig:two}.
In general we have
$$\{\text{endpoints}\}\subset \{L\text{-endpoints}\} \subset \{B\text{-endpoints}\}.$$ Thus we pose the following natural problem.
\begin{figure}
\centering
\begin{tikzpicture}[scale=5]
\draw (0.2,0.75+0.2)--(0.8,0.75+0.2);
\draw (0.5,0.62+0.2)--(0.8,0.62+0.2);
\draw (0.5,0.56+0.2)--(0.8,0.56+0.2);
\draw (0.7,0.53+0.2)--(0.8,0.53+0.2);
\draw (0.7,0.51+0.2)--(0.8,0.51+0.2);
\draw[domain=270:450] plot ({0.8+0.065*cos(\x)}, {0.685+0.2+0.065*sin(\x)});
\draw[domain=90:270] plot ({0.5+0.03*cos(\x)}, {0.59+0.2+0.03*sin(\x)});
\draw[domain=270:450] plot ({0.8+0.015*cos(\x)}, {0.545+0.2+0.015*sin(\x)});
\draw[domain=90:270] plot ({0.7+0.01*cos(\x)}, {0.52+0.2+0.01*sin(\x)});
\draw (0.8, 0.48+0.2)--(1,0.48+0.2);
\node[circle,fill, inner sep=1] at (0.8,0.48+0.2){};
\node[circle, inner sep=0] at (0.8,0.48){};
\end{tikzpicture}
\hspace{20pt}
\begin{tikzpicture}[scale=4]
\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);
\draw[dashed] (0,2/3)--(2/3,2/3)--(2/3,0);
\draw[dashed] (2/3,1)--(2/3,2/3)--(1,2/3);
\draw[dashed] (0,0)--(1,1);
\draw[thick] (0,0)--(1/3,2/3)--(1/2,5/9)--(2/3,2/3)--(1,1);
\node[circle,fill, inner sep=1] at (2/3,2/3){};
\end{tikzpicture}
\caption{$B$-endpoint which is not an $L$-endpoint.}
\label{fig:one}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}[scale=5]
\draw (0.1,0.75)--(0.6,0.75);
\draw (0.4,0.62)--(0.6,0.62);
\draw (0.4,0.56)--(0.6,0.56);
\draw (0.4,0.53)--(0.6,0.53);
\draw (0.4,0.51)--(0.6,0.51);
\draw[domain=270:450] plot ({0.6+0.065*cos(\x)}, {0.685+0.065*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.03*cos(\x)}, {0.59+0.03*sin(\x)});
\draw[domain=270:450] plot ({0.6+0.015*cos(\x)}, {0.545+0.015*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.01*cos(\x)}, {0.52+0.01*sin(\x)});
\draw[thick] (0.4,0.47)--(0.6,0.47);
\node[circle,fill, inner sep=1] at (0.4,0.47){};
\node[circle,fill, inner sep=1] at (0.6,0.47){};
\begin{scope}[yscale=-1,xscale=-1,yshift=-0.94cm,xshift=-1cm]
\draw (0.1,0.75)--(0.6,0.75);
\draw (0.4,0.62)--(0.6,0.62);
\draw (0.4,0.56)--(0.6,0.56);
\draw (0.4,0.53)--(0.6,0.53);
\draw (0.4,0.51)--(0.6,0.51);
\draw[domain=270:450] plot ({0.6+0.065*cos(\x)}, {0.685+0.065*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.03*cos(\x)}, {0.59+0.03*sin(\x)});
\draw[domain=270:450] plot ({0.6+0.015*cos(\x)}, {0.545+0.015*sin(\x)});
\draw[domain=90:270] plot ({0.4+0.01*cos(\x)}, {0.52+0.01*sin(\x)});
\end{scope}
\end{tikzpicture}
\hspace{20pt}
\vspace{10pt}
\begin{tikzpicture}[scale=4]
\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);
\draw[thick] (0,0)--(1/3,2/3)--(2/3,1/3)--(1,1);
\draw[dashed] (0,2/3)--(2/3,2/3)--(2/3,0);
\draw[dashed] (1,1/3)--(1/3,1/3)--(1/3,1);
\node[circle,fill, inner sep=1] at (0.5,0.5){};
\draw[dashed] (0,0)--(1,1);
\end{tikzpicture}
\caption{Example of a $B$-endpoint which is not an endpoint, where $X$ is not an arc.}
\label{fig:two}
\end{figure}
\begin{problem}
For which $\underleftarrow{\lim}\{I,f_i\}$ is every $B$-endpoint an endpoint?
\end{problem}
In this section we show that every B-endpoint is an endpoint in the case when all $f_i$
are zigzag-free or long-zigzag and leo. Later we study chainable continua for which all inhomogeneities are endpoints. It will be important to restrict to the cases in which endpoints are B-endpoints, see the proof of Theorem~\ref{thm:FisE}.
\begin{lemma}\label{lem:endpts}
Assume that every map $f_i$ from $\underleftarrow{\lim}\{I, f_i\}$ is zigzag-free. Then $x\in \underleftarrow{\lim}\{I, f_i\}$ is an endpoint if and only if it is a $B$-endpoint. Specially, all three definitions of endpoints are equivalent.
\end{lemma}
\begin{proof}
Assume $x$ is not an endpoint, so there are subcontinua $A, B\subset \underleftarrow{\lim}\{I, f_i\}$ such that $x\in A\cap B$ and $A\setminus B, B\setminus A\neq\emptyset$.
Let $A_i=\pi_i(A), B_i=\pi_i(B)$, $i\in\N_0$ be coordinate projections. They are all intervals, $x_i\in A_i\cap B_i$ for every $i$, and there exists $N\in\N$ such that $A_{i}\setminus B_{i}, B_{i}\setminus A_{i}\neq\emptyset$, for all $i>N$.
Since $f_{i+1}$ does not contain a zigzag, there exists an interval $(l_{i+1}, r_{i+1})\ni x_{{i+1}}$ such that $f_{i+1}|_{[l_{i+1}, r_{i+1}]}\colon [l_{i+1}, r_{i+1}]\to A_{i}\cup B_{i}$ is one-to-one and surjective, see Figure~\ref{fig:nzz}.
So we can find an arc
$A:= [a_N,b_N] \overset{f_{N+1}}{\longleftarrow} [a_{N+1},b_{N+1}] \overset{f_{N+2}}{\longleftarrow}[a_{N+2},b_{N+2}] \overset{f_{N+3}}{\longleftarrow} [a_{N+4}, b_{N+4}]\overset{f_{N+4}}{\longleftarrow}\ldots$
such that $x\in \Int(A)\subset \Int(A_{N}(x))$, so $x$ is not a $B$-endpoint.
