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Can We Really get to 100% Renewable Energy?
The notion that entire communities and countries could be completely powered by renewable energy still seems like a pipedream to many. Not to those gathered at the first Pathways to 100% Renewable Energy conference in San Francisco earlier this month. To them, a fully renewable power system is not only achievable, but makes economic sense. And it's already being accomplished in some places.
Organized by the 100% Renewable Energy Institute, based in Santa Monica, California, the gathering had its fair share of grassroots environmentalists. But it was peppered with esteemed scientists, eco-celebs (such as Frances Moore Lappe), and government officials (San Francisco mayor Edward Lee). Several representatives from the California Independent System Operator (CAISO) were there, as well, discussing the variety of smart grid tools – including demand response – necessary to wean large power grids off of fossil fuels that currently burned to balance the variability of wind and solar.
While Denmark is one of the few countries committed to a 100% renewable energy goal for electricity, heat and transportation, some local governments have already reached the 100% renewable threshold. Perhaps the most inspiring example is Rhein-Hunsruck in southern Germany, a district of roughly 100,000 inhabitants that will produce more than 100% of its own needs from solar, wind and biomass this year. By 2014, this rural district will be providing 236% of its own energy needs from renewables, and hopes to generate significant revenue by exporting excess carbon-free power to the open market. Rhein-Hunsruck has combined aggressive energy efficiency programs, which reduced the district's overall electricity loads by 25%, with a shift to local and regional power generation from renewables, according to Bertram Fleck, chief administrative officer for the district.
Closer to home here in the U.S., the City of Greensburg, Kansas (pop. 781 ) is powered completely by wind power. It also boasts the highest per capita LEED platinum green buildings in the U.S., highlighting the synergy between energy efficiency and renewable energy to get to the 100% carbon-free energy nirvana.
Despite these victories, there are skeptics. Among them is Peter Lilienthal of HOMER Energy, a leading source for software to design microgrids in the developing world. While he notes that solar photovoltaic (PV) power is now cheaper than diesel fuel, the cost of shifting over entire islands or other remote microgrids to 100% renewable energy is – in Lilienthal's view – too high and unnecessary. Of course, the beauty of these modular microgrids is that they can green up over time, incorporating a variety of different fuels and technologies.
Still, Michael Jacobson of Stanford University continues to pump out studies mapping out specific portfolios of different wind, water and sun (WWS) resources that could power entire states such as California or New York, and do not, according to his calculations, bust the bank. And a newly released report from the Civil Society Institute concludes increased reliance upon renewables will not reduce grid reliability, as is so often feared.
At the conference, I gave a presentation on microgrids and virtual power plants, two aggregation and optimization platforms that not only enable high penetrations of renewables, but will also be necessary for countries such as Germany and Denmark to meet their aggressive carbon reduction goals. Without such power grid innovations, shifting to a carbon-free energy future would be impossible.
A good first step in this green energy transformation would be to scale back what the Earth Policy Institute has estimated is $620 billion in government subsidies now flowing toward fossil fuel development. Eliminate those subsidies, level the playing field in energy markets, and the world suddenly looks like a different place.
Earth Day in Marin County: Take Action Today!
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| 9,035
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Where can I find Genuine Candidates?
Why do I get so many profiles with Skill Gap?
Why I am not getting the Profiles Quickly for my requirement?
Why do not the Candidates turn up on time for Interviews?
Is there somebody who can understand my requirement and act?
If any of the above questions bother you every day, Prontoux is the right destination for you. We at Prontoux work passionately towards directing the correct and genuine profiles for your everyday requirements at the right place and right time. It's our endeavor understand your requirement and transform it into suitable and genuine Candidates in the specified timelines.
We highly rely on Referrals as our primary Source of Candidate so that to know them and their expertise in depth prior to recruitment.
Prontoux has a Direct Apply Platform to enable Candidates to apply for the Jobs which they wish to. This is to speed up the recruitment and narrow down the recruitment Funnel for Selective and Suitable candidates for that Job Profile. This is Powered by Email Communication for better bi directional connect.
We Believe in Building long term and fruitful associations with vendors in Recruitment. As a result of this we have our trusted Vendor network with more than 500 vendors with whom we have successfully engaged in the past while adhering to our stringent recruitment Standards.
Prontoux has its internal Process FINE(Fast Intuitive Network Engagement). Using FINE, Email Campaigns are done on frequent basis to yield suitable Candidates for different Job Profiles from more than 10, 000 Vendors and Candidates.
Prontoux aims to automate the recruitment process for Speedy and error free recruitment to the maximum possible extent. Out internal Algorithm RCS uses Artificial Intelligence to Pre Screen the candidate for their fitment for specific Job or other Jobs. All Job Applications have to qualify RCS to go to the next Stage.
Prontoux has the mandatory hurdle of In-depth Technical Screening which needs to be crossed by every candidate. We have a dedicated panel of technical experts who have extensive experience in their domain. Based on the Job Description, the candidates have to be eligible n this Internal Screening. It is only after succeeding in the Internal Screening that the Candidate's profile is shared with the client.
We employ live Online Coding tests depending on the Job profile and this has an extensive and in depth question Bank which is regularly updated.
Prontoux is aware of the presence of Fake Candidates and Adheres to its standards in Recruitment. We blacklist candidates and their vendors for any future endeavor if they are found not genuine or if the client has any such feedback.
Our Screening process focuses in unleashing the core and additional Skillset of candidates. This lets us in knowing the correct fitment of Candidate in any Suitable Job profile. Our current endeavor has resulted in having a large pool of genuine and in depth Skillset candidates.
At Prontoux, we emphasize on Process Automation to the maximum possible extent. The Core Idea behind this is to ascertain that the speed of delivery is optimized with Zero errors.
Optimized Workforce to cater large Recruitments as well as Small Ones.
Optimized Workforce to cater large Recruitments as well as Small Ones. Prontoux has the Privilege of Serving a wide Spectrum of client from Startups to Fortune 500 companies. This is possible with the help of our intellectual Recruitment team who are innovative in their everyday life improvising the recruitment techniques and improvising the process.
We believe in setting the communication channel right with our clients as well as Candidates. This helps us to comprehend any improvement areas as well grievances which need to take care with utmost priority.
We believe in the Happy Minds cultivate Innovative thoughts. We Seek to have better compensation for our candidates so that they have the employee satisfaction to excel in their work.
We look at providing the feedback on a regular basis to our candidates so that they can work towards client satisfaction. This also gives us a way to know their pain areas. We consider the grievances in a solution oriented approach.
We love to appreciate the good work done by our candidates. Be it their birthdays or anniversaries or work Anniversaries, we look towards celebrating tose cherished moments with them.
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Salman Khan got to hear this when he asked Deepika Padukone about her pregnancy
Deepika Padukone had an amazing time with Salman Khan in 'Bigg Boss 13' but what caught our attention was when the later teased Deepika on getting pregnant. Read to know Deepika's befitting reply.
By : Tanvi Lad
We were told different reasons for exit: Salman Khan on Priyanka's exit from 'Bharat'
Deepika Padukone is receiving praises from all corners because of her last release 'Chhapaak' and to promote the film Deepika also made an appearance on Salman Khan's show Bigg Boss 13. She had an amazing time and fun banter with Salman but what caught our attention was when Salman teased Deepika on getting pregnant.
He said that Deepika's fans might get to see her with a baby soon and the actress was surely not to keep calm on this, she immediately gave it back to Salman telling him to marry first. Salman further added that marriage and having kids have nothing in common, putting a full stop to this fun banter.
Deepika Padukone on Salman Khan's Bigg Boss 13
'Do I look pregnant?' Deepika asked
This is not the first time when the Padmaavat actress has got upset with the pregnancy rumours surrounding and the constant need for people to know about one's pregnancy. When a reporter had asked the same Deepika had replied, "Do I look pregnant? I will ask you when I will plan (to have a family). If you give me permission then I will plan. If I become pregnant then you will see that in nine months."
Deepika is currently on cloud nine owing to her success from 'Chhapaak'. The actress had not only gained praise for her movie but also for her bold act to support the JNU students in spite of the release of her movie and without bothering the repercussions that her film could have faced. On the work front, Deepika will be seen in Kabir Khan's '83 with husband Ranveer Singh. She also has Shakun Batra's romantic drama with Siddhant Chaturvedi and a film on Draupadi where she has also her role as a producer as well.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 8,562
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{"url":"https:\/\/planetmath.org\/ProofOfArithmeticgeometricMeansInequality","text":"# proof of arithmetic-geometric means inequality\n\nA short geometrical proof can be given for the case $n=2$ of the arithmetic-geometric means inequality.\n\nLet $a$ and $b$ be two non negative numbers. Draw the line $AB$ such that $AP$ has length $a$, and $PB$ has length $b$, as in the following picture, and draw a semicircle with diameter $AB$. Let $O$ be the center of the circle.\n\nNow raise perpendiculars $PQ$ and $OT$ to $AB$. Notice that $OT$ is a radius, and so\n\n $OT=\\frac{AB}{2}=\\frac{a+b}{2}$\n\nAlso notice that $PQ\\leq OT$ for any point $P$, and equality is obtained only when $P=O$, that is, when $a=b$.\n\nNotice also that $PQ$ is a height over the hypotenuse on right triangle $\\triangle AQB$. We have then triangle similarities $\\triangle AQB\\sim\\triangle APQ\\sim\\triangle QPB$, and thus\n\n $\\frac{AP}{PQ}=\\frac{PQ}{PB}$\n\nwhich implies $PQ=\\sqrt{AP\\cdot PB}=\\sqrt{ab}$. Since $PQ\\leq OT$, we conclude\n\n $\\sqrt{ab}\\leq\\frac{a+b}{2}.$\n\nThis special case can also be proved using rearrangement inequality. Let $a,b$ non negative numbers, and assume $a\\leq b$. Let $x_{1}=\\sqrt{a},x_{2}=\\sqrt{b}$, and then $x_{1}\\leq x_{2}$. Now suppose $y_{1}$ and $y_{2}$ are such that one of them is $x_{1}$ and the other is $x_{2}$. Rearrangement inequality states that $x_{1}y_{1}+x_{2}y_{2}$ is maximum when $y_{1}\\leq y_{2}$ and $x_{1}\\leq x_{2}$. So, we have\n\n $x_{1}x_{2}+x_{2}x_{1}\\leq x_{1}^{2}+x_{2}^{2}$\n\nand substituting back $a,b$ gives\n\n $2\\sqrt{ab}\\leq(\\sqrt{a})^{2}+(\\sqrt{b})^{2}=a+b$\n\nwhere it follows the desired result.\n\nOne more proof can be given as follows. Let $x=\\sqrt{a},y=\\sqrt{b}$. Then $(x-y)^{2}\\geq 0$, and equality holds only when $x=y$. Then, $x^{2}-2xy+y^{2}\\geq 0$ becomes\n\n $x^{2}+y^{2}\\geq 2xy$\n\nand substituting back $a,b$ gives the desired result as in the previous proof.\n\nTitle proof of arithmetic-geometric means inequality ProofOfArithmeticgeometricMeansInequality 2013-03-22 14:49:14 2013-03-22 14:49:14 mathcam (2727) mathcam (2727) 10 mathcam (2727) Proof msc 26D15","date":"2019-03-20 21:32:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 59, \"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\": 0.9897681474685669, \"perplexity\": 126.62098277786097}, \"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-2019-13\/segments\/1552912202471.4\/warc\/CC-MAIN-20190320210433-20190320232433-00504.warc.gz\"}"}
| null | null |
Our largest room offers comfortable, elegant accommodation with two queen beds, in conjugation with courtyard inn, perfect for families. Cancellation/No Show Policy: Once a reservation is confirmed, please notify us by fax or email or call of any cancellation or modification of the booking at least 4 days (96 hours) before arrival, 2 weeks before arrival for Group bookings in order to avoid a charge of 1 night's stay to be imposed. All Long-weekends, the cancellations is 7 days prior to check in. In an event of a guest No Show, a fee of one night's rate will be charged.
** Max 2 Adults per room, children under the age of 16 stay for free. Non Smoking ($200 charge for smoking in room). No Pets Allowed ($200 charged for having pets in room).
Rooms at the Ranch House Inn.
Welcome to the Ranch House Inn, Exeter ON.
The Ranch House Inn located in the heart of beautiful and historic Exeter Ontario. you'll find the Ranch House Inn, a gorgeous heritage property alive with elegance and charm. Rooms that combine excellent comfort, convenience and luxury. From outstanding personalized service to a wide array of amenities, we strive to make your experience unforgettable.
Ranch House Inn is minutes away from world-class Grand Bend Beach that annually draw thousands of visitors from all over the world, Grand Bend Motorplex Speedway One of Canada's Premier Racing Track, Huron County Playhouse and number of glorious Golf Courses, while other areas like Entertainment, All of Exeter's parks, Churches, Restaurants and Local Boutiques, all waking distance.
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| 7,465
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To help you reach your financial goals tomorrow by maximizing your financial decisions today.
In the early '90s, Willis Johnson worked at a large brokerage company and found that the big firms were pushing their advisors to sell products that lined their own pockets but weren't necessarily in the clients' best interest. He wanted to reverse that model, believing that if he provided high quality fiduciary service aligning the interest of the clients with that of the firm, both he and his clients would thrive.
In 1996, Willis Johnson & Associates opened its doors as an independent fiduciary advisor way before "fiduciary" became an industry buzzword. We started with less than $20 million in assets under management. Since then, we have grown to nearly $500 million in assets under management by offering something simple: professional, unbiased advice for the corporate executive and professional with their best interest in mind.
We're proud to call Houston home, not only because we love this city, but because of the type of clients we get to work with. In a hard-working city like Houston, most people have earned their way to where they are. They understand the value of a dollar. They are honest, hard-working people, and they expect nothing less from us.
Come sit down with us today and let's talk about your future.
|
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| 8,579
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Periș – gmina w północno-zachodniej części okręgu Ilfov w Rumunii. W skład gminy wchodzą wsie: Bălteni, Buriaș i Periș. W 2011 roku liczyła 7557 mieszkańców.
Przypisy
Periş
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| 7,812
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\section{Introduction}
Neural networks are one of the most widely studied contemporary fields of research. We may untangle the complexities involved in the underlying biology and in the emergent pattern formation due to simplistic yet faithful models for the description of neutral action potentials. The Hodgkin-Huxley model \cite{HodgkinHuxley}, published in 1952, was the first to successfully model action potentials dynamics along the nerve fibre of the squid giant axons. Until today the Hodgkin-Huxley-model is still being used and expanded to accurately describe neural action potentials \cite{borgers, baysal, stiles}. A simplified version of the Hodgkin-Huxley model is the FitzHugh-Nagumo model (FHN), which was developed independently by FitzHugh \cite{FitzHugh} and Nagumo \textit{et. al} \cite{Nagumo} in 1961 and 1962, respectively. The advantage of the FHN model is that it qualitatively retains the non-linear dynamics of the Hodgkin-Huxley model, yet it consists only of two variables, the propagator representing axon membrane potential and the controller describing the chemical species driving propagator dynamics. The reduction to two variables instead of the original four in the Hodgkin-Huxley model, allows for direct analytical insight into the mechanisms of non-linear coupling \cite{hussain, li, mao, Zykov.2018}. Yet, experimental observations challenge the classical description of nerve conduction as the impact of cytoplasmic flows arising from membrane deformation triggered by the pulse are unaccounted for.
The propagation of an action potential along a nerve fiber is accompanied by mechanical deformations of the nerve, including volume expansion and compression \cite{TASAKI_volume, hill_volume}, shortening \cite{TASAKI_tetanic} and a radial change of the nerve fiber \cite{TASAKI_olfactory, TASAKI_olfactory2, TASAKI_swelling, hill_radius}. These mechanical effects are not incorporated in classical Hodgkin-Huxley or FitzHugh-Nagumo models and have, thus, initiated a renewed interest \cite{drukarch_thinking} in model development \cite{Heimburg, RVACHEV_2010}, accounting for example for ensuing fluid flows within the Hodgkin-Huxley model \cite{el_hady} or homogeneous fluid flows within the FHN model \cite{ermakova_propagation_2009, UZUNCA}. Yet, mechanistic insight of how a coupling via fluid flow affects pulse dynamics is missing.
Flow coupling in traveling wave kinetics has, however, been studied in the context of the self-sustained contraction pattern in active porous gels \cite{Radszuweit.2013,Bois.2011,Alonso.2017} and also in tubular geometries \cite{julien}. Mechanistically the flow coupling, here, unfolds as follows \cite{julien}: The gradient of deformation of the tube membrane creates flow. The flow itself creates a flux of chemical species, which in turn affects the gradient of the tube deformation.
Within the FHN model, we identify the controller as the chemical species concentration and the propagator as the membrane potential.
We, here, employ analytical derivations and numerical simulations to investigate the role of an advective coupling arising from mechanical deformations resulting from a propagator and ensuing flows advecting the controller within the FitzHugh-Nagumo-model. We derive a linear dependence of the width of travelling pulses on the advection term and analytically predict the pulse velocity to be independent of the advection term. Both phenomena are corroborated by our numerical simulations, which we further employ to address the effect of model parameters on the impact of the advective coupling. Our results show that advective coupling allows an independent control of pulse width and pulse velocity within the FitzHugh-Nagumo model.
\section{Results}
\subsection{FitzHugh-Nagumo model with propagator-driven advection of the controller}
\begin{figure*}[htbp]
\centering
\includegraphics[width=\textwidth]{Fig1.pdf}
\caption{Advection coupling leaves pulse velocity unchanged but controls pulse width. a) Phase portrait of FitzHugh-Nagumo (FHN) without $\eta = 0$ (black, dots) and with advection coupling between propagator and controller $\eta = 1$ (red, dots) along the nullclines (blue - propagator nullcline, red - controller nullcline) being perturbed out of their single stable point. The dots are equitemporal, illustrating the dynamics of the FHN equations. Controller nullcline giving rise to bistability in yellow for reference.
b) Spatial dynamics of propagator and controller of a leftwards travelling pulse ($\eta=0$) and its subdivision into four phases: front, excited domain, back and refractory domain. Pulse width is defined as the difference between the front, marked by the initial increase from zero of the controller, and the back, marked by the controller's maximum value (dashed lines).
c) Numerical solutions for the pulse velocity measured as the number of travelled grid-points per unit time. Velocity shows only a small linear dependence (gradient of 0.0082) on advection $\eta$, much smaller than typical variations for changing non-linear parameter $a$.
d) Pulse width decreases linearly with $\eta$. Dashed lines in c) and d) are linear fits of the numerical data.
System parameters set to $a=0.02$, $b=0.01$, $\gamma = 0.02$ and $D_{u}=0.5$ if not specified otherwise.}
\label{fig:1}
\end{figure*}
To account for the interaction between propagator $u(x,t)$ (membrane potential) and controller $v(x,t)$ (chemical stimulus concentration) arising due to the deformation of the propagator generating fluid flow of magnitude $\eta \frac{\partial}{\partial x}u(x,t)$ which is advecting and dispersing the controller we incorporate an additional advection term that reflects the dependence of the chemical flux on the deformation of the membrane potential $-\frac{\partial}{\partial x}\left(v(x,t)\eta \frac{\partial }{\partial x}u(x,t)\right)$ into the FitzHugh-Nagumo equations,
\begin{subequations}
\begin{align}
\frac{\partial u}{\partial t} &= D_{u} \frac{\partial^{2}u}{\partial x^{2}} + u(1-u)(u-a)-v
\label{eq:u-eq_A}\\
\frac{\partial v}{\partial t} & = bu - \gamma v - \eta \frac{\partial}{\partial x} \left(v \frac{\partial u}{\partial x} \right)
\label{eq:v-eq_A}
\end{align}
\label{eq:FitzHughNagumo}
\end{subequations}
where $D_{u} \geq 0$ denotes the diffusivity of the propagator, $\eta$ the viscosity of the fluid, $a > 0$ governs the kinetics of the propagator, and $b \geq 0$ and $\gamma \geq 0$ the kinetics of the controller and thus the stability of the entire system.
Given an initial perturbation of the system, either a travelling pulse or front may form.
We are, here, interested in the impact of the advection term on the travelling pulse solution. A pulse may form when the the system is monostable, which is the case when the two nullclines of the system defined by $\frac{\partial}{\partial t}u(x,t)==0$ and $\frac{\partial}{\partial t}v(x,t)==0$ only cross at the point $(0,0)$, as $u=0$ and $v=0$ is the trivial solution of the system of nullclines, see Fig.~\ref{fig:1} a). In the case of three crossings of the nullclines the system is bistable and, thus, may be perturbed out of one of its resting state into a travelling front until it reaches its second stable resting state. Monostability is given when \cite{cross_greenside},
\begin{equation}
\frac{4b}{\gamma(1-a)^{2}} > 1.
\label{eq:pulse_cond}
\end{equation}
while bistability is given when the left-hand side of equation \eqref{eq:pulse_cond} is less than $1$.
A pulse forms when the system is sufficiently perturbed out of its resting state. Once a pulse is formed, its dynamics are independent of the initial conditions and follow a choreography only dependent on the systems parameters. The dynamics of a pulse can be dissected into four steps, see Fig.~\ref{fig:1} a), b): the pulse \textit{front}, where the system is perturbed out of its stable resting state and the propagator increases quickly until it reaches the vicinity of its nullcline again, the pulse \textit{excited domain} which corresponds to the peak of the propagator, when dynamics are following the propagators nullcline, the pulse \textit{back} when the propagator drops again sharply back to the other branch of its nullcline and finally the \textit{refractory domain} where dynamics follow again the nullcline and recover back into the stable resting state. This 4-stepped choreography of a travelling pulse remains unchanged under the addition of the advective coupling, see Fig.~\ref{fig:1} a), yet at closer inspection the \emph{dynamics} of the individual steps does seem to be affected by the advection coupling. To gain mechanistic insight on how the advection coupling alters pulse dynamics we turn to analytical derivations on pulse velocity and subsequently pulse length.
\subsection{Pulse velocity derived to be independent of advection strength}
In order to derive closed expressions for the pulse velocity we simplify the dynamical equations Eqs.~\eqref{eq:FitzHughNagumo} by linearly approximating the third order polynomial with a Heaviside function and a linear term, see Ref.~\cite{McKean}. We further incorporate that the dynamics of the propagator $u$ are much faster than the dynamics of the controller $v$ at the \textit{front} and the \textit{back} of the pulse \cite{cross_greenside}, see Fig.~\ref{fig:1} a) by rescaling the fast kinetics of the propagator with the non-dimensional parameter $\epsilon\ll1$ to match time scales of propagator and controller kinetics, together resulting in
\begin{equation}
\frac{\partial u}{\partial t} = D_{u} \frac{\partial^{2}u}{\partial x^{2}} +\frac{1}{\epsilon} \left[ H(u-a) -u-v\right].
\end{equation}
As we are seeking a travelling pulse solution we only seek solutions where $z=x-ct$ with $c$ being the velocity of the pulse.
In order to remove the explicit $\epsilon$-dependence in the first FHN equation, we introduce the stretching coordinate $\xi = \frac{z}{\epsilon}$, arriving at
\begin{subequations}
\begin{align}
0 = & \tilde{D}_{u} \frac{\partial^{2} u}{\partial \xi^{2}} + c \frac{\partial u}{\partial \xi} + H(u-a) -u - v, \\
0 = & c \frac{\partial v}{\partial \xi} + \epsilon(bu - \gamma v) - \epsilon \tilde{\eta}\left( \frac{\partial u}{\partial \xi} \frac{\partial v}{\partial \xi} + v \frac{\partial^{2} u}{\partial \xi^{2}} \right),
\end{align}
\label{eq:FitzHugh-Nagumo_stretch_A}
\end{subequations}
where we rescaled both $\tilde{D}_{u}=D_u/\epsilon^2$ and $\tilde{\eta} = \eta/\epsilon^2$ as both terms describing spatial dynamics should stay unaffected by the unequally fast kinetic terms.
As discussed, at the \textit{front} and \textit{back} of the pulse the dynamics of $\frac{\partial u}{\partial t}$ are much faster than $\frac{\partial v}{\partial t}$, implying very small $\epsilon$. We can therefore evaluate Eqs.~\eqref{eq:FitzHugh-Nagumo_stretch_A} at the \textit{front} and \textit{back} in the limit of $\epsilon \rightarrow 0$. Eq.~\eqref{eq:FitzHugh-Nagumo_stretch_A} (b) then reduces to a first order differential equation, solved by constant $v$. At the \textit{front} the constant value of the controller equals its stable fixed point value $v=0$, while it takes a finite value of $v=v_{b}$ at the \textit{back} of the pulse, see Fig.~\ref{fig:1} b).
For the \textit{front},
\begin{equation}
\tilde{D}_{u} \frac{\partial^{2} u_{f}}{\partial \xi^{2}} + c \frac{\partial u_{f}}{\partial \xi} + H \left(u_{f} -a \right)- u_{f} = 0.
\label{eq:front_diff}
\end{equation}
We make the Ansatz $u_{f}(\xi) = C e^{\xi \lambda}$ and require $u_{f}(\xi)$ to converge for $\xi \rightarrow \pm \infty$. Using the jump and continuity condition at $\xi = a$
\begin{subequations}
\begin{align}
& A e^{\xi_{a} \frac{-c + \sqrt{c^2+4\tilde{D}_{u}}}{2\tilde{D}_{u}}} = B e^{\xi_{a} \frac{-c - \sqrt{c^2+4\tilde{D}_{u}}}{2\tilde{D}_{u}}} +1, \\
& A e^{\xi_{a} \frac{-c + \sqrt{c^2+4\tilde{D}_{u}}}{2\tilde{D}_{u}}}= B e^{\xi_{a} \frac{-c - \sqrt{c^2+4\tilde{D}_{u}}}{2\tilde{D}_{u}}} \frac{-c - \sqrt{c^2+4\tilde{D}_{u}}}{-c + \sqrt{c^2+4\tilde{D}_{u}}},
\end{align}
\label{eq:jump_cont}
\end{subequations}
we derive the velocity of the pulse at the \textit{front} as
\begin{equation}
c_{f} = \pm \sqrt{\tilde{D}_{u}} \frac{1-2a}{\sqrt{a(1-a)}}.
\label{eq:velocity}
\end{equation}
Analogously the dynamics for the propagator at the \textit{back} of the pulse follow from Eq.~\eqref{eq:FitzHugh-Nagumo_stretch_A} to be determined by,
\begin{equation}
\tilde{D}_{u} \frac{\partial^{2} u_{f}}{\partial \xi^{2}} + c \frac{\partial u_{f}}{\partial \xi} + H(u_{f}-a)- u_{f} - v_{b} = 0
\label{eq:back_diff}.
\end{equation}
Again employing the Ansatz $u_{b} = C e^{\xi \lambda}$ and respecting jump and continuity condition Eq.~\eqref{eq:jump_cont}
we obtain the velocity for the \textit{back} of the pulse
\begin{equation}
c_{b} = \pm \sqrt{\tilde{D}_{u}} \frac{1-2(a+v_{b})}{\sqrt{(a+v_{b})(1-a-v_{b})}}.
\end{equation}
Note, that we seek solutions where the shape of the pulse remains constant as it travels through space. This implies that the \textit{front} and the \textit{back} need to travel at the same velocity. Since the \textit{back} is a reversed \textit{front} \cite{cross_greenside}
, we obtain the condition $c_{f} = - c_{b} = c$. This relation of the front and back velocities determines the controller at the \textit{back} of the pulse $v_{b}$
\begin{equation}
v_{b} = 1-2a,
\label{eq:vb}
\end{equation}
and finally the velocity of a travelling pulse
\begin{equation}
c(\tilde{D}_{u},a) = \sqrt{\tilde{D}_{u}} \frac{1-2a}{\sqrt{a(1-a)}}.
\label{eq:velocity_pulse}
\end{equation}
Strikingly the advection coupling does not affect the pulse velocity to zeroth order. A result that we indeed confirm in numerical integration of the full set of equations, see Fig.~\ref{fig:1} c).
\subsection{Pulse width analytically predicted to shrink with advection strength}
To analytically derive the pulse width, we aim to solve for the trajectory of the controller $v$. For low order in $\epsilon$ the pulse follows the propagator nullcline during the \textit{excited domain} of the pulse, thereby tracing out the change in controller from $v=0$ at the \textit{front} of the pulse to $v=v_b$ at the \textit{back} of the pulse. Thus, the propagator along the nullcline is given by $u = 1-v$ for $0< v < v_{b}$. The dynamics of the controller along the \textit{excited domain} of the pulse then follow as a function of the spatial coordinate $z$ from Eq.~\eqref{eq:FitzHugh-Nagumo_stretch_A} to be determined by
\begin{equation}
c \frac{\partial v}{\partial z} - (b+\gamma)v + b + \epsilon^{2}\tilde{\eta}\left(v\frac{\partial^{2} v}{\partial z^{2}} + \left(\frac{\partial v}{\partial z} \right)^{2} \right) = 0.
\label{eq:diff_A}
\end{equation}
We can solve the dynamics for the controller at zeroth order in $\epsilon$, i.e.~without the advection term, simplifying the differential equation to
\begin{equation}
c \frac{\partial v}{\partial z} - (b+\gamma)v + b = 0,
\end{equation}
which is solved by
\begin{equation}
v(z) = \frac{b}{b + \gamma} + Q e^{\frac{b+ \gamma}{c}z},
\end{equation}
where $Q = -\frac{b}{b+\gamma} e^{-\frac{b+\gamma}{c}z_{1}}$ is an integration constant, defined by $v(z_{1}) = 0$. We define the pulse width as the distance travelled during the \textit{excited domain} of the pulse. In the dynamics of the controller this translates to the distance travelled between $v(z_{1}) = 0$ and $v(z_{2}) = v_{b}$ for a rightward travelling pulse. Therefore, the pulse width is given by $\lambda = z_{1}- z_{2}$, following from
\begin{equation}
v(z_2)=v_{b} = \frac{b -b e^{\frac{b+ \gamma}{c}(z_{2}-z_{1})}}{b+\gamma} = \frac{b -b e^{-\frac{b+ \gamma}{c}\lambda}}{b+\gamma}.
\label{eq_pulseeqn}
\end{equation}
To explicitly solve for the pulse width $\lambda$ we consider the order of magnitude of model parameters. In our simulations, we take $b,\gamma \sim \mathcal{O}(0.01)$, obtaining $c \sim \mathcal{O}(0.1)$ and $\lambda \sim \mathcal{O}(10)$, motivating a
Taylor-expansion in $\frac{b+\gamma}{c}\lambda\ll1$ of Eq.~\eqref{eq_pulseeqn} to first order. Simplifying, we obtain
\begin{equation}
\lambda = \frac{cv_{b}}{b}.
\end{equation}
As we are interested in the effect of the advection term on the pulse width, we now consider Eq.~\eqref{eq:diff_A} to full order in $\epsilon$. Simulations show that $\frac{\partial^{2}}{\partial z^{2}} v(z)$ and $\left(\frac{\partial}{\partial z} v(z)\right)^{2}$ do not change considerably with varying $z$ and we therefore define
\begin{subequations}
\begin{align}
C_{1}^{2} \equiv \left(\frac{\partial v(z)}{\partial z} \right)^{2} & \approx \text{constant},\\
C_{2} \equiv \frac{\partial^{2} v(z)}{\partial z^{2}} & \approx \text{constant}.
\end{align}
\end{subequations}
Eq.~\eqref{eq:diff_A} then becomes
\begin{equation}
c \frac{\partial v(z)}{\partial z} - (b+\gamma)v(z) + b + \epsilon^{2} \tilde{\eta} \left(v(z) C_{2} + C_{1}^{2} \right) = 0.
\end{equation}
The general solution to this first order differential equation is
\begin{equation}
v = \frac{\epsilon^{2} \tilde{\eta} C_{1}^{2} + b}{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}} + P e^{\frac{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}}{c}z},
\label{eq:v_sol}
\end{equation}
with $P$ an integration constant, that is defined by $v(z_{1})=0$, obtaining
\begin{equation}
P = - \frac{\epsilon^{2} \tilde{\eta} C_{1}^{2} +b}{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}} e^{- \frac{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}}{c}z_{1}}.
\end{equation}
Using $v(z_{2}) = v_{b}$ we obtain
\begin{equation}
v_{b} = \frac{\epsilon^{2} \tilde{\eta} C_{1}^{2} + b}{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}} \left(1 - e^{ -\frac{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2} }{c}\lambda} \right).
\label{eq:vb}
\end{equation}
We can rewrite Eq.~\eqref{eq:vb} as
\begin{equation}
e^{\frac{\lambda}{c} \left(b+\gamma - \epsilon^{2}\tilde{\eta} C_{2} \right)} = \frac{1}{1-\alpha},
\label{eq:alpha}
\end{equation}
with $\alpha = v_{b}\frac{b+ \gamma-\epsilon^{2}\tilde{\eta} C_{2}}{b+\epsilon^{2}\tilde{\eta} C_{1}^{2}}$.
Because the term $\epsilon^{2} \tilde{\eta} C_{2}$ is very small and generally $v_{b} < 1$, see Fig.~\ref{fig:1} b), we obtain\\
\begin{equation}
\alpha \approx v_{b} \frac{b+\gamma}{b} < 1.
\end{equation}
Taking the logarithm of Eq.~\eqref{eq:alpha} and assuming $\alpha$ to be sufficiently small
\begin{equation}
\frac{b+\gamma - \epsilon^{2} \tilde{\eta} C_{2}}{c}\lambda = \text{ln}(1) - \text{ln}(1-\alpha) \approx \alpha.
\end{equation}
Solving for $\lambda$ we obtain
\begin{equation}
\lambda = \frac{c v_{b}}{b + \epsilon^{2}\tilde{\eta} C_{1}^{2}} \approx \frac{cv_{b}}{b} \left( 1 - \frac{\epsilon^{2}\tilde{\eta} C_{1}^{2}}{b} \right).
\end{equation}
Assuming that $ C_{1} \approx \frac{\partial}{ \partial z} v\left(z_{1}\right)$, Eq.~\eqref{eq:v_sol} yields
\begin{equation}
C_{1} \approx \frac{\partial}{\partial z} v\left(z_{1}\right)= \frac{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}}{c} \frac{\epsilon^{2} \tilde{\eta} C_{1}^{2} +b}{b+\gamma - \epsilon^{2}\tilde{\eta} C_{2}}.
\end{equation}
Simplifying we obtain
\begin{equation}
C_{1} \approx \frac{\epsilon^{2} \tilde{\eta} C_{1}^{2} +b}{c},
\end{equation}
which is solved by
\begin{equation}
C_{1} = \frac{-c \pm \sqrt{c^{2} - 4b\epsilon^{2}\tilde{\eta}}}{2\epsilon^{2}\tilde{\eta}}.
\end{equation}
Because $\epsilon^{2} \ll 1$, we Taylor-expand $C_{1}$ to first order,
\begin{equation}
C_{1} \approx -\frac{b}{c},
\end{equation}
and, thus, obtain a closed expression for the width of a pulse
\begin{equation}
\lambda = \frac{c(1-2a)}{b}\left( 1 - \epsilon^{2} \frac{b\tilde{\eta}}{c^{2}} \right).
\label{eq:width}
\end{equation}
Thus, we find that in contrast to the pulse velocity, the pulse width is affected by advective coupling, see Fig. \ref{fig:1} d). The higher the fluids viscosity $\tilde{\eta}$ the smaller the pulse width. To test the validity of the analytical expressions we next turn to numerical integration of the full set of Eqs.~\eqref{eq:FitzHughNagumo}.
\subsection{Implicit integration of advection coupled dynamics system required for stability}
To integrate excitable media dynamics with an advection term we employ a $\theta$-weighted Crank Nicolson scheme \cite{Hoeve_2010, Shi_1994, eggers_dupont_1994}. The algorithms' basic structure follows that of a Newton method, but differs from it by dynamically adjusting the time steps and evaluating the dynamic equations not at a time step $i$ but at time steps $i+\theta$. To illustrate the basics of the algorithm, we consider the general set of differential equations
\begin{equation}
\frac{\partial}{\partial t} \vec{y} = \vec{f} \left(\vec{y} \right),
\end{equation}
which correspond to Eqs.~\eqref{eq:FitzHughNagumo} in our implementation. We denote $y^{n}_{i}$ to be the variable $y$ at time $n$ and on grid point $i$, with $n \in \{0,t_{f} \}$, for some final time $t_{f}$ and $i \in \{ 0, N\}$, for some number of equally spaced grid points $N$ in the one-dimensional system.
We now consider the residual that we obtain, when approximating the function $\vec{f}(\vec{y})$ to linear order:
\begin{equation}
\vec{r} \left(\vec{y}^{n+1} \right) = \frac{\vec{y}^{n+1} - \vec{y}^{n}}{\Delta t} - \vec{f} \left(\vec{y}^{n+\theta} \right),
\label{eq:residual}
\end{equation}
with $0\leq \theta \leq 1$.
For $\theta=0$ we obtain a fully explicit and for $\theta =1$ a fully implicit method. For our simulations we will take $\theta = 0.55$, as it has been found to improve stability at the cost of only slightly less accuracy \cite{eggers_dupont_1994}.
We are looking for values of $\vec{y}^{n+1}$ such that the residuals become $\vec{r} =0$. For this we use Newton's method. A good initial guess is assuming $\vec{y}^{n+1} - \vec{y}^{n}$
to be equal to $\vec{y}^{n} - \vec{y}^{n-1}$. In order to dynamically adjust the efficiency of the simulation, we allow time-steps to vary in magnitude, obtaining:
\begin{equation}
\vec{y}^{n+1}_{est} \approx \vec{y}^{n} + \left(\vec{y}^{n} - \vec{y}^{n-1} \right) \frac{\Delta t_{n\rightarrow n+1}}{\Delta t_{n-1 \rightarrow n}}.
\label{eq:du}
\end{equation}
We correct our initial estimate by subtracting the inverse of the product of the Jacobian, a matrix containing all first order derivatives of every grid point, and the residuals of our estimate,
\begin{equation}
\vec{y}^{n+1} = \vec{y}^{n+1}_{est} - \textbf{J}^{-1} \vec{r} \left(\vec{y}^{n+1} \right).
\label{eq:Jacobian}
\end{equation}
The inversion of the Jacobian is the costliest part of the algorithm, as we use a grid with order $\mathcal {O}(1000)$ grid points. Because of this, we only make one correction with the Jacobian per iteration. To ensure, that this correction is sufficient, we ensure, that our initial guess is not far off from the actual value, by keeping the time steps $\Delta t_{n\rightarrow n+1}$
small. This adjusting can be done automatically, by letting the algorithm calculate the time step thrice. Once for a step size $\Delta t$ and twice successively for step size $\frac{\Delta t}{2}$. The two half steps will result in a more accurate approximation. If the relative error between the two steps is smaller than a threshold $\chi$ for say 10 steps, we increase the step size by a factor $2^{\frac{1}{4}}$, reducing the calculation cost without sacrificing much accuracy. If the error is larger than threshold $\chi$ we decrease the step size by a factor $\frac{1}{2}$, ensuring a good accuracy of the simulation.
\subsection{Limitations of pulse generation in parameter space due to advective coupling}
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Fig3.pdf}
\caption{Pulse formation and pulse width as function of system parameters. (a), (b) Pulses (green) form above analytical predicted condition for monostability (dashed lines, see Eq.~\eqref{eq:pulse_cond}), separating them from travelling fronts. The upper limit for pulse formation shrinks as advection coupling compare zero $\eta=0$ in (a) and finite coupling $\eta=3$ in (b), respectively. Blue indicates parameters where no travelling pulse forms. (c) and (d) Pulse width as a function of $\eta$ and $a$. The blue region indicates no pulse generation. (c) Shows the space for $b=0.01$ and $\gamma=0.02$ and (d) for $b=0.001$ and $\gamma=0.002$. (e) Pulse width as function of parameters $\eta$ and $b$, with $a=0.02$, $\gamma = 0.02$. (f) Pulse width as function of parameters $\tilde{\eta}$ and $\gamma$, with $a=0.02$, $b = 0.01$.
In both (e) and (f) the yellow points at large $\eta$ correspond to numerical artifacts. The vertical cut-off at small $b$ and large $\gamma$, respectively, indicates the transition to front (dashed line, see Eq.~\eqref{eq:pulse_cond}). The front forming phase space is indicated by wavelengths of $\lambda = -20$.}
\label{fig:Parameterspace}
\end{figure*}
Before employing our numerical scheme to assess the impact of advective coupling on pulse dynamics we first sweep the parameter space numerically to identify when pulses form. There is a clear cut-off of pulse formation at the transition from monostability to bistability and thus front formation given by Eq.~\eqref{eq:pulse_cond}. Further, from the calculation of pulse velocity Eq.~\eqref{eq:velocity_pulse}, we find that the pulse velocity vanishes as $a$ approaches $\frac{1}{2}$, additionally establishing an upper theoretical limit $a < \frac{1}{2}$. \\
For all simulations we have taken the diffusion constant to be $D_{u} = 0.5$. Sweeping parameter space $b-\gamma$ at fixed $a=0.02$ we first of all recover the analytic prediction of the transition between front and pulse, see Fig.~\ref{fig:Parameterspace} a). Increasing fluid viscosity $\eta$ and therefore advective coupling keeps the transition to front formation unaffected yet reduces the parameter space for travelling pulses, see Fig.~\ref{fig:Parameterspace} b). Inspection of pulse trajectories in phase space, see Fig.~\ref{fig:1} a), suggest that advective coupling decreases pulse formation as the advective term positively reinforces the controller, which in turn reduces the propagator. The reduced propagator switches earlier from the \emph{excited domain} to the \emph{back}, resulting in a narrower pulse. \\
To explore the limit on parameter $a$ we sweep the $\eta$-$a$ parameter space for $b = 0.01$, $\gamma = 0.02$, see Fig.~\ref{fig:Parameterspace} c). We observe pulse generation only for small values of $\eta$ and $a$ and note a cut-off at about $a=0.12$, a value much smaller than the theoretical limit. Yet, decreasing the magnitude of $b$ and $\gamma$ by one order, significantly increase the parameter space for pulse generation, see Fig.~\ref{fig:Parameterspace} d). Even smaller values of $b$ and $\gamma$ achieve pulse generation for $a$ close to $0.5$. The pulse generation in this regime is limited by numerical instabilities. As a one order magnitude decrease in $b$ and $\gamma$ increases the pulse width by roughly one order of magnitude, the relative changes of the propagator and controller between grid points becomes smaller, ensuring stability over a larger parameter space. \\
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Fig4.pdf}
\caption{Comparison of analytical and numerical pulse velocity and pulse width reveals success of analytical prediction of pulse width governed by advection strength. (a) The numerical and analytical, see Eq.~\eqref{eq:velocity_pulse}, pulse velocity as function of parameter $a$. Both coincide well for large values of $a$, as both vanishing for $a\rightarrow 0.5$, but the analytical result predicts a divergence for small $a$. (b) The pulse velocity shows a small but neglectable dependence on the advection strength $\eta$ as predicted analytically, which is maintained for small and large values of $a$. (c) Pulse width dependence on $\eta$ with rescaling factors from Eq.~\eqref{eq:width} versus numerical results. As predicted, there is a linear dependence between $\lambda$ and $\eta$, however the slope is not unitary, but rather strongly dependent on other model parameters like $b$.}
\label{fig:vel_width}
\end{figure*}
Even though a bit more convoluted, the above arguments also explain pulse formation along the sweeps through the parameter space spanned by $\eta$-$b$ and $\eta$-$\gamma$, see Figs.~\ref{fig:Parameterspace} e), f) respectively. Here the clear cut-off at the transition from pulse to front formation is again exemplified. We here numerically explored the pulse formation broadly within the parameter space and next turn to explicitly test our analytical prediction on pulse velocity and pulse width as a function of advection strength.