\end{proof}
\begin{figure}
\centering
\begin{tikzpicture}[scale=5]
\draw (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw (0,0)--(0.6,0)--(0.6,0.6)--(0,0.6);
\draw (1,0.4)--(0.4,0.4)--(0.4,1);
\draw[thick,red] (0,0)--(0.4,0);
\node[red] at (0.3,-0.07) {\small $A_{{i+1}}$};
\draw[thick,purple] (0.4,0)--(0.6,0);
\draw[thick,blue] (0.6,0)--(1,0);
\node[blue] at (0.8,-0.07) {\small $B_{{i+1}}$};
\draw[thick,red] (0,0)--(0,0.4);
\node[red] at (-0.1,0.3) {\small $A_{{i}}$};
\draw[thick,purple] (0,0.4)--(0,0.6);
\draw[thick,blue] (0,0.6)--(0,1);
\node[blue] at (-0.1,0.8) {\small $B_{{i}}$};
\node at (0.9,0.9) {\small $f_{i+1}$};
\node[circle,fill, inner sep=1] at (0.5,0){};
\node at (0.5,-0.06) {\small $x_{{i+1}}$};
\draw[thick] (0.25,0.1)--(0.3,0)--(0.4,0.41)--(0.6,0.58)--(0.7,1)--(0.75,0.9);
\end{tikzpicture}
\caption{Since $f_{i+1}$ is zigzag-free, for minimal interval $J\subset I$ such that $f_{i+1}(J)=A_i\cup B_i$ it holds that $f_{i+1}|_J$ is one-to-one.}
\label{fig:nzz}
\end{figure}
The previous theorem can be generalized to the class of long-zigzag maps which are locally eventually onto (leo).
\begin{observation}\label{obs:leo}
If $f$ is a leo map, and $A\subset \underleftarrow{\lim}\{I, f\}$ is a proper subcontinuum, then $|\pi_i(A)|\to 0$ as $i\to\infty$ (see \eg \cite[Lemma 2]{BrBr}).
\end{observation}
\begin{theorem}\label{thm:llzzb}
If $f\colon I\to I$ is long-zigzag leo map, then every $B$-endpoint in $\underleftarrow{\lim}\{I, f\}$ is an endpoint.
In particular, the sets of $B$-endpoints, $L$-endpoints, and endpoints in $\underleftarrow{\lim}\{I,f\}$ are the same.
\end{theorem}
\begin{proof}
The proof follows as the proof of Lemma~\ref{lem:endpts}. We only need to find $N\in\N$ such that $|A_i\cup B_i|<\eps$ for all $i>N$. However, this is always possible by Observation~\ref{obs:leo}, since $f$ is leo.
\end{proof}
\begin{remark}
There exist long-zigzag maps (not leo) for which some $B$-endpoints are not endpoints, see \eg Figure~\ref{fig:two}. However, every such $B$-endpoint must be an endpoint of its non-degenerate $0$-basic arc, as we show in the next proposition. We define double spiral points first and then show that in the inverse limits of long-zigzag maps they do not exist.
\end{remark}
\begin{definition}\label{def:doublespiral}
$B$-endpoint $x\in \underleftarrow{\lim}\{I, f_i\}$ which is not an $L$-endpoint is called a {\em double spiral point}.
\end{definition}
\begin{proposition}\label{prop:doublespiral}
If $f\colon I\to I$ is long-zigzag, then $\underleftarrow{\lim}\{I, f\}$ does not contain double spiral points.
\end{proposition}
\begin{proof}
Denote by $\eps>0$ the constant from the definition of a long-zigzag map. Assume that $x=(x_0, x_1, \ldots)\in \underleftarrow{\lim}\{I, f\}$ is a $B$-endpoint which is contained in the interior of a non-degenerate arc. That means that there exist non-degenerate arcs $A, B\subset \underleftarrow{\lim}\{I, f\}$ such that $\{x\}=A\cap B$. Denote by $A_i=\pi_i(A)$ and $B_i=\pi_i(B)$ for $i\geq 0$ and note that every $A_i, B_i$ are intervals in $I$, $x_i\in A_i\cap B_i$, and $f|_{A_i}, f|_{B_i}$ are surjective. Furthermore, since $A\cup B$ is an arc, we can take without loss of generality $|A_0\cup B_0|<\eps$.
Since $f$ is long-zigzag, for every $i\in\N$ we can find an interval $x_i\in U_i\subset A_i\cap B_i$ which is mapped homeomorphically onto $A_0\cap B_0$.
As in the proof of Lemma~\ref{lem:endpts} we conclude that $A_0(x)$ is non-degenerate, and $x\in \Int A_0(x)$, which is a contradiction.
\end{proof}
Note that for double spirals in Figure~\ref{fig:doublespiral1} and \ref{fig:one}, there exists a homeomorphic continuum with a different inverse limit representation $\underleftarrow{\lim}\{I,f\}$ for which there are no double spirals. Thus we ask the following question:
\begin{question}\label{question1}
Given $X=\underleftarrow{\lim}\{I, f\}$, is it always possible to find continuous maps $g_i\colon I\to I$ such that $X$ is homeomorphic to $\underleftarrow{\lim}\{I, g_i\}$ and such that every $B$-endpoint in $\underleftarrow{\lim}\{I, g_i\}$ is an $L$-endpoint (that is, can we find a representation of $X$ in which there are no double spirals)? Can we do this requiring $g_i=g$ for every $i\in \N$?
\end{question}
\begin{example}\label{ex:leo+doublespiral}
There exists a leo interval map $f$ (with infinitely many critical points) so that $\underleftarrow{\lim}(I,f)$ has a double spiral, see Figure~\ref{fig:doublespiralleo}.
Namely, the intervals $[a_i',b_i']\subset I$ are such that $f([a_{i+1}',b_{i+1}'])\subset[a_i',b_i']$ for every $i\in\N_0$, so $A:=[a'_0,b'_0] \overset{f}{\longleftarrow} [a'_1,b'_1] \overset{f}{\longleftarrow}[a'_{2},b'_{2}]\overset{f}{\longleftarrow}\ldots$ is a well-defined subcontinuum of $\underleftarrow{\lim}\{I,f\}$. The map $f$ is constructed such that $A$ is a double spiral obtained in the similar way as in Figure~\ref{fig:doublespiral1}. There are intervals $[a_i,b_i]\subset [a_i',b_i']$ for every $i\in\N_0$ for which $f|_{[a_i,b_i]}$ is one-to-one for every $i\in\N_0$, $f{[a_{i+1},b_{i+1}]}\subset [a_i,b_i]$, and $\diam f^{i-1}([a_i,b_i])\to 0$ as $i\to \infty$. It is not difficult to see that $x=(x_0,x_1,x_2,\ldots)\in A$ for which $x_i\in[a_i,b_i]$ for every $i\in\N_0$ is a double spiral point.