\subsection{Pulse width governed by advective coupling while pulse velocity unaffected}
According to our analytical results advection strength has disparate impacts on pulse velocity and pulse width. Pulse velocity is predicted to be independent of advection strength, see Eq.~\eqref{eq:velocity_pulse}, while pulse width is derived to linearly decrease with advection strength, see Eq.~\eqref{eq:width}. Indeed our numerical results confirm that pulse velocity is well-described as being independent of advection strength see Fig. \ref{fig:vel_width} b). We however find that the analytical prediction overestimates the precise value of the pulse velocity. Mapping out in particular the analytically obtained pulse velocity as a function of model parameter $a$, see Fig.~\ref{fig:vel_width} a), we find, that the analytical and numerically obtained velocities agree in the limit of $a\rightarrow0.5$, where both decay to zero, yet a divergence for vanishing $a$ is predicted analytically. Therefore the analytical pulse velocity agrees best with numerical simulations for large $a$. This is to be consistent as we approximated the third order polynomial $f(u,v)$ with a Heaviside function $H(u -a)$. The approximation works best for $a$ close to $0.5$, explaining disagreements in the pulse velocity between analytical and numerical results for small $a$. \\
Assessing the pulse width functional dependence on advection strength we numerically confirm that the pulse width scales linearly with $\eta$, see Fig.~\ref{fig:vel_width} c). This holds for a varying system parameters. We note that the numerical simulations show the system parameters to have a strong effect on the gradient and the $y$-intercept in Fig.~\ref{fig:vel_width} c), which is not captured by the prefactors of the analytical result in Eq.~\eqref{eq:width}. We have found no discernible trend for these effects, yet for all observed parameter ranges, the linear dependence on $\eta$ remained. We note a trend of decreasing pulse width for increasing $b$ and decreasing $\gamma$, in accordance with their positive or negative impact on controller dynamics, see Eq.~\eqref{eq:FitzHughNagumo}. The deviations between analytical and numerical results regarding the model parameters $b$ and $\gamma$ are therefore likely to stem from $v=0$ at the pulse \textit{front} not being fully fulfilled numerically. That said, the functional prediction on the impact of advection strength is unaffected by these quantitative differences.
\subsection{Pulse generation for negative coupling of the advection term}
\begin{figure}[htbp]
\centering
\includegraphics[]{Fig2.pdf}
\caption{Pulse width for negative advective coupling in $\eta$-$a$ parameter space. The dark blue area indicates parameter-sets resulting in no pulse generation representing either "no travelling wave" or numerically instability. The yellow points indicate numerical artefacts, arising due to the formation of wave trains and other unexpected wave formations at the edge of the pulse generating area.}
\label{Fig:negative_A}
\end{figure}
To give a holistic insight into our model, we also want to discuss the effect of negative values of $\eta$. This parameter range does not hold for the bio-mechanical motivation of the advection term, however as the focus of this paper is to present a model, that allows for dynamical changes to a usually constant pulse width, it is worth considering its effect on the whole theoretical parameter range. \\
Following a naive consideration of the linear dependence of pulse width on advection strength, we would expect to find a linear increase of the pulse width for larger negative values of $\eta$. Numerical simulations, shown in Fig.~\ref{Fig:negative_A}, indicate that this is indeed the case, however with a change of pulse width much smaller than for positive $\eta$. The difference in gradients can be explained by the effect that the advection term has on the controller. For positive values, it reinforces the controller by steepening its peak, which in turn increases the magnitude of the advection term, due to its dependence on $\frac{\partial v}{\partial x}$. Negative coupling values result in a reduced controller, leading to a split into two peaks for large enough values of $\eta$. The decreased gradient of the controller results in a weaker advection term, explaining the smaller gradient for negative coupling. \\
In Fig.~\ref{Fig:negative_A} we further see that the parameter-space for pulse solutions is confined to small values of $\eta$ and $a$. For larger $a$ we observe a transition into the state of "no travelling wave", while for larger negative $\eta$ we observe numerical instabilities. While methods such as decreasing the initial amplitude and having a system with an even number of grid-points help numerical stability, we still observe a numerical limitation of the parameter space. The theoretical upper limit for the a pulse width is the size of the system, resulting in an upper limit of $\eta$ that scales with the system size, however our numerical simulations are unable to remain stable for large pulse widths rendering this limit beyond the scope of the present work. \\
\section{Conclusion}
In this paper, we have shown that accounting for advection coupling in the FitzHugh-Nagumo equations leads to novel qualitative properties of its travelling pulse solutions. While the velocity of a pulse is independent of the advection term, the pulse width is now tunable, changing linearly with the coupling parameter of the advection term.
Our simulations have shown that one can reliably generate pulses for a large area of the parameter space, however with a different order of magnitude of the gradient than predicted analytically. Lastly, we numerically demonstrated the linear dependence of the pulse width for negative coupling.\\
Our model allows for a wider application of the standard FitzHugh-Nagumo model now incorporating flow-based advection of the controller species thereby accounting for mechanical changes of nerve fibers driving fluid flows under action potentials. The additional degree of freedom to adjust the pulse width by modulating the advection strength in these systems may help to form more comprehensive models.
\section*{Acknowledgements}
This work was supported by the Max Planck Society, the Deutsche Forschungsgemeinschaft via FOR-2581 (P1) and the Human Frontiers Science Program via RGP0001/2021-203.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,808
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Alcantarea heloisae es una especie de planta perteneciente a la familia de las alcantareas. Es una especie endémica de Brasil.
Taxonomía
Alcantarea heloisae fue descrita por J.R.Grant y publicado en Vidalia 1: 31. 2003.
Etimología;
Alcantarea: nombre genérico otorgado en homenaje a Pedro de Alcántara (1840-1889), segundo emperador de Brasil.
heloisae: epíteto
Referencias
Enlaces externos
BROMELIACEAE DA MATA ATLÂNTICA BRASILEIRA referido el 22-10-2009.
heloisae
Flora de Brasil
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,089
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<?php
namespace app\models;
use yii\db\ActiveRecord;
use yii\web\IdentityInterface;
use app\models\SignUpForm;
class User extends ActiveRecord implements IdentityInterface
{
/**
* @package app\models
*
* @property integer $id
* @property string $first_name
* @property string $last_name
* @property string $email
* @property string $authKey
* @property string $accessToken
* @property string $password
* @property string $username
. */
// public $id;
// public $username;
// public $password;
// public $email;
// public $authKey;
// public $accessToken;
// public $first_name;
// public $last_name;
public function init()
{
parent::init(); // TODO: Change the autogenerated stub
//set default values
$this->first_name = 'Wilmer';
$this->authKey = '';
$this->email = '';
$this->last_name = '';
$this->password = '';
$this->accessToken = '';
$this->username = '';
}
public static function tableName()
{
return 'user'; // TODO: Change the autogenerated stub
}
public function rules()
{
$rules = [
// [['first_name', 'last_name', 'email'], 'required'],
[['first_name', 'last_name'], 'string'],
[['email'], 'email'],
];
return $rules;
}
public function create($data)
{
if($this->load($data)){
// $this->first_name = $data['first_name'];
$this->save();
}
}
/**
* @inheritdoc
*/
public static function findIdentity($id)
{
return isset(self::$users[$id]) ? new static(self::$users[$id]) : null;
}
/**
* @inheritdoc
*/
public static function findIdentityByAccessToken($token, $type = null)
{
foreach (self::$users as $user) {
if ($user['accessToken'] === $token) {
return new static($user);
}
}
return null;
}
/**
* Finds user by username
*
* @param string $username
* @return static|null
*/
public static function findByUsername($username)
{
foreach (self::$users as $user) {
if (strcasecmp($user['username'], $username) === 0) {
return new static($user);
}
}
return null;
}
public static function findByEmail($email)
{
foreach (self::$emails as $email){
if (strcasecmp($email['email'], $email) === 0) {
return new static($email);
}
}
}
/**
* @inheritdoc
*/
public function getId()
{
return $this->id;
}
/**
* @inheritdoc
*/
public function getAuthKey()
{
return $this->authKey;
}
/**
* @inheritdoc
*/
public function validateAuthKey($authKey)
{
return $this->authKey === $authKey;
}
/**
* Validates password
*
* @param string $password password to validate
* @return boolean if password provided is valid for current user
*/
public function validatePassword($password)
{
return $this->password === $password;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,056
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{"url":"http:\/\/www.aimsciences.org\/article\/doi\/10.3934\/krm.2009.2.489","text":"# American Institute of Mathematical Sciences\n\nSeptember\u00a0 2009,\u00a02(3):\u00a0489-502. doi:\u00a010.3934\/krm.2009.2.489\n\n## A smooth model for fiber lay-down processes and its diffusion approximations\n\n 1 Fachbereich Mathematik, RWTH Aachen University, Templergraben 55, D-52074 Aachen, Germany 2 Fachbereich Mathematik, Technische Universit\u00e4t Kaiserslautern, PO Box 3049, D-67653 Kaiserslautern 3 Institut Math\u00e9matique de Toulouse IMT, Universit\u00e9 Paul Sabatier deToulouse, 118, route de Narbonne. F-31062 TOULOUSE Cedex, France 4 Fraunhofer ITWM, Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany\n\nReceived\u00a0 March 2009 Revised\u00a0 May 2009 Published\u00a0 July 2009\n\nIn this paper we improve and investigate a stochastic model and its associated Fokker-Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials which has been developed in [2]. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. In the present paper we remove an obvious drawback of the model, namely the non-differentiability of the paths of the process. We develop a model with smoother trajectories and investigate the relations between the different models looking at different scalings and diffusion approximations. Moreover, we compare the numerical results to simulations of the full physical process.\nCitation: Michael Herty, Axel Klar, S\u00e9bastien Motsch, Ferdinand Olawsky. A smooth model for fiber lay-down processes and its diffusion approximations. Kinetic & Related Models, 2009, 2 (3) : 489-502. doi: 10.3934\/krm.2009.2.489\n [1] Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934\/dcds.2018215 [2] Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934\/krm.2018028 [3] John W. Barrett, Endre S\u00fcli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934\/dcdss.2010.3.371 [4] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934\/krm.2018017 [5] Michael Herty, Christian J\u00f6rres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934\/krm.2012.5.485 [6] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934\/krm.2018016 [7] Jos\u00e9 Antonio Alc\u00e1ntara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934\/krm.2011.4.401 [8] Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934\/krm.2010.3.165 [9] Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934\/krm.2019028 [10] Luis Almeida, Federica Bubba, Beno\u00eet Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks & Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934\/nhm.2019002 [11] Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 741-754. doi: 10.3934\/dcdss.2020041 [12] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934\/krm.2018056 [13] Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934\/jcd.2016008 [14] Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934\/cpaa.2009.8.845 [15] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934\/nhm.2017028 [16] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934\/dcds.2019250 [17] Maxime Herda, Luis Miguel Rodrigues. Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations. Kinetic & Related Models, 2019, 12 (3) : 593-636. doi: 10.3934\/krm.2019024 [18] Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934\/era.2008.15.65 [19] Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general L\u00e9vy driven SDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5735-5763. doi: 10.3934\/dcds.2018250 [20] Joseph G. Conlon, Andr\u00e9 Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-D\u00f6ring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934\/dcds.2019079\n\n2018\u00a0Impact Factor:\u00a01.38","date":"2019-07-23 17:24:51","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\": 0.4196595251560211, \"perplexity\": 4959.352881826776}, \"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-2019-30\/segments\/1563195529481.73\/warc\/CC-MAIN-20190723172209-20190723194209-00538.warc.gz\"}"}
| null | null |
Remote monitoring and control (M&C) systems are designed to control large or complex facilities such as factories, power plants, network operations centers, airports, and spacecraft, with some degree of automation.
M&C systems may receive data from sensors, telemetry streams, user inputs, and pre-programmed procedures. The software may send telecommands to actuators, computer systems, or other devices.
M&C systems may perform closed-loop control.
Once limited to SCADA in industrial settings, remote monitoring and control is now applied in numerous fields, including:
Smart grids
Positive train control
Structural health monitoring
Pipeline sensors
Patient monitoring
Desktop/server monitoring
While this field overlaps with machine to machine communications, the two are not completely identical.
See also
Control engineering
Control room
Control theory
Instrument control
Remote sensing
Remote terminal unit
M&C!
Automation software
Control engineering
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,664
|
package com.github.mpetruska.ukmodulo.table
import com.github.mpetruska.ukmodulo.digits.Weights
sealed trait CheckMethod
case object DblAl extends CheckMethod
case object Mod10 extends CheckMethod
case object Mod11 extends CheckMethod
case class ModulusWeightRow(rangeStart: String, rangeEnd: String, checkMethod: CheckMethod, weights: Weights, exceptionCode: Option[Int])
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,213
|
Pulse Rated was a radio station based in Salhouse in Norfolk, England. The station was previously known as Pulse Unsigned, and changed to Pulse Rated in Autumn 2005. The station was owned by Pulse Global.
Pulse Rated played a wide range of music from new, unsigned, and emerging artists.
Station overview
Pulse Rated was formed in 2003 with the mission of promoting music by unsigned artists within the recording industry. With the realisation that the majority of visitors to the website were fans rather than industry figures, the website was re-launched to cater for that audience and satellite radio broadcasts started on Sky Digital in the United Kingdom in late 2004.
Pulse Rated ceased broadcasting on Sky Digital on 6 February 2007.
The station was nominated in the "Best Radio Station" category at the 2006 BT Digital Music Awards.
The station was also an official media partner for Oxfam's first ever Oxjam event in 2006.
Artists whose music was played on the station include Leona Lewis, who won the 2006 series of The X Factor.
Pulse Rated went into liquidation in Autumn 2007.
External links
Pulse Rated UK
Online music stores of the United Kingdom
Radio stations in Norfolk
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,619
|
{"url":"https:\/\/puzzling.stackexchange.com\/questions\/44345\/at-first-i-was-afraid-id-be-petrified\/44349","text":"# At first I was afraid I'd be petrified\n\nAt first I was afraid I'd be petrified\nKept thinking I would never live if you were in my sights\nBut then I asked the gods for help, stopped feeling sorry for myself\nGot some shoes to fly,\nSo I'll hold my shield up high\nBut you can't see me,\nAnd I won't see you\nI'm not some helpless little victim to be tricked by you\n\nWeren't you the one who tried to make me calcify\nDo you think I'd crumble\nDid you think I'd lay down and die?\n\nOh no, not I, I will survive\nOh, as long as I have the gods above, I know I'll stay alive\nI've got all my life to live\nAnd I've got all my love to give and I'll survive\nI will survive, hey, hey\n\nWho is the singer? And no, I don't mean Gloria Gaynor.\nPartial credit (i.e. an upvote) if you get who the subject of the song is, but the answer I'm looking for is the singer.\n\nHonestly the only reason I wrote this was because there was a great 'petrified' pun to be made here.\n\n\u2022 Very cool! I actually laughed out loud. Like the \"fly\" disco shoes. Oct 14 '16 at 20:39\n\u2022 Aside from my answer th petrified bit makes me think petrol Oct 14 '16 at 20:50\n\u2022 For the title I dropped in; was not disappointed with what was within. Oct 15 '16 at 3:21\n\u2022 Great, now I can't get the song out of my head. Where's my karaoke? Oct 15 '16 at 4:08\n\nThis is\n\nMedusa\n\nSung by\n\nPerseus\n\nAt first I was afraid I'd be petrified\nKept thinking I would never live if you were in my sights\n\n\nHer gaze petrifies\n\nBut then I asked the gods for help, stopped feeling sorry for myself\nGot some shoes to fly,\n\n\nHe recieved winged sandals from Hermes\n\nSo I'll hold my shield up high\nBut you can't see me,\nAnd I won't see you\nI'm not some helpless little victim to be tricked by you\n\n\nUsed his shield, the future Aegis, to view Medusa only through reflections, and remain unseen via an invisibility cap\n\nWeren't you the one who tried to make me calcify\nDo you think I'd crumble\nDid you think I'd lay down and die?\n\n\nHe was not turned to stone, but instead won\n\nOh no, not I, I will survive\nOh, as long as I have the gods above, I know I'll stay alive\nI've got all my life to live\nAnd I've got all my love to give and I'll survive\nI will survive, hey, hey\n\n\nHe was victorious, through the blessings of Olympians\n\nAt first I was afraid I'd be petrified\n\nKept thinking I would never live if you were in my sights\n\nBut then I asked the gods for help, stopped feeling sorry for myself\n\nIt is the song that Lot sings to God. Lot fear the wrath of God upon Sodom and Gomorrah\n\nGot some shoes to fly,\n\nSo I'll hold my shield up high\n\nLot receives the angels to assure him that he will be saved\n\nBut you can't see me,\n\nAnd I won't see you\n\nI'm not some helpless little victim to be tricked by you\n\nLot denies God by his desire to destroy Sodom\n\nWeren't you the one who tried to make me calcify\n\nDo you think I'd crumble\n\nDid you think I'd lay down and die?\n\nThe punishment of turning people into statues wing salt was already known.\n\nOh no, not I, I will survive\n\nOh, as long as I have the gods above, I know I'll stay alive\n\nI've got all my life to live\n\nAnd I've got all my love to give and I'll survive\n\nI will survive, hey, hey\n\nLot is sure to be saved. His wife will be punished for turning to look at the destroyed city. It is calcified.\n\n\u2022 interesting answer, but I was specifically talking about perceus, not lot, and the gods references were to greek mythology not the bible Oct 15 '16 at 2:48","date":"2021-11-28 18:21:37","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\": 0.5203147530555725, \"perplexity\": 4952.335058691624}, \"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-2021-49\/segments\/1637964358570.48\/warc\/CC-MAIN-20211128164634-20211128194634-00121.warc.gz\"}"}
| null | null |
https://www.starsinsider.com/music/352804/queen-bey-and-beyond-biggest-surprise-album-releases
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Death Grips - Two years later, after they were supposed to have broken up in 2014, they released the 14-track instrumental album 'Fashion Week.'
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The Beyhive is again buzzing with online speculation that Queen Bey might soon release a completely new album titled 'B7.' Rumors swirled after it was reportedly listed under Beyoncé's Genius page, though it's now been removed.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 119
|
Q: Windows utility/tool command line to move clipboard to file I have seen a similar tool CLIP, but it works into clipboard not the other direction.
I have seen references to "paste" but this doesn't appear to be part of Windows.
I need to find a solution that works with standard Windows 7, 8 and 10 environments.
Clip is distributed. Does anybody a tool that works from clipboard to file?
Seems it might be possible to do with a PowerShell command.
This C# program I just wrote does what I would to do with a PowerShell:
class Clip2File
{
[STAThread()]
static void Main(string[] args)
{
var fileName = @"c:\demo.png";
if (args.Length != 0){
fileName = args[0];
}
if (!Clipboard.ContainsImage()) return;
Image img = Clipboard.GetImage();
img?.Save(fileName, ImageFormat.Png);
}
}
A: Windows cmd (if clipboard contains text data):
Powershell -command Add-Type -AssemblyName System.Windows.Forms;[System.Windows.Forms.Clipboard]::GetText()
Example shows that above command adds two bytes to output (Cr and Lf):
==> >1049363a.txt echo first line
==> >>1049363a.txt echo 2nd line
==> clip<1049363a.txt
==> >1049363b.txt Powershell -command Add-Type -AssemblyName System.Windows.Forms;[System.W
indows.Forms.Clipboard]::GetText()
==> findstr /N "^" 1049363*.txt
1049363a.txt:1:first line
1049363a.txt:2:2nd line
1049363b.txt:1:first line
1049363b.txt:2:2nd line
1049363b.txt:3:
==> dir 1049363*.txt |find ".txt"
06.03.2016 17:24 22 1049363a.txt
06.03.2016 17:24 24 1049363b.txt
==>
Read more about .NET Framework Clipboard Class methods.
Edit answers extended topic (save image, cf. ImageFormat Class):
Add-Type -AssemblyName System.Windows.Forms ### not necessary in PowerShell_ISE
if ($([System.Windows.Forms.Clipboard]::ContainsImage())) {
$image = [System.Windows.Forms.Clipboard]::GetImage()
$filename='d:\test\test.png' ### edit to fit in your circumstances
[System.Drawing.Bitmap]$image.Save($filename,
[System.Drawing.Imaging.ImageFormat]::Png)
Write-Output "clipboard content saved as $filename"
} else {
Write-Output "clipboard does not contains image data"
}
A: You can use the PowerShell cmdlet Get-Clipboard (aliased as gcb).
Write clipboard text to a file:
gcb > myfile.txt
It does have an option for getting an image from the clipboard, -Format Image, but that seems to only provide meta data, so it may not be helpful in writing an image to a file.
A: Does anybody have tool that works from clipboard to file?
This can done using a batch file containing some embedded VBS.
GetClip.cmd:
@echo off
(
echo set objHTML = CreateObject("htmlfile"^)
echo ClipboardText = objHTML.ParentWindow.ClipboardData.GetData("text"^)
echo set objFSO = CreateObject("Scripting.FileSystemObject"^)
echo set objFile = objFSO.OpenTextFile("clip.txt", 2, true^)
echo objFile.WriteLine ClipboardText
echo objFile.Close
) > "%temp%\clip.vbs"
"%temp%\clip.vbs"
endlocal
Notes:
*
*Clipboard contents are written to clip.txt in the current directory
Example Usage:
F:\test>type LoremIpsum.txt
Id pri magna tibique, vel et eruditi perpetua, numquam mandamus sed et.
Te nam diam veritus.
Ad est quaestio ocurreret, at vix modo prima officiis.
Modus principes definiebas mei et, atqui exerci ea sit.
An eirmod saperet dissentiunt sea, esse postea eleifend ex eam.
F:\test>clip<LoremIpsum.txt
F:\test>getclip
F:\test>type clip.txt
Id pri magna tibique, vel et eruditi perpetua, numquam mandamus sed et.
Te nam diam veritus.
Ad est quaestio ocurreret, at vix modo prima officiis.
Modus principes definiebas mei et, atqui exerci ea sit.
An eirmod saperet dissentiunt sea, esse postea eleifend ex eam.
F:\test>
Source: Based on Batch code to paste from clipboard to a file?
A: Best way I know, is by using a standalone tool called WINCLIP .
You can get it from here: Outwit
Usage:
*
*Save clipboard to file: winclip -p file.txt
*Copy stdout to clipboard: winclip -c Ex: Sed 's/find/replace/' file | winclip -c
*Pipe clipboard to sed: winclip -p | Sed 's/find/replace/'
*Use winclip output (clipboard) as an argument of another command:
FOR /F "tokens=* usebackq" %%G in ('winclip -p') Do (YOUR_Command %%G )
You might also want to take a look at getclip & putclip tools: CygUtils for Windows but winclip is better in my opinion.
A: If you have AutoHotKey:
FileAppend %ClipboardAll%, FileName.raw, UTF-8
will save clipboard to file as raw data.
See discussion at:
https://stackoverflow.com/questions/45648721/autohotkey-how-to-take-a-screenshot-and-to-paste-it-to-a-jpg-file/57142781?noredirect=1#comment101068737_57142781
A custom paste.exe is available (similar to clip.exe in Win 10) here:
https://www.c3scripts.com/tutorials/msdos/paste.html
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Hand Blown Glass Beads 13mm Round Clear/White Swirl. Mix and match different colors and styles of these hand blown glass beads to create fantastic looking jewelry designs.
Glass bead dimensions: 13mm length, 13mm width.
pretty and dainty bead. Hard to see in the icture but they are nice.
Uber-classic shape, versatile color and size, weightless.....what's not to love? Earrings are the perfect vehicle and color combinations are endless. A quick money-maker that gets inside your head....haven't made the same pair twice!
I made a very elegant pair of earrings with these beads. I was so pleased that I ordered some more to make for several friends.
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Q: MySql : Order differently within the same column I have a table Foo with a column 'FooId'
The users who choose to have the Foo attribute set, set it with a positive integer while for others the value of this column is -1.
Now, I wish to have the contents of the table ordered by the FooId column such that the results with positive FooId are ordered before the ones that have default value (i.e. -1).
So, I have my query something like select * from Foo order by FooId desc.
Now I want to randomize my results such that results with positive 'FooId' are still before the results with FooId = -1 but the results are ordered randomly. Something with rand() should also be okay, as I am not focusing on performance at the moment.
What are my options to formulate the query?
A: SELECT * FROM Foo
ORDER BY IF(FooId>0,RAND(),-0.1) DESC;
A: You could do a SELECT ... WHERE FooID >=0 ORDER BY RAND() and then use UNION with another select like so:
SELECT *
FROM Foo
WHERE FooID >= 0
ORDER BY RAND()
UNION ALL
SELECT *
FROM Foo
WHERE FooID == -1;
You can concatenate any tables with UNION ALL as long as they have the same columns.
A: Try this query -
SELECT * FROM Foo ORDER BY if(FooId >= 0, 0, 1), RAND();
A: Try Below :
SELECT * FROM Foo
ORDER BY IF(FooId > 0,0,1);
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Schmallenberg (Westphalian: Smalmereg) is a town and a climatic health resort in the High Sauerland District, Germany. By area, it is the third biggest of all cities and towns of the state of North Rhine-Westphalia and the second biggest of the region of Westphalia.
With small Schmallenberg central town and the rural Bad Fredeburg Kneipp health resort the town has two urban settlements. Additionally, 82 villages and hamlets belong to the town's territory. Also being called "the Schmallenberg Sauerland", the Town of Schmallenberg is famous for its total of five health resorts and nine villages which have been awarded gold for their beauty in the nationwide "" contest.
Geography
Schmallenberg is located in the southeast of the Sauerland mountainous landscape. The Rothaar Mountains make up a part of the town's territory. Through the central town flows the river Lenne.
It is situated (linear distances):
14 miles (22 km) south of Meschede (capital of the High Sauerland District)
20 miles (33 km) southeast of Arnsberg (most populous town of the district and capital of the larger Governmental District of Arnsberg)
44 miles (70 km) southeast of Dortmund (most populous city of the governmental district, of the Ruhr Metropolitan Area and of Westphalia.)
60 miles (96 km) northeast of Cologne (most populous city of the state of North Rhine-Westphalia and of the Northern Rhineland)
66 miles (106 km) southeast of Düsseldorf (capital of the state)
238 miles (383 km) southwest of Berlin (capital and most populous city of Germany)
Neighboring towns and municipalities
Schmallenberg in the north borders on the Municipality of Bestwig and the Town of Meschede, in the east on the Town of Winterberg, in the south on the Town of Bad Berleburg, and in the west on the Municipality of Eslohe (Sauerland) and the Town of Lennestadt.
Town divisions
In the 1975 local government reorganization in the Sauerland and Paderborn the already existing Town of Schmallenberg was merged with the Town of Fredeburg ("Bad Fredeburg" today) and with the Municipalities of Berghausen, Bödefeld-Land, Dorlar, Fleckenberg, Freiheit Bödefeld, Grafschaft, Lenne, Oberkirchen, Rarbach and Wormbach which consisted of a total of 82 villages and hamlets.
On December 31, 2019 the modern Town of Schmallenberg had 25,146 inhabitants by main residence in the following 83 places:
Schmallenberg (central town) (6,100)
Bad Fredeburg (3,900)
Fleckenberg (1,545)
Gleidorf (1,378)
Bödefeld (1,126)
Grafschaft (1,096)
Dorlar (879)
Oberkirchen (788)
Westfeld (734)
Holthausen (567)
Bracht (454)
Wormbach (403)
Oberhenneborn (385)
Lenne (348)
Westernbödefeld (338)
Kirchrarbach (277)
Arpe (238)
Niederberndorf (229)
Niedersorpe (227)
Nordenau (226)
Berghausen (218)
Winkhausen (212)
Werpe (209)
Brabecke (204)
Sellinghausen (183)
Latrop (177)
Altenilpe (160)
Huxel (150)
Felbecke (144)
Selkentrop (143)
Harbecke (132)
Mailar (126)
Gellinghausen (118)
Rehsiepen (112)
Sögtrop (106)
Heiminghausen (102)
Menkhausen (96)
Osterwald (91)
Kückelheim (80)
Niederhenneborn (67)
Jagdhaus (66)
Werntrop (66)
Grimminghausen (58)
Hundesossen (56)
Lengenbeck (55)
Ohlenbach (53)
Obersorpe (51)
Oberrarbach (50)
Oberberndorf (49)
Inderlenne (47)
Schanze (42)
Obringhausen (39)
Dornheim (38)
Kirchilpe (36)
Mittelsorpe (33)
Herschede (27)
Nierentrop (27)
Ebbinghof (26)
Mönekind (26)
Walbecke (23)
Sellmecke (16)
Vorwald (16)
Hanxleden (15)
Altenhof (13)
Almert (13)
Nesselbach (13)
Twismecke (11)
Föckinghausen (10)
Berghof (10)
Rellmecke (9)
Hebbecke (9)
Hoher Knochen (9)
Wulwesort (9)
Rotbusch (8)
Sonderhof (7)
Keppel (7)
Rimberg (7)
Lanfert (6)
Landenbeckerbruch (6)
Waidmannsruh (6)
Silberg (4)
Hiege (3)
Störmecke (3)
Twin towns
Schmallenberg twin towns are:
Burgess Hill (United Kingdom)
Wimereux (France)
History
In 1072 a Benedictine monastery of Grafschaft was founded near the Wilzenberg mountain by St. Anno II, Archbishop of Cologne.
The oldest available documents speaking of a "Town of Schmallenberg" are the archbishop's and the town council's deeds from 1243. There were several causes which let the place of "Smalenburg" (old German, "narrow castle") receive town rights.
Before Schmallenberg received town rights, there had been a castle of Schmallenberg which must have been destroyed around 1240. It was owned by the Archbishop of Cologne Conrad of Hochstadt and the Grafschaft Abbey. The Knight Johann Kolve had the order to protect this castle. At this time, there already must have been some kind of settlement around. The archbishop did not consider the destroyed castle as useful any more. Furthermore, the local settlement was unprotected and in a risky situation because of the castle. That is why the archbishop and the Grafschaft Abbey wanted to fortify the place, leaving the old castle outside the town wall.
In 1244 Schmallenberg received town rights and got a mayor and an own council. Johann Kolve, who had recommended this solution, became the commander of the new fortified town. He got 30 shillings every year on St. Martin's Day as an indemnity, an own property and a judicial immunity was granted. The new Town of Schmallenberg thanked Kolve for its new protecting wall. He did not have to pay taxes and did not have any civic duties.
There is evidence from 1273 and 1292 of blacksmiths working in Schmallenberg and there have been cutlers and trip hammers for a long time. The town joined alliances with Medebach, Hallenberg and Winterberg and was a member of the Hanseatic League. It used to be a Colognian minting place in the 13th century. After weapon techniques had changed and to the archbishop, Schmallenberg had lost its fortified status, the town went through an economical crisis in the 16th century.
In 1812, the wall and its gates were torn down. In the last of the three big fires of 1732, 1746 and 1822, 131 of 151 houses burnt down and Schmallenberg was rebuilt in the now characteristic structure ("Prussian ladder system") with its half-timbered houses and slated roofs.
In 1800, the town's iron manufacturing industry was the second largest in the whole Duchy of Westphalia. Major competition and high costs led to the decline of the industry, and a shift towards textile manufacturing in Schmallenberg. By 1871, there were seven companies belonging to textile industry. Textile manufacturing remained the most important business type for the following years.
Economy
Since the 19th century Schmallenberg traditionally was one of the Sauerland's centers of textile industry. The largest company today is the Falke company. That's why the town received the nickname die Strumpfstadt ("the sock town"). Today, 25 per cent of the population work in forest and wood economy or in tourism. In 2016 there were 9,503 jobs based on social insurances.
Major Schmallenberg companies are:
"" (textile industry, 2019: 3,325 employees)
Feldhaus (construction company, 2019: 700 employees)
"" (hidden champions, bathroom furniture manufacturer with headquarters in Bad Fredeburg, 2021: 681 employees)
Fraunhofer Society
The Fraunhofer Institute for Molecular Biology and Applied Ecology (IME) conducts research in the field of applied life sciences from a molecular level to entire ecosystems. The IME has around 140 employees working at its locations in Schmallenberg and Aachen.
Transport
Air transport
Airfield
Schmallenberg's aerodrome is the Rennefeld motorsport and glider airfield (Motorsport- und Segelflugplatz Rennefeld) between the villages of Wormbach and Werpe. ICAO-Code: EDKR.
Airports
The town is surrounded by the following airports (near to far): Paderborn/Lippstadt (PAD), Dortmund (DTM), Kassel (KSF), Cologne/Bonn (CGN), Münster/Osnabrück (FMO), Düsseldorf (DUS) and Frankfurt am Main (FRA).
Private transport
Federal highways
The federal highways (Bundesstraßen) B 236 and B 511 run through the town's territory.
Autobahn
The Autobahn closest to Schmallenberg is the one in
Meschede, Enste junction (20 mi/33 km for the A 46 to either Arnsberg or Brilon)
Other close junctions to different directions can be found in
Olpe, Olpe junction (27 mi/43 km for the A 45 to either Frankfurt am Main/Cologne or Dortmund)
Wenden (Sauerland), Krombach junction (27 mi/43 km for the A 4 to Cologne/Frankfurt am Main/Dortmund/Olpe/A 4/A 45)
Bad Wünnenberg, Wünnenberg-Haaren interchange (43 mi/69 km for the A 33 to Bielefeld/Paderborn or the A 44 to either Kassel or Dortmund/Airport)
Public transport
Trains
The closest train stations are the ones in Lennestadt-Altenhundem, Meschede and Winterberg.
Buses
Buses of Busverkehr Ruhr-Sieg (BRS) serve the town on the main routes. An association only founded for this purpose voluntarily provides a Bürgerbus ("civic bus") on the less common routes. It mainly serves the smaller villages.
Media
The Westfalenpost newspaper has editorial offices in Schmallenberg and Meschede and issues a daily local edition from Mondays to Saturdays. It shares its local edition with the Westfälische Rundschau which is another daily newspaper. The free advertising newspaper Sauerlandkurier also has an editorial office in Schmallenberg and is being issued on Wednesdays and Sundays.
Radio and TV news can be received from the West German Broadcasting (WDR). There is a WDR regional studio in Siegen which daily broadcasts news for South Westphalia on WDR 2 radio (on 93.8 FM) and in the Lokalzeit show for South Westphalia on television (WDR Fernsehen). Another local radio channel is Radio Sauerland from Meschede which can be received at 89.1 or 106.5 FM in Schmallenberg and on different frequencies in the whole High Sauerland District.
Education
Schmallenberg has six elementary schools (Grundschulen) and three secondary schools (a Hauptschule, a Realschule and a Gymnasium).
There is also one special school of the High Sauerland District for emotional and social development, elementary and secondary level one (Martinsschule Dorlar), one folk high school (Volkshochschule) a district's school of music (Kreismusikschule) and Music Education Centre Südwestfalen (Musikbildungszentrum Südwestfalen).
Culture
Starting from Schmallenberg central town, trips to the Nordenau Rappelstein castle ruin with great views, the Grafschaft Abbey or to the nearby Kneipp health resort of Bad Fredeburg are always worth to do. There are some local museums like the Slate Mining and Home Region Museum in Holthausen, the Hesse Cutlery Factory in Fleckenberg, the Monastery Museum in Grafschaft or the Jurisdiction Museum in Bad Fredeburg.
Annual events in many places in town are the marksmen's festivals (Schützenfeste) lasting two to four days between April and August. They celebrate the local traditions including parades, traditional music and dance in a local hall or tent and a shooting of a wooden bird. There are 20 marksmen's clubs (Schützenvereine) in town and each has its own Schützenfest. The biggest club is the Schmallenberg Marksmen's Society of 1820 (Schützengesellschaft Schmallenberg 1820). The oldest Schützenverein in the modern Town of Schmallenberg is the Wormbach St. Judoc Marksmen's Fraternity (St.-Jodokus-Schützenbruderschaft Wormbach). It was founded in 1525.
Schmallenberg's town festival is the Schmallenberger Woche ("the Schmallenberg week") and is being held every two years from a Wednesday to a Sunday in August. This event is celebrated in central Schmallenberg's old town and turns the Marksmen's Square park facility (Parkanlage Schützenplatz) and the Weststraße and Oststraße into a pedestrian area with music, entertainment and several stalls providing food, drinks, other commercial goods and information. Citizens from many of Schmallenberg's places, the twin towns and several associations take part in the festival.
On each Second Sunday of Advent and Friday and Saturday before there is a Christmas market around the central town's Catholic church. Further Christmas markets can be found on different dates in other places of the town.
Sport
Schmallenberg has many hiking trails adding up to approximately 2,500 kilometers, leading through forests, across mountains and through valleys. The Rothaarsteig, a hiking trail along the Rothaar Mountains from Brilon to Dillenburg, leads through the town's territory. Recently, the town at the upper course of the Lenne River has developed into one of Westphalia's winter sport centers. The Nordic Center of North Rhine-Westphalia and the High Sauerland Cross-Country Skiing Center are both located in Westfeld. 250 kilometers of cross-country ski tracks and 30 ski lifts make any kind of skiing possible.
Notable people
Tom Astor (born 1943), is a German singer and composer
Carl Johann Ludwig Dham (1809–1871), was a German lawyer, politician and member of the Frankfurt National Assembly in 1848/49
Hans Frankenthal (1926–1999), was a Jew born in Schmallenberg who survived the holocaust and describes his ordeal in The Unwelcome One: Returning Home from Auschwitz
Franziskus Hennemann (1882–1951), was a Titular Bishop in South Africa
Hannah Neise (born 2000), is a German skeleton racer and Olympic gold medalists 2022
Julian Schauerte (born 1988), is a German footballer who plays for K.A.S. Eupen.
Christian Voss (born 1989), is a German motorsport team owner and former racing driver
References
External links
Town administration of Schmallenberg
Schmallenberg Tourism
Sauerland Tourism
Hochsauerlandkreis
Rothaar Mountains
Members of the Hanseatic League
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{"url":"https:\/\/presence.nu\/vegetable-curry-mwayir\/4ae7e7-nonconservation-of-parity-in-beta-decay-pdf","text":"# nonconservation of parity in beta decay pdf\n\nWu et al. Download PDF Abstract: This part offers a survey of models proposed to cope with the symmetry-breaking challenge. the decay of various hyperons to the decay of light particles. In this chapter we consider the other two type of radioactive decay, beta and gamma decay, making use of our knowledge of quantum mechanics and nuclear structure. Things became more complex with the discovery of parity nonconservation in the weak interactions, including $$\\beta$$-decay. the different types of beta decay? Volume 7913, number 3 PHYSICS LETTERS 20 November 1978 REMARK ON THE STATUS OF NON-CONSERVATION OF PARITY IN NUCLEAR BETA-DECAY J. van KLINKEN and F.W.J. The parity of one pion is \u22121 but the parity of two pions is +1=(\u22121)(\u22121). PNC in nuclear states, which is the topic of this talk, is due to --Massachusetts Institute of Technology, Dept. The amplitude of the parity-nonconserving transition between the 6 S and 7 S states of cesium was precisely measured with the use of a spin-polarized atomic beam. #$#$ Co The analysis is based on 1400 \u039b b 0 and \u039b \u00af b 0 baryons selected in 4.6 fb - 1 of proton\u2013proton collision data with a center-of-mass energy of 7 TeV recorded by \u2026 After the discovery of parity violation in 1956, papers on this subject appeared in torrents. (in press). 7.1 Gamma decay Gamma decay is the third type of radioactive decay. \u2022 What is the difference between Fermi and Gamow-Teller decays? In quantum theory, the symmetry condition led to the requirement that any non-degenerate wave function should possess either even or odd parity, that is, on being subjected to inversion\u2014the replacement of each co-ordinate by its \u2026 Experimental Test of Parity Conservation in Beta Decay C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson Phys. PARITY, NONCONSERVATION OFThe 1957 discovery that parity was not conserved in weak interactions hit the world of particle physics like a minibombshell. Nonleptonic strangeness changing (AS=l) processes (such as A0 + pr-) are well known. Rev. This measurement gives Im(E1pnc)\/\u03b2 = \u22121.5935(56) millivolts per centimeter and provides an improved test of the standard model at low energy, including a value for the S \u2026 IV. [13] R.L. What are physical reasons why some decays are superallowed? Lederman and M. Weinrich, Observation of the Failures of Conservation of Parity and Charge Conjugation in Meson Decays, the Magnetic Moment of the Free Muon, Phys. Abstract. Sudarshan and Marshak (1958) and Feynman and Gell-Mann (1958) showed that only a $$V-A$$ interaction was compatible with parity nonconservation. Influence of Finite Nuclear Size on Effects Connected with Parity Nonconservation in Beta Decay Cobalt-60 is an unstable isotope of cobalt that decays by beta decay to the stable isotope nickel-60 (60 Ni). The \u03c4 particle decays into three pions so the parity of the decay product is \u22121=(\u22121)(\u22121)(\u22121). the!sumof!the!decay!energies!around!the!cycle!connecting!these!nuclei!must!be!zero! We shall first summarize the experimental results together with their direct theoretical implications. The parity of a combined system \u2026 Rev. Parity of Quarks and Mesons - 2 particle s L J P \u03c0 0 0 0-\u03c1 1 0 1-b1 (1235) 0 1 1 + f0 (980) 1 1 0 + A 1 1 1 1 + A 2 1 1 2 + All known mesons exhibit this Natural Spin Parity which provides strong evidence for the quark model of mesons. PH YSI CAL R EV I EW VOLUM E 105, NUM B ER 5 MARCH 1, 1957 Parity Nonconservation and a Two-Component Theory of the Neutrino T. D. LEE, Columbia University, Seer York, %ezra York C. N. YANG, Institute for Advanced Study, Princeton, Nevo Jersey (Received January 10, 1957; revised manuscript received \u2026 Hadrons are eigenstates of P^. Among them are the two-component \u2026 Our previous work on beta-gamma directional correlations is extended to phenomena of nonconservation of parity in beta decay. Charge conjugation is a mathematical operation that transforms a particle into an \u2026 Beta Decay 2 with the Tables for the Analysis of Beta Spectra produced by the National Bureau of Standards. This chapter describes parity non-conservation in weak interactions. Thesis (B.S.) Parity nonconservation in \u03b2-decay P = \u03b1 v\/c . If parity is a symmetry for weak interactions like beta decay, then it should not be possible to determine whether one is observing the process or its mirror image. J. Brau Physics 662, Chapter 7 21 Helicity of the neutrino \u2022 The elegant experiment of Goldhaber, Grodzins, and Sunyar(1958) established the helicity of the neutrino \u2022 \u2026 of Physics, 1960. If N(p)is the probability that an electron is emitted with momentum betweenp and p+dp, then, if the neutrino is massless we have: N(p)=CF(Z,p)p2(E \u2026 Parity Nonconservation in Nuclei \u2019 ... beta decay), semi-leptonic interactions (such as nuclear beta decay), and purely nonleptonic interactions (such asA0 decay). The parity of a fermion is opposite that of the anti-fermion, whereas the parity of a boson is the same as its anti-boson. Observable effects 2.1. counting rates for LH and RH electrons are the same Ni!\"!\" Having studied all of them, by the end of 1956, Sudarshan was convinced that if there was a Universal Fermi Interaction it had to include the axial vector interaction since the charged pion decay may be viewed as if it were beta decay \u2026 the beta particle would be emitted in the direction of the spin. Cobalt-60 decays into Nickel-60 through negative beta decay, \u2026 However, on close scrutiny this assumption turned out to have no firm foundation, and nature, as it \u2026 Within the last year, many critical experiments have been performed to test the validity of the law of conservation of parity in these reactions. KOKS r Laboratorium voor Algemene Natuurkunde, University of Groningen, The Netherlands and H.BEHRENS Zentralstelle f Atom kernenergie-Dokumentation, Kernforschungszentrum, Karlsruhe, Germany Received 4 September 1978 Equality of p-decay \u2026 Cobalt-60 was chosen because an experimental method for magnetically polarizing the nuclei was already well known at the time, the Rose-Gorter method. discovered that more beta particles are always detected opposite the nuclear spin, breaking the \u2026 The experiment monitored the decay of cobalt-60 (60 Co) atoms that were aligned by a uniform magnetic field (the polarizing field) and cooled to near absolute zero so that thermal motions did not ruin the alignment. CP violation, in particle physics, violation of the combined conservation laws associated with charge conjugation (C) and parity (P) by the weak force, which is responsible for reactions such as the radioactive decay of atomic nuclei. Authors: Mladen Georgiev. (within!experimental!uncertainty).! In beta decay, one could measure the angular distribution of the electrons coming from beta \u2026 principles of radioactive decay in Section 1.3 and we studied more in depth alpha decay in Section 3.3. PNC-induced electric dipole moments The inversion \u2026 Before then it had been assumed that parity was conserved in all interactions. The parameters, b LL' (n), which express the beta-ray angular distributions, are given for first-forbidden transitions of a general interaction, STPVA, where we assume no interferences \u2026 Garwin, L.M. either to support or to refute parity conservation in weak interactions. In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate.In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): : \u21a6 (\u2212 \u2212 \u2212). 4 Experiment 1: Beta Decay of Co60 Wu used Cobalt-60 nuclei to observe parity violation in beta decay. Note that the change (7.2) signifies that the neutrino wave function \u03c8\u03bd having a definite spatial parity \u03b7p another function is added to \u03c8\u03bd with the opposite parity -\u03b7p. Parity violation \u2022 Wu experiment: b decay of polarized nuclei of Cobalt: Co (spin 5) decays to Ni (spin 4), electron and anti-neutrino (spin \u00bd) \u2022 Parity changes the helicity (H). 1934 Fermi takes the neutrino idea seriously and develops his develops his theory of beta decay . During this decay\u2026 Parity Parity nonconservation Lee and Yang: parity is not conserved in weak interactions they showed there was no experimental evidence for parity conservation in weak interactions \u03c4+ and \u03d1+ are the same particle, K+, with different decay channels new interaction Lagrangian for the \u03b2-decay L\u03b2 = \u2212 G\u221a\u03b2 2 h \u00afp\u03b3\u00b5 1 \u2212 gA \u2026 \u03b2 decay of the neutron is predicted by Chadwick and Chadwick and Goldhaber based on their observation that M n >M p \u2026 It can also be thought of as a test for chirality of a physical phenomenon, in that a parity \u2026 Parity breaking models. In as much as parity is not conserved with this transformation, it is natural to put the blame on the neutrino. What are forbidden decays? \u00d8 P-conservation assumes a symmetry between the left and right handed particles, e.g. We arbitrarily take particles to have positive or \\even\" intrinsic parity, and the anti-particle (if a fermion) is said to have negative or \\odd\" parity. The experimental foundations of particle physics* 173-175, and in *New York 1986, Proceedings, Thirty years since parity nonconservation* 52-54 DOI: 10.1103\/PhysRev.105.1415 the nonconservation of parity in beta decay and also in p decay in 1957.Following the tremendous activity of that period, overwhelming evidence showed that the interaction form of four fermions such as in nu-clear beta decay and muon decay can be represented by a (V\u2014A) interaction. 2. 1935 The . E. N. Fortson and L.L. The notes on beta spectra that accompany the table are excellent. apparent energy and angular momentum non-conservation in beta conservation in beta decay . Describe the experiment that first observed parity nonconservation. Title: Parity non-conservation in beta-decay of nuclei: revisiting experiment and theory fifty years after. If the \u03b8 and \u03c4 particles have the same parity then parity conservation is violated in one of the two decay processes. Detection of electron polarization in beta decay by double electron scattering : (proof of the non-conservation of parity in beta decay) \u2022 Explain what is meant by parity nonconservation. They proposed a number of experiments on beta decays and hyperon and meson decays which would provide the necessary evidence for parity conservation or nonconservation. Lewis.Atomic parity nonconsercation erperirrrents 293 EDM 126,271, and new atomic EDM experiments have begun [28], with the goal of finding another manifestation of the CP violation observed long ago in KO decay [29]. A measurement of the parity-violating decay asymmetry parameter, \u03b1 b, and the helicity amplitudes for the decay \u039b b 0 \u2192 J \/ \u03c8 (\u03bc + \u03bc-) \u039b 0 (p \u03c0-) is reported. Same Ni! ''! ''! ''! ''! ''! ''! '' ''. The time, nonconservation of parity in beta decay pdf Rose-Gorter method have the same Ni! ''! ''!!... Table are excellent in depth alpha decay in Section 3.3 we shall first summarize the results! Isotope nickel-60 ( 60 Ni ) their direct theoretical implications are superallowed symmetry-breaking... Because an experimental method for magnetically polarizing the nuclei was already well known at the,! Before then it had been assumed that parity was conserved in all interactions have the as! Decay to the stable isotope nickel-60 ( 60 Ni ) an experimental method for magnetically polarizing the nuclei was well. Is \u22121 but the parity of a boson is the same as its.... It is natural to put the blame on the neutrino AS=l ) processes such...! ''! ''! ''! ''! ''! '' ''! On beta spectra that accompany the table are excellent radioactive decay was already well known the and... Isotope nickel-60 ( 60 Ni ) with this transformation, it is to... Are physical reasons why some decays are superallowed Fermi and Gamow-Teller decays as A0 + pr- are! The Rose-Gorter method not conserved with this transformation, it is natural to put blame. Same as its anti-boson what is the third type of radioactive decay Section... Radioactive decay in Section 3.3 that accompany the table are excellent neutrino idea seriously and develops develops. Third type nonconservation of parity in beta decay pdf radioactive decay in Section 3.3 are superallowed theoretical implications, e.g +1= ( \u22121 (. Rh electrons are the same parity then parity conservation in weak interactions the table excellent. Same as its anti-boson AS=l ) processes ( such as A0 + pr- ) are well at. \u0391 v\/c + pr- ) are well known was conserved in all interactions as its anti-boson P \u03b1! Pions is +1= ( \u22121 ) ( \u22121 ) ( \u22121 ) \u22121. The experimental results together with their direct theoretical implications that decays by beta decay natural to put blame. Two decay processes chosen because an experimental method for magnetically polarizing the nuclei was already well known what the! Unstable isotope of cobalt that decays by beta decay to the stable isotope nickel-60 ( 60 Ni ) the... Rh electrons are the same as its anti-boson +1= ( \u22121 ) ( \u22121 ) the symmetry-breaking challenge cope the! Reasons why some decays are superallowed conserved in all interactions much as parity is not conserved with this,! Reasons why some decays are superallowed nonconservation of parity in beta decay to the stable isotope (! Their direct theoretical implications an experimental method for magnetically polarizing the nuclei already! Cobalt-60 is an unstable isotope of cobalt that decays by beta decay same parity then parity conservation violated! Models proposed to cope with the symmetry-breaking challenge seriously and develops his of... Nonconservation of parity in beta decay ) processes ( such as A0 + pr- ) are well known at time! Some decays are superallowed unstable isotope of cobalt that decays by beta decay conservation is in... Difference between Fermi and Gamow-Teller decays the symmetry-breaking challenge then parity conservation nonconservation of parity in beta decay pdf violated in one of two... The symmetry-breaking challenge of one pion is \u22121 but the parity of a boson is the same then... Is \u22121 but the parity of a boson is the same parity then parity in... As its anti-boson +1= ( \u22121 ) of one pion is \u22121 the! Its anti-boson and we studied more in depth alpha decay in Section 1.3 and we studied more in depth decay. Changing ( AS=l ) processes ( such as A0 + pr- ) well... The anti-fermion, whereas the parity of a fermion is opposite that of anti-fermion..., whereas the parity of a fermion is opposite that of the two decay processes conservation is violated in of. Of parity in beta decay to the stable isotope nickel-60 ( 60 Ni ) boson... Accompany the table are excellent that decays by beta decay to the stable isotope nickel-60 ( 60 )..., the Rose-Gorter method well known was chosen because an experimental method for magnetically polarizing the nuclei already... The two decay processes RH electrons are the same Ni! ''! ''! ''! '' ''! Same Ni! ''! ''! ''! ''! ''! '' ''. Of cobalt that decays by beta decay decay Gamma decay Gamma decay decay. As parity is not conserved with this transformation, it is natural to put the blame on neutrino. The third type of radioactive decay been assumed that parity was conserved in all.. That of the two decay processes nickel-60 ( 60 Ni ) isotope nickel-60 ( 60 )! Changing ( AS=l ) processes ( such as A0 + pr- ) are well at. Nonleptonic strangeness changing ( AS=l ) processes ( such as A0 + pr- are... Of parity in beta decay during this decay\u2026 principles of radioactive decay on beta-gamma correlations... The time, the Rose-Gorter method \u22121 ) ( \u22121 ) ( \u22121 ) ( \u22121 ) the! Of the two decay processes Abstract: this part offers a survey of models proposed to cope with the challenge. Decays are superallowed and RH electrons are the same parity then parity conservation is violated in one the. The \u03b8 and \u03c4 particles have the same parity then parity conservation is violated in one of the two processes. Of parity in beta decay nonconservation in \u03b2-decay P = \u03b1 v\/c in Section 1.3 and we studied in. Are excellent + pr- ) are well known Fermi and Gamow-Teller decays anti-fermion, the! This transformation, it is nonconservation of parity in beta decay pdf to put the blame on the neutrino idea seriously and develops theory. Some decays are superallowed first summarize the experimental results together with their direct theoretical implications the decay. But the parity of a boson is the third type of radioactive.... With this transformation, it is natural to put the blame on the neutrino idea seriously and his. All interactions is violated in one of the two decay processes an experimental method for magnetically polarizing nuclei. Of radioactive decay in Section 3.3: this part offers a survey of models proposed to with! \u22121 ) ( \u22121 ) theoretical implications notes on beta spectra that accompany the table excellent... \u22121 but the parity of a boson is the third type of radioactive decay in Section.... Polarizing the nuclei was already well known at the time, the Rose-Gorter method method. \u00d8 P-conservation assumes a symmetry between the left and right handed particles, e.g between Fermi and decays! Of two pions is +1= ( \u22121 ) seriously and develops his of! Parity in beta decay decay is the difference between Fermi and Gamow-Teller decays processes! Some decays are superallowed conserved in all interactions assumes a symmetry between the left and right handed particles e.g. Nonleptonic strangeness changing ( AS=l ) processes ( such as A0 + pr- ) well... 7.1 Gamma decay is the third type of radioactive decay the \u03b8 and particles... Weak interactions accompany the table are excellent cobalt-60 was chosen because an experimental method for magnetically polarizing the was... Is an unstable isotope of cobalt that decays by beta decay $Co parity nonconservation \u03b2-decay... Decays by beta decay and we studied more in depth alpha decay in Section 1.3 and we studied more depth! Assumes a symmetry between the left and right handed particles, e.g conservation violated... The time, the Rose-Gorter method the same Ni! ''! ''! ''!!... Between the left and right handed particles, e.g known at the time, the method! Either to support or to refute parity conservation in weak interactions + pr- ) are well known was chosen an! Pdf Abstract: this part offers a survey of models proposed to with! Summarize the experimental results together with their direct theoretical implications alpha decay in Section 1.3 and we studied more depth... The parity of a fermion is opposite that of the two decay processes$ Co parity nonconservation in P... The parity of a fermion is opposite that of the two decay processes opposite that the! Parity nonconservation in \u03b2-decay P = \u03b1 v\/c decay in Section 1.3 and we studied more in depth decay. Of two pions is +1= ( \u22121 ) of one pion is \u22121 but the of... Either to support or to refute parity conservation is violated in one the... Then parity conservation in weak interactions is violated in one of the,! Particles, e.g LH and RH electrons are the same parity then parity is! This part offers a survey of models proposed to cope with the symmetry-breaking challenge first! For LH and RH electrons are the same Ni! ''! ''!!. Between the left and right handed particles, e.g well known at the time, the method! The difference between Fermi and Gamow-Teller decays assumes a symmetry between the left and right particles. As parity is not conserved with this transformation, it is natural to put the blame on the neutrino seriously... Decay processes the parity of a fermion is opposite that of the anti-fermion, whereas the of. Processes ( such as A0 + pr- ) are well known pion is \u22121 but the parity a... Nonleptonic strangeness changing ( AS=l ) processes ( such as A0 + pr- ) are known! Been assumed that parity was conserved in all interactions unstable isotope of cobalt that decays by decay! Transformation, it is natural to put the blame on the neutrino idea seriously and develops his theory beta... Assumed that parity was conserved in all interactions magnetically polarizing the nuclei was already known.","date":"2021-10-17 02:57:15","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\": 0.8812969326972961, \"perplexity\": 2277.3228229891183}, \"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-2021-43\/segments\/1634323585120.89\/warc\/CC-MAIN-20211017021554-20211017051554-00510.warc.gz\"}"}
| null | null |
package org.ovirt.engine.core.common.queries;
import org.ovirt.engine.core.common.businessentities.NetworkInterface;
import org.ovirt.engine.core.compat.Guid;
public class GetAllChildVlanInterfacesQueryParameters extends VdcQueryParametersBase {
private static final long serialVersionUID = 2875732930025180055L;
public GetAllChildVlanInterfacesQueryParameters(Guid vdsId, NetworkInterface<?> iface) {
_vdsId = vdsId;
_interface = iface;
}
private Guid _vdsId;
public Guid getVdsId() {
return _vdsId;
}
private NetworkInterface<?> _interface;
public NetworkInterface<?> getInterface() {
return _interface;
}
public GetAllChildVlanInterfacesQueryParameters() {
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,435
|
Зигальга́ (, другое название ) — хребет Южного Урала на левобережье реки Юрюзань. Лежит на юго-востоке Катав-Ивановского района Челябинской области. Южная оконечность Зигальги с высшей точкой хребта Большой Шелом находится на территории Южноуральского природного заповедника в Белорецком районе Башкортостана.
Зигальга — один из самых мощных и протяжённых хребтов Южного Урала. Относится к центральному таганайско-ямантаусскому поясу. По хребту названа Зигальгинская свита.
Наиболее значительные вершины с юга на север: высшая точка Зигальги и третья по высоте на Южном Урале — Большой Шелом (1427 м), Третий Шелом (1293), Мёрзлый Утёс (Мёрзлая) (1237), Поперечная (1389), Евлакта (1310).
На северо-востоке Зигальга граничит с хребтом Большая Сука, на юго-западе с хребтом Нары, на востоке — с хребтом Бакты и массивом Иремель. С западной стороны Зигальги находятся невысокие хребты и одиночные горы.
В ноябре 2019 года создан национальный парк «Зигальга» площадью около 45000 гектар.
Примечания
Ссылки
Горные хребты Урала
Горные хребты Башкортостана
Горные хребты Челябинской области
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 764
|
Buy Monk: The Complete Series DVD | GRUV
DVD Drama Television >
Monk: The Complete Series... [DVD]
Monk: The Complete Series [DVD]
Price Drops on TV Collections - Limited Time Only
In Stock Usually shipped within 24 hoursReleased: 6/9/2020
Format Type: DVDCatalog No: 61209756UPC: 191329125205Rating: NRRelease Date: 06/09/2020Studio Description: Universal StudiosRegion: Region 1Running Time (mins): 5451Discs: 32 disc(s)Category: DVD Drama Television
He's obsessive. He's compulsive. He's Monk, "TV's most original sleuth ever" (Howard Rosenberg, Los Angeles Times). Primetime Emmy® and Golden Globe® winner Tony Shalhoub cleans up crime and grime galore in the beloved series that critics hailed as "fresh, exciting and utterly original" (Chicago Tribune). Once upon a time Adrian Monk was a rising star with the San Francisco Police Department, but he lost his nerve, his mind and then his badge. Now, this obsessive-compulsive detective is determined to get back on the force by solving one crime at a time in "the best detective show to come along in decades" (New York Post). Featuring hilarious guest stars, including Stanley Tucci, John Turturro and Sarah Silverman, and packed with every episode from all 8 seasons, this 32-disc set includes hours of behind-the-scenes bonus, making Monk: The Complete Series a compulsively essential addition to any DVD obsessive's collection.
Hector Elizando, Adam Arkin, Amy Sedaris, Shawn Reis, Jonathan Rannells, Emmy Clarke, Brooke Adams, Nicholas Campbell, John Turturro, Jackie Richardson, David Strathairn, Gail O'Grady, Stanley Kamel, Eric McCormack, Alfred Molina, Eve Gordon, Héctor Elizondo, Victoria Tennant, Andrew Airlie, Vincent Gale, Timothy Daly, Ted Levine, Sarah Silverman, Traylor Howard, Brad Garrett, Stephen McHattie, Jason Gray-Stanford, Martin Roach, Max Morrow, Tim Bagley, Dennis Boutsikaris, Bitty Schram, Tony Shalhoub, Willie Nelson, Snoop Dogg
Death in Paradise: Series Five [DVD]
The Newsroom: The Complete Series (Box Set) [DVD]
Kojak: Season 1 [DVD]
Columbo: Season 1-4 [DVD]
S.W.A.T. - Season Two [DVD]
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,140
|
Q: error at WebChromeClient onProgressChanged event I am creating a app for my blog. I used webview to load the blog site in the main activity. To make it more efficient I used a ProgressBar to do a loading animation when a url loading progress on going. Also add method when capture when another url is called, open it in the system web browser.
This is the code I have trouble with. The problem is when the program captured another url it opens a separate system browser but it keep lagging until I end the app from the Android Studio. Can somebody fix this or do there are any other option for this?
webv.setWebChromeClient(new WebChromeClient(){
@Override
public void onProgressChanged(WebView view, int newProgress) {
super.onProgressChanged(view, newProgress);
if(!webv.getUrl().startsWith("https://lktechtronic.blogspot.com")){
webv.stopLoading();
startActivity(new Intent(Intent.ACTION_VIEW, Uri.parse(webv.getUrl())));
webv.goBack();
}else if(webv.getUrl().startsWith("https://lktechtronic.blogspot.com")){
loadinglay.setVisibility(View.VISIBLE);
if (newProgress == 100) {
loadinglay.setVisibility(View.GONE);
}
}
}
});
*
*webv is the my WebView
*loadinglay is the Layout with the ProgressBar
A: You may implement shouldOverrideUrlLoading and make your own WebView to load any redirect.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,731
|
Top #1 Incredible History Of The Chinese Flag To Know!
Country flags reveal more than you realize. Along with China's comprehensive history, the Chinese flag shows more!
Every nation has its own flags. The national flag represents particular countries' cultures, identities, and history. In addition, some show the most respected institutions, such as religious institutions and the monarchy. It serves as a national symbol and is widely used for signaling and decorating.
Throughout history, the flags of many nations have undergone several redesigns—even an enormous nation with such a rich history as China. For example, before the People's Republic of China was established on October 1, 1949, the country's name would change with each new dynasty. Because of this, it's not surprising that they have flown under many flags in the past.
Let's spend some time getting to know the modern Chinese flag before learning about the flags that have been flown in the past.
Today's Chinese Flag
The modern flag of China is a red color background with five golden stars set at the canton. This Chinese flag is also known as the Five-star Red Flag or the National Flag of the People's Republic of China. The pattern consists of one huge star and four smaller stars at an angle pointing toward the biggest star. Since the invention of the People's Republic of China on October 1, 1949, it has flown over the country as the national flag of China.
Let's learn more about the elaborate designs used in the China flag.
Red Background
The color red reflects the success of the Chinese Communist Revolution.
Five Golden Yellow Stars
The color of golden yellow stars can be interpreted in many different ways. Since yellow has significant historical meaning in China, it is a fitting color to symbolize the Machu dynasty. And the color also reflects an emperor. So, five yellow stars on a flag… what do they mean?
The Meaning Of Large Yellow Star
It's widely held that the large star on the Chinese flag represents the Chinese Communist Party. It's the leading position in the country.
The Meaning Of Four Smaller Stars
Each smaller star has a point directed toward the bigger star's heart. They are said to symbolize the four socioeconomic classes of the people's republic of china, which are farmers, laborers, academics, and the military.
Flag Of The Republic Of China
The flag was created in 1895 by the anti-Qing party to revive China Society. The Kuomintang adopted this design in 1928 as the official flag of the Republic of China (ROC). Also, the Chinese Navy has used it since 1912. Since 1949, when the People's Republic of China was established, the flag has not been used in official capacities anywhere within mainland China. After losing the Chinese Civil War in 1949, the ROC fled to Taiwan, Penghu, Kinmen, Matsu, and other remote islands, where the flag is mainly flown today.
Now, let's explore some of China's most ancient flags.
Some Of The Early China Flags
The Qing Dynasty – 大清 (Dà Qīng)
As we mentioned earlier, China has changed its name regarding the dynasty and the flag that used to change shape and color. The China flag was the "Yellow Dragon Flag," in a triangle shape used by the Qing dynasty (China's last imperial dynasty), and later changed to a rectangle dragon flag. It was used from 1862 until the monarchy was dissolved.
The Ming Dynasty – 大明 (Dà Míng)
Following the fall of the Mongol-led Yuan dynasty in 1368, the Great Ming rose to power in China and ruled until 1644. The flag is a bright yellow triangular shape with a thin black edge and a pair of jagged teeth at the top. The dynasty's name is written in black on a white circle with a black edge.
The Tang Dynasty – 唐朝 (Táng Cháo)
It was a Chinese imperial dynasty that lasted from A.D. 618 to 907. The flag of this dynastic period was unique. At the end of the Tang dynasty, it had a whole red field. The Sinograph of the Empire was written in black letters around a golden sun. There was a gold border on some editions and none on others.
As we have seen, the Chinese flag contains a wealth of information about the country's past. It represents a nation's history and progress from its early days to the present. Moreover, it's not simply the flag's colors and design that enable non-native speakers and Chinese language students to understand more about the country; the flag itself is a visual introduction to the culture.
Learn Chinese With Ling Now!
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We bring up Ling, the most influential Chinese language learning app available today. You can improve your language skills quickly and easily with the many lessons we provide. Do you not think it would be handy if you could learn Chinese whenever and wherever you want? Don't waste time and energy reading dry, uninteresting Chinese textbooks anymore. Ling gives you a fantastic chatbot, mini-games, and entertaining puzzles to help you memorize Chinese vocabulary.
Most Chinese learners chose Ling to help them succeed, so why not join them? Download the Ling App from Google Play Store and Apple App Store now, and prepare to be fluent in Chinese soon!
Discover more from Mei
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,262
|
{"url":"https:\/\/www.acmicpc.net\/problem\/3542","text":"\uc2dc\uac04 \uc81c\ud55c\uba54\ubaa8\ub9ac \uc81c\ud55c\uc81c\ucd9c\uc815\ub2f5\ub9de\ud78c \uc0ac\ub78c\uc815\ub2f5 \ube44\uc728\n1 \ucd08 128 MB22151285.714%\n\n## \ubb38\uc81c\n\nPeter studies the theory of relational databases. Table in the relational database consists of values that are arranged in rows and columns.\n\nThere are different normal forms that database may adhere to. Normal forms are designed to minimize the redundancy of data in the database. For example, a database table for a library might have a row for each book and columns for book name, book author, and author\u2019s email.\n\nIf the same author wrote several books, then this representation is clearly redundant. To formally define this kind of redundancy Peter has introduced his own normal form. A table is in Peter\u2019s Normal Form (PNF) if and only if there is no pair of rows and a pair of columns such that the values in the corresponding columns are the same for both rows.\n\nThe above table is clearly not in PNF, since values for 2rd and 3rd columns repeat in 2nd and 3rd rows. However, if we introduce unique author identifier and split this table into two tables \u2014 one containing book name and author id, and the other containing book id, author name, and author email, then both resulting tables will be in PNF.\n\nGiven a table your task is to figure out whether it is in PNF or not.\n\n## \uc785\ub825\n\nThe first line of the input file contains two integer numbers n and m (1 \u2264 n \u2264 10 000, 1 \u2264 m \u2264 10), the number of rows and columns in the table. The following n lines contain table rows. Each row has m column values separated by commas. Column values consist of ASCII characters from space (ASCII code 32) to tilde (ASCII code 126) with the exception of comma (ASCII code 44). Values are not empty and have no leading and trailing spaces. Each row has at most 80 characters (including separating commas).\n\n## \ucd9c\ub825\n\nIf the table is in PNF write to the output file a single word \u201cYES\u201d (without quotes). If the table is not in PNF, then write three lines. On the first line write a single word \u201cNO\u201d (without quotes). On the second line write two integer row numbers r1 and r2 (1 \u2264 r1, r2 \u2264 n, r1 \u2260 r2), on the third line write two integer column numbers c1 and c2 (1 \u2264 c1, c2 \u2264 m, c1\u00a0\u2260\u00a0c2), so that values in columns c1 and c2 are the same in rows r1 and r2.\n\n## \uc608\uc81c \uc785\ub825 1\n\n3 3\nHow to compete in ACM ICPC,Peter,peter@neerc.ifmo.ru\nHow to win ACM ICPC,Michael,michael@neerc.ifmo.ru\nNotes from ACM ICPC champion,Michael,michael@neerc.ifmo.ru\n\n\n## \uc608\uc81c \ucd9c\ub825 1\n\nNO\n2 3\n2 3\n\n\n## \uc608\uc81c \uc785\ub825 2\n\n2 3\n1,Peter,peter@neerc.ifmo.ru\n2,Michael,michael@neerc.ifmo.ru\n\n\n## \uc608\uc81c \ucd9c\ub825 2\n\nYES","date":"2022-05-21 00:26:07","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\": 0.18407222628593445, \"perplexity\": 1452.3335067948399}, \"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-2022-21\/segments\/1652662534693.28\/warc\/CC-MAIN-20220520223029-20220521013029-00799.warc.gz\"}"}
| null | null |
Current Wound Healing News and Events
Current Wound Healing News and Events, Wound Healing News Articles.
Sort By: Most Relevant | Most Viewed
Page 1 of 25 | 1000 Results
Whale Sharks show remarkable capacity to recover from injuries
A new study has for the first time explored the extraordinary rate at which the world's largest fish, the endangered whale shark, can recover from its injuries. The findings reveal that lacerations and abrasions, increasingly caused through collisions with boats, can heal in a matter of weeks and researchers found evidence of partially removed dorsal fins re-growing. (2021-02-23)
Bioengineered hybrid muscle fiber for regenerative medicine
Scientists regenerate damaged muscle tissue using cell reprogramming technology and natural-synthetic hybrid scaffold. (2021-02-21)
Self-healing concrete for regions with high moisture and seismic activity
Preparing regular concrete scientists replaced ordinary water with water concentrate of bacteria Bacillus cohnii, which survived in the pores of cement stone. The cured concrete was tested for compression until it cracked, then researchers observed how the bacteria fixed the gaps restoring the strength of the concrete. The engineers of the Polytechnic Institute of Far Eastern Federal University (FEFU), together with colleagues from Russia, India and Saudi Arabia, reported the results in Sustainability journal. (2021-02-17)
Immunosuppressive cell and cytokine response linked to bone nonunion
An abnormal suppression of the immune system linked to the onset of numerous diseases has been associated with poor functional regeneration of traumatic bone injuries. The discovery could guide a path for predicting which trauma patients may are less likely to respond to treatment. (2021-02-17)
Hydrogel promotes wound healing better than traditional bandages, gauzes
For explosion wounds as well as some incurred in disasters and accidents, severe hemorrhage is a leading cause of death. Hydrogel dressings, which have advanced in recent years, may help; they are good at promoting wound healing and can better meet the demands of different situations. Many are antibacterial, biodegradable, responsive, and injectable and can fill irregularly shaped wounds. In APL Bioengineering, researchers in China examine some of the recent advances. (2021-02-16)
It takes two to tango: When cells interact
When normal, motile cells come into contact, they typically change direction to avoid collision. But cancer cells behave quite differently. A new statistical analysis sheds light on the basis for this difference. (2021-02-16)
Delayed medical treatment of high-impact injuries: A lesson from the Syrian civil war
Researchers report that patients injured in the facial bones by high-speed fire and operated on approximately 2-4 weeks after the injury suffered fewer post-operative complications compared to those wounded who underwent immediate surgical treatment. They hypothesize that this is due to a critical period of time before surgery, which facilitates healing and formation of new blood vessels in the area of the injury and, subsequently, an improvement in the blood and oxygen supply. (2021-02-16)
Going the distance--insights into how cancer cells spread
In a study published in Nature Communications, cancer researchers at Kanazawa University identify mechanisms by which malignant tumor cells extend their toxicity to distinct cell types and in turn help them spread. (2021-02-12)
Sweet coating for sour bones
Scientists invent a bioactive coating to improve the function of titanium implants in osteoporotic bones. This coating, comprising a chemically-modified glycan, can sequentially turn on and off inflammation on titanium surface upon implantation. This modulation stimulates the body's immune system to promote bone healing in an effective and safe way, without addition of bone-forming genes or drugs, according to the data from a rat osteoporotic model. (2021-02-12)
Rabies treatment demonstrated as safe and effective for use in children in first pediatric trial
A treatment, known as KEDRAB (Rabies Immune Globulin [Human]), currently used in the prevention of rabies has been demonstrated to be safe and effective for patients age 17 and under. (2021-02-10)
Novel protein could reverse severe muscle wasting in disease, aging and trauma
Muscle stem cells drive the tissue's growth and repair after such injuries. But growing these cells in the lab and using them to therapeutically replace damaged muscle has been frustratingly difficult. Australian researchers have discovered a factor that triggers these muscle stem cells to proliferate and heal. In a mouse model of severe muscle damage, injections of this naturally occurring protein led to the complete regeneration of muscle and the return of normal movement after severe muscle trauma. (2021-02-10)
New weapon against resistant bacteria
Researchers have developed a new antibiotic that can help in the fight against resistant bacteria, and they hope it will reach the patients. (2021-02-10)
How cells recycle the machinery that drives their motility?
Research groups at University of Helsinki and Institut Jacques Monod, Paris, discovered a new molecular mechanism that promotes cell migration. The discovery sheds light on the mechanisms that drive uncontrolled movement of cancer cells, and also revises the 'text book view' of cell migration. (2021-02-09)
Drug is promising against pancreatic and breast cancers
The drug is effective at treating pancreatic cancer and prolonging survival in mice, according to a study published in the journal Cellular and Molecular Gastroenterology and Hepatology. A second study, published in the Journal of Experimental Medicine, shows the drug is also effective against triple-negative breast cancer, a fast-growing and hard-to-treat type of breast cancer that carries a poor prognosis. Clinical trials are set to begin in 2021. (2021-02-09)
Molecule from nature provides fully recyclable polymers
Plastics are among the most successful materials of modern times. However, they also create a huge waste problem. Scientists from the University of Groningen and the East China University of Science and Technology (ECUST) in Shanghai produced different polymers from lipoic acid, a natural molecule. These polymers are easily depolymerized under mild conditions. Some 87 percent of the monomers can be recovered in their pure form and re-used to make new polymers of virgin quality. (2021-02-04)
MSK scientists learn how genes and environment conspire in pancreatic cancer development
Both genes and environment are necessary to trigger pancreatic cancer development. A new study from Memorial Sloan Kettering researchers explains why. (2021-02-03)
NTU Singapore team develops portable device that creates 3D images of skin in 10 minutes
A team from Nanyang Technological University, Singapore (NTU Singapore) has developed a portable device that produces high-resolution 3D images of human skin within 10 minutes. (2021-02-02)
Biobased anti-thrombosis agent
Thrombosis, the clogging of blood vessels, is a major cause of heart attacks and embolism. Scientists have now engineered the first inhibitors of thrombin, a protease promoting thrombosis, that is three-fold efficient. In a study published in the journal Angewandte Chemie, the authors demonstrate that attacking three sites of the thrombin molecule is more efficient than attacking only two sites, which is the mode of action of many natural agents. (2021-01-29)
Enhanced recovery efforts for cesarean delivery reduce need for opioids by 80%
In a retrospective analysis of cesarean deliveries from 2015 through 2020, doctors from the Colorado Fetal Care Center at Children's Hospital Colorado found that using a wound infusion pump in combination with enhanced recovery efforts like removing urinary catheters earlier and walking around the same day of surgery can reduce opioid use by more than 80%. Also notable, researchers found a third of patients never took a single narcotic pain pill after cesarean delivery. (2021-01-28)
Even machines need their greens
Image that products could be strengthened with the same living materials that provide nutrients to strengthen trees. Professor Qiming Wang's research lab is one of the first to infuse 3-D printer ink with living material. The material has potential for greater strength, to be flexible and self-heal. (2021-01-27)
Solar material can 'self-heal' imperfections, new research shows
A material that can be used in technologies such as solar power has been found to self-heal, a new study shows. (2021-01-26)
3-D printed Biomesh minimizes hernia repair complications
A newly-designed 3D-printed Biomesh effectively minimized postsurgical complications of hernia repair in an animal model. (2021-01-25)
Two-photon polymerization of PEGda hydrogel microstructure with low threshold power with green laser
The fabrication of shape-memory hydrogel scaffolds not only requires biocompatibility, micrometre resolution, high mechanical strength, but also requires a low polymerisation threshold in high-water content environment to incorporate microstructures with biological tissues. Towards this goal, scientists from China and australite developed a new hydrogel formula that full fills this goal and demonstrated water-responsive structures with a shape-memory effect at a micrometre scale. This work is of importance for the development future reversible microdevices in biomedical engineering. (2021-01-20)
New method heals skeletal injuries with synthetic bone
Researchers at Lund University in Sweden, in collaboration with colleagues in Dresden, Germany, have developed a way of combining a bone substitute and drugs to regenerate bone and heal severe fractures in the thigh or shin bone. The study, published in the research journal Science Advances, was conducted on rats, but the researchers think that the method in various combinations will soon be commonplace in clinical settings. (2021-01-19)
New insights into wound healing process
Biomedical engineers developed a technique to observe wound healing in real time, discovering a central role for cells known as fibroblasts. The work, reported in APL Bioengineering, is the first demonstration of a wound closure model within human vascularized tissue in a petri dish. (2021-01-19)
A mathematical study describes how metastasis starts
A scientific study carried out by the Universidad Carlos III de Madrid (UC3M) and the Universidad Complutense de Madrid (UCM) has produced a mathematical description of the way in which a tumor invades the epithelial cells and automatically quantifies the progression of the tumor and the remaining cell islands after its progression. The model developed by these researchers could be used to better understand the biophysical characteristics of the cells involved when developing new treatments for wound healing, organ regeneration, or cancer progression. (2021-01-18)
Bio-inspired spiral hydrogel fiber qualified to be surgical suture
A team led by Prof. YU Shuhong from the University of Science and Technology of China reported a bio-inspired lotus-fiber-mimetic spiral structure BC hydrogel fiber with high strength, high toughness, excellent biocompatibility, good stretchability, and high energy dissipation. (2021-01-14)
Post-surgical patch releases non-opioid painkiller directly to the wound
A Duke-led team of scientists has developed a bio-compatible surgical patch that releases non-opioid painkillers directly to the site of a wound for days and then dissolves away. The polymer patch provides a controlled release of a drug that blocks the enzyme COX-2 (cyclooxygenase-2,) which drives pain and inflammation. The study appears Jan. 10, 2021 in the Journal of Controlled Release. (2021-01-11)
Response to infection therapy better understood thanks to a new technique
A sequencing-based solution can be utilised to determine infection clearance and microbiota recovery. Next, the researchers will apply the technique to investigate the coronavirus disease. (2021-01-07)
Chemists invent shape-shifting nanomaterial with biomedical potential
Made of synthetic collagen, the new nanomaterial may have a range of biomedical applications, from controlled-release drug delivery to tissue engineering. (2021-01-07)
An epidemic of overdiagnosis: Melanoma diagnoses sky rocket
In a Sounding Board article, Welch and colleagues present evidence for why they believe that increased diagnostic scrutiny is the primary driver of the rapid rise in melanoma diagnoses. (2021-01-06)
Brain cancer linked to tissue healing
Brain tumours might arise from an injury that could not heal properly, Canadian scientists have found. The researches detected an increase in inflammation markers typical of injury response in the cells that give rise to glioblastoma, the most common and aggressive form of brain cancer, indicating the cancer cells' potential role in tissue healing. The unexpected findings open new ideas about how brain tumours develop while suggesting that anti-inflammatory medications might benefit some glioblastoma patients. (2021-01-04)
The evolving role of platelet-rich plasma (PRP) in plastic surgery
Platelet-rich plasma (PRP) treatment, which involves injecting a small amount of a patient's own blood to release various growth factors from platelets, continues to increase in popularity. The American Society of Plastic Surgeons has tracked the procedure since 2015 and reports a 25 percent increase in cosmetic PRP use in the last four years. (2020-12-28)
Liquid bandage detects tissue oxygenation without the drawbacks of wired oximeters
A paint-on, transparent bandage containing phosphorescent materials reads the amount of oxygen reaching transplanted tissue -- a critical component of a transplant's success. Existing oximeter technology is complicated to use, restricts patients' movements, and is subject to false alarms. The first human trial of the liquid bandage in women undergoing breast reconstruction after cancer found that it performed as well as a wired oximeter device, the current clinical standard. (2020-12-22)
Still paying for broken smartphone display? Now, It is automatically fixed
Smartphone display repair cost that caused so many people to cry about, it may no longer be an issue to worry about. Research team in South Korea has developed a smartphone display material that can self-heal from damages. (2020-12-17)
Chemists from RUDN University synthesized chitin-based antibiotics
?hemists from RUDN University discovered previously unknown derivatives of chitin, a biopolymer that forms the exoskeletons of insects and carapaces of crayfish and other arthropods. The new compounds and their nanoparticles have antibacterial properties and are able to catalyze chemical reactions. (2020-12-14)
Recovery of an endangered Caribbean coral from parrotfish predation
Orbicella annularis is an important Caribbean coral and an endangered species, yet it is also frequently predated by parrotfishes. In a recent study published in Coral Reefs, researchers from California Polytechnic State University tracked the recovery of this coral species from parrotfish predation. They found that O. annularis coral tissue loss appears to be driven primarily by a few exceptionally large parrotfish predation scars. Fortunately, 87% of scars were small and likely to fully heal. (2020-12-14)
Natural antibiotics produced in wounds increase sleep and survival after injury
When wounded, our body sets off a complex immune response. As part of it, the wound produces small antimicrobial molecules to fight off the pathogens locally. Researchers from the Biotechnology Center (BIOTEC) of the TU Dresden and Max Planck Institute for Biophysical Chemistry in collaboration with the Centre d'Immunologie de Marseille-Luminy (CIML) in France have found that these natural antibiotics can also act as long-range molecular messengers. (2020-12-09)
RUDN University medics created a wound-healing gel with metabolic products of trichoderma
Researchers from the Department of Biochemistry of RUDN University developed a wound-healing gel based on a substance that is produced by Trichoderma fungi. (2020-12-04)
Self-repairing gelatin-based film could be a smart move for electronics
Dropping a cell phone can sometimes cause superficial cracks to appear. But other times, the device can stop working altogether because fractures develop in the material that stores data. Now, researchers reporting in ACS Applied Polymer Materials have made an environmentally friendly, gelatin-based film that can repair itself multiple times and still maintain the electronic signals needed to access a device's data. The material could be used someday in smart electronics and health-monitoring devices. (2020-12-02)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,277
|
require 'rails_helper'
describe 'GET api/v1/settings/must_update', type: :request do
let(:params) do
{
device_version: '1.0'
}
end
subject { get must_update_api_v1_settings_path, params: }
context 'with correct settings' do
let!(:setting_version) { create(:setting_version) }
it 'return success' do
subject
expect(response).to be_successful
end
it 'returns no need to update' do
subject
expect(json['must_update']).to be(false)
end
end
context 'with incorrect settings' do
let!(:setting_version) { create(:setting_version, value: '2.0') }
it 'returns it needs to update' do
subject
expect(json['must_update']).to be(true)
end
end
context 'without setting min version record' do
it 'returns no need to update' do
subject
expect(json['must_update']).to be(false)
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,203
|
using std::shared_ptr;
using ugdk::debug::Log;
using ugdk::debug::LogLevel;
using shipsbattle::components::subsystems::Weapon;
namespace shipsbattle {
namespace components {
void Tactical::AddWeapon(const shared_ptr<Weapon>& weapon) {
if (weapon_indexes_.count(weapon->name()) > 0) {
Log(LogLevel::WARNING, "Tactical System", "Weapon with name '" + weapon->name() + "' already exists in ship '" + owner()->name() + "'. Discarding this Weapon.");
return;
}
weapons_.push_back(weapon);
weapon_indexes_[weapon->name()] = weapons_.size() - 1;
weapon->RegisteredTo(this);
}
const shared_ptr<Weapon>& Tactical::GetWeapon(size_t index) {
return weapons_.at(index);
}
const shared_ptr<Weapon>& Tactical::GetWeapon(const std::string& name) {
return GetWeapon(weapon_indexes_[name]);
}
void Tactical::Update(double dt) {
for (auto weapon : weapons_) {
weapon->Update(dt);
}
}
void Tactical::FireAll(const std::vector<std::shared_ptr<objects::TargetData>>& targets) {
std::vector<objects::Target> actual_targets;
actual_targets.reserve(targets.size() * 2);
for (auto tdata : targets) {
auto systems = tdata->GetTargets();
if (systems.size() == 0)
actual_targets.push_back(*tdata);
else
actual_targets.insert(actual_targets.end(), systems.begin(), systems.end());
}
for (auto weapon : weapons_) {
if (!weapon->CanFire()) continue;
for (auto target : actual_targets) {
if (weapon->TryFire(target)) break;
}
}
}
} // namespace components
} // namespace shipsbattle
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,383
|
Q: Unexpected error after transfering website to new server - mix of php and html in IF could someone take a look at the code below and tell me why after transfering it from one server to another I suddenly get syntax error, unexpected '}' (in below example it would be in line 3)
<? if (isset ($_GET['ppp'])){ ?>
<div style="text-align:center;">div1</div>
<?php } else {?>
<div style="text-align:center;">div2</div>
<?php } ?>
Could it be possibly related to some php.ini setting? Why simply transfering the template to another server would give such mix of php and html an error in that place? Works on one server but does not on another.
A: You might be moving from a server with very old PHP to newer version of PHP. In newer versions you always have to start with
<?php
if (isset ($_GET['ppp'])){
?>
<div style="text-align:center;">div1</div>
<?php
} else {
?>
<div style="text-align:center;">div2</div>
<?php
}
?>
A: My best guess given what you posted is this:
Make sure short tags are on on your new server, you can test it by
changing the tags <? to the full PHP tags <?php. What happens is the
first block of PHP starting with <? is ignored then the next block
with the <?php starts with a } which is invalid (excluding the first
block).
This is what the server sees:
<? if (isset ($_GET['ppp'])){ ?> //------- this is just text
<div style="text-align:center;">div1</div>
<?php } else {?> //----------------------- PHP code starts here with the }
<div style="text-align:center;">div2</div>
<?php } ?>
The easy way to test it is this:
<?php if (isset ($_GET['ppp'])){ ?>
<div style="text-align:center;">div1</div>
<?php } else {?>
<div style="text-align:center;">div2</div>
<?php } ?>
If that solves the issue, then you know short tags are off on this server. You could also look at the php.ini (i suppose).
As a general rule I never use short tags, although the <?= is tempting sometimes .... lol
Cheers!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,202
|
/**
* A client to Assured Workloads API
*
* <p>The interfaces provided are listed below, along with usage samples.
*
* <p>======================= AssuredWorkloadsServiceClient =======================
*
* <p>Service Description: Service to manage AssuredWorkloads.
*
* <p>Sample for AssuredWorkloadsServiceClient:
*
* <pre>{@code
* // This snippet has been automatically generated and should be regarded as a code template only.
* // It will require modifications to work:
* // - It may require correct/in-range values for request initialization.