Moreover, we set $f([a'_0,b'_0])=I$. Then for every interval $J\subset I$ such that $[a_n',b_n']\subset J$ it holds that $f^n(J)=I$. We can set $f|_{[a_i',b_i']}$ to be of slope $3$ or more, and $f|_{[b_i',a_{i-1}']}$ to be of slope greater than $1$ for every $i\geq 1$. Therefore, if $J$ does not contain any $[a_n',b_n']$, then the diameter of $f^k(J)$ increases as $k$ increases. We only need to find $k\in\N$ for which $f^k(J)$ contains some $[a_n',b_n']$ and conclude that $f^{k+n}(J)=I$.
\end{example}
\begin{figure}
\begin{tikzpicture}[scale=8]
\draw[thick] (1,1)--(9/10,0)--(8/10,1)--(0.77,0.91)--(0.71,0.99)--(0.68,0.9)--(0.6,0.8)--(0.57,0.715)--(0.53,0.765)--(0.5,0.68)--(0.43,0.6)--(0.4,0.54)--(0.38,0.565)--(0.35,0.5)--(0.29,0.43)--(0.26,0.385)--(0.25,0.395)--(0.22,0.35)--(0.18,0.29)--(0.16,0.252)--(0.155,0.258)--(0.14,0.22)--(0.115,0.18);
\draw[dashed] (0.9,0.9)--(0.8,0.9);
\draw[dashed](0.14,0.14)--(0.14,0.22)--
(0.22,0.22)--(0.22,0.35)--(0.35,0.35)--(0.35,0.5)--(0.5,0.5)--(0.5,0.68)--(0.68,0.68)--(0.68,0.9)--(0.9,0.9)--(0.9,0);
\draw[dashed](0.115,0.115)--
(0.115,0.18)--(0.18,0.18)--(0.18,0.29)--(0.29,0.29)--(0.29,0.43)--(0.43,0.43)--(0.43,0.6)--(0.6,0.6)--(0.6,0.8)--(0.8,0.8)--(0.8,1);
\draw[dashed] (0,0)--(1,1);
\draw (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);
\draw[dashed] (0.8,0)--(0.8,0.8);
\draw[dashed] (0.68,0)--(0.68,0.68);
\draw[dashed] (0.6,0)--(0.6,0.6);
\draw[dashed] (0.5,0)--(0.5,0.5);
\draw[dashed] (0.43,0)--(0.43,0.43);
\draw[dashed] (0.35,0)--(0.35,0.35);
\draw[dashed] (0.29,0)--(0.29,0.29);
\draw[dashed] (0.22,0)--(0.22,0.22);
\draw[dashed] (0.18,0)--(0.18,0.18);
\draw[dashed] (0.14,0)--(0.14,0.14);
\draw[dashed] (0.115,0)--(0.115,0.115);
\draw[dashed] (0,0.9)--(0.9,0.9);
\draw[dashed] (0,0.8)--(0.8,0.8);
\draw[dashed] (0,0.68)--(0.68,0.68);
\draw[dashed] (0,0.6)--(0.6,0.6);
\draw[dashed] (0,0.5)--(0.5,0.5);
\draw[dashed] (0,0.43)--(0.43,0.43);
\draw[dashed] (0,0.35)--(0.35,0.35);
\draw[dashed] (0,0.29)--(0.29,0.29);
\draw[dashed] (0,0.22)--(0.22,0.22);
\draw[dashed] (0,0.18)--(0.18,0.18);
\draw[dashed] (0.68,0.9)--(0.68,1);
\draw[dashed] (0.5,0.68)--(0.5,0.8);
\draw[dashed] (0.35,0.5)--(0.35,0.6);
\draw[dashed] (0.22,0.35)--(0.22,0.43);
\draw[dashed] (0.14,0.22)--(0.14,0.29);
\node[circle,fill, inner sep=1] at (1,1){};
\node[circle,fill, inner sep=1] at (0.9,0.9){};
\node[circle,fill, inner sep=1] at (0.8,0.8){};
\node[circle,fill, inner sep=1] at (0.77,0.77){};
\node[circle,fill, inner sep=1] at (0.71,0.71){};
\node[circle,fill, inner sep=1] at (0.68,0.68){};
\node[circle,fill, inner sep=1] at (0.6,0.6){};
\node[circle,fill, inner sep=1] at (0.57,0.57){};
\node[circle,fill, inner sep=1] at (0.53,0.53){};
\node[circle,fill, inner sep=1] at (0.5,0.5){};
\node[circle,fill, inner sep=1] at (0.43,0.43){};
\node[circle,fill, inner sep=1] at (0.4,0.4){};
\node[circle,fill, inner sep=1] at (0.38,0.38){};
\node[circle,fill, inner sep=1] at (0.35,0.35){};
\node[circle,fill, inner sep=1] at (0.29,0.29){};
\node[circle,fill, inner sep=1] at (0.26,0.26){};
\node[circle,fill, inner sep=1] at (0.25,0.25){};
\node[circle,fill, inner sep=1] at (0.22,0.22){};
\node[circle,fill, inner sep=1] at (0.18,0.18){};
\node[circle,fill, inner sep=1] at (0.14,0.14){};
\node[circle,fill, inner sep=1] at (0.115,0.115){};
\node[circle,fill, inner sep=1] at (1,0){};
\node[circle,fill, inner sep=1] at (0.99,0){};
\node[circle,fill, inner sep=1] at (0.91,0){};
\node[circle,fill, inner sep=1] at (0.9,0){};
\node[circle,fill, inner sep=1] at (0.8,0){};
\node[circle,fill, inner sep=1] at (0.77,0){};
\node[circle,fill, inner sep=1] at (0.71,0){};
\node[circle,fill, inner sep=1] at (0.68,0){};
\node[circle,fill, inner sep=1] at (0.6,0){};
\node[circle,fill, inner sep=1] at (0.5,0){};
\node[circle,fill, inner sep=1] at (0.43,0){};
\node[circle,fill, inner sep=1] at (0.35,0){};
\node[circle,fill, inner sep=1] at (0.29,0.){};
\node[circle,fill, inner sep=1] at (0.22,0){};
\node[circle,fill, inner sep=1] at (0.18,0){};
\node[circle,fill, inner sep=1] at (0.14,0){};
\node[circle,fill, inner sep=1] at (0.115,0){};
\node[circle,fill, inner sep=1] at (0.57,0){};
\node[circle,fill, inner sep=1] at (0.53,0){};
\node[circle,fill, inner sep=1] at (0.4,0){};
\node[circle,fill, inner sep=1] at (0.38,0.){};
\node[circle,fill, inner sep=1] at (0.25,0){};
\node[circle,fill, inner sep=1] at (0.26,0){};
\node[circle,fill, inner sep=1] at (0.16,0){};
\node[circle,fill, inner sep=1] at (0.155,0){};
\node at (-0.05,1) {\scriptsize $1$};
\node at (0,-0.03) {\scriptsize $0$};
\node at (0.15,-0.03) {\scriptsize $a'_5$};
\node at (0.19,-0.03) {\scriptsize $b'_5$};
\node at (0.23,-0.03) {\scriptsize $a'_{4}$};
\node at (0.3,-0.03) {\scriptsize $b'_4$};
\node at (0.36,-0.03) {\scriptsize $a'_3$};