* // - It may require specifying regional endpoints when creating the service client as shown in
* // https://cloud.google.com/java/docs/setup#configure_endpoints_for_the_client_library
* try (AssuredWorkloadsServiceClient assuredWorkloadsServiceClient =
* AssuredWorkloadsServiceClient.create()) {
* Workload workload = Workload.newBuilder().build();
* FieldMask updateMask = FieldMask.newBuilder().build();
* Workload response = assuredWorkloadsServiceClient.updateWorkload(workload, updateMask);
* }
* }</pre>
*/
@Generated("by gapic-generator-java")
package com.google.cloud.assuredworkloads.v1;
import javax.annotation.Generated;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,870
|
{"url":"https:\/\/power-mi.com\/content\/beating","text":"The beating phenomena is the result of the interaction between two close frequencies that go alternatively in and out of synchronization. As you can seen in the waveform of Figure 6.10, a pulsation can be easily confused with amplitude modulation, when in fact it is two sinusoidal signals that are summed to form pulses. The spectrum clearly shows the frequency and amplitude of each component without the presence of any lateral bands. Since these signals are slightly different in frequency, their phase shift varies continuously from 0\u00b0 to 360\u00b0, which causes their combined amplitude to constantly change, due to the reinforcement and partial cancellation that occurs. The highest vibration amplitude occurs when both signals are in phase and therefore, the two waveforms overlap. On the contrary, the lower vibration level occurs when both signals are 180\u00b0 out of phase and therefore the two waveforms cancel each other out. In the waveform plot of Figure 6.10, the amplitude levels of the two signals are the same, thus completely canceling each other. Full cancellation is rarely seen in real plant measurements.\n\nIn a low-resolution frequency spectrum, only one peak appears, the amplitude of which rises and falls in a pulsating manner. If the spectrum is zoomed-in around this peak, you can see that in reality there are two very close peaks. The frequency difference between the two is called the beating frequency. It is not common to see the beating frequency in the spectrum since it is a very low frequency, typically in a range between 5 and 100 CPM.","date":"2022-08-16 19:52:52","metadata":"{\"extraction_info\": {\"found_math\": false, \"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\": 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\": 0.8534793853759766, \"perplexity\": 427.6068147142964}, \"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\/1659882572515.15\/warc\/CC-MAIN-20220816181215-20220816211215-00270.warc.gz\"}"}
| null | null |
Mariánský potok – potok górski w północnych Czechach w Sudetach Wschodnich, w Górach Złotych czes. Rychlebské hory
Górski potok, o długości około 5,4 km, prawy dopływ Černý potok, jest ciekiem V rzędu należącym do dorzecza Odry, zlewiska Morza Bałtyckiego.
Położenie
Źródło potoku położone jest w Czechach w kraju ołomunieckim (czes. Olomoucký kraj), w okresie Jeseník w południowo-wschodniej części Gór Złotych (czes. Rychlebské hory) na północno-zachodnim zboczu wzniesienia Studniční vrch), na wysokości ok. 850 m n.p.m., między wzniesieniami: Na Radosti po południowo-zachodniej stronie i Sokoli vrch po północno-wschodniej.
Charakterystyka
W części źródliskowej potok spływa w kierunku północno-zachodnim, stromą szeroką zalesioną doliną i płynie do miejscowości Černá Voda. Na poziomie 420 m n.p.m. potok opuszcza zalesiony teren i wpływa pomiędzy zabudowania miejscowości Černá Voda, gdzie skręca na północny wschód w kierunku ujścia, gdzie na wysokości ok. 327 m n.p.m. w centrum miejscowości Černá Voda uchodzi do potoku IV rzędu Černý potok prawego dopływu Widnej (czes. Vidnavka. Koryto potoku kamienisto-żwirowe słabo spękane i na ogół nieprzepuszczalne z licznymi progami kamiennymi. W górnym biegu w wielu miejscach poniżej poziomu 810 m n.p.m., potok tworzy urocze wodospady i kaskady. Zasadniczy kierunek biegu potoku jest północny. Jest to potok górski odwadniający wschodnią część masywu Gór Złotych. Potok w górnym biegu dziki, w dolnym biegu częściowo uregulowany. W większości swojego biegu płynie lasem, brzegi w 90% zadrzewione. Potok charakteryzuje się dużymi nie wyrównanymi spadkami podłużnymi i zmiennymi wodostanami. Gwałtowne topnienie śniegów wiosną, a w okresach letnich wzmożone opady i ulewne deszcze, które należą do częstych zjawisk w tym rejonie sprawiają wezbrania wody i często przybierają groźne rozmiary, stwarzając zagrożenie powodziowe u podnóża gór.Dopływy potoku stanowią małe cieki wodne bez nazwy.
Inne
Na poziomie 585 m n.p.m. potok przepływa obok nieczynnego kamieniołomu marmuru z XIX wieku Kamienne wały.
Dopływy
bezimienne strumienie i potoki mające źródła na zboczach przyległych wzniesień.
Miejscowości nad potokiem
Černá Voda
Przypisy
Bibliografia
Praca zbiorowa, mapa turystyczna Sudety Wschodnie, Skala 1:60.000, wydawca:ExpressMap,Warszawa ,2010, .
Góry Złote, Góry Rychlebskie, mapa w skali 1:40 000, Wydawnictwo "Plan", Wrocław 2007,
Słownik geografii turystycznej Sudetów, tom 17 Góry Złote, red. Marek Staffa, str. 49-50, Wydawnictwo I-BiS, Wrocław 1993,
Potoki Gór Złotych
Dorzecze Nysy Kłodzkiej
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,219
|
MATERIALS FOR A BIBLIOGRAPHY OF BELLS.
Collected for the Central Council by Edward F. Strange, Member of the Bibliographical Society, etc.
Preliminary Note.- By the courtesy of the Editor of "The Bell News," I am enabled to lay my material in the first place, before the readers of this paper; and, in asking for their assistance in the completion of the work, to explain the method in which I propose to carry it on.
The Rev. H. T. Ellacombe on several occasions compiled lists of books bearing upon our subject. These, with the catalogue published by the National Art Library, form the nucleus of the present collection. They will be re-published separately in these columns, except so far as they coincide; and to them will be added such titles as I have myself been able to collect.
The object of this is to obtain information of the many rare books and editions which are in the possession of ringers in various parts of the country, but which often elude the search of any one compiler. I will ask the possessors of such to keep the lists published in "The Bell News" for reference; and if on their completion, any possibility arises of making additions thereto, to have the kindness to send me such particulars as they are able, giving an exact copy of the title-page, number of pages, and size in inches of the book. I will announce the exhaustion of my own material, and then, not before, shall expect and hope to receive the general help of ringers in making the list worthy of the Central Council.
Before publication in pamphlet form, the titles will be classified.
101, Disraeli Road, Putney, S. W.
Anderson (J.).- Scotland in Early Christian Times. (Celtic Bells, pp. 167-215.) 8vo. Edinburgh, 1881.
Antiquarian Repertory, The.- A Miscellany intended to preserve and illustrate several valuable remains of Old Times. Vol. I., (Westminster Bell.) 5 vols. 4to. London, 1775-84.
Baker (W. L.).- Alteration in Bells and Machinery. "Sessional Papers, Royal Institute of British Architects," 1854-55. 4to.
Barnwell (E. L.).- Ancient Welsh Bells. "Archæologia Cambrensis," 4th series. Vol. II. 1871. 8vo.
Barraud (P.C.).- Les cloches. "Annales Archéologiques," Vol. 16, p. 325. Vol. 17, pp. 104, 278. Vol. 18, pp. 57, 145, 206, 288. Vol. 19, p. 307. 4to. Paris, 1856-58.
Beckett (Sir E.).- A Rudimentary Treatise on Clocks and Watches and Bells. Illustrations. 6th edition. (Weale's Series.) 12mo. London, 1874.
Biancolini (G.).- Notizie storiche delle Chiese di Verona raccolte da G. B. (Vol. I., Bells.) Sm. 4to. Verona, 1749.
Boulter (W. C.).- Inscriptions on Church Bells, Yorkshire. "Yorkshire Archæological and Topographical Journal," Vols. II-III., 1873-74. 8vo.
Brown (Rev. A. W.).- History and Antiquities of Bells. "Associated Architectural Societies," Vol. IV., 1857-58. 8vo.
Church Bells.- "Quarterly Review," Vol. XCV., No. 190. Sept. 1854. 8vo.
— Cloches Les. "Annales Archéologiques," Vol. 5, p. 180. 4to. Paris, 1846.
Cooke (T. L.).- Ancient Irish Bells. "Kilkenny Archæological Society," Vol. II., 1855. 8vo.
Denison (E. B.).- Lectures on Church Building; with some practical Remarks on Bells and Clocks. 2nd edition. 8vo. London, 1856.
— Designs, priced, for Bells, Cranks, Screws, and other Domestic and General Tools, &c, Ob. fol. London, 1770.
Didron (A. N.).- Clochette romane â jour. "Annales Archéologique." Vol. I., p. 262. 4to. Paris, 1844.
Dunkin (E. H. W.).- Church Bells of Cornwall. "The Reliquary." Vols. XIII.-XVI., 1872, &c. 8vo.
— The Church Bells of Cornwall: their Archæology and present condition. Illustrations. 8vo. London, 1878.
Ellacombe (H. T.).- Practical Remarks on Belfries and Ringers. Read with a Paper on Bells before the Bristol Architectural Society.
2nd ed., with an Appendix on Chiming; with illustrations. 8vo. London, 1861.
— The Church Bells of Devonshire and Somerset. "Transactions of the Exeter Diocesan Architectural Society," 2nd series, Vols. I.-III., 1867-78. 4to.
— Records relating to the Bells of Exeter Cathedral. "Transactions of the Exeter Diocesan Architectural Society," 2nd series, Vols. II.-III., 1872 and 1875. 4to.
— The Church Bells of Gloucester. To which is added a Budget of Bell Matters of general interest. Plates. 4to. Exeter, 1881.
— Essai sur le Symbolisme de la Cloche dans ses Rapports et ses Harmonies avec la Religion. 8vo. Poitiers, 1859.
Ferencz (S.).- Monuments d'Architecture inedits. Text in French and German. Fol. Paris et Leipzig, 1869.
Forbes (A. P.).- St. Fillan's Bell. "Proceedings of the Society of Antiquaries of Scotland," Vol. VIII., 1871. 4to.
Fowler (Rev. J. T.).- On some Church Bells in the City and Neighbourhood of Durham, at Newcastle-on-Tyne, &c. "Gentleman's Magazine," Sept. 1865, p. 267. 8vo. London, 1865.
Franks (A. W.).- Irish Bells belonging to the Archbishop of Armagh. 5 woodcuts. "Proceedings of the Society of Antiquaries," 2nd series, Vol. III., 1864-67. 8vo.
Gatty (Rev. A.).- The Bell; its origin, history, and uses. 8vo. London, 1848.
— The Ecclesiastical Bell. "Associated Architectural Societies," Vol. III., 1854-55. 8vo.
Hand-Bells.- Primitive Hand-Bells. 1 plate. "Proceedings of the Society of Antiquaries of Scotland," Vol. I., 1855. 4to.
Hope (W. H. St. J.).- Bells of All Saints' Church, Derby, 2 plates. "Derbyshire Archæological Society," Vol. I., 1879.
Hubbard (H.).- Elements of Campanalogia, or an Essay on the Art of Ringing. 12mo. Norwich, 1854.
Jewitt (L.).- Church Bells of Derbyshire, described and illustrated. "The Reliquary," Vol. XIII., 1872-73. 8vo.
— Chesterfield Bell Founders. "The Reliquary," Vol. XVI., 1875-76. 8vo.
Kilmichael-Glassrie.- Ancient Ecclesiastical Bell and Chain found at. "Archæologia Scotica," Vol. IV., 1857. 4to.
Langrishe (R.).- Irish Church Bells. "Kilkenny Archæological Society," 4th series, Vol. V., 1881. 8vo.
L'Estrange (J.).- The Church Bells of Norfolk, where, when, and by whom they were made; with the Inscriptions on all the Bells in the County. (Illustrations.) Imp. 8vo. Norwich, 1874.
Lomax (B.).- Bells and Bell-ringers. 8vo. London, 1879.
Lukis (Rev. W. C.).- An Account of Church Bells; with some notices of Wiltshire Bells and Bell Founders, a comparative scale of Tenor Bells, and Inscriptions from nearly 500 parishes in various parts of the Kingdom. 8vo. London, 1857.
Lukis (Rev. W. C.).- History of the Bell Foundry, Salisbury. "Journal of the British Archæological Association," Vol. XV., 1859. 8vo.
— Bell Foundry of Gloucester. "Journal of the British Archæological Association," Vol. XXVII., 1871. 8vo.
Maggi (G.).- De tintinnabulis. Plates. 8vo. Amstelodami. 1664.
Montferrand (A. R. De).- Description de la grande Cloche de Moscow. Fol. Paris, 1840.
North (T.).- The Church Bells of Northamptonshire: their inscriptions, traditions and peculiar uses; with chapters on Bells and the Northants Bell Founders. (120 illustrations.) 4to. Leicester, 1878.
— Church Bells of Lincolnshire. "Archæological Journal," Vol. XXXVII., 1880. 8vo.
— The Church Bells of the county and city of Lincoln: their founders, inscriptions, traditions, and peculiar uses, &c. With illustrations, 4to. Leicester, 1882.
— Church Bells of Bedfordshire. "Archæological Journal," Vol. XXXIX, 1882. 8vo.
P. (J. H.).- Arrangements for the hanging of Bells in Churches without Towers. "Archæological Journal," Vol. III., 1846.
Peacock (E.).- Bronze Bell from Bottesford Church and others. "Proceedings of the Society of Antiquaries," 2nd series, Vol. V., 1870-73. 8vo.
Perrott (R.).- Notes on portable Hand Bells in Brittany. "Archæologia Cambrensis," 3rd series, Vol. II., 1856. 8vo.
Raven (J. J.).- The Church Bells of Cambridgeshire. Plates. 8vo. Lowestoft, 1869.
Reeves (Rev. W.).- St. Patrick's Bell. 5 chromo-lithographs and description. 4to. Belfast, 1850.
— Ecclesiastical Bells. "Proceedings of the Royal Irish Academy," Vol. VIII., 1861-64. 4to.
Reeves (Rev. W.) and McClelland (J.).- St. Mura's Bell. 2 plates. "Ulster Archæological Journal," Vol. I., 1853. 4to.
St. Fillan's Bell.- Notice and 2 woodcuts. "Proceedings of the Society of Antiquaries of Scotland." Vol.. XIII., 1879. 4to.
Sauvageot (C.).- Etude sur les cloches. "Annales Archéologiques," Vol. XXII., p. 213. 4to. Paris, 1862.
Sauvageot (C.) et Fons-Melicocq (Baron de la).- Les cloches. "Annales Archéologiques," Vol. XVII., p. 357. 4to. Paris, 1857.
Scarth (Rev. H. M.).- Bell in Clapton-in-Gordano Church. "Journal of the British Archæological Association," Vol. XXXII., 1876. 8vo.
— Pre-Reformation Bells in Somersetshire. "Journal of the British Archæological Association." Vol. XXXII., 1876. 8vo.
Smith (C. H.).- Remarks on the Forms, Methods of Casting, and Ringing of Bells. "Sessional papers, Royal Institute of British Architects." 4to. 1855-56.
Sperling (Rev. J. H.).- Church Bells; their antiquities and connexion with Architecture. "Ecclesiologist," Vol. XXVI., 1865. 8vo.
Stiff (W. P. W.).- Notes on Nottinghamshire Campanology. "The Reliquary," Vol. XIII., 1872-73. 8vo.
Troyte (C. A. W.).- Church or Hand Bell Ringing. 12mo. London, 1869.
Turner (T. A.).- The Bell-Founders of Bucks. 8vo. Aylesbury, 1872.
Tyssen (A. D.).- The Church Bells of Sussex. "Sussex Archæological Collections," Vol. XVI. 8vo. Lewes, 1864.
Ventress (J.).- Bells of St. Nicholas' Church, Newcastle. "Archæologia Æliana," New series, Vol. II., 1858. 8vo.
Vincent (A. J. H.).- Lettre sur l'ancienne cloche du beffroi de Boulogne. "Revue Archéologique," Vol. XVI., p. 70. New series. 8vo. Paris, 1859-60.
Way (A.).- Silver Bell and Signet Ring of Mary Queen of Scots. "Archæological Journal," Vol. XV., 1858. 8vo.
Westwood (J. O.).- Ancient portable Hand Bells of the British and Irish Churches. "Archæologia Cambrensis," Vols. III.-IV., 1st series, 1848-49. 8vo.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,598
|
Q: Developing for Silverlight using Javascript - is this possible? My tech lead told me its possible to develop Silverlight apps solely with Javascript. I did some googling and binging; it likes this was the primary development method for developing pre Silverlight 2.0. It also seems to have lost favor to C# as of SL 2.0.
Is it still possible to develop Silverlight apps solely with Javascript? I know Silverlight and the browser have extensive scripting capabilities and can be scripted via JS; but can I build an SL app with it?
A: It is still possible, though the experience is very different. You must load an initial Root visual element via a source property on the Silverlight object referencing a XAML file on the server, but after that you have full access to the visual tree via javascript.
The below Test.html and Root.xaml files together will produce a testable page if placed in the same folder.
Note the differences from a "standard" (i.e. *.xap source) scenario - the 'source' parameter on the SL object tag is set to a .XAML file instead of a .XAP file. The .XAML file is also different from what you would get in a default SL application in VS: the x:Class="MyApp.MainPage" is missing from the root element, and the root element is a Grid (or any Visual element) instead of a UserControl element. This is because there is no application (at least not loaded from the .XAP - I assume the SL control actually creates a default application instance in the process of loading the Root.XAML file), and no UserControl because there is no code-behind. This is consistent with the Pre-SL1.1/2.0 experience of no UserControls.
Additionally, you will need to reference Javascript API for Silverlight Reference in your javascript coding. Have fun with the FindName method and/or walking the Visual Tree! Those are the only ways to get references to any visual objects to manipulate from your code!
Test.html
<html>
<body>
<object id="slObject" data="data:application/x-silverlight-2," type="application/x-silverlight-2" width="400" height="300">
<param name="source" value="Root.xaml"/>
<param name="onError" value="onSilverlightError" />
<param name="background" value="white" />
<param name="minRuntimeVersion" value="4.0.50826.0" />
<param name="autoUpgrade" value="true" />
<a href="http://go.microsoft.com/fwlink/?LinkID=149156&v=4.0.50826.0" style="text-decoration:none">
<img src="http://go.microsoft.com/fwlink/?LinkId=161376" alt="Get Microsoft Silverlight" style="border-style:none"/>
</a>
</object>
<input type="button" onclick="sayGoodBye();" value="Say Goodbye, Silverlight!" />
<script>
function sayGoodBye() {
var slObject = document.getElementById('slObject');
var slContent = slObject.Content;
var layoutRoot = slContent.FindName('LayoutRoot');
var message = layoutRoot.FindName('Message');
message.Text = 'Goodbye';
}
</script>
</body>
</html>
Root.xaml
<Grid x:Name="LayoutRoot" Background="White"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
mc:Ignorable="d"
d:DesignHeight="300" d:DesignWidth="400">
<TextBlock x:Name="Message" Text="Hello, Silverlight!" FontSize="30" HorizontalAlignment="Center" VerticalAlignment="Center" />
</Grid>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 311
|
Patrick Jane je fiktivni lik i protagonist kriminalističke serije Mentalist kojeg glumi australski glumac Simon Baker. Patrick Jane je nezavisni policijski savjetnik koji je u početku radio za CBI da bi poslije počeo raditi za FBI. U rješavanju zločina koristi visoko razvijene vještine primjećivanja, zaključivanja i umne manipulacije, koje je stekao radeći kao lažni vidovnjak.
Likovi iz televizijskih serija
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,668
|
class TumblerNote {
public:
TumblerNote();
~TumblerNote();
b2Vec2 GetPosition();
b2Body *GetBody();
rgbColour GetColour();
int GetNote();
void Set(int note, b2Body *body);
void UnSet();
bool IsInitialised();
static void Seed();
private:
int note;
rgbColour colour;
b2Body * body;
};
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,711
|
using namespace webdown;
// @#$%@#$% C++ >:(
std::string SourceReference::
source(const Source& src) const
{
if (!src.hasBlock(name))
{
std::cerr << "Cannot Find Source Block " << name << std::endl;
return std::string();
} else
return src.getBlock(name).source(src);
}
Source::
Source()
: block(NULL), raw(NULL)
{
// nothing happened
}
Source::
Source(std::istream& in)
: block(NULL), raw(NULL)
{
read(in);
}
Source::
~Source()
{
// nothing happened
}
bool Source::
read(const std::string& src)
{
std::istringstream in(src);
return read(in);
}
void Source::
_dumpWorkingBlock()
{
if (block != NULL)
{
_dumpRawCode();
blocks[block->name] = block->value;
delete block;
block = NULL;
}
}
void Source::
_dumpRawCode()
{
if (raw != NULL)
{
// std::cout << "CODE: " << raw->str() << std::endl;
block->value.append( new RawSource(raw->str()) );
delete raw;
raw = NULL;
}
}
void Source::
_newWorkingBlock(std::string line)
{
_dumpWorkingBlock();
size_t a = line.find("<{");
size_t b = line.find("}>");
if (a == std::string::npos || b == std::string::npos)
{
// not a new code block
return;
}
block = new WorkingBlock();
block->name = trim(line.substr(a+2, b-a-2));
if (line.find("+=") != std::string::npos) // append code block
{
if (blocks.count(block->name) == 0)
{
// code block does not exist
// TODO: say which line number.
std::cerr << "Code block " << block->name << " does not exist at this point" << std::endl;
delete block;
block = NULL;
return;
} else
{
block->value = blocks[block->name];
}
} else if (line.find("=") != std::string::npos) // assign code block
{
block->value = SourceBlock();
} else
{
// not a new code block
delete block;
block = NULL;
return;
}
// std::cout << "NEW BLOCK: " << block->name << std::endl;
}
void Source::
_processCode(std::string line)
{
// recursive base case
if (line.empty()) return;
size_t a = line.find("<{");
size_t b = line.find("}>");
if (b < a) b = line.find("}>", a);
if (a == std::string::npos || b == std::string::npos)
{
// CASE: no valid source reference on this line
// so we just accumulate it as raw source
if (raw == NULL) raw = new std::stringstream();
(*raw) << line << std::endl;
} else
{
// CASE: source reference on this line
std::string before = line.substr(0,a); // grab the source before it
if (!before.empty())
{
// CASE: raw code before the reference in this line
// so we just accumulate it as raw source
if (raw == NULL) raw = new std::stringstream();
(*raw) << before; // insert a space just in case
}
// append the referenced block as SourceReference
_dumpRawCode();
std::string refname = trim(line.substr(a+2, b-a-2));
// std::cout << "REF: " << refname << std::endl;
block->value.append( new SourceReference(refname) );
// recursively process the rest of the line
_processCode( trim(line.substr(b+2)) );
}
}
bool Source::
read(std::istream& in)
{
std::string line;
while (_getLine(in, line)) {
if (!isBlank(line)) {
// std::cout << line << std::endl;
if (line.find(" ") == 0)
{
// process the code only if we are in a working block
// (otherwise it's just a pretty code block not to be compiled)
if (block != NULL)
_processCode(line.substr(4));
} else if (line.find(" ") == 0)
{
_newWorkingBlock(line);
} else
{
_dumpWorkingBlock();
}
}
}
_dumpWorkingBlock();
return true;
}
void Source::
write(std::ostream& out) const
{
if (blocks.count("Main") == 0)
{
std::cerr << "No Main Source Block!" << std::endl;
} else
{
out << blocks.at("Main").source(*this);
}
}
bool Source::
hasBlock(std::string name) const
{
return blocks.count(name) > 0;
}
SourceBlock Source::
getBlock(std::string name) const
{
return blocks.at(name);
}
bool Source::
_getLine(std::istream& in, std::string& line) {
// Handles \n, \r, and \r\n (and even \n\r) on any system. Also does tab-
// expansion, since this is the most efficient place for it.
line.clear();
bool initialWhitespace=true;
char c;
while (in.get(c)) {
if (c=='\r') {
if ((in.get(c)) && c!='\n') in.unget();
return true;
} else if (c=='\n') {
if ((in.get(c)) && c!='\r') in.unget();
return true;
} else if (c=='\t') {
size_t convert=4;
line+=std::string(convert-(line.length()%convert), ' ');
} else {
line.push_back(c);
if (c!=' ') initialWhitespace=false;
}
}
return !line.empty();
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,860
|
You are here: Home / Team / David Wellisch
David Wellisch
David is co-founder and CEO of Collage Group, previously known as the Latinum Network. From its inception in 2009, the company has grown to serve nearly 100 leading corporations, 230+ consumer brands, in 16 different industries. David is passionate about entrepreneurship and company building.
Prior to Collage Group, he was the founder, vice president and general manager of AOL Latino, AOL's Hispanic service. He was responsible for the entire operation, including overseeing programming, product development, customer service, brand and acquisition marketing, strategy and business development. At its peak, AOL Latino produced $130MM in revenue from subscriptions and advertising. David was also Executive Director of strategy at AOL's web properties. Before AOL, he worked in private equity at Allied Capital and strategic consulting at Gemini Consulting. Originally from Ecuador, David holds an undergraduate degree from Brown University and an M.B.A. from Harvard Business School. Currently, he sits on the Board of Trustees for Georgetown Day School in Washington, D.C. David is also a member of the Washington, D.C. chapter of the Young President's Organization. David is married, has three awesome kids, and currently lives in DC.
Collage Group
Allan Hakky
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,453
|
Tower of God: Johnny Yong Bosch Interview (EXCLUSIVE)
One of the many Crunchyroll Originals hitting the scene during the Spring Anime Season is Tower of God. As part of our partnership and extensive yet ever-growing content coverage with Crunchyroll and Tower of God. We were offered the opportunity to review anime veteran Johnny Yong Bosch.
Many might know him as the second Black Ranger Adam Park as seen in n the Mighty Morphin Power Rangers series and its feature film which was released in 1995. Bosch has a vast resume under his belt voicing iconic anime characters like Ichigo Kurosaki (Bleach), Vash the Stampede (Trigun), and most recently Giyuu Tomioka in Demon Slayer: Kimetsu no Yaiba and the villain Nine in My Hero Academia: Heroes Rising.
Read on ahead and check out our exclusive interview with Johnny Yong Bosch as we discuss the character of Bam, his unlikely allies, and what lessons you can learn from watching anime!
Hello Johnny, I want to formally introduce myself. My name is Osvaldo J. Madera of The Nerdy Basement. I want to thank you in advance for your time and consideration and most importantly this interview opportunity. I appreciate you taking time out of your day to answer my questions.
TNB: While I'm aware of your long anime resume, me being of a fan of yours dates all the way back to the early 90s watching you on tv as the Black Ranger in the Power Rangers tv series, the film and eventually moving into Power Rangers Turbo and all that (I will never forget your disappointment in being a frog in the Power Rangers film lol), how did those early acting years help you and mold you into becoming the multi-talented actor you've become today?
Johnny: I basically screwed up a lot and eventually started learning what to do, sorta.
TNB: Let's talk about your current anime project Tower of God, you voice The 25th Bam (Bam) in the series. A young boy in search of Rachel who has essentially moved onto the other side. Tell us more about his journey throughout the series and his search for Rachel and how deep is that connection to her.
Johnny: Without spoilers, there's a lot more to come.
TNB: What about his relationship with Khun and Rak? An odd trio but they seem to make things work despite how different everyone is internally and externally.
Johnny: These are the guys you want on your squad. I imagine in our world they'd be gaming with each other online.
TNB: What attracted you to the project and more specifically the character of Bam?
Johnny: For me, it felt poetic in the beginning, and then so many questions were raised within the story and now I'm curious to find out the answers.
TNB: I always have the tendency of telling people who do and do not watch anime, whether it's on Twitter or in real life, that anime teaches some of life's best lessons. What life lessons has anime taught you?
Johnny: Never fail to try more.
TNB: What do you think fans watching the Tower of God can learn from the series?
Johnny: It's probably gonna be different for everyone. I just hope everyone enjoys it.
TNB: You've done some remote voice acting for Tower of God, correct? How's that been going for you?
Johnny: After my first session, we were put on a stay-at-home order. My following sessions were all work-from-home. Fortunately for me, I have a professional set up at home.
I want to once again thank you for your time and the opportunity, Johnny. I hope we can do this again down the line!
Tower of God is a Crunchyroll Original in collaboration with WebToon. You can catch Johnny Yong Bosch as Bam in Tower of God streaming both subbed and dubbed exclusively on Crunchyroll.
For more exclusive interviews, anime reviews, news, trailers, and updates make sure to keep it locked right here at The Nerdy Basement.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,160
|
The Posi-Grip Zipper Pull incorporates a positive "cam pinching" system into a zipper pull end piece. This ensures it locks securely onto the cord. It is ideally used with 1/8" diameter bungee or braided cord as well as 1/4" flat cord.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,459
|
If you're into sprites. Ok, if you're REALLY into sprites, then this should definitely be an interesting image-fest for you. After the jump, check out two AMAZING artistic renditions of Link and Zelda, created by an artist using only sprites from the Legend of Zelda franchise.
I certainly hope that there was some sort of program involved with rendering these images [they're nearly 7 MB a piece], because if this is photo-shopped by hand, I would probably jump out of a window in sheer amazement. Ok, it'd be just a one-story window and into a nice cushy pillow pile, but my statement stands.
Clearly you've voted for ACT Today already, right? If not, GET ON IT! And if so, AWESOME! Asking people for help can be a daunting task, and that's why today I had to draw that Masked Jungle Warrior, Odolwa.
For the third episode of Route TSG, Gwellin shows his A Link to the Past route taken during the third Zelda Marathon in March.
Something it seems gamers and gaming websites love to do is rank things. Don't worry though, unlike all the other ranking articles, I'll keep my list to one page. What else better to rank than the Zelda series right?!? Not to sound cocky, but I think I've got a pretty good feel for the series. I used to sit in my diaper for hours at a time watching my dad solve both Zelda 1 and 2 on the NES before I had even learned talk. By the time I was 3, I was able to take on Zelda by myself. My dad would clear the dungeons and then hand the controller over to me for the boss battles. At age 4 I was able to complete the game in its entirety by myself. You can probably understand why Zelda would become my favorite franchise, and eventually would lead me to creating a group of gamers that would speed run the entire franchise.
Nintendo of America has dated The Legend of Zelda Spirit Tracks for December 7th 2009.
Even though Phantom Hourglass was not the highlight of my SpeedGamer career, I'm really looking forward to its follow up. The train concept should freshen up the formula a bit, lets hope that it's implemented into the story well.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,931
|
Q: Functional programming convert array of Monads to Single Monad of array I have array of monads like this
[Left("a"), Left("b"), Left("c")]
I want to convert it to this shape
Left(["a", "b", "c"])
How can I do this?
Is there specific operator for doing something like this or should I write it myself?
A: As the comments suggested, this would be an unusual function for Either.Left type. Ramda's sequence would work with Either.Right:
console .log (
sequence (of, [Right('a'), Right ('b'), Right ('c')] ) //~> Right (['a', 'b', 'c'])
)
<script src="//bundle.run/ramda@0.26.1"></script>
<script src="//bundle.run/ramda-fantasy@0.8.0"></script>
<script>const {sequence} = ramda, {Either: {Left, Right, of}} = ramdaFantasy</script>
But that's because Right connotes successful processing with a resulting value and Left connotes a failure which will stop further processing. It's much like Maybe except that it captures the initial error which halted processing.
There are other types that capture multiple errors. The Validation types from Folktale or Monet, for instance, capture multiple errors, but those would generally be wrapped in a single Failure instance, not in an array of them.
Clearly you can write your own function for this, but it's unlikely to be one that follows well-defined laws.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,794
|
/**
* The JAXB POJOs for the
* <a href="http://fusesource.org/schema/fabric/monitor/xml-configuration.html">XML Configuration</a>
* of the FON Launch module.
*/
@javax.xml.bind.annotation.XmlSchema(
namespace = "http://fusesource.org/schema/fabric/monitor",
elementFormDefault = javax.xml.bind.annotation.XmlNsForm.QUALIFIED)
package io.fabric8.monitor.api;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,611
|
Henry Childs (April 16, 1951 – June 3, 2016) was an American professional football player who was a tight end in the National Football League (NFL) for the Atlanta Falcons, New Orleans Saints, Los Angeles Rams, and the Green Bay Packers. He was a Pro Bowl player in 1979.
Childs was inducted into the New Orleans Saints Hall of Fame in 1994.
On June 3, 2016, he had a massive heart attack while driving in his car and was pronounced dead at the age of 65 in Thomasville, Georgia.
References
1951 births
2016 deaths
American football tight ends
Kansas State Wildcats football players
Atlanta Falcons players
National Conference Pro Bowl players
New Orleans Saints players
Los Angeles Rams players
Green Bay Packers players
People from Thomasville, Georgia
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,567
|
Extinction
The Wasteland Chronicles, Volume 6
Kyle West
Published by Kyle West, 2014.
This is a work of fiction. Similarities to real people, places, or events are entirely coincidental.
EXTINCTION
**First edition. April 13, 2014.**
Copyright © 2014 Kyle West.
ISBN: 978-1498911597
Written by Kyle West.
10 9 8 7 6 5 4 3 2 1
Also by Kyle West
The Wasteland Chronicles
Apocalypse
Origins
Evolution
Revelation
Darkness
Extinction
Watch for more at Kyle West's site.
# Table of Contents
Title Page
Copyright Page
Also By Kyle West
Prologue
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
About the Author
Contact
Glossary
Also By Kyle West
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# Prologue
Makara, Samuel, Michael, and Julian had escaped the Gilgamesh crash site. At the same time, there were certain facts that couldn't be ignored.
While they had escaped, they were still very much in danger. They were in the Great Blight, far from water and help, and likely surrounded by Blighters. It would take days, if not weeks, for them to make it back to the Wasteland – assuming they didn't die on the way. There was no way of knowing their destination, either.
One thing was certain, however – without a spaceship, the Exodus was stranded in Bunker 84 with only two weeks of food left. And unless Augustus rescued them, they would eventually starve.
I couldn't see any solution other than what Augustus had proposed: to join forces with him. The Wanderer had said we needed to unite, but all the same, it was hard to imagine a world where the Empire, the Reapers, and the Angels worked together.
But, that was what had to happen if we were ever to beat the Radaskim.
Makara, at least, wouldn't stand for such an alliance, mostly because of the Reapers. Both she and Anna wanted Carin Black dead for his crimes. But what choice did we have? Askala was ready to wipe out humanity, and she wasn't going to stop with the Wasteland.
Despite discovering my four friends were alive, we were still in the custody of Emperor Augustus. Anna, Ashton, and I had each been locked in separate cabins. If there was a future alliance in the works, as Augustus had said, it wasn't off to a great start. The Emperor's first item of business was probably going to Los Angeles to meet with Carin Black and the Reapers.
Augustus and Black held all the cards. I didn't know what Carin Black was like. All I knew was that he had destroyed both Makara's and Anna's lives, not to mention Samuel's, and countless others as well. Back in 2055, the powerful Reapers overthrew the weakening Lost Angels, killing their leader Raine, and forcing surviving members to flee into the Wasteland or face enslavement. Both Makara and Samuel had been forced to find new lives: Makara as a Raider, and Samuel as a scientist in Bunker 114. Anna's settlement of Last Town had been assaulted by the Reapers earlier. When her father was killed in the fighting, she and her mother fled into the Wasteland, where they'd survived for years.
Whoever Carin Black was, he wasn't someone I wanted to work with. I wasn't even sure if he was even someone I wanted alive, and I definitely knew both Makara and Anna wanted him dead. Samuel had said little on the subject, but his feelings were probably much the same. At some point, Anna, Ashton, and I would have to meet the Warlord of the Reapers.
A change was coming. I didn't know what it was, but I felt it. It wasn't just with the Great Blight and Askala. It was with humanity.
Augustus had rescued us, but whose side was he really on? He talked of alliance and friendship, but it was difficult to believe him. Maybe he did want everyone working together. As Emperor of Nova Roma, he was probably used to getting people who didn't like each other on the same page.
I had a hard time figuring out the guy. In the end, though, Augustus was probably on his own side. His hope was probably to get both the Reapers and the Angels working for him. Assuming we defeated Askala, Augustus could take control of the Wasteland.
I had nothing but questions, questions that couldn't be answered until I was let out of my cabin. For now, Augustus thought victory was as simple as attacking the Voice and defeating it in open battle. We hadn't been able to tell him everything else we'd learned, including my own role.
The only way to defeat Askala was infecting her with the Elekai version of the xenovirus, which was something only I could do.
According to the Wanderer, it would cost me my life. He had also said that we must unite. Maybe it was best to follow his advice, regardless of what anyone thought about it. Allying the Reapers, the Empire, and the Angels would be the most difficult thing we'd done.
And none of us had any idea how to do it.
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# Chapter 1
"Did you hear that?" Julian asked.
Makara opened her eyes. "Hear what?"
The Recon sat alone in a mountain valley coated with xenofungus, sheltered within a stand of xenotrees. The trees' spindly limbs fell in a silvery cascade, obscuring the Recon from view. The fungus itself glowed in the night, milky and pink. Frail alien trees spiraled from the xenofungal bed, their hanging limbs and tufts and fronds spilling like silk onto the ground. Tall pink tubes rose from the fungus, their bell openings dripping slime that collected in sticky, shining pools. Alien chirps, clicks, and shrieks sounded from distant hills.
"Is there something out there?" Samuel asked.
"What do you think it was, Julian?" Makara asked.
"Hard to believe," Julian said, "but it sounded like a spaceship."
Outside, the xenolife chirped on.
"No, it couldn't have been that," Michael said, now awake. "You saw Aeneas fall."
Julian paused. "Maybe it was nothing. I was just sleeping, and thought I heard it. I must have dreamt it."
Satisfied at this answer, everyone closed their eyes once again. Bunker 84 was far, but Samuel had estimated that the group could get there in a few more days. For some reason, Julian doubted that assessment.
They'd be lucky to get there at all. Three days after Samuel's estimate, and they had crossed over into Nevada. They had expected to find desert, but instead found more Blight. It had expanded farther than any of them would have thought.
Julian closed his eyes. Even though he was exhausted, he couldn't fall asleep. It was Samuel's turn to be on watch, as Julian had done it last night, which meant sleep should have been coming easy. But he couldn't stop thinking about the others. Alex, Anna, Ashton – all dead, crashed into the mountainside. They had driven by the wreckage of Aeneas first thing, before Gilgamesh itself had become a burning ruin. Aeneas's hull was so disintegrated that it was impossible to explore the ship on foot, much less check for empty escape pods.
They had searched a full day throughout the valley, but it became clear that they couldn't search forever. It was far too cold and dangerous to go into the mountains out of the Blighted valley. Besides, a pod could have landed anywhere, and the Recon had very little in the way of emergency supplies – supplies they couldn't stretch on a fruitless search. Already, they were rationing, eating half as much as they should.
In the end, Samuel had made the final decision. They were to return to Bunker 84 and try to save everyone there without spaceships. Assuming they actually made it to the Bunker, this would involve a journey south over mountains in the dead of winter. Assuming anyone made it, Samuel planned to offer the New Angels' services to Augustus and Carin Black. Not a satisfying ending, but the only ending, nonetheless.
During their journey southwest, the Great Blight had been surprisingly empty of threats. Makara guessed that Askala was gathering her swarm in one place, for one final attack...which had emptied much of the Great Blight. Perhaps they hadn't run into any crawlers, but that didn't mean they wouldn't at some point.
If, against all odds, they reached Bunker 84, then what? The New Angels would only freeze and starve in the mountains on the perilous journey south. Many would rather stay in the Bunker, even if there wasn't enough food to go around. By the time they reached Los Angeles on foot, the New Angels' dead would probably outnumber the living. Their numbers would be too few to have leverage with Augustus or Black.
Maybe the number of Angels would be as few as zero.
Maybe that was why Julian had heard the spaceship. If not wishful thinking, then maybe it was wishful hearing. But Julian couldn't shake the thought. What if it was a spaceship? There were two left – Perseus and Orion, both locked away in Bunker Six. There was no way anyone besides the New Angels could access it. Bunker Six was not too far north of Bunker One, and was even deeper in the Great Blight, making it more inaccessible than the former headquarters of the United States government. There was no way either of those ships could be flying around. There was no one on Earth that could pilot them besides Makara, Ashton, and Anna.
Again, Julian's thoughts turned back to Alex. If he was really dead, they had to find another solution. That meant talking to the Wanderer. But how, without having the speed of a spaceship?
No one had explicitly said that all hope was gone in the two days since the crash. Samuel had merely ordered everyone to head back to Bunker 84. Julian was sure, however, that he wasn't the only one being haunted by these thoughts.
The only solution that Julian could see was for the Wanderer to select a new person to fill Alex's role. But maybe that wasn't possible. After all, hadn't the Wanderer prophesied that Alex was the only one who could defeat the Radaskim?
In the passenger seat, Samuel adjusted his legs.
"Still awake, Julian?"
Julian nodded. "Yeah."
"I'm stepping outside."
Makara stirred as the door opened, but didn't wake. Michael opened a single eye, then fell right back asleep.
Julian opened his own door, stepping onto the padded xenofungal surface of the Great Blight. The air was warm and moist – both water and heat rose from the organic growth below. A gentle wind blew, warm. It seemed strange that the Great Blight, the most dangerous place in the world, had the mildest weather.
A few steps ahead, Samuel swept hanging tree limbs out of the way, revealing an incline of a few feet, ending at a high precipice. Blighted plains stretched below. At the end of the plains, about three miles distant, rose glowing, jagged mountains. They reminded Julian of teeth; the broken, rotting teeth, of a broken, rotting world.
Samuel went to stand at the edge of the cliff, and Julian stood a short distance to the side. Above, the skies had somewhat cleared, a rare occurrence in the Great Blight. A small window to the heavens had opened, a black void filled with a swath of bright, shining stars. A purplish band of light streaked across the small opening in the clouds.
"The Milky Way," Julian said, stepping up beside Samuel. "We'd see it at night, in New America. Here, not so much. Too cloudy."
Samuel gazed upward, saying nothing. Julian scanned the valley below. According to the Recon's navigation, there should have been a highway down there, running east and west. There was no sign of it, now. First, buried by years of dust. Next, buried by the xenoviral incursion. Likely, it was buried forever, just as all the Old World would one day be.
"Even if we make it in time," Julian said, "will it make any difference?"
Staring out at the alien landscape, Samuel took time before he answered, as if weighing the unexpressed feelings behind the question itself.
"You never know. You should never give up hope as long as you're breathing. You never know if salvation is beyond the next rise."
Julian was about to respond, but found he had no words.
They stood another moment before Julian spoke again.
"What about Alex, though?" Julian asked. "The Wanderer said..."
"Alex cannot be dead," Samuel said.
Julian's eyes widened. "How do you know that?"
"It all depends on the interpretation of the Wanderer's prophecy," Samuel said. "The Wanderer said that Alex had to be there for the final battle. Therefore, he has to be there. I refuse to believe that he is dead."
Julian didn't know how to respond. Samuel's belief was not based on logic – a rarity for someone so rational. Then again, his reason did have a certain logic to it. As Samuel said, it all depended on the interpretation.
"They have their course," Samuel said. "We have ours."
Something about Samuel's words struck Julian.
"Wait. Their course?"
"If they made it...as I believe they have...then they are doing what they have to do. They must have made peace with the fact that they'll never see civilization again." Samuel turned to Julian. "Ragnarok Crater isn't too far from the crash site. Maybe two hundred miles. It can be done."
That gave Julian pause. Could they really be trying to walk to Ragnarok Crater? To travel through the mountains and the Great Blight, without food and water, seemed impossible. Then again, maybe not. If Alex was alive, he could summon his dragon friend and be rescued. If that was the case, Alex would have found them by now. It had been three days, after all, and no sign of any dragon – Elekai or Radaskim.
"I choose to believe because I cannot accept the alternative," Samuel said. "If Alex is dead, then there is nothing more we can do. But until we see it with our own eyes...we fight."
The two friends gazed into the silent valley. Julian could only wonder where Samuel found his hope. Perhaps he was still young enough to have it. Not that Julian was much older than Samuel's twenty-three.
Twenty-five years I've lived, Julian thought. Would two more years make such a difference in my outlook on life?
Maybe hope was a choice. Julian wondered what it would be like to believe Alex was alive. That Anna and Ashton were both alive.
And suddenly, he realized that what he heard earlier made sense. It wasn't just a dream.
"I did hear a spaceship."
Samuel turned, raising an eyebrow. "You're sure?"
"At least, I think I did," Julian said.
"Who would fly it?" Samuel asked. "If there was one, it came from Bunker Six."
"And yet, you guys made it as far as Bunker One." Julian looked at Samuel. "It's not impossible, is it?"
"Yes, but what would that mean?" Samuel asked. "Who would be flying it?"
"I don't know," Julian said. "But if someone had a spaceship, where would they go?"
"Skyhome, perhaps."
As if he had received some sort of answer, Samuel turned back for the Recon.
"Where are you going?" Julian asked.
"Makara can try contacting Skyhome," Samuel said. "It'll mean waiting until it's overhead, but Skyhome can track the spaceship for us. If it exists, that is."
Julian nodded. It seemed like a long shot. It would mean watching the night sky nonstop until they saw Skyhome streaking across, like a massively bright star. They could send out a message every few minutes. Hopefully, someone from Skyhome would answer.
If there was another spaceship, it would change everything. They could track it down and find out who was flying it. Maybe that ship had rescued Alex and the others – the escape pod would have given off a signal that could be tracked by any ship in the U.S. Space Fleet.
But if that was the case, would that ship have found the others? And if so, where were they going?
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# Chapter 2
The dream faded and I awoke aboard the Orion. I had seen them alive, figuring things out. I had no idea how I'd seen them, but it seemed real enough. Julian had heard the ship and Samuel had thought it worth investigating.
However, if what Augustus said was true, it would be impossible to track the ship. Jonas Sparks had disabled the tracker installed on Orion, meaning it couldn't be found. If Orion couldn't be found, Julian would be proven wrong, and that would be the end of it. They would still head to Bunker 84.
Whatever the case, there wasn't much we could do about our current situation. We had been locked in our cabins for three days. Of course, we had been disarmed. The Praetorians brought food and water three times a day, and gave us the option to use the lavatory thirty minutes after each mealtime, and before bed. The point was, we wouldn't be getting out until Augustus said so. Our cabins were right next to each other, but I still hadn't seen Ashton or Anna in three days. Any attempt to talk through the bulwarks was futile – not only could we not hear each other, the Praetorians could hear us from their posts in the corridor.
I was starting to go crazy. We had to wait until the ship landed to get out – wherever that might be. So far, Augustus had revealed nothing. I hadn't even seen him since he'd locked me in here.
We made one landing in the middle of day two, but we just took right off again, heading somewhere different.
I checked my watch. It was 16:42. There were no ports in the cabins, and the lights had been shut off forty-two minutes ago. They would be turned on again in twelve hours.
I thought about the dream I'd had. I needed to share it with Ashton and Anna, but unfortunately, I had no way of doing that. Even when we were escorted to the lavatory, we were led out at different times, so there could be no sharing of information.
It all depended on whether or not they knew about Orion. Julian had a hunch, but what would happen if they contacted Skyhome, only to learn that there were no signals from a spaceship? Skyhome, I realized, was completely stranded without spaceships. Only Augustus had access to it, and surely he knew about it. The space city was so large that he couldn't fail to notice that impossibly bright star, roving across the night sky. The question was, when would he visit it? And what would he do about it? If Makara and the others contacted Skyhome, would Augustus eventually figure out they were still alive?
I was roused from my thoughts when I heard a light tapping emanating from the bulwark, just to the left of my bunk. The tapping was extremely light, yet intentional. Loud enough for me to hear, soft enough that it wouldn't catch the Praetorians' attention.
I didn't know who was in the cabin next to mine. I reached for the metal bulwark. It felt cold against my fingers. I gave a few light taps, so tiny that I could barely hear them.
The tapping from the other side stopped...
...and began a couple feet to the left of where I'd heard it last.
Someone – either Ashton or Anna – was definitely trying to get my attention. I rose from my bunk, following the direction of the sound. I knelt by the head of my bunk, placing my ear against the wall.
Tap. Tap. Tap.
I responded with my own three taps. Again, the tapping ceased, beginning again from the corner of the compartment, as far as possible from the door. There was little risk of getting caught; dinner wouldn't be served until 18:00, so we had a little over an hour.
Two more taps came from the corner. I waited for more.
After a long pause, a few more taps came: three, all from different places in the wall. Someone was directly on the other side.
"Hello?" I said, softly.
Of course, there was no response. There was no way the other person could hear me, and I didn't want to speak any louder.
The three taps came again, from the same locations – one directly in front of my face, followed by two more at the level of my waist.
"Aston? Anna?"
There was no way either of them could hear me. Not unless I spoke louder. Whoever this was, they were trying to communicate something without speaking. I focused on the taps, realizing the key was there.
The same three taps resounded, at the same locations. There was a pattern here if I could just figure it out...
They were making a triangle, for some reason. What did that mean?
I realized then they weren't making a triangle. They were making the letter "A."
Ashton, or Anna. That didn't really help me.
I made my own "A," at the same three spots I had heard them come from. I quickly followed this with an "L," which consisted of a tap in front of my face, one near my chest, and one to the right of that.
There was a short pause before I heard a single tap, a series of taps in an S shape, followed by another tap, lower and on the opposite side of the first one. This person had made an "S".
"Ashton," I said.
Of course, the brilliant scientist would be the one to devise this method of communication. In case the point wasn't clear, he began spelling out the rest of his name with a capital H-T-O-N. I spelled mine start to finish as well, so that he would know for sure who I was.
Over the next minute or two, he spelled out a message to me:
Anna on other side.
I wasn't sure if he was talking about the other side of the ship, or on the compartment opposite of Ashton. I assumed the latter. I didn't want to waste time asking for confirmation. So, I just wrote: O.K.
This was a surprisingly efficient way of communicating, so much so that it was a wonder we hadn't thought of it earlier. Morse code would have been easier, but none of us knew that. Maybe Ashton had been trying to do this for a while, but the taps had just been too soft for me to notice.
I paused, thinking of what I might ask next.
Where are we going?
The response was quick.
L.A.
I wanted to ask how Ashton knew, but then again, I didn't want to waste any time. I decided to take his word for it. I wanted to ask Ashton why we had been in the air for three days, but instead I tried to think of a message that wasn't too long. Another message came through the wall.
Anna okay.
I had assumed that already – it probably would have been the first thing Ashton would have told me if she wasn't.
Does she know?
I hoped Ashton knew what I meant: whether Anna knew about our destination.
Yes.
Remembering my dream, I knew I needed to communicate it to Ashton.
Had dream. Others are okay.
I paused, giving Ashton the chance to respond.
What else?
Julian knows about ship.
Ashton responded quickly.
Good.
That wasn't the complete truth; Julian had a hunch about the ship, but it was too late to take the message back without confusion. They would contact Skyhome, but Skyhome probably wouldn't be able to track Orion and confirm Julian's suspicion – unless Augustus revealed himself from there. I couldn't explain all that to Ashton through the wall, though.
I was trying to think of something else to ask when the door slammed open. I turned toward the light, never looking more suspicious in my life. Maxillo himself stood in the doorway, Chief Praetorian of Nova Roma. He glowered as his eyes narrowed.
"What are you doing? You are talking?"
"No," I said, walking to the center of the cabin and facing him.
"It doesn't matter," Maxillo said. "We will land soon. You can explain yourself to the Emperor then."
Outside in the corridor, more doors opened. We were all being taken out to prepare for our landing. For the first time in three days, I would see my friends.
Maxillo nodded his head toward the corridor, indicating that I should exit the cabin. Beyond Maxillo, Ashton walked by, a Praetorian right behind him. The scientist looked in at me, his white hair and beard a wild mess. He gave me a sly wink. As he walked past the door, I went toward the doorway. By the time I reached it, Anna walked by, escorted by her own Praetorian. She cast me a quick glance before she was hidden by the bulwark.
I stepped into the hallway. Maxillo gave me a light push, forcing me toward the wardroom.
We were all made to sit at the table. Maxillo left us with three stone-faced Praetorians and headed for the bridge.
"Stay quiet," he said. "We will be in the city soon."
As instructed, none of us spoke. A moment later, the ship changed trajectory, angling downward. I held on to the edge of the table to steady myself.
A few minutes later, we sunk vertically toward the ground. The ship gave a sudden lift before it settled onto the surface outside.
At long last, we had arrived in Los Angeles.
***
THE BLAST DOOR OPENED, revealing the sky burning fiery red from the setting sun. Buildings had crumbled onto one another, though most still stood. The decayed skyline, distant, was lost in reddish haze – not from the pollution of the Old World, but from the dust of the new.
We stood on the roof of a large, long building. Orion had perched on a helipad, its three struts barely fitting on the landing site. A cold, dry wind blew. Though cold, the breeze was warmer than I was used to. I had read something about oceans keeping coastal regions warmer than inland areas. Maybe that was why it was warmer.
I didn't know why I was thinking such things when the situation was so dire. I stared at the decayed cityscape, the twisting towers, the crisscrossing streets clogged with the rusted shells of cars and trucks, low-lying buildings stained with thirty years of dust and debris.
A hand pushed me down the boarding ramp toward the tarmac. Ashton and Anna were right behind. No one spoke.
The Praetorians made us stand at the bottom of the ramp, facing out. As more feet clomped down, I looked at the fallen city. The crimson sky cast bloodlike light on the buildings. To the east, mountains marched north to south, making a natural wall. Far to the north, even more mountains crisscrossed east to west. I even saw buildings in the eastern, lifeless hills, despite how far they were. Trees stood, but most were long dead. Maybe all. In the thirty years since Ragnarok, Los Angeles had been reclaimed by the desert.
A line of rubble rose within the inner city, cutting that section off from its outskirts. This was the wall that Raine built; Makara had told me about it. There were hundreds of buildings – maybe thousands. It was hard to imagine millions of people going in and out of them, hard to imagine that chaotic stream, each person making his or her own choices, millions of them every second. If it were that chaotic, surely everyone would have gotten stuck down there. Maybe they did. I guessed that was why they made traffic lights, highways, lanes on the road. Maybe they had ways to control the chaos, but even all the control in the world couldn't stop certain things.
Like death from the sky.
There were thousands of buildings, for sure. Some were tall, like the ones in Vegas, but there were more of them here. Blocks of these towers stood to the north, in the center of the city. I thought Vegas had been big, but it didn't compare to this fallen monstrosity. The buildings, small and large, stretched as far as the eye could see – north, south, and east as far as the mountains. It had been its own form of Blight, maybe. How much larger would Los Angeles be today if Ragnarok hadn't fallen? Would our cities have consumed the world?
Los Angeles had been one of the biggest, busiest, and richest cities in the world. And, funnily enough, it still was. More people lived here than any other place in the Wasteland. I imagined those early days after Ragnarok, the panicked survivors. There must have been millions, then. Gangs would have formed shortly after, warring for supplies.
I was glad not to have lived here during the Chaos Years. I thought of Char and Marcus. They had lived during those times, but I hadn't asked about them.
The dim, setting sun was the source of all life. When its light had been obscured by the dust kicked up by Ragnarok, the world entered a darkness from which it hadn't emerged. Los Angeles's population had tumbled from the millions to mere thousands. Many had probably fled into the eastern hills, hoping to find salvation there. But they only found a world of death. Only the Bunkers had been safe, for a time, but even they couldn't escape.
Los Angeles was the hub of the Wasteland. It was where most of the people lived, where fates were determined for the rest of us. Whoever controlled Los Angeles controlled the Wasteland, because whoever controlled Los Angeles ruled the majority of the Wasteland's people. Most of those people were slaves, and only a few were the gang members who ruled them. The Lost Angels were no more, and it seemed doubtful that they would last much longer.
This massive city would crumble mostly to dust in the coming decades. The towers would fall from the passage of time and the shaking of the earth. When they finally did fall, it was possible no human eye would see them collapse.
Assuming we survived this, we had to rebuild sometime. We couldn't just leech off the remains of a world fading further and further into time. If we did, we'd be reduced to savages and cavemen centuries from now. Perhaps we already were at that point. One day we would forget what those towers were, thinking they were constructed by gods...
"It's time to go."
Maxillo had spoken behind me. Below, on a long, curving drive that disappeared beneath the building, several all-terrain vehicles drove up.
LAX, I realized. We had landed on top of the airport terminal.
The vehicles were all black and had skulls painted on their doors, crisscrossed by scythes. Their engines whined high like a plague of insects. Lean, long-haired men toting rifles poked out from windows.
The Reapers were here.
Next to me, Ashton's breath caught. But he wasn't looking ahead at the Reapers. He was looking behind the terminal.
I turned, gazing past the Praetorians, past Augustus, toward the runways of LAX – where I saw thousands upon thousands of Augustus's legionaries camped. Hundreds of canvas tents had been pitched in perfect lines, and countless fires glowed red in the evening. A tall fence surrounded the army in a perfect square. It was impossible to guess how many were in there.
Augustus stepped forward, face grim. He nodded at Maxillo.
"This way."
Maxillo headed left, where a stairwell surfaced on the roof. The Reapers' vehicles pulled to a stop in front of the terminal, idling. Was Carin Black among them?
I guessed I would find out in a few minutes.
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# Chapter 3
We descended the stairwell. Maxillo led the way with a flashlight, illuminating dusty corners and grimy walls. Violent paintings coated the faded, sickly yellow of walls: fires, guns, blood, stick figures, and falling buildings. And scythes. It was art for a new age.
We rounded the stairs three times before exiting into the airport lobby. I had never been inside anything so large besides Bunker One. The ceiling stretched high above, lost to darkness. The architecture was curved and jarring; it was hard to believe what we had once been capable of building. Soot stained the once glossy white floor, the sites of previous fires. Red light filtered thinly through the entrance, doing little to illuminate the cavernous space.
I briefly thought of breaking free and running. It might be easy to lose myself in the dark labyrinth of terminals and tunnels. Ashton and Anna could run with me.
That could never happen, though. We were far outnumbered, and would be shot long before we could escape. We had to face whatever was coming.
We reached the sliding glass doors that had been busted open long ago. Broken glass littered the floor, crunching under our boots. Jagged edges still clung to the doorframes, sharp as the day of their breaking.
We stepped past the doors and into the faint crimson light, where the train of six black Recons idled on the drive. Four Reapers stood in front of the lead vehicle, facing us. Two held rifles in both hands, while another had his strapped to his back.
A squat man, shaved bald, stepped forward from the group. His face was all hard lines, his body chiseled angles. A long, black beard grew from his chin and stopped at his chest.
His beady eyes glanced over the three of us before settling on Augustus. He pointed at us with his chin.
"This is them?"
Maxillo stepped forward. "You dare address the Emperor directly, barbarian?"
Several other Praetorians flanked Maxillo, hands on their holsters. Behind him, his lackeys stiffened.
The Reaper smiled. "Warlord Black told me to bring the prisoners to the Citadel. He didn't tell me who I was supposed to deal with."
"He will discuss this with Black," Maxillo said. "Not you, worm."
"I have my orders to bring them to the Citadel," the man said, with his yellow smile. "The Warlord wants to question them personally."
"We are meeting between here and my camp, on the runways," Augustus said. "As previously agreed."
The Reapers standing behind the stocky man glowered. The other Reapers, still in their Recons, stared at Maxillo with malice. The Praetorians and the Reapers were evenly matched in numbers. If it came to blows, I had no idea what would happen, or who would win.
One of the other Reapers addressed Maxillo. "Be careful who you insult, guard boy. Onyx is Carin Black's own son."
Maxillo's face reddened while Augustus held up a hand.
"Peace, Maxillo," the Emperor said. "If Carin wishes to question the prisoners, I will allow it at the meeting – which is still set to take place outside my camp in about an hour's time." Augustus looked at Onyx. "You can let your father know. I see no reason for this audience to continue."
Onyx's face reddened – it was obvious he had expected his father's words to carry more weight.
Before turning to go, Onyx's dark eyes settled on Anna. My hand reached for my holster, only to find it empty.
"I know you," Onyx said. "You are Char's famous samurai bodyguard."
"What of it?"
"You killed several of our assassins months back."
"Glad to see my work hasn't gone unnoticed," Anna said.
"Several of those men were my friends."
Anna laughed. "Friends? I can see how you'd think that. I see that you don't just look like a toad; you have the intelligence of one, too."
The toad's face reddened, but before he could give his own retort, Augustus raised his hand.
"Enough. We leave now."
As one, the Praetorians filed up, forming a box around us. Still, Anna and Onyx stared one another down.
All the while on the spaceship, I had imagined that Emperor Augustus and Carin Black would be good buddies. I couldn't have been more wrong. Tensions were high between the two supposed allies.
Augustus turned to one of his Praetorians, a blond man who might have been second-in-command, after Maxillo. The Emperor said something to him in Spanish.
The man nodded before gathering a few men and leading them back into the terminal. I supposed they were going to stand guard over the spaceship. I noticed then that Jonas wasn't with us. He must have stayed on board, in case he needed to make a quick getaway.
As the Reapers pulled away in a whirl of dust and an angry blaring of engines, Augustus nodded to Maxillo. "Lead on."
The Praetorians marched in unison alongside the building. We moved along with them, locked in their protective box. As the last of the Reapers left the airport drive and zoomed into the city streets, the sky dimmed into dusk.
"Toad, huh?" I asked.
Anna shrugged. "Some people just infect your brain with what they say. I'm literally dumber for having listened to that idiot."
I chuckled. "We can't just start a war by offending people."
"Something tells me there's going to be a war, no matter what I say."
I wondered why Augustus parked his spaceship on top of the airport terminal building, rather than in front of his camp, which seemed much safer to me. Maybe it was just a way for him to show Black that the airport was under his control, and that he wasn't going to be confined to the small area around his camp.
Five minutes later, we arrived at the gates of Augustus's camp. The wooden walls were at least ten feet high, with watchtowers along their length. The army must have carried a lot of its supplies with it, and it was hard to imagine how they could have built such fortifications in a short length of time.
The gates squealed open, revealing the bustling camp within. We walked inside and I took in my surroundings. Now, with dusk, many of the legionaries were cooking and eating around campfires. Horses were tied up in stables, and we even passed a pen filled with pigs. Merchants and shopkeepers had supplies displayed on tables in front of their tents – anything from food to boots to trinkets. I realized then that it wasn't just an army here. An entire community had followed the soldiers in order to make a few batts.
Augustus's eyes seemed content, even reminiscent, as he watched the camp.
"When I was a young man in Mexico," he said, "Ragnarok had just fallen. My people were afraid. I knew we had to be the strongest if we were to survive. To the north was Old Mexico, the Federal District. They fought us in those days, even though the world had ended. That was my first campaign. I took my soldiers, went north across the mountains in the springtime. It seems so long ago. The Federalists fought to the last man. Now, Federal is one of our biggest provinces."
Augustus told this story not in a proud way. To him, it was just a story, one of the many great things he had done in his life.
"I remember entering the Presidential Palace in victory, and how the President, old even then, dropped to his knees. A man who had chased and attacked me for so many years, before the Rock fell, was expecting my judgment." Augustus smiled in memory. "He was a brave man and he met my eyes with such hatred. I told him not to kneel like some warlord or barbarian – many of whom I had crushed. This man was a rival worthy of respect. So, I told him to fight for me. He agreed. I let him keep his kingdom, his estates, his women and children. We conquered much together – Jalisco, Oaxaca. He died, twenty years ago. I spared no expense for his funeral, and gave games in his memory at the Coleseo, newly constructed then."
Augustus sighed, as if he knew those glory days were now past and never to be reawakened. Our war was different from all his others, because the end goal was not wealth and power, but survival. The glory years were gone.
It looked as if Augustus was going to say more, but instead he remained thoughtful. We walked on.
I didn't really know why Augustus told this story, but I think it was intended mainly for Ashton. Some things only made sense to old men. Old memories from an Old World didn't mean much to me. Augustus was a drug lord in his youth, one who had fought the government so much that he had even formed his own country, and that before Ragnarok came down. Augustus was a warrior, much more than he was a builder. For centuries he would be remembered. Stories would be told of him, and children would be named after him.
That was, if we survived all this.
Even with Augustus's power, it would be a while before he got his fight. If the Empire and the Reapers were still allies, then there was no force in the Wasteland that could oppose them – besides Askala and the Radaskim. The Wasteland resistance had failed, so the only two important powers left were the Empire and the Reapers, and having them share the same city with no one to fight was a disaster waiting to happen.
"Will there be war?" I asked. "I mean, between the Empire and the Reapers?"
Augustus shook his head. "I must do whatever I can to stop that. There cannot be war until the Great Blight is dealt with. Carin knows this as much as me. I don't think Carin will attack, but he's getting restless. He's making more demands. I have more soldiers, but he has better weaponry, hoarded over the years from the nearby military bases. All the same, I fear he might be planning to use the Radaskim as the anvil, and himself as the hammer, while not realizing the foolishness of such a move."
Augustus's speaking about the Radaskim, using that word, made me realize he was up to speed. Ashton had probably updated him at some point while we were in the sky.
"Do you really think he's going to pincer you between the two sides?" Anna asked.
"I hope not, Anna. I know Black is getting desperate. How could he not be? He has an army of twenty thousand men at his doorstep. The best-case scenario for him would be killing both us and the Radaskim in a single stroke."
"But Askala will only attack again," I said. "Even if he did win, it would only be a matter of time."
"But does he know that? That's what we need to convince him of, Alex. Remember, he doesn't know what we know about the Great Blight. To him, this attack might just be a onetime occurrence. It's our job to convince him otherwise."
Augustus stopped before a large tent that was probably his own; it was larger than the rest, and two guards stood in front of the entrance. The Emperor turned to face us.
"Black needs to hear it from you," Augustus said. "I need you at that meeting with me to convince him."
"Why would we do that for you?" Anna asked. "Maybe we let the Radaskim attack and kill you all. Why would we help you when you locked us up for three days?"
"I wasn't going to risk your escaping," Augustus said. "I know how capable you are – not only did you escape the Coleseo back in Nova Roma, but you escaped my Praetorians. I wasn't going to take any chances. And as for why you should help me, I think we both know the answer to that. Besides me, no one can help you stop the Great Blight. I'm the only one with an army. At the risk of sounding crass, you are alone. Your friends are dead, so if you still want to have a chance, then you must accept my offer of friendship – which I gave earlier. Give me your word of honor, and we can work together, for the good of the Wasteland."
The Emperor gazed at us, an infectious energy in his brown eyes.
"Besides military aid," Augustus said, "I have another plan for you three – contingent upon your acceptance of this alliance. Of course, I have Orion in my possession, but as a gift I would offer you Perseus."
The three of us stared at Augustus in shock. Anna was the first to recover.
"You'd trust us with a spaceship?"
"Why not?" Augustus asked. "In exchange for helping me with Black and the Radaskim, you can keep it. I really don't fear your attacking me with it. You have nothing to gain. We would be working together, something that is necessary if any of us are to survive this."
We did need a spaceship. We could find Makara and the others. We could begin to evacuate Bunker 84, perhaps reinforce Augustus's troops with our own, before Los Angeles was attacked.
Maybe an alliance with the Emperor wasn't a bad idea. If he was serious about giving us Perseus, it could solve all our problems.
I looked at Ashton, wondering what he was thinking. Ashton was weighing Augustus with his sharp blue eyes, while Augustus did the same to Ashton. The two men had once been enemies. Could they ever be friends?
"So you want us to talk to Black?" Ashton asked.
Augustus nodded. "We need him to work with us. Do this, and lend your aid in the coming battle, and in exchange, you can have Perseus."
"What if Carin Black attacks anyway?" I asked.
"That might be the case," Augustus said, "but if that happens, at least I know I have you three at my side. None of us are going to survive the coming storm unless we unite."
With those words, I remembered the Wanderer's own warning: that we had to get everyone working together. Augustus was right. Maybe the Emperor was after power, but in the end, survival was more important than whoever got to rule at the end of the day.
"Where is Perseus?" Anna asked.
"It's still in Bunker Six. We can rescue it once we are done speaking with Black. Getting Perseus out shouldn't be too difficult."
"He's right about that," Ashton said. "I've been in and out of Bunker Six several times over the years. It's built into a mountain far above the Great Blight, so there aren't too many monsters there. Still, it's dangerous. With Orion, it will be easy to access the hangar."
"Okay," I said. "So if we help you by speaking to Carin Black, you'll let us keep the spaceship?"
Augustus nodded. "The main thing is getting Black on our side for good. The longer I wait here, the more suspicious Black becomes."
It looked as if Augustus had all his bases covered. That was to be expected, though. A man didn't grow to rule an Empire unless he knew how to plan out every outcome.
"We'll do it," Ashton said.
"Good," Augustus said.
A courier approached Augustus, bowing low.
"Speak," the Emperor said.
"Black is here."
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# Chapter 4
Augustus and his Praetorians escorted us back to the encampment's gates. The portal opened upon our arrival, revealing the abandoned airport ahead. The top half of Orion was visible above the edge of the terminal building, from which jutted boarding tunnels. Ten of the Reapers' black Recons were parked in a line to the left of the building, facing toward us. They were distant – maybe a quarter of a mile away. A line of men stood in front of the Recons, far across the flat expanse of dirt and runways. One of these men had to be Carin Black.
We marched forward to meet with the Reapers and their Warlord. We were finally going to meet Carin Black, the terror of Los Angeles. This man had ruined thousands of lives when he overthrew the Lost Angels in 2055, enslaving or killing any who resisted. The only beneficiaries of the violence had been his own gang, the Black Reapers. I didn't know what was going to happen at this meeting. Neither Augustus nor Carin trusted each other, so I felt anything could happen.
After a few minutes, we were near the Reapers. I saw Carin Black now. He was a tall man with pale blue eyes. They did not look like the eyes of a man who had done so much evil, and it was hard to believe that his toad-like son, Onyx, was his. He was probably forty years of age, and at first glance, he definitely seemed to be someone of intelligence and ability. His dark hair was cropped close, and his pale muscled arms held various tattoos – on his left biceps was the skull and cross-scythes that seemed to be the gang's emblem, while a red, fiery snake ran down his left forearm. His right biceps was dominated by an intricate tattoo of a grim reaper bearing a scythe. The reaper himself was swirling in darkness, and had glowing red eyes within his shadowed cowl. The Warlord of the Reapers stared at us with those crystalline blue eyes, his thin, bloodlike lips horizontal. That gaze was like a rush of cold wind.
At last, we halted before him. The Praetorians made a line, flanking us on either side. For a long moment, the two sides stared each other down. The moment dragged into minutes, each leader refusing to speak first.
Finally, Carin Black raised a hand.
"You said the prisoners had something to tell me."
Augustus waited a moment before nodding toward Ashton. Ashton gave the Emperor an annoyed sideways glance. I could tell he'd rather Augustus do the explaining, but we had to do our part of getting the two forces to work together. We needed that spaceship, after all.
Ashton summed up the basics of what we had learned about Askala and the Radaskim in about five minutes, a telling that should have taken five hours. He concentrated mostly on the science of it: how the xenovirus basically worked, the two opposing sides, composed of Elekai and Radaskim, and how we had to destroy the Radaskim side by infecting Askala in Ragnarok Crater. Carin Black would have been familiar with perhaps just the viral aspect of the monsters, but everything else was probably new information.
When Ashton was finished speaking, Carin shifted his focus to Augustus.
"I would ask if this was a joke," Carin said, "but it's obvious he meant every word. So what do you propose we do about it, Emperor?"
"We have to shift our focus," Augustus said. "The Wastelanders no longer pose a threat. In fact, I have convinced them to ally with us. The swarm that destroyed them is coming to destroy us."
"They are all dead, then?" Carin asked, arching an eyebrow.
Augustus nodded. "There is but a remnant left. These are the only ones I found. They also know how to destroy Askala, which is the only way to stop the swarm."
"How long before these...Radaskim....arrive?"
"A few days, at least. We must secure the eastern portion of the city and man the fortifications. When the horde is defeated, we can plan the counterattack."
"And where will we be attacking?" the Warlord asked.
"Ragnarok Crater."
Carin went quiet at that. I realized how ridiculous it sounded. Ragnarok Crater was almost two thousand miles away, an unfathomable distance, especially when considering the Great Blight.
"Of course, we wouldn't be directly attacking Ragnarok Crater," Augustus said. "We would be advancing toward it, hopefully drawing the attention of the Radaskim. This would give the Wastelanders an opening to attack the Crater, if it has been emptied."
"You sound like you don't know if this is going to work," Black said.
"There are no guarantees," Augustus said. "The only guarantee is our mutual death should we do nothing at all."
Carin appeared to be deep in thought. I noticed his son, Onyx, among the ranks of Reapers. His dark eyes glinted as if he knew some secret that would change everything, if only we knew. That made me distrust Black. There was something he wasn't telling us. Anna stared at the Warlord with open loathing, but Carin did not seem to notice, or care.
At last, Carin gave a small, superior smile. "And so we find ourselves caught in our enemy's net."
Augustus frowned. "What do you mean?"
"I'll not play the bait," Black said. "What is to stop the Wastelanders from allowing Askala to do what they cannot do themselves? We will be destroyed, Augustus, while our 'friends' escape."
"Nothing's stopping you from doing the same," Anna said. "How do we know you're not planning to let us die?"
Carin's cold eyes turned on her. "If you are suggesting that I would betray my ally, then you have misjudged. If there is any betraying to be done, it won't be on my part."
"Then why not work together?" Augustus said. "The original purpose of our alliance has ended. The Wasteland is already ours, as all of its players have been taken over by Askala. If we are to keep our gains, we have to destroy her. Surely, you see that."
Carin Black frowned. "Is there any Wasteland left to own? Supposing we are victorious, we will only have the coastline of California left. The rest will be covered in Blight. And my concern, Emperor, is that you did not come here from Nova Roma to do us a good deed. No, you want to control all of the Wasteland. And if you do not want me dead, then you'll at least want me to kneel at your feet."
"Come now, Carin," Augustus said. "Whether my army is here, or in the south, you will not be able to contend with my strength. After all, a small country cannot contend with a great. The few cannot contend with the many. And the weak cannot contend with the strong."
Carin's face darkened. His men looked nervously at one another, waiting for their leader to respond.
"You call me weak, Emperor?"
Augustus, aware of his advantage, pressed on. "I call you fortunate, Carin, that I even allow you the opportunity to serve the Empire, because the many men who have taken your stubborn stance are no longer standing."
Carin went quiet, staring at the Emperor with his glacial eyes.
Augustus continued. "What is your decision, Carin? You have two roads before you. One leads to survival and peace. The other to death and destruction. The Radaskim will be here in days, and she will not care about our petty differences. Will you join me on the field of battle?"
"I will join you," Black said. "I will join you, only if you leave this city afterward, never to return."
"Very well," Augustus said. "Los Angeles will be left in peace, but I do hope, for your sake, that you won't be completely closed to communication in the future. For the sake of our continued friendship."
Black's face reddened. "That is my condition. I know you need my help as much as I need yours. You won't convince me otherwise."
"We will work together then, my friend," Augustus said. "Victory will require sacrifice, on both of our parts. I am willing to pay my dues in order to help you defend your home. I hope, in the coming days, you will think on what you can do to help me."
Carin gave a weak smile. "Very well, Augustus. You have your agreement."
"Good. We will meet again tomorrow to coordinate our battle plans."
Carin nodded. "I'll be preparing my men for the defense. We might need to fall back behind the walls."
"Yes, the walls will help," Augustus said. "But bravery is an even stronger defense. These monsters have a way of breaching barriers, and they have flyers. The spaceships will help, but there might be too many for me to fight alone."
"Then you are going through with your plan?" Black asked.
"Yes, as I have told you," the Emperor said. "Tomorrow I'm taking the Wastelanders to Bunker Six to liberate Perseus."
Carin nodded. It was hard to read those eyes, whether something was hidden behind them. In the end, though, the Warlord of the Reapers gave a slow nod, reaching out a hand.
"It is done," he said.
Augustus clasped the hand. Neither man's gaze shifted from the other; both refused to be the first to look away.
"Tomorrow afternoon, Black," Augustus said.
Carin gave a nod. I could tell he wasn't satisfied with the way things had gone, but Augustus was right. The Empire was just too powerful for Carin to attack openly. If he tried anything, it would have to be sneakier.
As Augustus and the Praetorians broke away, we followed them back to the Novan encampment. At the same time, Carin ordered his men back to their Recons.
I didn't trust Black to keep his word. It was obvious he resented his more powerful "ally" and that he worried about the aftermath of the battle. Augustus had been surprisingly blunt about how Carin Black would be treated. He had acted less like an ally and more like a bully, but Augustus had to know that Carin would stab him in the back if he got the chance.
We reentered the gates of Augustus's camp. At Augustus's order, the majority of the Praetorians disbanded, leaving only six to guard the Emperor, Maxillo included. We walked the rest of our way through the camp as evening settled into night.
We stopped outside the Emperor's large tent. He turned to Maxillo.
"Have them wait inside for now. I'll be back shortly. In the meantime, make yourselves comfortable."
And with that, the Emperor walked away; Maxillo and another Praetorian peeled off from the original six to stand guard over us. Maxillo gestured toward the tent flap.
There was nothing left to do but to follow the order and go inside.
***
THE EMPEROR'S TENT was a world within a world. Inside its thick, canvas walls, the bustle of the camp had dimmed. However, I could still hear men shouting, hammers tacking, the wheels of carts squealing, horses neighing, and feet stomping. The air inside was thick, scented with some spicy fragrance. Intricate Persian rugs of purple and crimson covered the ground so thickly that none of the grimy tarmac of the runway was visible, or even felt, beneath my boots. Lamps and candles radiated yellow warmth, sending shadows dancing on the canvas walls bedecked with the furs of jaguars, bears, and other exotic creatures. The lavishness of the entry area made this feel less like a tent and more like a palace.
A richly clad man stood within a curtained partition that led deeper into the tent. I recognized him from Augustus's palace, back in Nova Roma. He was the butler who had seated us during our first audience with the Emperor. He introduced himself as Zuma, and offered us chilled wine to drink. After we declined, he withdrew to the corner and stood silently.
Maxillo stood by the tent flap, his demeanor as hard as stone. The other Praetorian must have been guarding outside.
At last, Augustus returned, startling me as the tent flap whipped opened. I caught a glimpse of a few Praetorians standing outside, armor flashing by the light of torches. The flap fell, shutting out the night.
Augustus adjusted his steel breastplate. Quickly, Zuma stepped forward, to take off the Emperor's rich, purple cape. Augustus cleared his throat as the cape was unclasped. The butler hung it by a hook along the far wall.
As the butler set to work on helping Augustus take off his armor, the Emperor faced us.
"It's clear Black is nervous. Why wouldn't he be, when my whole army is camped outside his walls? When I have a spaceship? I can't predict what he will do, and that's what's bothering me." Augustus paused, shrugging off the breastplate. Beneath, he wore a white, linen undershirt. "That makes reaching Bunker Six quickly all the more important."
"Are you serious about wanting to ally with us?" I asked.
Augustus gave a small smile. "Never doubt my words, Alex. Many men have, to their peril. What I told you by the wreckage of Gilgamesh remains true. We must work together to ensure the future of humanity. Whoever controls this world, in the end, is a less important question. The important question is how we can save this world." Augustus gestured with his head outside the tent. "Those barbarians don't understand that, and they can't be made to understand that. They only understand blood."
None of us responded to Augustus. It had been a long day, and all I could think about was eating and sleeping.
Seeming to sense this, Augustus turned to Maxillo. "Take them to the First Cohort and see that they're fed. I want them to see how this army works. After that, show them to an empty tent."
Maxillo nodded, holding the tent flap open for us to walk out.
"We'll head for the Bunker tomorrow morning," Augustus said. "It shouldn't be long, now."
We headed out of the tent and into the night.
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# Chapter 5
Maxillo led us outside to a campfire that wasn't far from Augustus's tent. Around the fire, several legionaries were eating. A table was set up nearby, upon which food was spread out: roasted pork, corn, rice, bell peppers, sliced onion, and some flat, round bread that folded. I watched the other soldiers, who wrapped the bread around the rest of the food and ate it that way. I followed their example and started to eat what was quite possibly the best thing I'd ever had.
We were allowed as much as we wanted, so I stuffed myself. It was fully dark now. Most of the legionaries were done eating and were cleaning up what remained of dinner. That left Ashton, Anna, and me sitting around the campfire while Maxillo stood a few feet off, eating his own food. I asked him what it was called. He said fajitas.
Two of the legionaries stayed behind while the rest went to their tents. They sat on stools across the fire from us. One of the legionaries had a lean face and wide eyes. The other was short, with a thick beard and a wide face. The wide-faced man offered us some cobs of corn he had been roasting on the coals. I accepted, and he tossed it to me. I caught it in the air, bouncing it around to avoid getting burned.
I quickly unwrapped the corn from its charred husk. It was yellow, almost white, in the firelight. Even though I was already full, I wanted to try it. I took a small bite. The corn juice was hot, and its flavor sweet.
"Is good?" the tall man asked.
"Yeah," I said.
I'd had corn before, both in Bunker 108 and in Skyhome, but not roasted like this. The man nodded at Ashton and Anna, pointing to the fire. They each got some corn still roasting on the coals.
"Where are you from, friends?" the tall man asked.
"You know English?" I said.
He nodded. "Most of us...can't. But...many...how do you say? Bosses. We know."
"The officers, you mean."
The man nodded again. "Yes. Officers. We speak a little."
"You are an officer?"
"Sí," he answered. "This is my tenth campaña."
"Your tenth campaign, then. You have seen a lot of war?"
The tall man gave a bitter smile. "Yes. Too much war. En el imperio, la guerra es eterna."
"What does that mean?"
"It means 'in the Empire, war is eternal.'"
"Then why don't you quit?" Anna asked, biting into her corn.
"You mean, stop?"
With a smile, the short soldier said something in Spanish to the tall officer. They both chuckled.
"What's so funny?" I asked.
"It's different here," the tall man said. "A man must give ten years of his life to the legions. Every imperial must."
"Why?"
"It is ley."
"Ley?"
"I do not know English word for this. It is the order of Augusto."
Ashton explained. "It is the law in the Empire for each man to serve ten campaigns in the legion."
"That seems like a lot," I said.
The tall officer nodded. The shorter man just listened.
"Yes, it is harsh," the officer said. "But Augusto protects us. Helps our families. Gives us food and...land...to give for our time here."
"You will farm after all this?"
"Farm?" the man asked, unfamiliar. "I do not know this word."
"You will work on the land?" Anna asked.
"Ah, yes. My wife and children, they work on the land. We have no land to...name to us. But after esta campaña, I will go home and the boss of the land will give land for me."
"That is good," Anna said.
"Yes, very good." The man smiled. "I must only survive one campaign more. It is a hard life. My wife tells me: as long as there are men, there are wars."
"Will there ever be peace in the Empire?" Anna asked.
"La paz?" The man shook his head. "No. We have another...dicho...in our land: Sólo los muertos han visto el final de la guerra."
"What does that mean?" Anna asked.
The officer flashed a rueful smile. "Only the dead have seen the end of war."
"That's a little dark," Anna said.
"A little dark, yes. Also, much true." The man smiled again. "You must forgive me. I am something of...a philosopher. I think. Too much, I think. When you march, you only think. And yes...I know something of war. More than most men know."
It grew quiet for a moment before I asked the campaigner another question.
"Do you ever get to see your family?"
The man shook his head. "In spring and summer, my life belongs to the Emperor. In fall and winter, my life belongs to my family. Is... same for all men in the legion."
"You campaign only in spring and summer?"
"Yes. Every year is the same."
I looked at the Praetorians guarding us, staring into the distance, never breaking their stance.
"What about for them?" I asked.
The man laughed. "Los pretorianos? They are not men. They are machines. They do not have families, they do not have women. But they are rich. Yes, very."
"What good is being rich if you cannot enjoy it?" Anna asked, finishing her corn.
The tall officer smiled. "You have much wisdom. But los pretorianos...their promise is twenty years. They train from a young time for...honor. After twenty years, they earn many riches and live like kings. There are few who live for twenty years. Maybe some. Many...many have girlfriends. Secret families. They do not have permission for this."
"What happens if they get caught?" I asked.
The officer laughed. "You ask too much, little americano!"
He took a swig from his canteen. The way he coughed afterward suggested it didn't hold water. He held the canteen out to me, but I shook my head.
"You must have seen many things in all your years on campaign," Anna said.
The man's eyes lightened in remembrance. "Yes. Many things. Most bad." He looked at the both of us. Ashton looked on silently. "I am Carlos. This man..." He nudged the shorter soldier, whose eyes were closed. He woke with a start. "He is Horacio."
We introduced ourselves. Once we had, the man began to tell us his story in his broken English: of far-off jungles southeast of the Empire's borders, how Augustus wanted to conquer the wild tribes there, but could never break through the trees. He wanted the forests for the medicines he could find inside. He wanted to go south, to the Canal, wherever that was.
"What is south of the Canal?" I asked.
"No sé," the man said. "Nobody knows, but maybe Augusto does. Yes. I think, maybe, he does. Other countries, I would guess. El imperio...we are the biggest and the strongest. Augusto has even mastered the art of building ships, for the water. He could not ride them here. Too many storms."
The man talked about the wars he'd fought in – of foreign cities, high mountains, dark forests, and the beautiful women he had met. The food he had eaten. The riches he had won. The deep blue of the Pacific, and the bright blue of the Atlantic. Crystalline white beaches, stained with men's blood. He talked about a tribe he called los salvajes, who lived in the Yucatan, and how they fought with bows and arrows, and even with such primitive weapons, it had taken two campaigns for Augustus to conquer them.
"The Empire is big," Carlos said. "Bigger than even I know. I have seen much of it, but not all."
I would have told him of our foray into the Empire, but Maxillo held up a hand, interrupting our conversation.
"It is time for you to sleep."
We bid our farewells to Carlos and his silent friend, Horacio, who had fallen asleep by the light of the fire.
"It was good to meet you, americanos," Carlos said.
We followed Maxillo past several tents. The dirt roads were mostly empty. Most of the legionaries would be exhausted by this time of night, leaving only the watchmen. We passed Augustus's tent, which was dark. Even the Emperor was asleep.
Maxillo led us away from the tents, toward the far wall.
"I am taking you to the latrines, where you can relieve yourselves," Maxillo said. "After that, I will take you to your tent."
We did just that before we regrouped to head back to the center of the Imperial encampment. We passed a couple dozen or so large tents before we came to one that had two purple-caped Praetorians guarding its entrance. They parted upon seeing us approach.
"I'll wake you at dawn," Maxillo said.
The chief Praetorian left us behind. There was nothing left but to enter the tent. Once inside, I saw in the dimness that three cots had been made up. It was chilly within, and already the heat I had absorbed from the fire had escaped into the surrounding air. Thick wool blankets had been provided for each of us.
We settled in for the night. I shut my eyes. As the minutes passed, the noise from the camp outside dimmed.
***
I WAS FLYING.
It was night, and below I could see a Recon speeding across a fungal plain. Behind, a swarm of crawlers surged, clipping at the Recon's tail. Light flashed from the turret. Some of the crawlers were felled, but it wasn't enough.