\node at (0.44,-0.03) {\scriptsize $b'_{3}$};
\node at (0.5,-0.03) {\scriptsize $a'_2$};
\node at (0.54,-0.035) {\scriptsize $a_{2}$};
\node at (0.58,-0.03) {\scriptsize $b_{2}$};
\node at (0.62,-0.03) {\scriptsize $b'_{2}$};
\node at (0.68,-0.03) {\scriptsize $a'_{1}$};
\node at (0.715,-0.035) {\scriptsize $a_{1}$};
\node at (0.775,-0.03) {\scriptsize $b_{1}$};
\node at (0.81,-0.03) {\scriptsize $b'_{1}$};
\node at (0.9,-0.03) {\scriptsize $a'_{0}$};
\node at (0.94,-0.03) {\scriptsize $b_{0}$};
\node at (0.985,-0.035) {\scriptsize $a_{0}$};
\node at (1.02,-0.03) {\scriptsize $b'_{0}$};
\end{tikzpicture}
\caption{Example of an interval leo map $f$ for which the inverse limit has double spirals.}
\label{fig:doublespiralleo}
\end{figure}
\iffalse
\begin{proof}
Obviously, the basic arc of the point $(1/2,1/2,\ldots)$ is degenerate. Furthermore, $(1/2,1/2,\ldots)$ is not an endpoint due to the following argument. Consider $[1/3,2/3]\ni 1/2$ and let $\eps=1/15$. Then if $1/2\in J$ and $f^N(J)=[1/3,2/3]$, point $1/2$ separates $f^{-N}([1/3,2/5])$ from $f^{-N}([3/5,2/3])$. Thus $f^{-N}$ is not $\eps$-crooked with respect to the point $1/2$.
Since all proper subcontinua of $\underleftarrow{\lim}(I,f)$ are arcs (since all the critical point are periodic - {\bf This needs to be explained more. Are you sure about this? We don't have to go that far, we can try to construct an arc which contains $(1/2,1/2,\ldots)$ in the interior. For that $f$ also needs to be defined more precisely.}) we conclude that $(1/2,1/2,\ldots)$ is a double spiral.
\begin{figure}
\begin{tikzpicture}[scale=5]
\draw[thick] (0,0) -- (1/6, 1) -- (1/3, 1/6) -- (0.4,0.4)--(0.42,1/3) -- (0.45, 0.45);
\draw[thick] (1,1) -- (5/6, 0) -- (2/3, 5/6) -- (0.6,0.6) -- (0.58, 2/3) -- (0.55,0.55);
\draw (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);
\draw[dashed] (1/3, 0) -- (1/3, 1);
\draw[dashed] (2/3, 0) -- (2/3, 1);
\draw[dashed] (0, 1/3) -- (1, 1/3);
\draw[dashed] (0, 2/3) -- (1, 2/3);
\draw[dashed] (0, 0) -- (1, 1);
\draw[solid, fill] (1/2,1/2) circle (0.012);
\node at (0.1,0.9) {\small $f$};
\end{tikzpicture}
\caption{Example of an interval map $f$ for which Proposition~\ref{prop:counterdoublespiral} holds.}
\label{fig:doublespiral}
\end{figure}
\end{proof}
\fi
\begin{remark}
The characterization of $L$-endpoints or double spirals in $\underleftarrow{\lim}\{I,f_i\}$ is tightly connected with the characterization of $\underleftarrow{\lim}\{I,f_i\}$ which are arcs. Such a characterization is in general tedious and technical, but in
Appendix~\ref{appB} we give an easily accessible and applicable characterization of arcs $\underleftarrow{\lim}\{I,f\}$ for piecewise monotone bonding maps $f$ which is easy to describe in terms of the dynamics of the bonding map.
\end{remark}
\section{Folding points}\label{sec:fp}
We say that a point $x\in\underleftarrow{\lim}\{I, f_i\}$ is a {\em folding point} if it does not have a neighbourhood homeomorphic to $S\times (0,1)$, where $S$ is a zero-dimensional set.
Raines proved in \cite[Corollary 4.8]{Raines} that, if $f\colon I\to I$ satisfies properties (i)+(ii) (see below), then $x\in\underleftarrow{\lim}\{I, f\}$ is a folding point if and only if $\pi_n(x)\in\omega(C)$ for every $n\in\N_0$.
\begin{enumerate}
\item[(i)] $f$ is piecewise monotone,
\item[(ii)] If $J\subset I$ is an interval, then $|A|\leq |J|$ for every component $A$ of $f^{-1}(J)$.
\end{enumerate}
By Corollary~\ref{cor:Raines}, every $f$ which satisfies (i)+(ii) is long-zigzag.
We will generalize Raines' result to the class of long-zigzag maps. That is indeed a larger class, since long-zigzag maps can have infinitely many critical points, or not satisfy property (ii). The proof crucially uses the fact that for long-zigzag map $f$ it holds that $\underleftarrow{\lim}\{I, f\}$ contains no double spirals.
The characterization of folding points obtained here will be used later in the proof of Theorem~\ref{thm:FisE}, with the assumption that $f$ is either zigzag-free or long-zigzag and locally eventually onto. In this setting we know that then point in the inverse limit is an endpoint if and only if it is an $L$-endpoint by Theorem~\ref{thm:llzzb}.
\begin{theorem}\label{thm:folding}
Assume $f\colon I\to I$ is a long-zigzag map. Then $x\in\underleftarrow{\lim}\{I, f\}$ is a folding point if and only if $\pi_n(x)\in\omega(C)$ for every $n\in\N_0$.
\end{theorem}
\begin{proof}
Denote by $\eps>0$ the constant from the definition of long-zigzag map, \ie for every $n\in\N_0$ and minimal $[a, b]\subset I$ such that $f^n([a, b])=[f^n(a), f^n(b)]$ (or $f^n([a, b])=[f^n(b), f^n(a)]$) and $|f^n(a)-f^n(b)|<\eps$ it holds that $f^n|_{[a,b]}$ is one-to-one. Denote by $x=(x_0, x_1, \ldots)\in \underleftarrow{\lim}\{I, f\}$.