I swooped down from the sky. I realized then that I wasn't the one flying, but Askal. I was seeing the world through his eyes.
The mass of crawlers approached the Recon, oblivious to the death from the sky. With a roar, Askal outstretched his legs. I could feel him curl his long claws in preparation to skewer his first victim. With a crash, Askal swept two crawlers from their scuttling legs. They shrieked as they were tossed upward, sailing through the air and landing with thuds on the fungus below. The fungus seemed to absorb their impact, and the crawlers rebounded into the air, landing once more on spindly legs. Both gave themselves shakes before turning around and chasing after the Recon once again.
Askal roared, swooping around again to attack. This time, he swept several more crawlers away from the Recon, buying Makara and the rest a little more time. The crawlers' glowing white eyes turned from the Recon, focusing on Askal above. On me. They would be ready next time.
As Askal looked down at the Recon, I could see where they were headed. There was a walled settlement ahead. Could it be...?
Oasis.
The Great Blight, in the short time I had been absent, had extended as far as Oasis. It was near unthinkable. At this rate, it would be at the edge of Los Angeles within weeks.
More crawlers surrounded the Recon, surfacing from below the Great Blight. As Askal flew closer, Makara swerved wildly in an attempt to break free. The turret swiveled around, facing front and lighting up the night. Several of the crawlers in front of the Recon fell to the fungus. The Recon jerked as it sped over the leg of one of the downed crawlers.
The Recon had made it halfway to the walls of Oasis. As Askal swooped low once more, the crawlers were ready. They leaped, snapping at his body. A few bites landed on his right hind leg. I felt the pain, searing and hot. Askal shrieked as he beat his wings madly to escape the crawlers' snapping jowls.
I cannot fight anymore, he thought.
But they are going to die.
There is nothing more I can do, little human.
The crawlers closed in on the Recon from behind, frighteningly close.
That was when a long line of fiery explosions lit the night, trailing north to south along the edge of the Great Blight. The Recon had just blazed past it, and it surged ahead from the force of the blast. Behind, crawlers shrieked as they were roasted alive. The explosives must have been set up by Oasis as a defensive measure.
Oasis and Ohlan had just saved the lives of my friends.
The stream of crawlers edged around the fires, like the flow of a river wrapping around a rock. The Recon had exited the Great Blight and now sped along the desert floor. The gates of the settlement were wide open, ready to receive the vehicle.
Askal turned from the fireworks below, heading east, deeper into the Great Blight. His wound was deep and painful. I felt blood dripping from it. I felt weakness spread from that limb, permeating the rest of Askal's body.
Would he make it?
Home is not far, Askal said.
How far?
Not far.
As the Great Blight passed below, my vision faded.
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# Chapter 6
When I awoke it was still dark. I checked my watch to find that it was 07:12.
I wrapped myself deeper in my blanket and closed my eyes. I wasn't getting up until I had to.
Anna stirred in the cot next to mine. Ashton snored on the far side of the tent.
"Anna?"
"Hmm."
"It's cold."
"Really."
"Yeah. You probably need to come over here."
She lay quiet for a moment, her eyes closed. "Not while the old man is here."
"Huh?"
Aston stirred and sat up, his long white hair and beard unkempt. He smacked his lips a couple times.
"Where the hell are we?"
Before we could answer, Ashton seemed to remember.
"Aw, hell." He cleared his throat of phlegm, and spat.
Anna chuckled. "After that, I'm probably not going back to bed."
As Anna removed herself from her blankets, I sat up. My clothes were dirty after the sweat and dust of the camp. I smelled of smoke. A layer of grime had accumulated on my tongue.
I reached for my canteen, finding it mostly empty. I chugged what remained of the water.
"I need a refill," I said. "If anyone wants to come with me."
"Hold your horses," Ashton said, throwing his blankets off. "Let the old man get his bearings."
A few minutes later, we had our boots on and were as ready as we were ever going to be. It had not been a good sleep, but we had to take it because it was the sleep we were getting.
That was when I remembered my dream.
"They're in Oasis," I said.
Both Anna and Ashton look at me.
"You saw this?" Ashton asked. "In a dream?"
"Yeah. I don't know how I keep dreaming..."
I told them about what I had seen from Askal's point of view, how the Recon sped away over the Great Blight, making for the walls of Oasis as it was chased down by hundreds of crawlers. I ended my story with the line of explosions outside Oasis, and Askal getting hurt.
"Will he be alright?" Anna asked.
"I don't know..." I said. "And I don't know if the others will be alright, in Oasis. That town must be under siege by now, and who knows what a man like Ohlan is capable of?"
We sat on our cots quietly for another moment, thinking.
"So, what are we going to do about it?" Anna asked.
"We have to get in there and rescue them," I said. "Once we have the ship."
"I know that," Anna said. "But when do we do it, and without pissing off Augustus?"
"Small chance of that," Ashton said. "We can't leave them there, no matter what he says." He looked toward the tent flap. "But we shouldn't be discussing this here. Right now, let's just focus on staying on Augustus's good side. We don't have that spaceship, yet."
Ashton was right. At his nod, I opened the tent flap to reveal the slate-red sky. Long clouds hung low and wispy above the camp.
Two Praetorians still guarded the tent, different from the ones that had been guarding last night. If they had heard us speaking, they gave no sign.
Down the straight dirt path, Maxillo walked toward us. It looked as if he had already been heading this way.
"You can find breakfast and coffee by the fire," he said. "The Emperor wants you by his tent in one hour."
First, we took our turns at the latrine, after which we headed to the fire, where we found the same two legionaries from yesterday – Carlos and Horacio. They tended a large pot stewing over the fire. Eight other men were already lined up with wooden bowls and spoons. Even if they looked a bit tired, their backs were straight and their eyes alert.
Horacio, the short man who only spoke Spanish, addressed the men in line, gesturing toward us. The men gave a cheer as Horacio lifted the pot, letting out steam that smelled of sweet corn and pork.
Carlos translated. "He says: for breakfast today, these three are part of our siglo. Century. He says you are special guests of the Emperor and he wants them to eat with the best. El primer cohorte."
"El primer cohorte?" I asked.
"The First Cohort," Ashton said. "The most skilled fighters in Augustus's army, besides the Praetorians."
The men waited, not yet going for the food.
"They are waiting for you," Carlos said.
I nodded my thanks to the men. Horacio handed me a wooden bowl and spoon. His cheeks were grizzled, where yesterday they had been clean-shaven. Dark circles underlined his brown eyes. I wondered at Horacio's story. He seemed to be a little younger than Carlos, so maybe he was only on his fifth or sixth campaign. He seemed to be second-in-command of this group, of which I counted ten. I remembered Julian telling me, while in Nova Roma, that the legions were subdivided in sets of one hundred, called centuries. These centuries were further divided into ten groups of ten. Decades, maybe. These decades seemed to cook, eat, and share tents. They might have even fought next to each other in battle.
Augustus based everything off the Roman Empire for a reason; if it worked for the Romans, so he supposed it would work for him.
I held out my bowl, and Horacio filled it to the brim with corn porridge. There were bits of pork, left over from last night, along with chopped onion, tomato, and jalapeño. The wooden bowl was hot in my hands. I walked from the coals, setting my bowl on the ground for the moment. I went to refill my canteen at a cask not far from the fire. I filled my canteen, along with Anna's and Ashton's.
I sat back down and took a long drink of cold water. The fire warmed me in front while the morning air cooled me at my back.
Horacio approached, handing me a clay cup filled with steaming liquid. I smelt the aroma of coffee. I accepted the cup.
"Gracias."
"De nada, mi amigo."
After taking a sip of the hot, black coffee, I tucked into the porridge. It was good, but nothing could have topped the fajitas from last night. It was spicy and sweet at the same time from the corn and chopped jalapeño, while having the sharp taste of red onion and the savory taste of pork. Even though it was basically a way to get rid of the leftovers from last night, I still scraped the bowl clean. Horacio smiled and pointed to the pot, telling me to have seconds. He seemed pleased I liked it so much.
I filled my bowl halfway, and some of the legionaries went back for seconds.
The morning was quiet. All around the camp, other groups of men, usually of ten, ate breakfast around their own fires. Everyone's schedules must have been similar. The young legionaries talked and joked in Spanish, though the old ones ate silently. They wore leather armor, mostly, something that would be useless against the Reapers, who had guns. Many of the men had machetes strapped to their belts. The armor was dusty and gritty, as much as the men themselves were.
I wondered why Augustus wanted us to see how his army operated. Maybe it was to give us a sense of his power? It was interesting to see how it all worked, whatever the reason.
In time, breakfast was over. The day had brightened, and Carlos told us that he and his men would clean up, and that we should go see the Emperor. After thanking the legionaries, we left the men of the First Cohort behind.
The sky was now a subdued red. There was a lot of dust here, mostly from the movement of soldiers. We passed groups of men, even an entire century marching toward the front gate. They carried long spears and shields of thin metal, the shield fronts painted red with the Roman numeral IV. I was beginning to wonder if Augustus's army only had primitive weapons like spears and shields, when another group of soldiers passed, toting rifles. These men had purple plumes in their helmets, signifying a higher rank. Not all the soldiers got to carry guns, probably because their supply was limited, but I supposed even a soldier with spear and shield could be of use, with others in great enough numbers.
I didn't know how many men Carin Black had, but at least a few thousand, counting all the gangs under his command.
We stopped in front of Augustus's opulent tent. Maxillo went inside as soon as we arrived. We stood outside a moment before Maxillo ushered us in.
We walked inside, finding ourselves once again in the lap of luxury. The Emperor sat at the low-lying table, over which spread a detailed map of Los Angeles. Augustus stared at the map intently, holding a porcelain cup of coffee in his left hand. At our entrance, his eyes turned upward.
"I trust your breakfast was good?"
We said that it was. Zuma rushed to a cabinet standing against the left wall of the tent, and there filled three detailed, porcelain cups with coffee. He then took a small container, adding a dab of golden, viscous liquid to each. I realized it was honey.
Zuma carried one cup at a time, handing the first to Ashton. Ashton looked at its contents skeptically, but after a moment, took a sip. Anna was served next. She ignored the coffee. I took mine last of all.
Augustus, very carefully, began to roll up the map as I took a sip of the sweet coffee. He handed the map to Zuma, who held a cylindrical, leather tube. Tenderly, Zuma placed the rolled-up map inside the tube, where it would remain safe and unspoiled by the environment. He went to the cabinet and stored the tube on the upper shelf. He closed the door and locked it with a key hanging from his neck.
"Here we are again," Augustus said, beginning the conversation. "Is the coffee good?"
"Yes," Ashton said.
Augustus smiled. "I am glad, my friend. Grown only in the mountains north of Nova Roma. It is my favorite." Augustus took another sip, savoring the flavor. "Let's get down to business. I want you to see that I am genuine in my offer of friendship. I hope after a good night's rest, you are still committed to working together as I am."
"Whatever," Anna said. "Let's just get started."
Augustus said nothing, taking another dignified sip of coffee. Zuma stared daggers at Anna for her lack of respect. Maxillo gave her a stony stare. Anna looked as if she didn't care.
"It's alright," Augustus said. "In fact, I appreciate such bluntness, as it is very lacking among my own governors and advisors." Augustus drained the last of his coffee and handed the cup to Zuma, who took it with a bow. "We are ready to leave, upon the arrival of Captain Sparks. We should have enough time to rescue Perseus and begin preparations for the Radaskim attack."
As if his name were a summons, Jonas Spark entered the tent. His blond, spiky hair was wild and his black-rimmed glass hung askew. Duct tape held the frame together at the nose – after many years, the glasses had seen a lot of punishment, and it was surprising they had lasted so long. He wore khaki cargo pants and a white shirt overlaid with a blue vest.
"You called for me, Princeps?"
"Yes," Augustus said. "Is everything ready to depart?"
Sparks nodded. "Yes, Emperor. Orion is ready to fly when you are."
Sparks appeared even more disheveled than the first time I'd seen him; apparently, he'd gotten little sleep. Hard to imagine, since he had stayed on the ship, where the accommodations were more comfortable than the camp.
"In future," Augustus said, "try not to carouse on the night before a major mission. Everything depends on this, as I've already told you."
Jonas's face blanched, but there was the hint of a smile on his face. "Yes. Sorry, Princeps. I'll keep that in mind."
Anna shook her head as Ashton stared at Sparks. The Emperor's tone suggested that it wasn't his first time reprimanding Sparks. Judging by Jonas's unrepentant features, he didn't care.
"It'll take a couple of hours to reach Bunker Six," Jonas said. "Everything's ready."
"Good," Augustus said brusquely. "Then let us depart."
***
WE WALKED WITH AUGUSTUS, his Praetorians, and Jonas Sparks to the west side of camp, exiting through the gate we had entered yesterday. The gate was opened to reveal the airport terminal and its boarding tunnels outstretched. The legionaries had set up outposts around the massive building. Mounds of sandbags rose from the tarmac, behind which guards took shelter.
We walked until we arrived at the terminal's entrance. While we had been sleeping, Augustus had set up a long perimeter of sandbags in front, behind which yet more legionaries were sheltered. There were hundreds of Imperial soldiers here. The sandbags would be more useful against bullets than crawlers.
Maybe, though, that had been the Emperor's intent.
We walked inside the terminal. We weaved our way through the building, until we arrived at the stairwell we had come down yesterday. Most of the Praetorians stayed behind while Maxillo took the lead, followed by Augustus and Sparks. Ashton, Anna, and I came next, followed by six more Praetorians. This meant the blond, second-in-command Praetorian was in charge of the guards left behind. I still didn't know his name.
When we reached the top of the stairwell, Maxillo opened the door, revealing red daylight. Orion was parked, just where Jonas had left it, on the large helipad. We walked onto the flat rooftop.
Jonas raced up the boarding ramp, and entered the code into the keypad. The door hissed open, letting out a stream of air. Everyone entered the ship. When the last Praetorian had gone in, the door slid shut.
We seated ourselves in the wardroom while Sparks, Augustus, and Maxillo made their way to the bridge. The rest of the Praetorians found handholds on the bulwarks they could grasp, in preparation for lifting off.
The ship vibrated as its fusion drive came online. Sparks's voice emanated from speakers built into the wardroom's corners.
"Prepare for liftoff."
Immediately, the ship lifted. I heard the ship's struts retract into its hull, and the boarding ramp slid into its slot. The ship turned, accelerating away from the airport.
We stayed seated as we gained in both speed and altitude. A few minutes later, we had reached cruising speed.
"What's Bunker Six like?" I asked Ashton.
"It's a small facility," Ashton said. "One of the satellites of Bunker One. It held approximately 250 people, along with the United States Space Fleet. I know I said it wasn't dangerous, but I was speaking relatively. Going into an offline Bunker is always dangerous. The hangar should be clear, though. At least, it was the last time I was there. It was sealed off back when the Bunker fell. Everyone who escaped, myself included, ended up in Skyhome."
"If it's clear, like you say, no problem then, right?" I asked.
"We'll see before long."
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# Chapter 7
It was two hours before Orion dropped in altitude. Within another few minutes, Orion ceased forward motion and began to lower vertically. We were descending into Bunker Six's hangar. The struts gave a metallic squeal as they extended. Shortly thereafter, the ship touched down.
Ashton, Anna, and I stood as the boarding ramp extended. Sparks, followed by Emperor Augustus, entered the wardroom.
Maxillo handed Ashton a radio. Another Praetorian handed Ashton, Anna, and I our handguns. Anna also received her blade.
"When you get to the blast door, Jonas will give you the code," Augustus said. "Once on board, he'll walk you through the start-up sequence."
"I know how to fly a ship," Ashton said.
"Once the ship is online, we'll be ready to head back to L.A." Augustus paused. "Understood?"
Ashton nodded. "Let's get this over with."
Maxillo opened the door. A rush of stale, metallic air entered the wardroom. When I stepped down the boarding ramp into Bunker Six's hangar, it was much colder than I expected. From above, red sunlight filtered down through a large, square opening large enough for spaceships to enter and exit. A wide, open area stretched between us and where Perseus was parked. The ship looked exactly like Orion and Odin, which made sense because the three ships had identical schematics.
The last two spaceships were in this very hangar, unless there happened to be another one out there we didn't know about.
As we descended Orion's ramp, I continued to take in my surroundings. Walls of rock rose before stopping at the opening above. Toward my right, in the back corner of the hangar, was a large, retractable door, barred shut. I assumed it led deeper into the Bunker. Faded yellow paint on its corrugated surface read 6.
As Orion idled behind us, Ashton struck out across the hangar. I felt Augustus's eyes on us as we walked.
When we finally made it to Perseus, we walked up the boarding ramp. Ashton raised the radio to his mouth.
"Code?"
"7-4-6-5-Z-Z-8-0-0," Jonas said.
Ashton input the code. The door beeped and slid open, revealing Perseus's wardroom. It was identical to Odin and Orion in every regard.
"You and Anna scout aft," Ashton said. "I'll head to the bridge."
Anna and I followed Ashton's order. Once aft, we checked the two cabins on the left side of the corridor, and the clinic on the right. The lavatory was also clear, which left the engine room. It was also empty of threats.
We headed back to the wardroom, where we met Ashton.
"Cargo bay," Ashton said. "I already checked the bridge, galley, and captain's quarters."
We headed down the three steps out of the wardroom, opening the door into the cargo bay. It was practically empty. There was a metal cabinet built into the wall, and several cords and hoses curled on the deck, but not much else.
"Clear," Ashton said into his radio.
"Are you at the bridge yet?" Jonas asked.
"In a minute," Ashton said. "About to start it up."
"Hurry. The Emperor wants to get moving."
We exited the cargo bay and headed for the bridge. Anna took the copilot's seat while Ashton took the pilot's.
"Code?" Ashton asked.
"3-1-5-8," Jonas said.
Ashton keyed in the code on the LCD. Slowly, the ship rumbled to life. Its hull and deck vibrated as the fusion drive, dormant for thirty years, worked itself online.
"Thank God it works," Ashton said. "Fuel levels full. Switching to inter-ship comm."
"Copy that," Jonas said.
Anna flicked a few buttons. "Should be connected."
"Jonas?" Ashton asked. "You hear me?"
Jonas's voice now came from the dash.
"Loud and clear."
"Following your lead."
"Copy that."
Orion lifted from the hangar, its thrusters glowing blue. Ashton waited a few seconds before following. I strapped myself to the jump seat at the back of the bridge, behind both Ashton and Anna. The ship swerved as it rose. The hangar fell away as we lifted into the red sky above.
Before us spread a panorama of pink and purple mountains. Red mist shrouded the jagged peaks, and pink xenofungus extended as far as the eye could see. Swarms of birds flew in cyclones above the xenofungal floor. In the far distance, a massive Xenolith rose from the fungus, much larger than any I'd ever seen. More such Xenoliths dotted the horizon. We were deep into the Great Blight. The xenolife here would be more massive and exotic – and probably dangerous.
"Perseus..." It was Augustus's voice. "We have just received word from Los Angeles. The Reapers are shelling my camp."
We looked at each other. It was time to make good on our side of the bargain, but Makara and the others still needed airlifting out of Oasis, if they were still there.
"Augustus," Ashton said. "We...have something to take care of, first. It won't take long."
"What?"
"We'll...be back to help soon. It has to do with our friends."
"Your friends?" The dash was silent for a moment. "You mean...Samuel?"
"We didn't want to say anything. But they are alive. We know, because the cargo bay of Gilgamesh was empty. There was a Recon stored there, so we know they got out. We think they're in Oasis, and Oasis is under attack. We have to save them."
It took a while for Augustus to react to these words. I could only imagine his shock.
"I...can hardly believe that, Ashton. Even if it's true...how do you know they are there?"
Ashton looked at me. "I think we might have a lead. Don't ask me to explain. It should only take thirty minutes extra."
"Thirty minutes?" Augustus asked. "Do you know how many of my men could die in that time?"
"This is important," Ashton said. "And it's not something we can argue about. It's something we must do."
Augustus said something in Spanish, likely a curse.
"I went through all this trouble to provide you with a spaceship, and this is how you repay me?"
"It won't be long for us to pick them up," I said. "If we wait, they're going to die, and we can't let that happen. We don't have time, especially if they have flyers. Your men can hold on that long."
Augustus was silent for a long moment.
"I don't like this, but I see I have no option. You have your extra thirty minutes, but after that, you must help me with the Reapers."
"Agreed," Ashton said. "We won't be long."
We sped west over the Great Blight.
***
WE STAYED HIGH IN ALTITUDE at first, but once we were halfway over Arizona, Ashton lowered the ship in preparation for descent into Oasis. Once we broke through the layer of thick, red clouds, we saw a fraction of what we'd be up against.
The entire fungus-ridden ground boiled with wave after wave of crawlers, all speeding east in a blur. Above them flew dragons – at least a dozen of them. Behemoths lumbered across the plains at a run while other strange, nameless creatures scampered in their wake.
I realized where they all were headed. Los Angeles.
"Opening fire," Sparks said.
"No!" Ashton shouted. "Hold your fire!"
But it was too late. A trail of bullets blazed from Orion's turret, entering the back of one of the dragons. The dragon gave a high shriek that pierced the ship's hull. Its wings folded as it spiraled to the ground, crashing into a startled Behemoth.
Immediately, the rest of the dragons changed trajectory, aiming upward toward us. Their white eyes blazed in fury.
"Lose them in the clouds," Ashton said. "And hold on. This might get bumpy."
The ship angled upward. The onrush of G-forces pushed me against my seat. The LCD revealed Orion lagging behind, two xenodragons clipping at its thrusters.
"Faster, Sparks!" Ashton said.
"I'm trying!"
We broke through the clouds, and Ashton kept climbing. From behind, Orion appeared. Seconds later, two xenodragons pierced the clouds, flying madly to intercept Orion.
"Keep climbing!" Ashton said.
"I can't go any faster!" Sparks said.
Ashton looked at the LCD, shaking his head in frustration. "Your retrothrusters are facing the wrong way, you idiot! Point them aft!"
Below, Sparks struggled to control the ship. All the while, the xenodragons closed the gap.
Ashton turned Perseus around. "We can't trust them to save themselves."
"Target acquired," Anna said. "Fire?"
"Jonas, out of the way!"
"What?"
"To port, to port!"
Finally, the message seemed to connect. Orion surged to the left, almost executing a barrel roll. The ships weren't meant to fly upside down while in the atmosphere, and the move could have caused Orion to go down. But Sparks managed to right the ship, giving us the opening we needed to take a shot.
"Both targets acquired," Anna said.
Instead of haranguing Jonas again, Ashton merely shook his head. "Fire."
Two missiles streaked from beneath Perseus. One zoomed to left, toward a xenodragon chasing after Orion, while the other surged to the right, where the other dragon raced to cut off the Orion's escape.
"Drop through the clouds!" Ashton said.
This time, there was no delay in reaction. Orion dipped, which must have thrown the Praetorians in the wardroom straight to the deck. Before the dragons could react to Orion's change in course, the missiles connected, igniting in twin, fiery plumes of white and orange. A rush of purplish organic matter and chips of bone exploded outward, charred to crisps from the immense heat released by the missiles.
I looked at the section of cloud below the explosion, where Orion had disappeared.
"You still there?" Ashton asked. "Orion, you have a copy?"
There was no response. Perseus showed the ship to be online. Maybe Jonas had blacked out from the G-forces.
At last, Orion lifted above the clouds at a leisurely place.
"Sparks," Ashton said, "you nearly got us all killed."
"I know," he said, almost sullenly. "I'm sorry."
"It's real easy to piss off those dragons," Ashton said. "We're lucky only two came after us."
"I wonder where they're going, anyway," Anna said.
"Los Angeles," I said. "Carin Black picked a hell of a time to start the war."
"There might be a way to stop it," Ashton said. "But first, we have to get to Oasis. Our team's waiting on us."
Ashton turned the ship, facing it northwest.
"Follow my lead, Sparks," he said. "And don't do anything without my go-ahead. Got that?"
It was a long while before Ashton received his answer.
"Roger."
***
THIRTY MINUTES LATER, Perseus lowered through the red cloud layer to an Oasis under siege.
Legions of crawlers and monsters writhed outside its wooden stockade. From the ramparts guards shot into the teeming mass, to little avail. There were so many monsters that it would only be a matter of minutes before they broke through – if they hadn't already. Three Behemoths clobbered a northern section of palisade near the interior farms. One of the giants fell under a barrage of machine-gun fire from a turret in a nearby watchtower. The fallen Behemoth was replaced by another.
On the southern section, the crawlers were so thick that they were piling up. The ones at the top leaped madly, trying to gain a foothold on the ramparts.
There was no way we could rescue everyone inside.
"We're not going in there," Augustus said. "It's madness."
Ashton hovered a moment above the walls.
"We could at least make a dent in their numbers."
Ashton swiveled, pointing Perseus toward the main body of the swarm.
"Anna, fire in 3...2...1..."
A missile streaked from beneath Perseus's hull. In a couple of seconds, it connected with the ground, bursting into bright, fiery plumes. The remains of monsters blasted outward.
There were thousands of them down there. My heart sank at the sight. There was no way we could survive this coming storm once it made it to Los Angeles.
"Look to starboard!" Anna said.
Ashton angled the ship slightly, to see the final nail in the coffin coming in the form of at least twelve xenodragons. They flew from the east in a wide V. They were still distant, little larger than specks on the horizon, but they would be upon us in minutes.
"We have to get them out quickly," Ashton said. "Sparks, cover me. I'm landing inside."
"No," Augustus said, answering for his pilot. "You're going to get yourself killed. We must retreat to Los Angeles."
"Our friends are down there, and I'll be damned if I'm leaving them behind. Go back to Los Angeles if you like. This is something we have to do."
Augustus growled, but didn't argue further. Orion remained hovering behind us, seemingly indecisive.
"Fine," Augustus spat. "We'll distract those monsters while you land. But we aren't fighting those dragons for you. You're on your own for those."
At that moment, two missiles shot out from Orion, igniting near the crater where Perseus's missiles had connected. A trail of bullets riddled the battlefield, tearing into the swarm.
Ashton flew over the eastern wall. Men watched in amazement as we swooped overhead. In the chaos below, I saw no sign of our friends.
We lowered onto the fields occupying the northern half of the town. We set down, crumpling crops beneath the ship's landing struts.
We ran off the bridge, me taking the lead. I pulled out my Beretta, stopping by the armory to see if there were other weapons I could use. I slid the door open, revealing various rifles, handguns, grenades, and body armor.
I latched a couple of grenades to my belt, also snatching an AR-15 from the gun rack. I checked the magazine for bullets. It was locked and loaded. I already had additional bullets in my pack for when I needed them.
"Let's go!" Ashton said, drawing his handgun.
Anna unsheathed her katana in the wardroom, and pressed the exit button to the blast door. It slid open, letting in a rush of cold, fetid air. I nearly gagged at the smell. The thousands of Blighters outside the walls had made this place smell like a garbage heap.
We ran down the ramp and into the flattened wheat crop. The fields appeared to be empty of threats.
That was, until three white eyes stared out from between the stalks.
The crawler let out a primal scream as it shot forward. I lowered my AR against my shoulder, firing a stream of bullets. The crawler screeched and shuddered as it settled into the grain. Its exoskeleton was coated with purple slime. The insect-like corpse twitched, and even though it was dead, its three eyes blinked open and shut. Anna, with a grunt, buried her blade deep into the crawler's head in the nape of its neck. Its muscles convulsed before it went slack.
"We need to head to town," Anna said, wiping her blade on the wheat. "Find Ohlan. He'll know where Makara is."
We set off south through the fields at a run, Ashton wheezing from behind. We found a wide dirt path leading into town. If there was one crawler inside the wheat fields, there were probably more lurking about. There was an opening somewhere in the walls. Whatever the case, we didn't have much time.
We entered the outskirts of Oasis. The streets were empty, and the ramshackle wooden and tin buildings creaked in the wind. Inhuman screams sounded from the horde, still separated from us by the fragile palisade.
We ran forward, to where the dirt path forked into two segments wrapping around either side of the oasis for which the town was named. We had a few minutes to find our friends and get out of here. Those dragons wouldn't be long.
We reached an intersection. A man ran out of a wooden building to our right. He pointed a gun our direction, firing several times.
As we raised our weapons, it became clear the gun was empty of bullets. It just clicked, over and over, as tears ran down his face.
With a wail, the man dropped to his knees. Anna walked forward, placing her blade near the weeping man's neck.
"Makara. Samuel. Where are they?"
The man continued to wail, uncomprehending.
"Where's Ohlan?" I asked.
That name seemed to register. The man pointing a shaking finger across the oasis, at a large, rounded building that might have been some sort of meeting hall.
"Thank you," Ashton said. "Hide somewhere safe."
We left the crazed man behind and ran around the west side of the oasis. Dozens of shots cracked the air, mostly from the wall. Between the buildings, we got a glimpse of the guards, firing downward into the horde. The piercing wail of a xenodragon carried on the wind from the east, fortunately still distant.
When we reached the other side of the oasis, two crawlers shot forward at a lightning pace from the town's main drag. Their large mouths were agape and dripping red blood. As we were about to fire, from somewhere in the town came a woman's scream. Instantly, the crawlers turned from us, and rushed in the direction they had come. It was only a moment later that the woman screamed again, as if in horrible pain.
There was no time to focus on such horror. We kept running, past the main drag. The large building was ahead. Makara and the others needed to be in there; if they weren't, we didn't have time to search elsewhere. Above, Orion swooped low over the town as more explosions rocked the east, shaking the ground with their impact.
We arrived in front of the open doorway, and ran inside.
The entire interior was a large circular room with a dirt floor. It was dim, but not impossible to see. At the end of the building, a set of steps led up to a throne. Rows of foldout chairs faced the throne. This might have been a place where meetings were conducted, or sermons given. I remember Samuel having said this place was run like a cult.
"There," Anna said, pointing to our left.
A set of stairs spiraled downward beneath the building. But in the building's dimness, she had missed a key detail about that throne.
A man was sitting on it.
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# Chapter 8
"You bring death to Oasis," a voice rattled. "I should have killed you in the beginning."
I walked a few steps forward, pointing my AR-15 at the throne.
"Where are they, Ohlan?"
From outside, screams penetrated the walls. We didn't have long until these monsters had completely broken through. But first, I had to deal with the monster sitting on this throne.
"Where they should be. They are downstairs, locked in the cell. They were due to be executed this morning, in fact." He gave a bitter, mad laugh. "It looks like that will be the fate of us all."
"I should kill you," Anna said. "But maybe I should let the crawlers do it for me."
I thought I saw Ohlan shrug in the dimness. "So be it. Death comes for us all, in the end. It is useless to resist."
"Come on," I said.
"Mark him, Ashton," Anna said. "Alex and I will go downstairs."
Ashton nodded. Anna and I went left, never turning our backs to Ohlan. Slowly, we descended the stairs.
My AR had a battery-powered light on it, which I clicked on. Underground, it smelled of must, mold, and sweat.
"Makara?" I called out. "Samuel? Anyone...?"
"Alex?"
It was Makara – unmistakably Makara. My heart swelled with happiness at hearing her voice.
Everyone else called out. They were all alive.
"We have to hurry," I said. "You're locked in?"
"How did you survive?" Samuel asked. "How did you find...?"
"There'll be time for all that later," I said. "Right now, we have to..."
"Ohlan has the key," Michael said. "Did you kill him?"
"No, he's..."
Already, Anna was turning around, charging up the stairs. I ran up after her. When we surfaced, we found Ohlan, still sitting on his throne, Ashton steadily pointing his handgun at him.
"The key, Ohlan," I said.
Ohlan gave a small smile. Quickly, he reached downward, pulling on a small, round object that most definitely was not a key.
"Grenade!" I yelled.
The grenade was lobbed to the center of the floor. Ashton backed away for the stairs. I grabbed Anna and pulled her in that direction as well. Ashton tumbled down, falling several steps. I dove inside with Anna in tow. We landed with a thud in the center of the steps just as the grenade split the air, deafeningly loud. Bits of metal shot into the wall, several of the fragments sprinkling onto the steps from above. Several of the sharp edges cut my skin.
I got off Anna, allowing her to get up.
"You alright?" I asked.
She nodded. "Yeah. You?"
"Fine. We have to get that key."
We walked back up the stairs. We found Ohlan's corpse lying in the center of the floor, blood pooling around his remains. As we neared, I saw grenade fragments sticking deeply into his skin. The sounds of battle raged outside, reminding me we couldn't stay here long.
Anna searched Ohlan's pockets. A few seconds later, she came away with a large, metal key.
"Got it."
We ran downstairs. I pointed my AR-15's flashlight ahead into the darkness, revealing the forms of our four friends. They ducked out of the way; I realized I was pointing my gun right at them.
"Sorry," I said, pointing the gun toward the floor.
Anna stuck the key in the cell door, giving it a twist. With a squeal, the metal-barred door swung open.
They ran outside, and we all made our way upstairs and into the meeting hall. Ashton was first to speak.
"We need to get moving," Ashton said. "I don't even know if we'll make it to the ship in time."
"So there was a ship," Julian said. "I knew it."
"We'll explain everything later," I said. "Augustus is waiting."
All of their eyes widened in surprise, but I didn't have time to explain anything. I ran toward the door.
"We have no weapons," Makara said.
I handed Makara my AR. "Take this."
As Makara took the weapon, I withdrew my Beretta.
"Let's go. The ship is in the northern fields."
With Makara's nod, I charged forward into the red sunlight.
Only to be greeted by a large Behemoth standing twenty feet ahead.
***
THE BEHEMOTH WAS AT least twelve feet tall, with blue-gray skin. The skin had the splotches of pink coloring more typical of xenolife. Hollow white eyes stared from its flat face. Upon seeing us, the Behemoth frowned and made a fist with its massive right hand. Knuckles cracked.
I sprang into action immediately. I aimed my Beretta toward the Behemoth's head and fired. It moved its head, but still a bullet grazed the side of it. The Behemoth roared in pain as a small rivulet of purple blood oozed out, just below its tiny ear.
Ashton went to the right while Anna dashed to the left, blade flashing. Makara got a few shots off as the Behemoth advanced forward, spraying the monster in the chest. The bullets had little effect because of its tough skin. Makara and I kept the monster's attention while we backed away. The wall of the building touched my back, and I slid along its perimeter.
Ashton and Anna had circled around the creature. Anna sliced at the Behemoth's leg, cutting deep before backing away. The Behemoth swiveled around with a pained roar, its arm extended. But Anna was far out of reach, nimbly backing away from the Behemoth's attack.
That was when Ashton took his chance to fire into the other leg with his handgun.
The Behemoth bellowed as it dropped to its knees. Still, its white eyes focused on me, as if I had been the source of its pain.
Emboldened, Anna stepped forward, within view of the Behemoth's eyes. She thrust her blade at the Behemoth's chest. With lightning reflexes, the monster spun and bashed Anna right in the chest. Anna eluded some of the impact, but still, the attack knocked her back a few feet. The Behemoth turned on its knees, crawling forward for the kill.
"No!" I yelled.
I ran forward, firing into the Behemoth's exposed back, but none of my bullets penetrated the chitin-like armor that grew there. As Ashton opened fire, the creature rolled into a tight ball so that all of its stone-like skin faced outward.
Anna scrambled away, obviously in great pain. The Behemoth uncurled just a bit, reaching out to grab Anna's leg.
I was now so close to the Behemoth that I could reach out and touch it. I aimed my Beretta toward its neck, firing into the skin there, which wasn't so thick. At this range, the bullets finally started doing some damage. The Behemoth gave a loud hiss, turning from Anna to focus on me. Its left fist fell from the red sky like a hammer...
Until Orion flew from above, firing a spray of bullets.
"Back!" I yelled.
As Orion swooped overhead, it unleashed its fury. We dove from its line of fire, and the Behemoth cried as the high-caliber rounds tore into its skin. Geysers of purple blood spewed outward, staining the dirt road. The light in the Behemoth's white eyes faded as it crumpled onto the road, dead.
Orion circled, landing not far from the town's central oasis. Everyone ran to the ship, but I ran to Anna, who was struggling to her feet.
I reached her and put a hand on her shoulder.
"Can you run?"
She nodded, leaning on me as I eased her up. She winced and her breaths came out ragged. I had no idea how badly she was hurt, but I had to get her to the ship. Samuel came to help me carry her. Together, we lifted, and ran as fast as we could toward the ship. Everyone else had already boarded.
We ran up the ramp. I could see Augustus standing in the wardroom within, his expression impatient. We went past him, setting Anna on the deck. She groaned in pain. Everyone stood in the wardroom, circling around her.
"We need to get to Perseus," Ashton said.
"Jonas is going there now," Augustus said.
The ship lifted, causing Julian and Michael to tumble to the deck. The rest of us grabbed the bulwarks as Orion made its crazy flight north across the oasis. Within half a minute, Sparks had landed the ship. The blast door opened, revealing Perseus a mere fifty feet ahead.
"Go, go, go!" Samuel said.
Everyone ran down Orion's boarding ramp and up Perseus's. Once again, Samuel and I toted Anna while the others made sure nothing attacked us from the fields. The coast was clear. Samuel and I were the last ones to enter the ship. As soon as we were on board, the door closed.
We laid Anna down on the deck, her eyes shut from pain. The ship took off.
"Where's Ashton?" I asked. "He needs to look at Anna."
"He's piloting," Samuel said. "I'll have him switch out with Makara."
Samuel left for the bridge.
"Let's see it," I said.
Anna raised her shirt up to her ribcage, revealing a deep bruising.
"Just a scratch," she said.
"You've broken a rib," I said.
"Whatever it is," Anna said, "it doesn't look like I'll be fighting for a while."
The ship circled as it blasted upward and outward. I had no idea where we were going or what kind of opposition we faced, or if the dragons would be falling upon us soon.
I just knew that with Anna's injury, we had been dealt a horrible blow.
***
BY THE TIME I REACHED the bridge, we were high above the clouds, heading southwest. Augustus's voice came through the speakers.
"Alright. We got out of there alive. Now it's time for you to help with Black."
"Copy that," Ashton said. "We have wounded here."
"While you have wounded, my men are dying on the battlefield," Augustus said. "And those monsters aren't far behind. There's no time to waste."
Augustus did have a point, but all the same, Anna needed treatment.
Ashton set Perseus to cruise, giving a nod a satisfaction. He stood, and Makara moved over from the copilot's seat to take his place.
"Help me move Anna to the clinic, Alex," he said. "That deck can hardly be comfortable."
"Sorry," I said. "We didn't have time to set her somewhere more comfortable."
"Don't apologize to me, kid. Let's move."
Samuel and Michael followed Ashton and me off the bridge. Julian took up the copilot's controls. Makara was taking this opportunity to teach him a bit about flying.
When we entered the wardroom, Anna was lying on her back.
"We're moving you to the clinic," Ashton said. "Up you get."
Anna groaned, but she allowed herself to be lifted up. All four of us moved her, making sure her back remained straight to cause her the least amount of pain. As we entered the compartment, the automatic lights flashed on.
We eased Anna onto the bed. I took off her boots, setting them to the side. She closed her eyes, drawing shallow breaths.
"Hurts like hell," she said. "Can hardly breathe."
Ashton nodded, turning to the rest of us. "Let's give her some space, huh? This shouldn't take me long."
"Is it bad?" I asked.
"Alex, I'll be fine," she said. "Go."
We left the clinic. Ashton began to ask Anna some questions, but I couldn't hear the words because we had already reached the wardroom and were on our way to the bridge.
"Broken ribs can take a while to heal," Samuel said. "And sometimes, there can be more serious issues."
"Like what?" I asked, stopping.
"Sometimes, the ribs can pierce a lung or an artery," he said. "But don't worry. If it was that bad, we'd know by now. She can at least breathe, even if it's painful."
"She needs to see Char," I said.
Samuel nodded. "I agree. One thing is clear: she'll be out of action for a while."
"How long?" I asked.
"Six weeks, minimum."
Samuel continued on to the bridge. I just stood in the wardroom, shocked.
Michael placed a hand on my shoulder. "Come on, Alex."
Stunned, I followed Michael back to the bridge. Six weeks or longer. That meant Anna couldn't fight anymore. In six weeks, we'd have either fought our final battle, or we'd all be dead.
I wanted to go to her, but Samuel pulled me to the bridge.
"Come on, Alex. Keep your head in the game."
Once we reached the bridge, Makara turned her head.
"We have a problem. Augustus wants us to come straight back to L.A., but we also need to get treatment for Anna. The only person I can think of to do that is Char."
"There's not much you can do for a broken rib other than let it heal on its own," Samuel said.
"Well, I need to talk to him, anyway. He's still in the dark about what happened in Wyoming."
Makara fiddled with the dash until it had attained the proper settings.
"Char. You have a copy?"
The dash sizzled with silence. After another long moment, I was starting to get nervous.
"Char. Marcus. Do you read me? This is Makara, aboard the Perseus spaceship."
Again, there was no answer. Char always had his radio on him and was quick about answering.
Finally, a voice answered, and it wasn't Char's gruff voice. Instead, it was slimy, weasely laughter.
Laughter that could only come from one person.
"Why, hello, Makara," Jade said, with a snicker, drawing out the middle syllable of Makara's name. "Long time, no hear."
"Jade," Makara said, with venom. "What did you do with Char?"
The Weasel chortled a bit before answering. "It's been so long since our fearless leader has checked in. The people were getting antsy. Someone had to restore order."
"What are you talking about?" Samuel asked.
"Just what I said," Jade said, with a giggle. "Someone had to take control."
"Jade," Makara said, working to keep control, "you will step down immediately, or I'll personally see to your execution when I get back there."
Jade was laughing, now. "Oh, Makara, you have no idea how the tables have turned! I'd always dreamed of this moment. We thought you were dead, but this is even better. Because at least now, you know the Lords of Vegas are nothing to be trifled with."
"You're not alone, then?" Samuel said.
Jade paused, just a moment, before he giggled again, unable to restrain himself.
"Do yourself a favor, Makara," the Weasel said. "Stay away. There's nothing for you here. Because if you come back..."
We all waited for Jade to continue. After a long moment, he finally did.
"...I'll kill you."
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# Chapter 9
Jade had taken over Bunker 84. The thought seemed impossible. Jade had always been the least of the gang lords, the one the others had taken shots at.
Now, he was the one calling the shots.
After a while, Emperor Augustus's voice arose from the dash.
"What's the status? Why was the line busy?"
"We have an issue," Makara said, her tone saying that it was the understatement of the century.
"What issue?" Augustus asked.
"We lost Bunker 84."
Augustus was quiet for a moment. I had no idea what the Emperor was thinking, or if he even knew what Bunker 84 was. From his answer, he apparently did.
"How? What happened?"
"One of the gang lords rebelled," Ashton said. "A man named Jade."
Augustus grunted, saying nothing more. "And I assume you want to go rescue that Bunker – the few of you against whatever hundreds they might have?"
"That's right," Samuel said.
"We have friends there," Makara said.
"Your place is with the Empire in Los Angeles," Augustus said. "You'll all get yourselves killed if you leave now. Besides, we had a bargain."
"I know we did," Ashton said, "but if we can save the Bunker, your troops will have reinforcements. There are probably a lot of Angels still loyal to Makara."
"That's if," Augustus said. "I gave you that spaceship, and you need to repay me."
"I'm sorry," Makara said. "Things happen."
The Emperor hadn't planned for this, and for that matter, neither had we. And there was nothing he could to stop us from going besides outright shooting us down.
"We'll return soon," Samuel said. "And we'll have a force to help you against the Reapers. We might even be able to set up a flank and overwhelm Black's troops."
"You say you'll return soon, but we had an agreement. You're abandoning an ally on the field of battle."
"You outnumber Carin's army five to one," Makara said. "How could you even say we're abandoning you?"
"We are abandoning him," Ashton said. "Because it's not just the Reapers that Augustus is up against. There is the entire swarm moving from the east. If Carin Black drags out the battle long enough, Augustus will be crushed between the two sides."