($\Rightarrow$) Assume first that there exists $\delta>0$ such that $(x_0-\delta, x_0+\delta)\cap\{f^n(C):n\in\N_0\}=\emptyset$ and study $0$-box $U:=\pi^{-1}_0((x_0-\delta, x_0+\delta))$. Let $A\subset U$ be a maximal connected set. Then we can write $A=\underleftarrow{\lim}\{A_i, f|_{A_i}\}$, where $A_i\subset I$ is a maximal connected set such that $f(A_i)=A_{i-1}$, for every $i\geq 1$, and $A_0=(x_0-\delta, x_0+\delta)$. Note that a priori $f|_{A_i}$ does not need to be a surjection onto $A_{i-1}$. However, note that $A_i$ does not contain a critical point $c\in C$, since otherwise $f^i(c)\in A_0$, which contradicts the assumption. We conclude that $\pi_0|_A$ is one-to-one onto $(x_0-\delta,x_0+\delta)$. Thus, $U=(x_0-\delta,x_0+\delta)\times S$, where $S=\pi_0^{-1}(x_0)\subset \underleftarrow{\lim}\{I, f\}$ is one-dimensional (otherwise $x_0\in f^n(C)$). By applying $\widehat{f}^n$ to $x$ we can argue analogously if $x_n\notin \omega(C)$. This covers one side of the proof. Note also that here we did not use the fact that $f$ is long-zigzag.
($\Leftarrow$) Let $x_n\in\omega(C)$ for every $n\geq 0$ and assume by contradiction that $x$ is not a folding point. Recall from Proposition~\ref{prop:doublespiral} that there are no double spirals so the $0$-basic arc $A_0(x)$ is non-degenerate. Also, we can assume that $x_0$ is contained in the interior of $A_0(x)$ (otherwise we find the smallest $i\in \N$ so that $x_i\in \Int(A_i(x))$). Take $0<\delta<\eps$ such that $\pi_0(A_0(x))\supset (x_0-\delta, x_0+\delta)$ and let $J\subset A_0(x)$ be such that $\pi_0(J)=(x_0-\delta, x_0+\delta)$. Denote by $J_n=\pi_n(J)$ for $n\in\N_0$. We will study the open neighbourhoods of $J$ given by $U_n:=\pi_n^{-1}(J_n)$ for $n\in\N_0$. Note that, since $n$-boxes form a basis for the topology, and after shrinking $\delta$ if necessary, we can find $N\in\N$ such that $U_n$ is homeomorphic to a zero-dimensional set times an arc for every $n\geq N$.\\
Fix $n\geq N$ and let us study the arcs of maximal length in $U_n$. Every such arc is of the form $B=\underleftarrow{\lim}\{B_i,f\}$, where $B_n=J_n$, and $B_{k+n}$ is maximal such that $f^k(B_{k+n})\subset J_n$. Note that $f|_{B_i}$ is a homeomorphism onto the image for every $1\leq i<n$. If $f|_{B_j}$ is also a homeomorphism for $j\geq n$, then endpoints of $B$ are projected via $\pi_0$ to $x_0-\delta$ and $x_0+\delta$. Since $x_0\in\omega(C)$, there exist $\underleftarrow{\lim}\{B_i,f\}$ for which there is $j\geq n$ such that $f|_{B_j}$ is not one-to-one. Let us take the smallest such $j$. Then, since $\delta<\eps$, $f|_{B_j}$ is not onto and $f$ maps endpoints of the arc $B_j$ to the same point. Moreover, since $f$ is surjective, for every $k\geq 1$ we can find $B_{j+n+k}$ such that $f|_{B_{j+n+k}}$ is onto $B_{j+n+k-1}$. Denote by $Q_n=\underleftarrow{\lim}\{B_i, f\}$; it must be an arc since $U_n$ is a zero-dimensional set of arcs. Moreover, by construction, the endpoints of $Q_n$ are projected via $\pi_n$ to the same point. Specifically, the endpoints are projected via $\pi_0$ to the same point (either $x_0+\delta$ or $x_0-\delta$).
Note that an arc $Q_n$ exists for infinitely $n\geq N$. Since the neighbourhoods $U_n$ of $A_0(x)$ shrink as $n$ increases, there exists a sequence of arcs $(Q_n)_{n\geq N}$ in $U_0$ such that $\partial Q_n$ converges to an endpoint of $A_0(x)$ as $n\to\infty$.
However, this implies that such $U_0$ is not homeomorphic to a zero dimensional set of arcs, a contradiction.
\end{proof}
\begin{remark}
In general, it can happen that $\pi_n(x)\in\omega(C)$ for every $n\geq 0$, but $x$ is locally homeomorphic
to a zero dimensional set of arcs, see for example Figure~\ref{fig:doublespiral1}. In that case $x$ is contained in a double spiral. In general it is very difficult to check if a double spiral point is a folding point or not, see \eg Example~\ref{ex:leo+doublespiral}. A positive answer to Question~\ref{question1} would give a nice way around that difficulty.
\end{remark}
\section{Retractability along $\omega(C)$ and endpoints}\label{sec:persistence}
We generalize the notion of {\em persistent recurrence} introduced by Blokh and Lyubich in \cite{BL}. Namely we define when an interval map is non-retractable and show that a long-zigzag leo map $f\colon I \to I$ is non-retractable along $\omega(C)$ if and only if the {\em set of endpoints} $\E$ equals the {\em set of folding points} $\F$ in $\underleftarrow{\lim}\{I,f\}$.
\begin{definition}\label{def:pullback} Let $f\colon I\to I$ be a map with critical set $C$.
Let $x=(x_0, x_{1}, \ldots)\in \underleftarrow{\lim}\{I, f\}$ and let $J\subset I$ be an interval.
The sequence $(J_n)_{n\in\N_0}$ of intervals is called a \emph{pull-back} of $J$ along
$x$ if $J=J_0$, $x_{k}\in J_k$ and $J_{k+1}$ is the largest interval such that $f(J_{k+1})\subset J_k$
for all $k\in\N_0$. A pull-back is {\em monotone} if $C\cap \Int( J_n)=\emptyset$
for every $n\in\N$.
\end{definition}
\begin{definition}\label{def:persrec}
Let $f\colon I\to I$ be a map with critical set $C$. We say that $f$ is {\em retractable along $\omega(C)$} if there exists a
backward orbit $x=(x_0, x_1, x_2, \ldots)$ in $\omega(C)$ and an open interval $U\subset I$ such that $x_0\in U$ and such that $U$ has a monotone pull-back along $x$. Otherwise, $f$ is called {\em non-retractable along $\omega(C)$}. We will often only write {\em non-retractable}.
\end{definition}
\begin{remark}
Note that the notion of non-retractable map does not require recurrence of critical points or of the critical set $C$. For example, the map in Figure~\ref{fig:twosided} has the property that $\omega(C)\cap C=\emptyset$, but $f$ is non-retractable.
\end{remark}
The motivation for studying when $\E=\F$ comes from the topology of inverse limits of infinitely renormalisable unimodal maps, where it is known that $\F=\E$, and $f|_{\omega(c)}$ is conjugate to an adding machine. Later, {\em strange adding machines} were discovered in non-infinitely renormalisable unimodal maps \cite{BKM}. That led to a hypothesis that $\F=\E$ might be related to embedded adding machines, or at least the property that $f|_{\omega(c)}$ is one-to-one, see \cite{Al}. However, examples in \cite{AlBr} showed the hypothesis to be false. In \cite{AABC} it is proven that the crucial notion for $\F=\E$ is {\em persistent recurrence}. It turns out that $\F=\E$ if and only if the critical point $c$ of a unimodal map is persistently recurrent. We generalize this to inverse limits of leo long-zigzag maps $f$ in the following theorem. The more general proof is even simpler than the proof of Theorem 4.13 in \cite{AABC}, once we understand the notions of folding points and endpoints well enough.
\begin{theorem}\label{thm:FisE}
Let $f\colon I\to I$ be a long-zigzag leo map with the critical set $C$. Then for $\underleftarrow{\lim}\{I, f\}$ it holds that all folding points are endpoints if and only if $f$ is non-retractable along $\omega(C)$.
\end{theorem}
\begin{proof}
Since $f$ is long-zigzag, we know that $x=(x_0, x_1, \ldots)\in \underleftarrow{\lim}\{I, f\}$ is a folding point if and only if $x_n\in\omega(C)$ for every $n\geq 0$ due to Theorem~\ref{thm:folding}. Also, since $f$ is additionally leo, we know that $x$ is an endpoint if and only if it is a $B$-endpoint (if and only if it is an $L$-endpoint) due to Theorem~\ref{thm:llzzb}. So, $x$ is not an endpoint if and only if there is $n\geq 0$ such that $(x_n, x_{n+1}, \ldots)$ is contained in the interior of its $0$-basic arc.
If $f$ is retractable, there exists a folding point $x=(x_0, x_{1}, \ldots)\in \underleftarrow{\lim}\{I, f\}$, an interval $J\subset I$ such that
$x_0\in \Int(J)$, and an infinite monotone pull-back $(J_n)_{n\in\N_0}$ of $J$ along $x$.
Therefore, $\underleftarrow{\lim}\{J_n, f|_{J_n}\}$ is an arc in $\underleftarrow{\lim}\{I, f\}$ and it contains $x$ in its interior, and
thus $x$ is not an endpoint.
For the other direction, let $f$ be non-retractable and assume that there is a folding point $x=(x_0, x_1, \ldots)\in \underleftarrow{\lim}\{I, f\}$ which is not an endpoint. Without loss of
generality we can assume that $x$ is contained in the interior of its $0$-basic arc. Otherwise, we use
$\hat f^{-j}(x)$ for some $j\in\N$ large enough. Let $A\subset A_0(x)$ be an open subset of the $0$-basic arc of $x$ such that $x\in \Int(A)$ and such that $A$ does not contain endpoints of $A_0(x)$. Then $(\pi_n(A))_{n\in\N_0}$ is a monotone pull-back of $\pi_0(A)$ along $x$, so $f$ is retractable, a contradiction.
\end{proof}
Appendix~\ref{appA} is a natural continuation of this section, but since it is considerably more technical, we moved it to the end of the paper. It gives a characterization of interval maps that are non-retractable along $\omega(C)$, through the dynamical properties of the maps.
\section{Endpoints and critical recurrence}\label{sec:recurrence}
Barge and Martin \cite{BM} give a characterization of endpoints in $\underleftarrow{\lim}\{I, f\}$ in terms of dynamics of map $f$. They prove, among other things, that if $f$ has finitely many critical points, a dense orbit, and if $\omega(C)\cap C=\emptyset$, then there are no endpoints. Assumption of dense orbit is indeed required, \eg we can take $f$ which generates a two-sided spiral as in Figure~\ref{fig:twosided} for an example that confirms that. In this section we establish a partial converse of the Barge and Martin result.
First, we show with the following example that only the condition $\omega(C)\cap C=\emptyset$ is not enough to obtain the converse of the statement above. Thus, we need to impose additional assumptions on $f$.
\begin{figure}
\begin{tikzpicture}[scale=5]
\draw (0,1/6) -- (1/3, 0) -- (2/3, 1) -- (1,5/6);
\draw (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);
\draw[dashed] (1/3, 0) -- (1/3, 1);
\draw[dashed] (2/3, 0) -- (2/3, 1);
\draw[dashed] (0, 1/3) -- (1, 1/3);
\draw[dashed] (0, 2/3) -- (1, 2/3);
\draw[dashed] (0, 0) -- (1, 1);
\draw[solid, fill] (1/9,1/9) circle (0.015);
\draw[solid, fill] (8/9,8/9) circle (0.015);
\node at (0.1,0.9) {\small $f$};
\node at (1/9,1/9) {\scriptsize $x$};
\node at (8/9,8/9) {\scriptsize $y$};
\end{tikzpicture}
\caption{Example for which $\omega(C)\cap C=\emptyset$ but $\protect\underleftarrow{\lim}\{I, f\}$ is an arc and thus has endpoints. Actually, endpoints are $(x,x,x, \ldots)$ and $(y,y,y,\ldots)$, $x,y\neq 1/2$.}
\label{fig:twosided}
\end{figure}
\begin{example}
There exists an interval map $f$ so that $\underleftarrow{\lim}(I,f)$ has no endpoints but $\omega(C)\cap C\neq\emptyset$, see map $f$ in Figure~\ref{fig:counter}. The three intervals $I_1=[0,1/3], I_2=[1/3,2/3], I_3=[2/3,1]\subset I$ are $f$-invariant and $\underleftarrow{\lim}(I,f)=\underleftarrow{\lim}(I_1,f|_{I_1})\cup \underleftarrow{\lim}(I_2,f|_{I_2})\cup \underleftarrow{\lim}(I_3,f|_{I_3})$. Note that $\underleftarrow{\lim}(I_1,f|_{I_1})$ and $\underleftarrow{\lim}(I_3,f|_{I_3})$ are disjoint Knaster continua and $\underleftarrow{\lim}(I_2,f|_{I_2})$ are two $\sin(1/x)$-continua that share the same bar. Furthermore, $\underleftarrow{\lim}(I_1,f|_{I_1})\cap\underleftarrow{\lim}(I_2,f|_{I_2})=\{(1/3,1/3, \ldots)\}$ is the endpoint of the Knaster continuum $\underleftarrow{\lim}(I_1,f|_{I_1})$ and the endpoint of one of the rays in $\underleftarrow{\lim}(I_2,f|_{I_2})$.