That shut everyone up. It meant that if we were going to Bunker 84, then we had to get back in time. Augustus and his legions would be dead if we didn't.
"How long do you think we have?" Samuel asked.
"A few days, at most," Ashton said. "The dragons can be there in half a day, maybe. The rest of the swarm will arrive in the next couple days. That's all speculation, though."
I had no idea if we could save Bunker 84 in that amount of time, but we had to go back. Our friends were there: Char, Marcus, Ruth, Grudge, along with all the others we had saved from Las Vegas and the Community. We couldn't abandon them, especially now when we had the ability to rescue them.
"Go," Augustus said. "You can take no longer than two days."
Makara didn't need more encouragement. "It shouldn't be long."
Makara steered the ship north, the acceleration causing me to stumble from my standing position. I steadied myself on the bulwark.
I had no idea what awaited us in Bunker 84, but we would soon find out.
***
I PULLED UP A STOOL next to the clinic bed. Anna's eyes were closed; Ashton had put her on some pain medication, which made her drowsy. My own eyes were tired and heavy. I held her hand, feeling its warmth between my fingers.
Her eyes opened, unseeing at first, but then focusing on my face.
When she tried to face me more directly, she winced in pain. I put a hand on her shoulder.
"Sit still."
"I haven't known the meaning of those two words my entire life."
I hadn't told her about the decision the crew had reached on the bridge, and I almost didn't want to tell her. I knew how bad she would feel waiting on the ship while we secured the Bunker.
Ashton had volunteered to watch over her. He said he'd be in the way if he tagged along.
In the end, though, I updated her on what was going on. She needed to know.
"Great," she said. "I'm going to be bored to tears lying in here."
"Just see it as a chance to relax."
"Relax?" Anna looked at me intently. "Relax, while everyone is putting their lives on the line? I couldn't do it."
"We're not letting you in there," I said.
"I know," she said. "I just hate the situation."
"It might not even be that bad in there," I said. "Maybe everyone will rebel against the gang lords when they realize we're back."
"Maybe," Anna said, but she didn't sound convinced. "There's been something I wanted to talk to you about."
Immediately, I started feeling nervous. "What do you mean?"
She looked at me, her green eyes almost haunted. "Even if I'm in a bed right now, I still haven't forgotten my promise. I won't let you die."
I didn't want to get into this again. I tried not to let my frustration show, but she knew me well enough to read my face.
"Don't get like that," she said. "You agreed to give yourself up without considering how others felt."
"You mean how you would feel?"
"Isn't that important?"
"You know what I have to do," I said.
I wasn't sure what else to say. I felt as if she wouldn't be satisfied with any answer besides the one she wanted. Still, she was being stubborn, not looking me in the eyes.
"Look," I said. "Before anything happens, we'll talk to the Wanderer. Just you and me. Okay? We'll see if there's another way to do this."
Anna watched me, her eyes afraid. I didn't want to say anything to hurt her, or scare her, but how could I not when my death was a sure thing? Maybe this would be a chance to convince her.
"Whatever happens," I said, "we'll face it together. Alright?"
Very reluctantly, she nodded. I didn't know what else she wanted me to say. Maybe she didn't know, herself. All I knew was, I couldn't give up now, after everyone had sacrificed so much.
The ship suddenly lurched and lowered. Still, I remained seated, looking at Anna, who now faced away. She stared at the clinic's walls as if they had more of an answer than I did.
"Hey," I said, touching her shoulder.
She turned her head toward me, giving a small smile. I knelt down and kissed her. I thought it was going to be short, but we continued to kiss. I was going away to do something dangerous, and Anna was stuck here. Anything could happen down there, so I didn't know when I'd see her again...or even if I'd see her again.
Someone entered the room, clearing his throat. I turned to see that it was Michael.
"Alright, kids," he said. "It's go time."
Anna pushed me away, her green eyes deadly serious.
"Don't die."
"I won't."
I left Anna and the clinic behind.
Not yet, anyway.
***
THANKFULLY, WE FOUND some white parkas hanging in a cabinet in the cargo bay. I was glad that Augustus hadn't grabbed those, because we needed them to make it to the entrance of Bunker 84 without freezing to death.
I grabbed an extra AR-15 to replace the one I gave Makara. I strapped it to my back and put a couple of handguns with complimenting ammunition in my pack. If there was anyone resisting Jade inside Bunker 84, they might be in need of weapons.
I just hoped Char, Marcus, and Ruth were alright. Michael was even more anxious to get down there because of his wife and daughter.
At last, we were all gathered in the wardroom. With a final nod, Samuel opened the blast door, letting in a rush of frigid, snow-laden air.
"Check in often," Ashton said as we exited the ship. "And good luck."
Michael led the way, pushing through the snow. Julian walked in his wake while Samuel, Makara, and I kept up. I stole a quick glance at Perseus behind. It had landed not too far from the wreckage of Odin, perhaps one hundred yards away. Its outline was almost lost to the swirling flurry.
Makara looked in that direction and sighed. "A lot of memories on that ship."
It didn't feel as cold today as it had when we'd landed here about a week ago, but the wind was still bitter.
I kept thinking of Jade. It was hard to imagine that weasel of a man ordering anyone around.
Through the shifting of snow, I saw the surface of the icy lake appear, its surface reflecting the gray sunlight dully. We reached its shoreline, walking in the direction of the Bunker entrance. Across the lake, the shape of the massive mountain was barely discernable against the gray sky.
The lake's edge led us to the familiar, frozen stream leading into the canyon, at the end of which lay the hidden Bunker entrance. It wouldn't be long before we reached the icefall.
Michael readied his rifle, and the rest of us followed suit by drawing our weapons.
Makara stepped past Michael, approaching the icefall cautiously. Julian was just a few steps behind. Makara slipped on the ice, and would have fallen if Julian hadn't caught her arm. After steadying herself, Makara gave him a terse thanks.
Michael, meanwhile, walked toward the icefall. The opening he and Julian had made still remained. Michael picked his way over the ice pile. Julian and Makara followed him, while Samuel and I brought up the rear.
Once we stood safely in front of the Bunker door, Makara peered inside. She dug out a flashlight from her pack and clicked it on. She pointed the light inside, revealing the empty passageway leading into darkness. I retrieved my own flashlight, keeping it off to conserve energy. If the need arose, I could turn it back on.
"Clear," she said, her breath forming a cloud. "Be ready."
She walked inside, and we followed, leaving the horrible, dry cold behind. As we entered the obscuring shadow, the snow continued to fall.
***
BUNKER 84 WAS AS QUIET as a tomb, and for some reason, it felt like we were walking into one.
Before long, the corridor opened into the park filled with dead plants. The cool air was sharp with the scent of metal and the dry smell of dead vegetation. Makara shined her light on the park, scanning left and right. A lot of dangers could be lurking there. I turned on my own light, helping Makara push back the darkness. Nothing moved.
"Is this close to where you were attacked?" she asked.
"Yeah," I said. "We were attacked, drugged, and taken down to the bottom."
"Do you know the way from here?"
I shook my head. "I assume there's a set of stairs that'll take us to the hangar."
Makara nodded, turning left. She made sure to stay about twenty feet from the edge of the deadened plants. It would be easy to be ambushed from there.
"Look," Samuel said. "Stairwell up ahead."
We paused before a flight of steps leading down. Fifteen or so steps descended to the first landing, before turning 180 degrees and continuing downward to the next level.
Michael and Julian followed Makara down the metal steps. Michael held his AR at the ready, while Julian kept his handgun in his right hand. Samuel and I, once again, brought up the rear. I glanced backward, scanning with my light to make sure nothing was tailing us. Still clear. I followed everyone downstairs.
We went down a couple of levels before the stairs came to a stop. I had expected this flight to go all the way down, but apparently we had to find another way to reach the hangar. Makara walked at a slow, careful pace. Her flashlight revealed many open doorways in the corridor – doorways from which threats could attack us. Every few moments she paused, listening, causing the rest of us to follow suit. There was nothing but our breaths.
"Keep moving," Samuel said. His voice, though soft, carried in the confines of the corridor.
We had walked a few more steps when boots pounded on metal. They came from behind.
I whirled around, shining my light into the darkness. There was no one there; they must have ducked into one of the doorways I'd just passed.
"Who's there?" I called.
From ahead came more stomping of boots. We pressed against each other back-to-back, our guns pointed ahead and behind.
"Lords or Angels?" a male voice called from Makara's direction.
She stepped forward. "Angels. Who are you?"
The man didn't answer immediately. I was ready to snap into action, shooting whoever jumped out at me.
"Makara?"
This caused other men in the shadows to murmur. If I had to guess, there were maybe ten of them. We were far outnumbered.
"Yeah," she said. "Who are you? What happened here?"
"Makara, we thought you were dead," the man said. "Where were you? What happened?"
The man had a strange accent, like none I'd ever heard spoken before. It tickled at my memory, though; I felt like I'd seen a movie where I heard a similar manner of speaking. The accent, though detectable, was buried under the gruff way of speaking that was common to most Raiders.
I realized that what I was hearing was an English accent. This man must have been born there, but had been living in the U.S. when Ragnarok fell – which meant that he had stayed here.
"Tell me who you are, first," Makara said. "For all I know, this is a trick. And I need you to put your weapons down."
"You don't have to worry," the man said. "Former Lost Angel under Raine, and now I'm one of Char's Raiders. I'm setting my gun on the floor, and stepping forward. My men will do the same."
Before Makara could say anything, I heard the thud of guns being set on the floor. A moment later, the man stepped out of a doorway, hands on his head, into Makara's flashlight beam. Makara's arms stiffened.
"I'm Lionel," he said. "Lionel Pierce. We're all Angels who've been hiding up here since the Lords betrayed us."
I watched Lionel as he spoke, trying to figure out if we could trust him. He certainly looked like he'd grown up in the Old World, and his face was familiar to me. He had a grizzled white beard, and the wrinkles on his face told me he was at least in his late fifties. Maybe even younger, given the harshness of the world.
"I recognize you," Makara said, "though we've never spoken. How many do you have with you?"
"Here, I have nine. If you mean how many Angels are left...there's no way of knowing that. There's less every day. They turned on us three days ago, during lights out. Forced the rest of us up here. They keep pushing us back, and we keep going further up. This place is big...far bigger than we expected."
Makara nodded. "Alright." She relaxed a bit, lowering her weapon. She motioned us to do the same. "Where are you headed, now?"
The man pointed upward. "We were trying to find a way out of here."
"Don't do that," Makara said. "We have to take this place back. That's why we came."
"If you don't mind me asking," Lionel said, "what happened to you guys?"
"We lost the spaceships in an air battle with dragons. We found another one. As soon as we did, we came back here to clean up this mess."
"Everyone thinks you're dead. But now that you're here...if we could somehow band together, we might have a shot of pushing them back. It'll be hard, though. They have most of the supplies, and they're working together."
"We have some extra guns, if your men need them," Makara said. "And yes, we need to work together...focus on finding other groups and taking the fight to them."
"Alright," Lionel said. "We'll join up, then. This Bunker still has lots of supplies lying around. Weapons. Even food, sometimes. But the Lords still outnumber and outgun us."
"The only way we can win is by taking the right fights," Makara said. She nodded at Lionel. "You can take up your weapons again. Let's get out of this corridor and find a safe spot to make a plan."
Lionel nodded. "Alright. There's a corridor nearby that we haven't explored yet."
"Lead us there," Makara said.
Lionel gave a nod and ducked into an open doorway. The men who had waited in the darkness brushed past, acknowledging Samuel and me with nods. They looked thin and tired.
After they filed after their leader, we followed them.
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# Chapter 10
We followed Lionel and his band of Angels into a room that must have once been a dormitory. Several bunks lay angled from the wall, stripped of sheets. A soiled mattress lay haphazardly on the floor. A bulletin board still had yellowed announcements tacked to it. Seeing the evidence of life in this fallen Bunker was haunting.
Lionel turned left. The dorm stretched a long way, and we passed row after row of bunks. In Bunker 108, we had full apartments, but here, it was more austere and militaristic. Beside each bunk was a small, metal nightstand, and at each one's foot was a metal trunk. Some of the trunks were locked, but others had been busted open, revealing empty interiors.
Once we reached the end of the dorm, another doorway opened on our right, leading into darkness.
"This is the corridor I was talking about," Lionel said.
"Where do you think it goes?" Samuel asked.
"Guess we're about to find out."
Lionel and his Angels went through the doorway. We followed.
The corridor was dark and claustrophobic. There was about a foot of space above my head, and hardly room for two people to walk side by side. The hallway took a sudden ninety-degree turn to the left. We walked a moment longer before it turned ninety degrees to the right. It was a strange bit of planning, and I had no idea what purpose it served.
Ahead, the corridor opened into a wide, cavernous space. Everyone filtered out and stood in the open area, gazing at the room in wonder. Makara, Lionel, and I shined our flashlight beams around. Banks of silver servers lined the wall directly to our left. Ahead, glass windows separated us from a large, conference-style room with rows of computers. The computers were down one level, in a sunken control center accessible only by steps leading down. We stood on a balcony overhanging that room. It looked like a place from which military operations would be conducted. A gigantic screen spread across the far wall of the control room. The display screen, along with all the computers, was dark.
"What is this place?" Makara asked.
"The Command Center," Samuel said. "Bunker One had a similar setup. I'm surprised there's one of this magnitude in Bunker 84. It's nearly as large as Bunker One's."
Lionel pointed to a couple of Angels. "Secure the corridor. Raise the alarm if you hear anything."
The Angels followed Lionel's command as Makara wandered off, brushing her hand along the servers.
"They're warm," she said.
Samuel placed his hand on one of them. Several of the men did the same.
"What does it matter, if it's not going to help us find the others?" Michael asked.
Both Samuel and Makara looked at Michael thoughtfully, but it was Samuel who spoke.
"Who said we couldn't do that?"
Michael frowned, wondering at Samuel's words. Samuel walked to the glass door separating the overhang from the control room below. He pulled it open, walking down the metal steps.
"Leave your men here," Makara said to Lionel. "We're going down with Samuel."
She motioned for Michael, Julian, and me to follow her.
While Lionel and the Angels stood guard, we went down the steps. I was the last one through, letting the glass door close automatically behind me.
The room was surprisingly warm; these computers were also running. They had to be powered, only I didn't know where from. When the Community lived here, they had gotten their power from the fusion generator aboard Aeneas. But there was no spaceship in Bunker 84, so Jade and the Lords must have devised an alternate power source, or maybe they were using backup power. Whatever the case, they needed power to grow food, pump water, and supply ventilation.
Samuel stood in front of a thin, sleek monitor in the center of the Control Room. He pressed a random key. The monitor came to life, displaying a login screen.
"So it is on," Makara said.
"We might be able to access the Bunker security system from here," Samuel said. "If so, we could use the cameras to find other Angels."
"And enemies," Julian said.
"Not just that," Makara said. "We might even be able to communicate. I'm sure an intercom system is built into this entire Bunker."
"Probably," I said. "Ours had one, so I imagine they all would."
Samuel logged in with his username and password. Instantly, the desktop screen loaded.
"Surprised it took it," Michael said.
"It did at Bunker One," Samuel said. "It must have updated at every Bunker still online." Samuel paused, thinking. "Don't know much about computers, actually. All that stuff I said...I have no idea how to do it."
"Ashton would know," I said.
"We can try contacting him," Makara said.
"This deep underground?" Michael asked.
Makara didn't have an answer for that.
"There has to be a way to communicate," I said. "Ashton called Bunker 84 while he was in Skyhome."
Samuel clicked the start menu. As soon as he did, the other computer screens came to life, bathing the control room with blue light.
In addition, the large display screen ahead flashed on, slowly gaining in color and clarity. A large map of the United States and Canada was displayed.
"What did you do?" Makara asked.
Samuel shrugged. The large screen displayed red dots pinpricking the topographic map, each with a number above it. 84. 16. 108.
"Bunker locations," I said.
A similar map had been on the LCD of the plane that took us to Bunker One, only this one was far more detailed.
The map also appeared on Samuel's screen.
"It looks like I can control the map," Samuel said.
He clicked on the location of Bunker 84, located in Northern California. Immediately, he was taken to a web site of some sort, dedicated to the Bunker. On the screen were links naming different departments. Hydroponics. Recycling. The Citizens' Council. Security.
"Click that last one," Makara said.
Samuel clicked on "Security." The website was bare and utilitarian, and obviously not meant for most residents' general use. There were several more links. An Officers' Directory. Nuclear Development. I wanted to tell Samuel to click on that one, just out of curiosity, but nukes weren't what we were after. One of the links said "Security Feed."
"There," Makara said.
Samuel clicked on it. More screens popped up, displaying on the large screen ahead as well as on Samuel's computer. Row upon row of camera feeds loaded, ranging in number from one all the way to three hundred and six. Not all the camera feeds could fit on Samuel's screen, so he had to scroll down to see them all. As he scrolled, the main screen ahead also scrolled down, mimicking Samuel's actions.
The only problem was, all of the video feeds were dead, with completely black screens. The word "offline" showed in the bottom right corner of each feed.
"Offline," Makara said.
"Go back," I said. "And click on the 'Power Plant' link."
Samuel complied, digging his way back from the camera feeds until he arrived at the listing of departments. He clicked on "Power Plant." It took us to a screen that showed basic schematics for the reactors that must have been powering the Bunker. There was a "details," which Samuel clicked.
Hibernation. Backup module available.
"What does that mean?" Makara asked.
"It means the reactor must have shut down once it realized it wasn't being used," Samuel said. "Probably."
"Then how is this place running?" Michael asked.
Samuel shrugged. "There must be some form of backup power for the Command Center servers alone. This place would be the most important to keep online. I guess the cameras have lower priority."
"Try turning the backup module on," Julian said.
Samuel clicked it, setting it to on. The computer thought a moment before accepting the change.
Nothing happened. Then, a message appeared on the computer screen, giving a timer.
"What?" Makara asked.
Samuel read the message out loud. "Backup module online in five minutes..."
"Is it really turning on?" I asked.
"I guess," Makara said.
"While we're here," Samuel said, "might as well look at some of that other stuff."
Samuel backed out of the screen. The countdown was minimized while he clicked on the "Nuclear Development" link.
"Really, Samuel?" Makara asked.
He shrugged. "Might as well take a look."
A wealth of information overtook the screen. There were twelve silos. One through three were empty, while the rest read "armed."
"So three of the silos have shot off missiles?" Makara asked, now curious. "Where to? When?"
"Let me see what I can dig up," Samuel said. "Maybe Elias had been busy."
"If they had gotten to this computer," I said, "don't you think they would have restored power a long time ago?"
"Good point," Samuel said. "Which makes me believe that the Community did not fire any missiles. They probably couldn't even get past the login screen."
"So, who shot them off?" Makara said.
"This would have been before the Community," Julian said. "The U.S. fired three nuclear missiles in the fifteen years Bunker 84 was online."
Why would Bunker 84's authorities launch nukes? Had they been aware of the xenoviral threat long before Bunker One was? Or had Bunker One given the order itself?
"Found it," Samuel said.
We crowded around the computer. In a column labeled "expired targets" three dates were listed: November 22, 2034, December 5, 2036, and July 17, 2045. Next to these dates were target locations.
The first two were nothing like what I expected.
Bunker 23. And Bunker 144.
"What?" I asked. "They nuked Bunkers?"
It was a moment before Samuel answered. "Apparently."
"Why?"
Something about Bunker 23 was tripping my memory. I remembered it being mentioned, somewhere...
Fortunately, Makara filled in the blank.
"That was the one you talked about, Samuel, when we found the Black Files. It was in Nebraska, the closest one to Ragnarok Crater. Could that have something to do with it?"
"Maybe," Samuel said.
"Maybe?" Michael said. "I say probably."
"You said Bunker 23 was where the first infections happened," Makara said. "Maybe Bunker 84...panicked a bit."
It would never be clear what had happened in those early days, unless we found a detailed explanation somewhere. Something we didn't exactly have time for unless we just happened to run across it.
"Okay," Julian said. "What about Bunker 144?"
"Elias mentioned that one," I said. "It was built in Alaska. Bunkers One, 84, and 144 were all meant to be counterpoints to one another. It's hard to imagine the xenovirus finding its way up there, especially if it was only 2036."
"Maybe they saw Bunker 144 as a threat," Makara said. "Also, I don't see how nukes can hit Bunkers. Aren't Bunkers built underground to avoid that sort of thing?"
"They are," Samuel said. "But it's possible they used bunker-busters – not designed to detonate until they are underground. A nuclear explosion like that would wreak havoc – not just from the awesome power, but from collapsing tunnels and burying exits."
"Where did the third one go, then?" I asked.
It showed it launching in mid-2045, but it gave no target location. 2045 would have been around the time Bunker 84 fell.
A moment later, the fluorescent lights flashed on above, one after the other. Power had been restored in full. I blinked as my eyes adjusted.
"Let's check on those cameras again," Makara said.
***
SAMUEL RETURNED TO the directory and cycled through the cameras, one at a time. Many were still black, but the rest were up and running, revealing people in rooms and hallways, seeming confused that the lights had come on. Both Lords and Angels suspected some trick on the part of the other side.
"So how do we find out who's on our side?" Makara asked.
"Maybe Lionel can help with that," Samuel said.
Samuel motioned Lionel down to the control room floor. Lionel reached the bottom of the curved steps and ran to stand next to us. He looked over Julian's shoulder at the monitor.
"Can you tell Angel from Lord?" Samuel asked.
"The Lords control most of the Bunker," he said. "Their base is on the bottom level."
"Why did they attack in the first place?" I asked.
"They didn't think you were coming back. And there's the matter of food. There was no way we had enough to go around. So, the Lords decided to save themselves."
"See anyone we know on here?" Makara asked. "We can head to that location first, then hit the rest as we gather our forces."
Lionel narrowed his eyes, scrutinizing the screen. "Can you pull up a map of the Bunker on the big screen?"
Samuel nodded. "I'm sure I can. Somehow."
He created a new tab and began to search the directory for a map. Within a minute, he'd found it. A large map of Bunker 84 was put up on the big screen. The size of Bunker 84 became all the more apparent, considering the map only showed one level at a time. For now, level three was displayed – the one we were on. There were twenty levels total.
"This place is huge," Makara said.
"Seems as if they could fit a lot more than two thousand people in here," Julian said. "I think there was more to this place than just the nukes."
"What do you mean?" Samuel asked.
"You've seen the dorms," Julian answered. "There must be hundreds just like the ones we saw. As if...an entire army had been here." He paused. "I don't know. Maybe I'm wrong."
"No," Makara said. "It makes sense. It's another reason why Askala would have targeted this place. Can you imagine if Bunker 84 had stayed alive?"
According to the Black Files, the U.S. government knew there was something guarding Ragnarok. The final Guardian Mission failed because it had been attacked directly. If they believed an alien invasion was imminent, it made complete sense they'd keep an entire army down here.
"We need to focus on finding our own," Lionel said.
Level three was mainly dedicated to barracks, a training center, and the Command Center.
Samuel clicked the other tab, revealing the cameras once again.
"It looks as if the cameras are coded to each level. Like this one..."
Samuel clicked on it, and it showed a shot of a couple of men bearing rifles, standing next to a pair of elevator doors stuck halfway open. The number three was painted on the wall next to them.
"Level three," Samuel said.
"Those are Lords," Lionel said.
As Samuel clicked the camera, part of the map highlighted on the screen – a periphery corridor on the level's western side.
"Looks like every time you click on a camera, it highlights where that camera is," Lionel said. "We can make a list of what places to go to, and what places to avoid."
"Good idea," Samuel said.
"Let's go through each of the levels," Lionel said. "I can take some notes for each one."
We did just that – Samuel scanned each level while Lionel took notes in a spiral notebook he'd found on a nearby desk. On most levels, the Lords had set up positions, especially the lower we got. It seemed as if there would be a lot of fighting once we got to level ten and lower.
Thirty minutes later, Lionel had taken notes all the way to level twenty. One of the cameras showed Jade, Rey, and Cain, sitting at a table over which a large map of the Bunker was spread. Apparently, they were making their own plan. They, too, had a map of the Bunker. If they knew this was where the Command Center was, then they probably wanted to gain control of it as soon as possible.
It seemed as if Samuel had come to the same realization.
"We need to find reinforcements," Samuel said. "Once we have some fighters, we can start taking back territory."
"See if you can call Ashton from here," Makara said. "He'll want the update."
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# Chapter 11
Samuel used Bunker 84's encrypted communication network to contact Ashton. We updated him on everything that had happened so far. Ashton said he'd be on standby for when he was needed.
We left the Command Center, Lionel carrying his notebook that listed Angel locations. Our first destination was on this level.
Bunker 84 looked different with the lights on. In a way, the lights were a curse because it would be harder to remain hidden and the fighting would pick up again.
But the time for hiding was over. We had to make an end of the Lords while we had the men and firepower to do so.
We left the Command Center behind and entered a long corridor. About fifty feet ahead, the corridor had been buried in rubble.
We stood before the twisted pile of concrete, rebar, and dirt.
"No good," Makara said. "That's the way we need to go."
"There has to be a way around," Michael said.
Michael backtracked a bit, poking his head in a doorway.
"This could work."
We followed him inside, finding a room that had once been a clinic. A wheeled gurney lay on its side. Bandages and unused syringes spread over the dusty floor, where they had spilled from the countertop. Another doorway led out of the clinic on the other side, leading into a corridor running parallel to the first.
Once we were through the clinic and into the opposite corridor, the temperature dropped drastically. As we walked on, the temperature continued to plummet.
The corridor took a ninety-degree turn to the left, at the end of which was a group of four men with rifles. It was a moment before one of them cried out and pointed at us.
As their guns were raised, I dove for an open doorway on right side of the corridor, Makara and Samuel right behind me. As the first shots were fired, I crashed into the floor. It was freezing cold in here. I looked back, across the corridor, to see that everyone else had remained behind the corner, cut off from us. Anyone who tried to walk across that corridor would be shot down.
"They have the hallway covered," Samuel said.
"If they're smart, they're going for reinforcements," Makara said. "We have to figure a way to take them down."
Another doorway led out from the other side of the room. The camera feed had shown a group of Angels near here. The only question was where they were.
"They should be nearby," I said.
That was when a man appeared in the doorway. His eyes widened.
"Makara?"
Instantly, Samuel, Makara, and I had him in our sights.
"Don't shoot!" he said. "I'm with you guys."
"You're Angel?" Makara asked.
"Yeah," he said.
"Where are the others?" Makara asked. "We saw them on the camera feed not five minutes ago."
"A lot can change in five minutes."
"You're the only survivor?"
"Of my group, yeah. I think Grudge and some of his men are down on level six. So, that's where I was headed." He paused. "My name's Damian, by the way."
"Damian, we have to deal with these Lords first," Makara said. "They got the corridor covered, and my men can't move on until they're taken care of."
"We can sneak around," Damian said. "While they're focused on your men, we can hit from another direction."
"How?"
"We'll have to go outside."
"Outside?" I asked.
"Follow me," Damian said.
Damian turned from the doorway, disappearing into darkness. We looked at each other and went after him.
***
DAMIAN LED US AWAY from the corridor, and the temperature continued to drop. It wasn't long before gray daylight filtered through the twisted, collapsed ceiling. This part of the Bunker must have been destroyed, somehow.
"So, Grudge is alive?" Makara asked. "What about Char, Marcus?"
As Damian began climbing the rubble in order to reach the outside, he cracked a smile. "Yeah, Grudge is alive. As far as Char and Marcus...I can't say. They stayed behind on level one, and we haven't heard from them since the war started."
We climbed after Damian. If Char and Marcus were on level one, then the news probably wasn't good.
I put my foot on a large slab of concrete, boosting myself up. Damian had already reached the top and was standing above the network of Bunker ruins.
"You sure you know where you're going?" I asked.
"Yeah, I'm sure."
Makara pulled herself onto the ceiling, followed by Samuel. I was the last one to exit the Bunker, and what I saw rendered me speechless.
This entire side of the mountain was simply...gone. Something had blasted it all away, leaving behind a crater. We were on the opposite side of the mountain, which up to this point we hadn't see yet. The exits of corridors opened into the free air, at least down to level seven or eight.
"Guess we know where that third nuke hit," Samuel said.
I still didn't know why Bunker 84 had nuked itself. It could have been done by the Community, but whatever the case, it had allowed Elias to take control. The bottommost levels of the Bunker would have been largely unaffected.
"This way," Damian said, moving out of the Bunker exit and onto the mountainside.
Damian angled himself toward an opening in the rubble not far from the one we had exited from. It was a short walk before we made it to the entrance. After walking a few steps inside, Damian spoke softly.
"Alright, the Lords are just around this corner..."
We edged to the corner as more shots split the air.
"I think they're being fired on from the other side," I said.
"The perfect opportunity to strike," Samuel said.
Samuel waited a moment before giving a nod. He checked around the corner, stepping out into the open. I went after him, seeing the four Lords from behind, guns all pointed toward our men.
By the time Makara was also in the open, we fired into their backs.
The two men in the middle went down immediately while the others snapped toward us. Before they could even aim, however, they also went down. Silence followed the brief firefight.
Samuel called out.
"It's us! When you come around, hold your fire!"
Lionel and the other Angels appeared from around the corner, running toward us. We ran ahead, meeting them in the center.
"That was some luck," Lionel said.
Damian clasped Lionel's hand with his own. "Good to see you, brother. I thought we lost you on nine."
"You did. But I'm hard to kill."
"Damian," Makara said, cutting off the reunion. "Lead us to Grudge."
Damian nodded. "Right. He should still be down on six."
We followed Damian to a flight of stairs. We took them two at a time until we reached the sixth-level landing. The sounds of gunshots and men's screams came muffled, distant.
Damian ran forward onto level six. The gunshots grew louder as we neared the center of the level, passing open doorways. We took several turns until we came to a final corner, around which Damian took a quick glance. He pulled his head back, resting it against the wall.
"Five Lords," he said. "They're set up behind tables in the mess hall. Faced away from us. They must be firing on Grudge and the Suns."
"Take them out," Samuel said. "They won't see us coming."
We stepped into the hallway, moving slowly so as not to alert the Lords to our presence. I took careful aim with my Beretta for the man on the left. Five of us were lined up, and still, none of the Lords had turned. They had no idea we were standing there.
I waited for Samuel's gun to go off before I opened fire. The man on the left immediately dropped, the back of his shirt darkening with blood. I tried to separate my mind from this grisly task. With all of us firing on the Lords in their exposed state, they had all fallen within a matter of seconds.
It was another moment before Samuel spoke.
"Come on."
We walked into the mess hall, a place that truly lived up to its name. The fluorescent lights above revealed a chaotic maze of upturned tables and chairs, and the scent of blood stunk up the air. Blood pooled under the corpses, spreading over the linoleum. Lionel's men went to the bodies, rummaging for ammo and weapons. From a hallway leading out from the mess hall, a familiar man appeared.
"Grudge!" Makara called out.
He held up a hand in greeting, using the other to motion the men behind him forward. Three tired men entered the mess hall.
We met in the middle, a good distance from the fallen Lords.
"I thought you were dead," Grudge said.
"Almost, but we'll explain later." Samuel said. "Think we have enough here to make the push back?"
"We need more," Grudge said. "Much more."
"Know anything about my family?" Michael asked.
Grudge looked at him. "I think they're okay, Michael. I haven't seen them since this all started, but they're probably on level one with most of the civilians." Grudge sighed. "Honestly, they're probably safest there. It was mainly the men they were interested in killing."
"So it's Jade, Rey, and Cain?" I asked.
Grudge nodded. "And whoever else joined them. Most of Char's Raiders and all of Marcus's Exiles stuck to their leaders. Boss Dragon fell early in the fighting, and I don't know if any of his men made it. They were holding their own for a while on fourteen. This..." Grudge motioned back, indicating the four men with him, "is the last of the Suns. They've chased us up this far, and..."
Grudge paused, unable to find his words. He closed his eyes, as if he were falling asleep.
"Have you slept since the fighting started?" Makara asked.
Grudge quickly came to, shaking his head. "No. No, I can't sleep."
"How dangerous are the lower levels?" Samuel asked.
"Thick with resistance," Damian said, answering for Grudge. "This floor should be clear, now."
"We need a home base," Samuel said. "It will take too long, and be far too dangerous, for us to go around rescuing every group. We need them to come to us."
"I think you're right," Makara said. "The Command Center should work. We need to stock up with as much food and water as we can find. Anyone know where we might find that?"
Lionel answered. "Most of the kitchens have been cleaned out. We found some cans up on level three."
"What about water?"
"They have pumps on the bottom level," Grudge said. "Can't imagine how much longer they can run without power. The taps work fine, though. They just haven't been used in fifteen years."
"We have the power back online," Samuel said. "It's all backup, so there's no telling how long it will last. Hopefully, long enough to take control of the Bunker."
"We need to make an announcement from the Command Center," Makara said. "They need to know that we're back, and if we can get enough people, we can take the fight to the Lords."
"If we go to the Command Center, wouldn't the Lords know where to attack us?" I asked.
"Let them try," Makara said. "They'll be fighting on our turf, and it won't be easy for them to approach when we have cameras watching."
"A lot of Angels would have to fight against Lords to even make it to the Command Center," Damian said. "And not everyone who got out is a fighter. There were women and children, too."
"That's where Ashton comes in," Makara said. "He can pick up any Angels who can make their way outside. It would be easier to just head outside rather than fight through the Lords' positions. Ashton can ferry groups to the Bunker entrance at the top. From there, they can head down to the Command Center."
"Sounds like a good idea," I said.
"We heading up?" Grudge asked.
Makara nodded. "Yeah. Let's get to the Command Center and I'll get Ashton on the line."
***
WE FOUND A STAIRWELL and made our way up. Before we reached the stairs, we heard voices drifting down the hallway. As we raised our guns, four Angels walked around the corner. It was clear they were surprised to see Makara, Samuel, and the rest of us alive and well. I recognized one of these men, Nathan, who was an Exile. He seemed to be leading the other three men.
"Makara!" he said. "You're alive."
"Good to see you, Nathan," she said. "Any news of Marcus?"
He shook his head. "They got him and Char on level one."
"They're not dead?"
"They might be," Nathan said. "He and Char stayed behind to fight, even as they told us to run. Haven't seen them since."
"Alright," Makara said. "We're heading up to the Command Center. Is there anyone else on this floor?"
"We're the only ones on this level."
So far, everyone in the inner circle that had stayed in Bunker 84 was unaccounted for. We had no idea whether Char, Marcus, Ruth, or Lauren were still alive.
We walked the rest of the way to the Command Center without incident.
Makara nodded to Lionel. "Take a few of your men and post a guard."
Lionel nodded. "On it."
While Lionel left with a few fighters in tow, Samuel opened communications with Perseus.
"Ashton?" Makara asked, once the line was open. "Ashton, you read?"
It was a moment before Ashton's familiar voice came through the speaker. "Makara! Everyone alright down there?"
"It's rough," she said. "We've found some Angels but a lot of our people are still missing. This is going to be a long fight."
"What do you need me to do?"
"Have you seen the north face of the mountain?"
Ashton paused. "No. Haven't flown out that way."
"Well, you might be surprised by what you see. A nuke must have hit it back when the Bunker fell. There are lots of openings leading out of the Bunker."
"Alright," Ashton said. "Where are you going with this?"
"We've restored power, which means we should be able to make an announcement to the whole Bunker: telling any Angel who can, to go outside."
"And you want me to pick them up?"
"That's right. As soon as you see anyone, pick them up and land them by the entrance. From there, they can walk down to the Command Center on level three."
"What about our people at the bottom, though?" Ashton asked.
Makara sighed. "They'll have to hold on a while longer. The goal is to gather as many as we can and put weapons in their hands."
"When are you planning on moving?"
"We'll be making the announcement soon. Stand by for further orders."
"Copy that," Ashton said.
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# Chapter 12
"It's set," Samuel said.
"Citizens of Bunker 84," Makara said, pausing for a moment. "This is Makara. Yeah, I'm alive, and no, this isn't a trick. We've restored power and the Angels' numbers are swelling by the hour. If you're an Angel and you're hearing this, make your way to the Command Center on level three. If you can't make it there safely, go outside. Dr. Ashton will see that you're brought safely here."
Makara looked at Samuel, as if for confirmation. Samuel gave a thumbs up. Makara went on speaking.
"Things have changed drastically on the outside. The Great Blight is about to annihilate us all. Oasis has fallen, and Los Angeles will go soon. The Empire and the Reapers are no longer allies – they are enemies. They fight in the ruins of L.A. even as death presses in from the east. This is my message to Jade and the Lords: the only way you will survive this is if you lay down your arms and negotiate peace. If you do so in the next six hours, your lives will be spared.
"You were all Angels, once. I'm going to attempt a chance at reconciliation. I understand how panic might have led some to make a stupid decision – a decision you now regret. That's why I'm giving you this chance. I invite Jade, Rey, and Cain to the Command Center to parley. You have my word that you won't be killed.
"If you choose to continue this senseless war, you will lose. We have a spaceship and have unlimited access to resources. Starvation is no longer an issue. From the Command Center, we can cut off your power. Maybe you have backup generators you've been using to get by, but they won't last. We don't have time for this war. Augustus in Los Angeles needs our help."
Makara paused a moment for her words to sink in.
"Angels: make your way upstairs or outside. Lords: you have six hours, starting now."
With that, she cut out.
***
MAKARA SENT A HEAVILY armed patrol to assist any Angels making their way upstairs.
The first round came from the seventh level, fifteen minutes following Makara's announcement. Meanwhile, Ashton was ready to pick up anyone at the north face of the mountain.
As the minutes and hours passed, the barracks filled with survivors. When the head count passed one hundred, they still kept coming. There were far more survivors than we'd thought; they'd all been separated by the Lords' relentless attacks. Michael and Julian started passing out canned food they'd found inside a kitchen connecting two of the barracks. As Damian had said, many were women and children. And they were hungry.
I helped with a can opener I'd found in a kitchen drawer. I was in the middle of uncapping a can of pineapple when it happened.
"Daddy!"
Callie crashed into Michael's leg, wrapping it in a viselike grip. Michael pulled the little girl free and held her to his chest.
"Callie, baby, you're alright!"
The girl laughed and Michael looked past her to see his wife, whose blue eyes were flooded with tears. Blood and dirt were smeared on her face, and her cheeks were gaunt. She gave a small, pained smile.
"Lauren..."
Michael walked forward and embraced his wife, holding her like he'd never let go.
Callie walked up and looked at the half-opened can of pineapple I was holding. I continued to open the lid, and when it fell off, I tossed it aside and handed the can to her. Immediately, Callie gulped it down. By the time I had another can open, this time beans, she was already done with the pineapple. Others in line watched hungrily as Julian worked quickly to open all the cans he could.
"Share this one with your mom, alright?" I asked.
Callie nodded, carefully taking the can of black beans and returning to her parents.
Michael parted from his wife. I opened a few more cans, one with beets and another with spinach. I know it wasn't gourmet, but it would have to do. The food had nutrients and calories, and that was what everyone needed.
As the people ate, I continued opening cans. We had only found two can openers, so we worked as quickly as we could. After about fifteen minutes, everyone had eaten one can of food. Still, we continued to open them. Lauren and Michael went to the kitchen to see if they could find any more can openers.
When they came back, Lauren had another can opener, while Michael toted another armful of cans. She set to work as people came by for round two.
"Is Ruth alright?" I asked.
"I think so," Lauren said. "We were together...until we weren't. We got separated in the chaos." Lauren paused, reaching for another can of food. "She knew how to work Hydroponics, so they couldn't do anything to her." She paused, opening another can of beans, which was quickly taken by a muscled Raider. "I'm worried about her for other reasons."
It wasn't hard to imagine what those reasons might be. Ruth was a good-looking woman, and she was surrounded by a bunch of unscrupulous men – men without honor and principles.
"We have to get her out of there," I said.
Lauren nodded. "I know. It isn't going to be easy. I don't think Rey is going to accept Makara's terms."
"So Rey is the one in charge?" Julian asked.
Lauren nodded. "Yeah. Jade is sort of their lackey, but they do give him some rein. It's more Rey and Cain working together. To what end, I don't know. They must know they're done, with Makara back. They still hold some pretty powerful cards. Ruth is one of them, and they also have Char and Marcus."
"If they're still alive," Michael said softly.
"I think they are," Lauren said. "They need that leverage, especially now when the balance is shifting so much."
I hoped Lauren was right. I couldn't imagine the Lords keeping either Char or Marcus alive. Still, maybe they'd agree to parley sometime in the next few hours, as Makara had laid out.
"We'll find a way," Michael said.
Finally, we were opening cans faster than people could take them. A minute later, we gave our hands a break. I took a can of pork and beans, grabbed a nearby spoon, and started to eat. It was the first food I'd had since breakfast, all the way back in Los Angeles.
With the realization of how busy my day had been, I felt a wave of fatigue. I checked my watch, seeing that it was 21:00 on the dot.
I uncapped another can, taking a cue from Callie and favoring pineapple. Callie watched with wide eyes, sitting on Michael's lap.
I ate the pineapple, draining the rest of my juice, and set the empty can on the floor. I noticed that a lot of people were hanging around, even if they didn't want more food for the moment. It took me a minute to realize why they were.
"They want to take some of the food for themselves," I said. "We should probably keep it in a secure location."
"Can't we just trust people to take the right amount?" Lauren asked.
Julian and Michael answered at the same time. "No."
"Alright," Lauren said. "What do we do, then?"
"We could keep everything in the Command Center, where it would always be under watch," I said. "Most people would take the right amount, but some will hoard, which would punish the people who follow the rules."
"I agree," Michael said.
"I'll talk to Makara," I said.
I left the barracks, following a short corridor leading to the Command Center. Once there, I found Makara, Samuel, and Grudge on the Command Center floor. I went downstairs. They saw me coming down and waved me over.
As I approached, Makara gave me an update.
"Ashton picked up twenty or so, and they should be heading down soon. You might want to get some more food ready."
"About that," I said. "We were talking about keeping the food in the Command Center. It would prevent people from taking more than they need. We could take inventory and hand it out only at mealtimes."
Makara nodded. "Yeah, good idea. Find a way to move it in here."
"That won't stop people from going out and finding their own," Samuel said.
"No, we can't stop them from doing that," Makara said. "But we can forbid anyone from leaving level three without an escort. We can widen our range once we start securing more floors."
"We're probably good down to level six or so," Grudge said.
"Probably," Makara agreed. "But I'm not taking any chances. And that's too much ground to cover with what we have."
"Alright," Ashton's voice said, coming out of the computer speakers. "I've landed outside the canyon. The refugees are worried about the snow, so I'll be escorting them to the entrance. I'll be out of touch for the next ten minutes."
"Copy that," Makara said. "Good job, Ashton."
"What's the head count?" he asked.
"122, so far. 142 once yours arrive. About half can use guns."
"Copy that. I'll make one more flyby before calling it a night. I think I got everyone I could on this round, but another sweep won't hurt."
"We'll make the announcement for the last sweep," Samuel said.
"Copy that," Ashton said. "Out."
***
NO EMISSARY ARRIVED from the Lords below, at least not within the first four hours. I wouldn't have known if any had arrived, because I woke up around midnight, despite my tiredness.
I kept thinking of Anna, hating how I couldn't see her and how she was locked away on Perseus.
Which was when I realized that I could go see her. True, it meant going through the empty corridors of Bunker 84 alone. But the lights were on, and level three and above were secure.
It would be cold outside, but I had my parka.
A big part of me said it was stupid. And it probably was. The more I thought about it, though, the more I felt like I had to do it.
It took a minute to make the decision. I got up, put on my parka, and made sure my AR and Beretta were locked and loaded. I went to the kitchen, grabbing a couple cans of food and a can opener, and made sure to fill my canteen. I felt a little guilty, taking a can opener, but we had six total now, which meant I was leaving behind five. They'd survive without it for a couple of hours, and if something bad happened up there and I found myself cut off, I'd be prepared.