A similar conclusion holds for $\underleftarrow{\lim}(I_2,f|_{I_2})\cap\underleftarrow{\lim}(I_3,f|_{I_3})=\{(2/3,2/3,\ldots)\}$. Therefore $\underleftarrow{\lim}(I,f)$ has no endpoints. However, $\omega(C)\cap C\neq\emptyset$ since the two critical points $1/3$ and $2/3$ are $2$-periodic.
\end{example}
\begin{figure}
\begin{tikzpicture}[scale=5]
\draw[thick] (0,1/3) -- (1/6, 0) -- (1/3, 1/3) -- (4/9,5/9)--(5/9,4/9)--(2/3,2/3)--(5/6,1)--(1,2/3);
\draw (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);
\draw[dashed] (1/3, 0) -- (1/3, 1);
\draw[dashed] (2/3, 0) -- (2/3, 1);
\draw[dashed] (0, 1/3) -- (1, 1/3);
\draw[dashed] (0, 2/3) -- (1, 2/3);
\draw[dashed] (0, 0) -- (1, 1);
\draw[dashed] (4/9, 4/9) -- (4/9, 5/9)-- (5/9,5/9)--(5/9,4/9) -- (4/9,4/9);
\draw[solid, fill] (4/9,5/9) circle (0.012);
\draw[solid, fill] (5/9,4/9) circle (0.012);
\node at (0.1,0.9) {\small $f$};
\end{tikzpicture}
\caption{Example of an interval map $f$ such that $\protect\underleftarrow{\lim}(I,f)$ has no endpoints but $\omega(C)\cap C\neq\emptyset$.}
\label{fig:counter}
\end{figure}
\iffalse
First we state the characterization of endpoints from \cite{BM} that we will use.
\begin{definition}
If $f:[c,d]\to [a,b]$ is a continuous surjection of the interval $[c,d]$ onto the interval $[a,b]$, $p\in [c,d]$ and $\eps>0$, then $f$ is \emph{$\eps$-crooked with respect to $p$} provided $p$ does not separate $f^{-1}([a,a+\eps])$ from $f^{-1}([b-\eps,b])$ in $[c,d]$.
\end{definition}
\begin{lemma}[Theorem 1.4 from \cite{BM}]\label{lem:BM}
Let $f:I\to I$ be continuous. Then $(p_0,p_1,\ldots)$ is an endpoint of $\protect\underleftarrow{\lim}\{I, f\}$ if and only if for each integer $i$, each interval $J_i$ with $p_i\in \Int(J_i)$ and each $\eps>0$, there is a positive integer $N$ such that if $p_{N+i}\in J_{N+i}$ and $f^{N}(J_{N+i})=J_i$, then $f^N$ is $\eps$-crooked with respect to $p_{N+i}$.
\end{lemma}
\fi
\begin{lemma}\label{lem:uncountably}
Assume that $f$ is zigzag-free and there is $c\in C$ such that $c\in\omega(c)$. Then $\underleftarrow{\lim}\{I,f\}$ has an endpoint. If additionally $\orb(c)$ is infinite, then $\underleftarrow{\lim}\{I, f\}$ has uncountably many endpoints.
\end{lemma}
\begin{proof}
Denote by $K_0\subset I$ an interval such that $f^{k_1}(c)\in\Int(K_0)$ for some $k_1\in\N$. If $f^k(c)\in\{0,1\}$ for every $k\in\N$, then $(0,0,\ldots)$ or $(0,1,0,1,\ldots)\in\underleftarrow{\lim}\{I,f\}$ and they are endpoints, see Theorem 2.9 in \cite{BM} for the detailed argument. We choose an arbitrary small interval $L_0$ such that $f^{k_1}(c)\in\Int (L_0)$, and an interval $K_{k_1}$ such that $c\in\Int(K_{k_1})$ and $f^{k_1}(K_{k_1})\subset L_0$.
Since $c\in\omega(c)$, we can continue inductively and for every $j\geq 2$ find $k_j\in\N$ such that $f^{k_j}(c)\in\Int(K_{k_{j-1}})$ (infinitely many if $\orb(c)$ is infinite). We choose an arbitrary small interval $L_{j-1}$ such that $f^{k_j}(c)\in\Int(L_{j-1})$, and an interval $K_{k_j}$ such that $c\in\Int(K_{k_j})$, $f^{k_j}(K_{k_j})\subset L_{j-1}$.
Since the intervals $L_j$ can be chosen arbitrarily small, and since $L_j$ is closed for every $j\in\N_0$, we can define $x_0=\cap_{i\geq 1} f^{k_1+k_2+\ldots +k_i}(L_i)\subset L_0$, and $x_{k_j}=\cap_{i\geq j+1} f^{k_{j+1}+k_{j+2}+\ldots +k_i}(L_i)\\
\subset L_j$, for $j\geq 1$. We claim that $x=(x_0,x_1,\ldots, x_{k_1},x_{k_1+1}, \ldots)\in\underleftarrow{\lim}\{I,f\}$ is an endpoint. Note that since $L_j$ can be chosen arbitrary, if $\orb(c)$ is infinite, we can construct uncountably many different such points $x$.
Since $f$ is zigzag-free, and thus every $f^n$ is zigzag-free (Lemma~\ref{lem:itzigzag}), then for every intervals $K,J$ such that $f^n(J)=K$ and $J$ minimal such (so there is no $J'\subset J$ such that $f^n(J')=K$), it holds that critical points of $f^n$ in $\Int(J)$ are mapped to $\partial K$.
Assume that there are subcontinua $A,B\subset\underleftarrow{\lim}\{I,f\}$ such that $x\in A\cap B$, and such that $A\not\subset B$, $B\not\subset A$. Denote the projections by $A_i=\pi_i(A), B_i=\pi_i(B)$ for $i\geq 0$. Thus $x_i\in A_i\cap B_i$ for all $i\geq 0$. There exists $N\in\N$ such that $A_i\setminus B_i, B_i\setminus A_i\neq\emptyset$ for all $i\geq N$. By the definition of $x$ and since $L_j$ can be chosen to be arbitrarily small, there are $k,j\in\N$ such that $f^k(L_j)\subset A_N\cap B_N$. It follows that there is a critical point $c\in\Int(A_{N+k}\cup B_{N+k})$ which is mapped to $\Int(A_N\cup B_N)$, and that is impossible since $f^k$ is zigzag-free. So, $x$ is an endpoint.
\end{proof}
\begin{theorem}\label{thm:endpt}
If $f$ is long-zigzag leo map, and there exists $c\in C$ such that $c\in\omega(c)$, then there exists an endpoint in $\underleftarrow{\lim}\{I,f\}$. If additionally $\orb(c)$ is infinite, then $\underleftarrow{\lim}\{I,f\}$ has uncountably many endpoints.
\end{theorem}
\begin{proof}
The proof follows analogously as in the previous lemma. Since $f$ is leo, it follows by Observation~\ref{obs:leo} that we can take $M\in\N$ such that $|A_M\cup B_M|<\eps$, where $\eps$ is the constant from the definition of a long-zigzag map and then proceed analogously as in the proof of Lemma~\ref{lem:uncountably}.
\end{proof}
\begin{remark}
In the proof of Lemma~\ref{lem:uncountably} we use in several places that $c\in \omega(c)$. It is natural to ask if the proof works in different setting.
The previous lemma and theorem also hold true if we assume that $c\in\omega(C)$ for every $c\in C$ under the zigzag-free and under the long-zigzag leo assumptions respectively, if we do the following. On the first occurrence replace $c\in\omega(c)$ with $c\in\omega(C)$ and then take any $c'$ such that $f^k(c')\in K$ and continue with that $c'$. In the next step we would have "since $c'\in\omega(C)$" etc. To get uncountably many endpoints in that case one needs to assume that every neighbourhood of every critical point contains at least two points from $\orb(C)$ as one can see following the preceding proof.
\end{remark}
\begin{example}
Besides some obvious examples of non-leo maps that give in the inverse limit countably infinitely many endpoints,
there also exist leo maps $f\colon I\to I$ such that $\underleftarrow{\lim}\{I,f\}$ has countably infinitely many endpoints, see \eg Figure~\ref{fig:leoexample}.
\begin{figure}
\begin{tikzpicture}[scale=5]
\draw[thick] (1,0) -- (1/2, 1) -- (5/16, 5/16) -- (1/4,1/2) -- (5/32, 5/32) -- (1/8,1/4)--(3/32,3/32);
\draw (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);
\draw (1/2,-0.02)--(1/2,0.02);
\draw (5/16,-0.02)--(5/16,0.02);
\draw (1/4,-0.02)--(1/4,0.02);
\draw (5/32,-0.02)--(5/32,0.02);
\draw (1/8,-0.02)--(1/8,0.02);
\draw (-0.02,1/2)--(0.02,1/2);
\draw (-0.02,1/4)--(0.02,1/4);
\draw (-0.02,1/8)--(0.02,1/8);
\draw[dashed] (0, 0) -- (1, 1);
\node at (0.5,-0.1) {\small $\frac{1}{2}$};
\node at (1/4,-0.1) {\small $\frac{1}{4}$};
\node at (0.115,-0.1) {\small $\frac{1}{8}$};
\node at (5/16,-0.1) {\small $\frac{5}{16}$};
\node at (5/32+0.01,-0.1) {\small $\frac{5}{32}$};
\node at (-0.1,1/2) {\small $\frac{1}{2}$};
\node at (-0.1,1/4) {\small $\frac{1}{4}$};
\node at (-0.1,1/8) {\small $\frac{1}{8}$};
\node at (0.1,0.9) {\small $f$};
\end{tikzpicture}
\caption{Locally eventually onto interval map for which the inverse limit has countably infinitely many endpoints.}
\label{fig:leoexample}
\end{figure}
Note that every $(x,x,x,\ldots)$, where $x\in I$ is a fixed point of $f$, is an endpoint (see \cite{BM}, Example 3 for the detailed argument), which gives countably infinitely many endpoints. Furthermore, all points $(x_0,x_1,x_2,\ldots)$ for which there exists $N\in \N$ so that $x_i\neq 1/2^n$ for all $i>N$ are contained in interiors of their $N$-basic arcs and are thus not endpoints. There is only countably many remaining points that can be folding points, they are of the form $\hat f^n((1/2,1/2^{2},1/2^3,\ldots))$ for every $n\in \Z$.
\end{example}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,713 |
Oscar Pistorius held over gun death
South African Paralympic star Oscar Pistorius has been arrested after his model girlfriend was shot dead at his home.
According to local media reports, Reeva Steenkamp, 30, may have been mistaken for an intruder at the property in the Silver Lakes Golf Estate in Boschkop in the capital Pretoria. There was speculation the shooting may have been a Valentine's Day surprise gone wrong.
Miss Steenkamp, described in her Twitter biography as a model, cover girl and law graduate, had tweeted about Valentine's Day yesterday, writing: "What do you have up your sleeve for your love tomorrow??? #getexcited #ValentinesDay".
Police spokeswoman Lt Col Katlego Mogale told the South African Press Association police were called to Pistorius's home following the shooting.
She said: "Paramedics declared the woman dead on the scene and police proceeded with their investigation. The woman sustained wounds to her head and the upper body." A 9mm pistol was recovered. According to unconfirmed reports four shots were fired.
Lt Col Mogale said a 26-year-old man was expected to appear at Pretoria Magistrate's Court later. The officer was quoted in local media as saying: "At this stage we are still conducting a preliminary inquiry. Statements are conducted with neighbours and people who were at the scene. A case of murder was being investigated."
Pistorius is known as the Blade Runner because of the ground-breaking prosthetics he uses for racing. He made history at the London 2012 Olympics when he became the first amputee sprinter to compete in the able-bodied Games, running in the 400m and 4x400m relay.
Tropika Island of Treasure – an entertainment show starring Steenkamp due to be broadcast this weekend – posted a tribute to her on its website. It says: "We are deeply saddened and extend our condolences to Reeva's family and friends."
Athletes joined fans in paying tribute to Miss Steenkamp. Jessica Ennis spoke of her shock on Twitter, describing the "the horrendous news" as "an awful tragedy". Denise Lewis added: "Not a good morning athletics fans – News reports on Oscar Pistorius are unbelievable". | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 457 |
Looks like one of the SpaceX Falcon 9 boosters exploded during launch (at 29 seconds in). Luckily the other 8 are designed to pick up the slack.
We watched the launch yesterday. At one point the video feed had the rocket aiming downwards and my girlfriend remarked that it appears to be burning up on it's way back to earth. Wonder if this explosion coincided with her remarks. After the explosion you can clearly see the thrust pattern change from 9 well-organized burns to a chaos of flames. | {
"redpajama_set_name": "RedPajamaC4"
} | 251 |
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