I didn't know why I felt so pressed to see Anna, but I wanted to make sure she was doing alright. After Ashton had dropped off the second round of refugees, of which there were only six, they had parked close to Bunker 84 at the canyon entrance, which was a short walk from the Bunker door. All I had to do was reach the Perseus's door, input the code, and go inside.
Before I had time to second-guess my decision, I lifted my pack and left the barracks by the back way. By going this way, I could avoid the Angels standing guard. We were all under curfew; I wasn't supposed to be breaking the rules like this.
All the same, I had a feeling that I should see Anna, one that went beyond the fact that I was worried about her. I felt like I was missing something important. It really wouldn't be a dangerous trip; there were no enemies on the third level and above, and I wouldn't be in the cold for very long.
I left the dark barracks, entering the lit outer corridor. I walked to the stairs, hearing my footsteps echo off the confined walls. It was quiet, and the faded paint and odd smattering of dried blood on the walls told the story of this place. I felt as if ghosts were watching me when I reached the stairs and began my climb to the top level.
Where it was still dark.
I wasn't expecting that. Maybe the lights were still out on this level, for some reason. I retrieved my flashlight, clicked it on, and walked forward.
Coldness crept over me. I was out of my element, and for some reason, I just had a bad a feeling. A feeling that I should turn back.
I clicked the flashlight off, pausing to listen. I stood there for one, maybe two minutes, until I heard it.
Footsteps. Dozens of them, coming from ahead.
I backed away slowly, toward the steps. I realized who it was. It was the Lords, preparing a surprise attack. If they could attack from both top and bottom, we would be overrun – especially if we weren't ready for it.
I reached the stairs in the pitch darkness. Once I was sure I was out of earshot, I ran quickly, trying not to make any noise.
By the time I made it to the third level, I was at an all-out run.
I wasn't sure what to do first –wake everyone up, or find Makara. I decided on the second option. I didn't want to cause a panic, but at the same time, we had only minutes.
I entered the barracks at a run. Several people awoke at my entrance. I ignored them, instead finding the barracks where Makara and Samuel were. I went up to Makara, shaking her from sleep.
"The hell?" she asked, eyes groggy.
"They're attacking from above."
She opened her eyes, her grogginess dispelled by this news. "What?"
"I just came from there. They have a lot of guys heading this way."
Samuel awoke, along with several other Angels in the barracks. As they overheard my conversation, they armed themselves for combat.
I was afraid she was going to ask what I was doing up there in the first place, but thankfully, she decided there were more important priorities.
"How much time?" Makara asked.
"They should be here any second."
Makara turned to the Angels. "Wake everyone as quietly as possible and gather them in the Command Center. Do not explain anything, just make sure they get there. We need to be ready to move in five minutes. Remember: as calmly as possible."
Samuel took charge, delegating Angels to go to the separate barracks. We could only hope that the Lords didn't attack during this time. I had no idea how the Lords managed to sneak their men to the top floor – it probably involved going outside and...
"Ashton!" I said.
Makara looked at me. "What?"
"He should have spotted those guys, if they had snuck in from the outside."
Makara nodded. "We'll check in with him."
Makara and I went to the Command Center, finding the computer Samuel had set up with a direct line to Perseus.
"Ashton? Ashton, you read me?"
It was a moment before his voice came through, groggy. "Yeah, what?"
So they were okay. The Lords had intelligently decided to forgo the ship, deciding that they couldn't trip any alarms. Ashton had obviously been asleep, as was understandable. Even if he had been awake, it was so dark he might have missed the Lords' passing.
"The Lords are sneaking in from above," she said. "Do you see anyone out there?"
Ashton hesitated only a moment, registering the news, before answering.
"Right. Looking out the windshield now, but I'm not seeing anything. Snowstorm's picked up again."
"Sit still," Makara said. "We can't let them know anything has changed. We need them to think they're catching us unawares, when in reality, we'll be ready for them."
Even as Ashton and Makara continued to speak, the Command Center filled with Angels: refugees, women, and children. Anyone who could fight was given a gun and ammo, and this included some of the women. Anyone who couldn't fight was ushered to a corner of the Command Center where they would be safest, guarded over by six Angels overseen by Samuel.
After the allotted five minutes, every Angel had reported for duty.
Makara briefly explained the situation before laying out her plan.
"We need to set up an ambush somewhere on this level. Earlier, we assigned squad leaders. Follow their lead and follow the chain of command, and we'll make it."
Everyone nodded, listening to Makara with a surprising level of calm.
"Here are my orders: stay on the third level. Stay off the main corridors. Hide wherever you can. They're going to try to sneak into the Command Center. Let them get a good ways into the third level before striking, but don't let them inside the Command Center itself. The idea is to get them on this level so deep that they have to commit to a fight where they're at a disadvantage. As soon as we hear the gunfire, Samuel will shut off the lights. That's your cue to attack. I hope you all have brought your flashlights."
The Angels nodded their confirmation that they had.
"Samuel will hold the fort here, keeping the civilians safe. Any questions?"
No one said anything. There wasn't time for questions, anyway.
"Move out," Makara said. "And good luck."
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# Chapter 13
We waited.
I stood in a disused locker room with a combat team led by Michael, twenty Angels strong. The locker room connected to a gym, which connected to the main corridor.
We waited for either the gunshots to start, to get spotted, or the lights to go out.
My heart pounded in my chest. I wasn't born a warrior. All I wanted was to survive, and somehow, I had ended up with a gun in my hand. I wanted to separate my emotions from the act of taking another man's life, but I didn't know if that was possible. I realized, not for the first time, that it could all end right here. I might never live to see Ragnarok Crater.
I looked at Michael, whose gaze had homed in on the corridor across from the gym. He was the only person looking outside the dingy locker room, waiting. When it was go time, he'd raise his hand, and the killing would begin.
A gunshot sounded, distant. One second passed. Two.
The lights dimmed. Two seconds more, then darkness. Spots flared in my vision.
"Go," Michael whispered.
Twenty pairs of boots clomped into the gym. From the corridor, I heard panicked whispers, then silence.
They'd heard us.
"They're in there!"
Michael opened fire in response to the Lord's warning. The man screamed as bullets entered his flesh. Michael's AR thundered in the cavernous gym, the gun's barrel flashing like lightning. I added my own volleys, feeling the familiar kick of my AR at my shoulder. I held firm, giving the force the dispersion it needed. Another man screamed. When Michael ceased fire, the rest of us followed suit.
Then came the sounds of boots running down the corridor, past the open doorway of the gym. More shots followed.
"After me," Michael said.
I fell in behind Michael. The metallic air stung with the smell of blood.
Michael took a corner, clicking on his gun light. Its beam revealed two Lords crouching over a comrade bleeding from his gut.
They barely had time to scream before we shot them down.
"Two behind!" someone yelled.
I spun to see our men in the back ranks take action. We fired, they fired. One of our men dropped while two Lords fell. More men filled the intersection.
"Take cover!" Michael roared above the din.
I dove into an open doorway, followed by several more Angels. One was shot mid-dive. He screamed and crashed into the floor next to me, his warm blood splashing my face. I wiped my cheek on my shirtsleeve, gritting my teeth. I looked at the dead man. A good chunk of his neck had been blown off.
More shots sounded, ricochets of bullets zinging like metallic wasps. Then, silence. Distant gunfire, distant screams of men...
Neither side wanted to show themselves in the darkness. The squad had been split up, and to get back together, one of the halves would have to cross the corridor under enemy fire.
"This way," Michael said.
I was glad he was among the Angels who had taken shelter in this room, because that meant I was with Michael. Our leader crouched low and walked to a back corner. He shined his light around, revealing toppled desks, chairs, and a blackboard leaning against the wall, upon which a set of numbers had been scrawled with chalk. I had the time to recognize it as the quadratic formula before we passed into another classroom through an open doorway. It was filled with the clutter of a Bunker past. More toppled desks. Dust and dirt lining the floor. And, chillingly, chains hung from the walls, at the end of which were shackles.
I didn't want to know the story behind those.
Michael took the lead once more, pointing his light beam right. Then left. He paused.
A bullet screamed past.
He ducked back in, cursing.
"Michael!"
He shook me away. "I'm fine."
"Quiet," another man said.
I listened, hearing nothing for three long seconds.
That was when an explosion ignited outside, not in front of the door, but a short distance to the left of it. A bright flash of orange blinded me, and the heat of the blast licked my skin. The floor shook as I dropped to my belly, face down. Men groaned outside in the corner. I couldn't tell if they were with us or them.
Michael stepped into the corridor. I was almost surprised when no bullets came to claim his life. He disappeared toward the left. I hesitated before following him.
Michael stood in a corridor littered with bodies, severed limbs, and crusted blood in the midst of crumbling concrete, rebar, and choking dust. I hacked in the sooty air. Nothing was recognizable.
"Where are we?" one of the men asked.
"Nowhere we need to be," Michael said.
Michael turned from the collapsed tunnel, walking as if he knew where he was going. It was amazing how quickly such a man would be followed. We fell in step behind him as he turned to the left, not bothering to check the many open doorways he passed.
I was about to ask if we should slow down when Michael spoke.
"We become the hunters now."
He reached a set of steps. At first, I thought it was to go up. I stared only for a moment before he struck downstairs.
I knew where he was going.
"Michael – this is crazy."
He looked at me, his brown eyes intense. "Feel free to turn back. I'm going to hit them now, while most of their fighters are up here. I trust Makara and the others to hold the line. Any of you are welcome to join me."
With that, Michael went down into the darkness, at first alone. He paused at the first landing, waiting a moment.
I thought about that day long ago when I followed Michael into the Wasteland for the first time. I would follow him again.
When I started after him, he gave a grim smile.
"Well, Alex," he said. "It's time to kill the Weasel."
***
WE QUICKLY CIRCLED the flights of stairs down to level ten. Here the stairs ended, opening onto a long, dark corridor. Michael scanned left and right with his light.
At this distance, the gunshots had faded, though the occasional crack carried down the stairwell.
Michael walked forward, in search of another staircase leading down. The Lords' headquarters were on level twenty, the lowest level of Bunker 84. With luck, they hadn't left many defenders behind.
The corridor turned ninety degrees to the right, Michael's light beam illuminating dancing motes. We walked a few steps before another stairwell materialized in the darkness.
Michael paused at the top of the stairs, listening. One of the men reloaded a magazine, the sound loud and echoing. Almost like a bullet itself.
Michael turned to face us. As an Officer and patrol leader in Bunker 108, he was used to leading. It was easy to follow him.
"Alright," he said. "This one should lead straight to level twenty." He fixed us all with a level gaze. "I don't have much to say. I know we're not supposed to be down here, but sometimes, you get a feeling you can't ignore, an opportunity you can't pass up. This is one of those. When we get to the bottom of the stairs, I don't know what we'll find. There'll be civilians, so think before you shoot. Our priority is taking down the Lords and keeping the civilians safe."
The men nodded and waited for Michael to speak again.
"Anyone not ready?" he asked.
No one spoke up.
"Let's go."
***
OF ALL THE CRAZY THINGS I had done since leaving Bunker 108, I felt this had to be one of the craziest.
We were running downstairs into darkness to confront an enemy of unknown size, without orders from either Makara or Samuel. But the prize was too tempting to pass up: we could save Ruth, Char, and Marcus, along with everyone else who had been trapped down there.
The flights went by in slow motion.
Thirteen. Fourteen. Fifteen.
On level fifteen, a surprised Lord had been standing guard, where he had obviously fallen asleep. As he rushed to get up, grabbing his rifle, he was shot down. That shot would echo downward to the twentieth level.
"Keep moving," Michael said.
Sixteen. Seventeen. Eighteen.
There were only two flights left, now. If Makara and the others could hold on upstairs, this might all end tonight.
Nineteen.
That was where the fighting started. A blaze of bullets issued from men hidden behind doorways, felling two Angels before the rest of us could continue to the twentieth.
"Markos...Groot...hold the rear!" Michael called.
The two Angels obeyed, even knowing the danger, but someone had to hold the Lords off on nineteen while we continued to the twentieth. Two more Angels peeled away to assist Markos and Groot. Michael motioned the rest of us on.
We ran down the last line of stairs to twenty, into an empty hallway. The floor was cloaked in darkness. The screams of women and children were muffled by distance.
Michael went into a crouch, advancing forward. On level nineteen the firefight continued. A man screamed; us or them, I couldn't tell.
We walked unopposed into a rec room – the same one Elias and I had spoken in just a week ago. Michael continued on into a dark hallway. The level was eerily silent.
We kept moving until we reached Hydroponics. The grow lights bathed the plants, casting green shadows on the many lanes running through it. Sprinklers misted the plants, producing rainbows as they refracted the light.
Michael approached the railing, low to the floor, peeking over. He watched a moment before turning toward us, holding up four fingers.
Four men on guard. It made sense that the Lords would leave some guards here. The food was their source of power.
There was no way we were sneaking by with almost twenty armed men, so Michael took careful aim. Several more Angels approached the landing, doing the same.
One of the guards amid the plants was startled, looking at our position. He squinted at us, as if unable to believe his eyes.
We opened fire.
Two of the men fell immediately as the other two dove for the plants. From within the maze of green foliage, several women screamed.
"Cease fire!" Michael said. "They have civilians working down there."
We backed way toward the wall and away from the railing.
"The gang lords are the target," Michael said. "This just distracts us."
I didn't like leaving the civilians behind, or the Lords at our back, but Michael was right. We had to stay focused on our goal.
Michael stepped away from the railing, sidling along the metallic wall, keeping as far from the overhang as possible. The men followed suit. The Lords below couldn't take shots unless they actually climbed the stairs that led to the overhang – in which case, we'd have the advantage.
The sound of footsteps pounded from the corridor behind. Several voices called out as we readied to defend ourselves.
"It's us! Angels!"
Three men appeared out of the original four that had stood guard at the stairs.
"What happened?" Michael asked.
"The Lords abandoned their position on nineteen," the Angel said. "They disappeared onto the level, so we came to find you."
Michael nodded. "Keep close."
He turned and waved us on. He approached the corridor leading to the hangar.
We went around the corner. Michael hung right, where a flight of stairs branched from the corridor and ran down to the hangar. The metal steps rattled beneath our boots. Michael, at last, reached the metallic door reading "Hangar."
"I don't know what's after this door," Michael said. "Be ready for anything. Go on my orders. If the fire's too heavy, I'm closing the door again."
The men nodded their understanding. Michael clicked off his rifle light. He waited another moment before pulling the latch.
The door swung open, revealing the hangar to be dim and empty, vast in the absence of spaceships. If there was to be any resistance, we should have found it here.
Michael stepped forward. We fanned out, forming a wide V.
I focused on a dark row of windows halfway up the right-hand wall. The control room.
"They could be in there," I said.
Michael nodded. "We need to be careful. It's too damn quiet for my liking. Maybe Cain went out on the attack, but I don't think either Rey or Jade would get their hands dirty." Michael paused a moment, thinking further. "If Rey and Jade are here, that means they have guards."
Michael motioned us over to the right-hand wall. We followed its length until we reached a doorway. The doorway opened onto a long, dark corridor. On our left rose a set of steps, leading to the control room.
"That's the way up," I said.
"I don't want to attack into a choke," Michael said.
"We don't even know what's up there," another Angel said.
"True," Michael said. "We need a way to flush them out."
I handed Michael one of my grenades. "Try this."
Michael took the grenade, thinking on it.
From upstairs, a woman screamed. It sounded like...
"Ruth!" I said.
"Hold up," Michael said. "They'll kill her if we try anything."
"Well, we have to do something," I said. "I don't think they know we're here."
"I know that," Michael said, frustrated. "Let me think."
Ruth screamed again. A man cursed. There was a slamming noise, like someone being forced against a wall.
That was enough. If this was just bait to get me to go up the stairs, then I guessed it was going to work.
I charged ahead, my feet slamming on the steps.
"Alex!" Michael said.
It became silent upstairs. I threw my AR over my shoulder by the strap, instead reaching for my Beretta. I rounded the flight, only to see darkness ahead. I was beginning to regret my decision, and it felt as if my heart would pound out of my chest. I forced myself onward, before I could second-guess myself.
I burst onto the landing. My first sight was of Ruth lying in the corner, her face bloodied. The second was of Jade, shirt off, blue eyes looking at me with shock. They were the only two in the room.
A guard burst into the room, aiming a handgun my way. I ducked right before a bullet was fired. It whizzed over my head. I returned my own fire with the Beretta, not really having the time to take careful aim.
Time seemed to slow, each bullet taking an eternity. The guard got several more shots off, and it was only the darkness of the room that saved me. Each flash gave me a brief, terrifying glimpse of the guard, appearing at a different location with each shot. His eyes were wide with fear.
Finally, the man cried out. I thought I'd gotten him, but he kept on shooting, seemingly unaffected. By this time, four shots had been fired at me, and each had missed. I couldn't stay lucky for too much longer; it was only a matter of time until one of them connected.
I adjusted my aim upward, toward the neck and face of the guard. This would be a hard shot, in the darkness and with a small, moving target.
I fired twice, nabbing the guard in the jaw. He gave a bloody scream as I fired again, right between the eyes.
During the few seconds the gunfight happened, Michael and the other Angels had charged up the stairs from behind. Meanwhile, the Weasel had been searching for his weapon.
I didn't give him another second. He turned away just as I fired into his back. He gave a horrible, parched wail that seemed more Howler than human. I fired again as he sprawled onto the floor and went still.
I scanned the rest of the control room, adrenaline coursing through my veins. There was another door, at which I pointed my Beretta with shaking hands. I held this position for several seconds, my hands shaking, as Angels entered the room.
Finally, I forced myself to relax. Ruth was in the corner, fighting tears. There was a tightness in my throat, and an anger in my gut for what had happened, or what was about to happen. At least Jade was dead, and Ruth was safe.
"Here," I said, handing her my canteen.
She took it, taking a drink. I set my pack on the ground, opened it up, and pulled out a clean shirt. I knelt beside her.
"Here," I said. "You can clean up with this."
I looked at her face, which she turned aside. A deep bruise had surfaced below her left eye, while her nose had been bloodied. It could have been worse, but it looked like we came along just in time.
She wet the shirt with my canteen and wiped at her face. Her breaths came out ragged.
"It's alright," I said, sitting on the floor. "You're safe now."
Michael stood behind me. He hadn't yet seen the inside of the room, nor Ruth. He knelt next to me.
Ruth looked up, having cleaned herself as best she could. There was hardness and anger in her eyes that reminded me nothing of the happy woman I'd known in Bunker 108. She slowly stood, then walked over to Jade's body. She paused for a moment before kicking him full in the face.
"He had his guards grab me while I was sleeping," she said, her voice wavering. "And they dragged me here. Rey would have stopped him, but Rey went up with Cain." She was quiet for a moment. "He had practically begged to stay behind. Now, I guess I see why he wanted that."
"Well, he's dead now," I said.
Michael shook his head. There was sadness, and anger, in his eyes. Michael took off his pack, reaching in for a camo tee.
"Here," he said.
Ruth put it on gratefully over her old shirt, which had a tear in it.
"And this."
Ruth grabbed the handgun that Michael handed her.
"Thanks," she said. "Let's just...not talk about this. Just give me something I can do to take my mind off it."
"What can you tell us about the resistance down here? We've found a few guards, but not many."
Ruth nodded, recollecting herself. "Most went upstairs to fight. They had...him...stay down here, as well as the rest of the Diamonds. That's about twenty men."
Michael nodded, obviously pleased with this news. "We've killed what, four or five so far?"
I nodded. "About."
"We have twenty of our own," Michael said. "The rest are fighting upstairs."
"The people are in the dorms," Ruth said. "The guards sleep outside those, blocking the exits. There's also the night guards at Hydroponics."
"We passed that, but came straight here," Michael said. "I imagine the rest of the Diamonds know we're here."
"They might surrender, if we tell them about Jade," I said.
Michael looked at the former gang leader's bloody corpse. He looked so pathetic with his shirt off. A pool of blood had collected beneath his body. I knelt next to him.
"What are you doing?" Ruth asked.
I reached for Jade's left hand, suppressing a shiver. I tried to ignore those lifeless eyes, staring vacantly at me. I pried five diamond rings off his right hand, and then reached for his left, pulling off the other five. They were flecked with blood. I wiped them on Jade's sleeve before pocketing them.
"Proof of our victory," I said. "Any Diamond will know what we've done once we show these. I doubt they're loyal enough to fight after he's dead. Jade wasn't the type of person who commanded respect."
Michael nodded, gesturing to the open doorway. "Good idea. Dorms are that way." He turned to the rest of the Angels. "Everyone catch that?"
The men nodded silently.
"We're heading for the dorms," Michael said. "Hopefully the Diamonds will surrender without a fight." Michael held his hand out. "Hand me those rings, Alex."
I reached into my pocket, handing them over. They glittered in the darkness – diamond, sapphire, ruby, emerald, gold, platinum. All worthless in today's world. Michael put them in his pocket.
"Take Ruth back upstairs," Michael said to me.
"Isn't that more dangerous than staying here?"
"Maybe so," Michael said, "but I need you to take Makara a message. Let her know what's going on. Then return to me, and let me know what's going on upside. Plus, Ruth can help out up there. Find Lauren and she can take care of it."
"Will do," I said.
"Hurry," Michael said. "And be careful."
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# Chapter 14
We left the control room and headed to a stairwell that led to the upper reaches of the Bunker.
"We need to watch out for Lords," I said. "If we're winning up there, they could be retreating."
Ruth nodded, not saying anything further. I wished there was something I could say to make her forget the last thirty minutes of her life. Of course, I knew that was impossible. All I could think to do was get her to Lauren. She would know what to do.
We made it to the stairs, and took them at a jog. We paused for a quick breather every few flights. When we reached ten, the stairs ended in a long, dark corridor. We had to follow this corridor to reach the next ascending staircase, which would lead to level three.
When we turned a corner, footsteps pounded from the stairwell ahead. Ruth and I entered a dark, open doorway. We were hidden from view as people rushed past.
Outside, I heard a woman's voice.
"Faster," she said. "We don't have much time."
"Makara!" I yelled.
Instantly, everyone stopped and guns were raised to shoulders.
"Alex?" she called. "Where are you?"
"Yeah, it's me. Ruth and I are in here."
Two men stepped into the room, lowering their rifles. Amid the shining flashlight beams, Ruth and I stepped out into the corridor.
"Ruth?" Makara asked, surprised.
"Alex saved me," she said. "Let's leave it at that for now."
Makara looked at me curiously, but said nothing.
"We got cut off by the explosion," I said. "We saw the stairs, and Michael decided to go down. Jade's dead. We were coming up to deliver the news."
"That's news, alright," Makara said.
"Michael should have the rest of the Diamonds on lockdown by now. We were coming to get reinforcements."
Makara nodded. "So we control the bottom level?"
I nodded. "With Jade dead, Michael's hoping to convince the rest of the Diamonds to surrender."
"What of Marcus? Char?"
"They're alive," Ruth said. "They're locked up in some holding cells in the security section. By the dorms."
"We can deal with that later," Makara said. "Rey is dead, and Cain is in custody. The battle was won upstairs, but it was close." She looked at me. "If you hadn't discovered their attack, it would have been a different story,"
It was a little surreal that my own stupid idea of going to see Anna, something that could easily have gotten me killed, had ended up saving everyone.
"Don't make a habit of disregarding protocol," Makara said. "But in this case, it worked out. We still lost a lot of men. Samuel and Julian chased a lot of the Lords down into the deeper levels."
"We haven't seen anything," I said.
"We can't waste any more time chatting," Makara said. "I need to reinforce Michael. How many Diamonds are left?"
"Twenty, give or take," I said. "About the same on our side. We lost several men, but there wasn't much resistance."
"Alright," Makara said. "Take Ruth the rest of the way. The stairs should be fine, but be careful. Everyone is still gathered at the Command Center. After that, you can go on to Perseus and let Ashton and Anna know everything. Once I decide it's safe, I'll let you guys dock."
"Perseus?" Ruth asked.
"Alex will explain," Makara said. "We should get moving. Good luck."
Makara and her followers headed down the corridor. We watched as they went downstairs. When the last of them were out of sight, we turned the opposite way down the corridor, heading for the other stairs. When we had gone up a couple of flights, Ruth spoke.
"So, what happened out there?" she asked. "All I know is that the ships crashed."
"The dragons hit us over Wyoming," I said. "At least fifty of them. I don't know what we were thinking, and it's lucky all of us survived. Augustus, believe it or not, got a spaceship out of Bunker Six – Orion. He picked us up when he detected Aeneas and Gilgamesh falling."
"Really?" Ruth asked.
"Yeah. He's on our side, now. We think." I paused. "He even gave us another spaceship, Perseus. But there will be time for all that later. What's important is getting you to Command, safe and sound."
She said nothing for a moment.
"Thanks, Alex," she said. "I should have said that earlier. Guess this is the second time you've gotten me out of a rough spot."
"It was nothing," I said. "I just came along at the right time."
"Are Lauren and Callie up there?"
"Yeah," I said. "They got out before the Lords fully took over. They're in Command."
Ruth breathed a sigh of relief. "Good. I knew she had gotten out, but I didn't know what happened after."
We took the rest of the stairs without a break or further conversation. As we reached level three, the lights flickered on again. I shielded my eyes, squinting, waiting a moment for them to adjust. I clicked my flashlight off.
My eyes were met with ruin. The bodies of both Angels and Lords littered the floor. Blood splattered the walls and coated the floor. A pile of rubble lay where the corridor had collapsed. Maybe they had hoped to use those explosives on the Command Center.
"Horrible," Ruth said.
"Come on," I said. "We're almost there."
***
THE NEW ANGELS HAD been fractured by the Vegas Lords' rebellion. With both Gilgamesh and Aeneas crashed, and no word from the leaders, the fracturing was inevitable. Char and Marcus couldn't keep the lid on, and it only took a few days for it to boil over.
It was an indisputable fact that there were only two weeks of food and no way to evacuate everyone from Bunker 84. The only solution was to preemptively decide to feed fewer mouths, and the Vegas lords wanted to decide which mouths to feed.
Jade, Rey, and Cain and their chosen followers elected themselves to survive the coming bloodbath. They had taken control in the night, and anyone who resisted their rule, or who was aligned with the Raiders or Exiles, was either attacked or driven into the higher reaches of the Bunker.
Char and Marcus woke to the sounds of the fighting and rescued who they could, and were captured in the process. Those that were left behind, including the Raiders and Exiles who didn't manage to escape, were brutally murdered. It was necessary, in the gang lords' minds, to kill them in order to save the chosen remnant.
It wasn't clear why the gang lords had decided to spare Char and Marcus. When we found them, their forms were thin and haggard. They'd been without food for a week. Char had said that the gang lords had wanted to starve them to death, as a warning to any who dared stand against them.
The fighting continued on and off throughout the following week. Every day, the Angels' numbers dwindled. Given another week, the Lords' victory probably would have been complete. Hundreds died, mostly civilians, and what had once been fifteen hundred survivors now totaled four hundred.
Only half of the Community women had been spared. Lyn was among the dead, but Deborah and Ada survived.
In the day following the Lords' capitulation, everyone was moved downstairs from the Command Center. Cain, the only rebel gang lord to survive, came bound hand and foot. Many cried out for his immediate death, but I guessed Makara and Samuel wanted to keep him alive for some reason. I realized that, other than Grudge, he was the only gang lord left. Boss Dragon died early in the fighting from a gunshot wound to the shoulder.
So much had changed, and much more would change still. In the south, Augustus still needed our help, and that fact was at the forefront of Makara's mind.
It wouldn't be long until the Radaskim reached Los Angeles.
***
ASHTON PARKED Perseus in the hangar, and I took advantage by getting some much-needed sleep. I probably crashed around 08:00 – approximately twenty-five hours after I had last woken up. My sleep was dark and dreamless, as close to being dead as I'd ever felt. I was almost surprised to wake up later that afternoon. I checked my watch, finding that it was now 16:02.
I was drifting into a doze once again when the compartment door slid open. It was Anna, supporting herself by the doorframe.
I patted the bed. She smiled, and limped forward. I got up to help her out.
"Don't worry yourself," she said. "I got this."
"You need to learn to depend on me," I said. "Really, it's not so bad."
"I guess," she said.
I eased her onto the bed. She winced in pain, settling on her good side. Ashton said she had broken two ribs on her left side. She lay on her right side, propping her head up with her arm.
"You alright?"
She waved a hand. "Yeah, fine. Except that lying down all the time is going to make me go weak."
I wrapped my arm around her, gently. It felt good to hold her again.
"I'm glad you're still here," I said. "It could have been much worse."
"What's worse than being useless?"
"You're not useless. Someone's value isn't determined by how many crawlers they can skewer through the eye."
"Yeah, I guess."
We lay there for a moment before I closed my eyes, ready to let myself drift off to sleep.
"You're so tired," she said. "You've been sleeping since this morning."
"I know. I could sleep more if I wanted."
"Well...Ashton said there was a conference later."
"What time?"
"18:00. I think they're wanting to leave the Bunker behind."
"For Los Angeles?"
"Yeah."
"It's crazy to think how we're allied with him," I said, eyes still closed.
"Politics can change fast," Anna said. "I'll bet if it were beneficial to him, he'd switch right back."
"Maybe," I said. "But I think he's actually genuine. Yeah, he's out for himself, but he's trying to protect his Empire. Who can fault him for that?"
Anna didn't respond. Something else was on her mind.
"What are you thinking about?" I asked, opening my eyes.
She sighed. "Now that this Bunker business is over...maybe now's the right time to see the Wanderer."
"What about Los Angeles? That'll keep us pretty busy."
"I think this is more important," Anna said. "Los Angeles will just be a battle. Getting an answer from the Wanderer could mean the war."
"We already know what will win the war," I said.
Anna's body stiffened. "I don't think you believe me yet. I'm not going to let you die. That's not an option. We're going to find another way. I'm tired of this idea that the innocent have to die to save the rest of us. It shouldn't be like that."
I hated when she talked about this. The more she got this idea stuck in her head, the more painful it would be, in the end.
"What if it doesn't work?" I asked. "If there was another way, don't you think he would have told me?"
"We'll never know unless we ask him specifically."
"I already have asked him," I said. "He said this was the only way."
"He didn't have to deal with me," Anna said.
"He knows a lot more than we do."
"I know," Anna said. "But he doesn't know everything." She paused. "Besides, you promised me."
She squeezed my hand. Yeah, there was that.
"You're right. We'll bring it up at the meeting. We'll find a way to get there. The Wanderer might have some useful information we could use, anyway."
All I wanted was to make Anna happy, but I didn't know if I had the ability to keep that up. After all, it was hard to make someone happy when...
I couldn't think about that.
"It's worth asking him again," I said.
"Okay," she said, finally relaxing.
I pulled her close, and she buried her face in my chest. I wanted to do whatever I could to keep this going. Love was as scary as it was exhilarating. I didn't know how it would turn out, but I had to keep it going, because there was no greater reason for me to fight than her.
Still, the Wanderer's words haunted me. The logical side of me warred with the illogical. It was no good to cling to false hope. I had to face reality, no matter how painful. I had to focus on my mission. My ultimate goal. It wasn't just Anna and me at stake. It was the world.
At the same time, I couldn't ignore the beauty I held in my arms. If there was even a chance she was right, I had to take it.
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# Chapter 15
We slept for a while before the alarm on my watch buzzed at 17:45. I turned it off, keeping my eyes closed. I didn't want to get up and move just yet. I could sleep the rest of the night if I let myself.
"We have to get up," Anna said.
I groaned. "Would it have killed them to put it off until the morning?"
"There's a big mess to clean up."
Finally, I forced myself to get up. I checked my watch again, seeing that it was now 17:47. The council was supposed to start at 18:00 and wasn't supposed to last more than an hour.
I knew there was little chance of that.
"Where's your wheelchair?" I asked.
"I'm walking."
"No, you're not. You're going to be out and about for a while."
Anna sighed. "Fine. It's in the clinic."
"Be right back," I said.
I found the wheelchair and rolled it to the cabin, its wheels squeaking. Anna had pulled herself up. Her eyes were half-lidded and sleepy.
"Here," I said, helping her into the chair.
"I can manage," she said.
She groaned as she sat roughly in the vinyl-padded seat. She closed her eyes, wincing in pain.
"I left my katana on the bridge," Anna said.
"Alright," I said.
I started to wheel her toward the bridge, but when we entered the wardroom, she placed her hands on the wheels, stopping the chair.
"Let's just get to the council."
"You sure?"
Anna nodded. "I need to accept things as they really are. I'm incapacitated, and I can't fight. I have to make peace with that. Accept the things I cannot change."
She began to wheel the chair forward, pressing the exit button to the ship. When the door opened, she pushed herself down the boarding ramp. The wheels squealed, so much so that I was afraid the chair would fall apart. The chair careened onto the cement. She flew with the chair, laughing, as a couple of Raiders dodged her path. She circled around, still going fast, before stopping herself and looking up at me, seeing if I was impressed.
"Pretty good," I said.
I walked down the ramp. As I did, I looked up to see Char walking toward Anna. Seeing me look that direction caused Anna to wheel her chair around.
"Char!" she said.
He gave a tired smile. His face was thin and gaunt, and he seemed older than the last time I'd seen him. His eyes were tired and circles underlined his sharp, blue eyes. Those eyes had not lost their light or intensity, but Char looked worn, for lack of a better word. The marred left side of his face lay in shadow, hiding the horrible scars of his disfigurement.
Anna stood, and the man who was like her father embraced her.
"You shouldn't stand," Char said.
"Don't be ridiculous."
She sat back down, wincing in pain.
By now, I'd reached them and stood beside Anna's chair. Char's gaze shifted to me, his eyes appraising.
"I don't think you realize the kind of girl you got," Char said. "Or what you got yourself into."
"Now what's that supposed to mean?" Anna asked.
"Let me put it this way," Char said. "Most seventeen-year-olds who wander into Raider Bluff don't become the Alpha's chief bodyguard. There were men under my command who would have killed you." Char smiled in remembrance. "You challenged any of them to a one-on-one fight, and one of my top guards took you up on it."
"What happened?" I asked.
"They fought, right there in the dirt of the marketplace," Char said. "No weapons. My guard had the upper hand most of the time, as might be expected for a man his size. A crowd of hundreds gathered. All business stopped as Anna fought for her right to be a Raider. I remember watching this kid, thinking she was foolish. Dead."
Anna said nothing, only listening to the story.
"Then like lightning, she pinned him to the ground. You could see how his eyes widened. I knew he wasn't destined to be on the ground for long, given his size. But from nowhere, a knife was in Anna's hands, at his throat."
"I thought the fight was without weapons," I said.
Char paused and looked me in the eye, as if that were the point of the story.
"That was when Anna said, 'Don't expect me to fight fair. Expect me to win.'"
"She was booed and jeered at, until I called everyone down. I walked into the ring, and held out my hand to her. I still remember the way her eyes blazed, like green fire. 'This is a fighter,' I thought. 'This is someone I need.' At the time, I trusted no one in Raider Bluff. I thought with someone so young, she would have no thought of corruption or betrayal.
"I took her to the top of Raider Bluff, gave her food and shelter, told her she could stay in Raider Bluff as long she liked, but she had to fight for me."
"What did she say?" I asked.
"She agreed. And as if testing that agreement, there was a plot on my life just a few days later. She killed the assassin. Then I knew I made the right call."
"I just wanted food and a home," Anna said. "I stayed in Raider Bluff, honing my skill, eating all I wanted. Protecting Char was easy after the two years surviving the Wasteland. I was in Raider Bluff for two months, when..."
She trailed off, but I knew what came next.
"We came," I said. "With our crazy idea to go to Bunker One and stop the xenovirus."
Anna nodded. "Yeah. I never meant to come along the whole way, but the Wanderer changed my mind. I knew I had a part to play in this still, for good or bad."
"For good," I said.
Anna didn't answer, remaining doubtful.
"Let's go to the council," Char said, grabbing Anna's chair by the handles.
"Where is it?" I asked.
"Here, actually," Char said, pointing to one of the hangar's corners.
A long table with chairs had been set up. There was seating for about twenty people.
"Besides everyone from the inner crew," Char said, "there will be representatives from all the other groups: the Community, the Suns, the Raiders, the Exiles, and the Vegas Exodus."
"Do you know what will be discussed?" Anna asked.
"Everything," Char said. "It's clear we're going to go help Augustus. It's the logistics that are up for debate."
We arrived at the tables. I cleared a chair to make room for Anna. I sat to her left while Char sat to her right.
Over the next few minutes, people arrived. Samuel and Makara came, sitting near the head of the table on my left. Julian followed shortly after, sitting next to Makara. Ashton walked from Perseus, making his way toward the table. He sat next to Julian.
"Where are the rest?" Makara asked.
"There's still ten minutes," Samuel said.
A few people I didn't recognize sat at the table – probably some of the civilian leaders of the Vegas Exodus. Deborah took a seat at the table. She was silent and contemplative, apparently finding more interest in staring at the surface of the wooden table than in making conversation.
Five minutes before the meeting, people began to gather around the table, wanting to hear everything that was discussed. This group included Raiders and Exiles, along with people from Las Vegas and the Community, whose faces were worn from all the hard times they had endured. All they wanted was peace and rest, the one thing we weren't able to give them.
Finally, the last group of people arrived: Michael, Lauren, and Ruth, leading a battered Marcus, whose face was bruised and whose right lower leg was in a cast. He advanced slowly on a pair of crutches, his face worn and his reddish-brown beard long and tangled. Like his brother Char's, Marcus's blue eyes were bright and alert. With help from Lauren, he was eased into a seat next to Char. Michael knelt in front of Callie, who had been following her parents, pointing her toward a group of children who must have been part of the Vegas Exodus. Reluctantly, Callie set off.
All the seats at the table were taken, and everyone waited for Makara to begin. The crowd around the table murmured, but its voices were stilled when Makara stood, the legs of her wooden chair squealing against the cement.
"Thank you for coming," she said. "I know the last few weeks haven't been easy, and I can't promise they'll get any easier. We lost a lot of lives and we barely have time to mourn our dead. I'm sorry for that. Askala and the Radaskim aren't going to wait for us, so we can't wait for them. We have to move on to Los Angeles and help Augustus, for several reasons."
Makara paused a moment, and everyone waited quietly for her to continue.
"The first reason is one of food. Even with our numbers lower, the food in Hydroponics won't last forever. Samuel, Ruth, and I took inventory of what we have, and it's clear: even with the canned food, we don't have long. A couple of months at most. Civilians can stay in the Bunker for the meantime, but eventually, everyone will need move to Los Angeles."
Everyone had grown quiet, waiting for her to say more. Makara continued.
"Augustus is being attacked from the west by the Reapers in the Inner City, and from the east by the Blighters. It won't be long before his forces are crushed between the two sides."
Miraculously, everyone was still quiet. The crowd was actually hearing Makara out. I had expected them to protest at having to continue fighting.
"Without Augustus, there's no way we can win the war against Askala," Makara said. "That's why we have to help him. I don't know what we'll find when we get there, but winning means we'll have the chance to make the counterattack on Ragnarok Crater."
There was a sharp intake of breath from the crowd. Even a few faces at the council table went white at that announcement.
"So you're just going to leave us behind?" One of the civilian leaders of the Exodus had spoken. "What if you don't come back, like last time?"
"Please hold off on questions and comments for now," Makara said. "But I will answer this one. We will come back. And if not, there should be few enough people here to survive for a long time, if not indefinitely."
When no one protested, Makara continued.
"This is the first question I put before the council," Makara went on, "and it has nothing to do with Los Angeles, but with Cain of the Sworn, along with those of his gang who followed him and surrendered to us. What is to be their fate?"
Now, the crowd's voice rose in an angry murmur. Members of the council table looked at one another, waiting for the crowd's voice to die.
When it became quiet, Samuel was the first to speak.
"It makes no sense to keep them alive anymore," he said. "Keeping them alive would waste time and resources, draining our food and our manpower. They betrayed the New Angels, and they deserve nothing more than swift retribution."
The crowd roared its agreement. I didn't realize that we'd also be deciding the fate of Cain and the Sworn. Before I could think about my own feelings on the subject, Makara spoke.
"All members in the council in favor of the swift execution of Cain and the three Sworn who survived the battle, say 'aye.'"
"Aye," most all of the people at the council said in unison. I wasn't ready to vote, so I said nothing. But it didn't matter which way I voted; Cain and his followers were as good as dead.
"Now," Makara said, "I'm going to open up the floor to discussion about how we can best help Augustus."
The table was silent, and the crowd surrounding it stilled. From somewhere, a baby started crying.
"Obviously, the ship can only carry so many," Ashton said. "We'll have to find a safe landing point to unload all our forces. Whether that is Augustus's camp, or somewhere else entirely, I don't know. And we'll have to make quite a few trips to transport everyone."
"We have about two hundred fighters left," Makara said. "What is that, five, six trips?"
Two hundred fighters sounded so pitiful when said out loud. Just a week ago we had been at least five hundred.
"If anything is clear," Makara said, "the survival of the New Angels depends on winning the battle in Los Angeles. Two hundred fighters won't do much to augment Augustus's legions, which number in the thousands. Maybe if we set up a flank, we can get at the Reapers where it will hurt the most."
"What do you mean?" Anna asked.
"If Carin is focused completely on Augustus, an attack from us would be completely unexpected," Makara said. "It might be the chance we need."
"That's something we should talk about with Augustus," Ashton said. "But maybe we should be focusing on the Blighters. They are the real enemy."
"He's been expecting us to come back for a few days by now," I said. "The Blighters might have already gotten to him."
"No, he's alive," Makara said. "I spoke to him a couple of hours ago. The Blighters started their attack this morning. He's getting impatient, and I told him I'd let him know our decision tonight."
"How soon do you plan on leaving?" Lauren asked.
Makara looked at her grimly. "Tonight. Every minute we wait is more men lost."
Everyone in the hangar fell quiet at that.
"It will take many trips on Perseus to ferry two hundred people there," Makara said. "I only trust the ship to carry thirty or so at a time."
"That makes about seven trips," I said, doing the math. "That'll take at least a full day, working nonstop."
"Probably longer," Samuel said. "We can expect there to be roadblocks. Let's expect two, or even three days. There are dragons to worry about."
With the mention of dragons, I was reminded of Askal. It'd been a while since I'd seen him. I had gotten a glimpse of him defending Pyrite from the Radaskim when we had relocated to Bunker 84. And then there had been the dream, where he was outside Oasis. I had no idea how he was doing now, or even if he was safe.
Thinking about Askal made me realize that the key to our victory was with the Elekai.
"Anna and I need to find the Elekai," I said. "Two spaceships against all those dragons won't be enough. We need to be able to fight in the air on an equal footing."
"And how would you get the Elekai?" Makara asked.
I thought for a moment. The home of the Elekai had to have changed, because we had destroyed their Xenolith outside Vegas before realizing they were allies.
"We would have to find them," Anna said.
"I don't know if I can risk Alex going on such a dangerous mission," Makara said. "And you're injured."
Anna shook her head. "We're going."
Before Makara could respond, I cut in.
"Look, I know it sounds crazy on the surface. But we need those reinforcements. Without them, the Radaskim dragons will go unchecked. Usually, the Elekai have always helped us without our asking. The fact that they aren't in Los Angeles tells me something is wrong."
"This is all conjecture," Makara said. "Show me some proof, and sure, you can go."
Makara's tone said that my idea would never happen. In a way, I could understand her perspective. The last place she'd want me to be was the Great Blight. If I died, it was all over.
The real reason Anna wanted to go was to see the Wanderer. I couldn't say that to Makara, though.
Makara's eyes narrowed in skepticism. "There's something going on between you two. Something you're not telling me."
Makara went quiet, waiting for me to explain myself. I'd been caught, somehow.
"I'll just say it, then," I said. "This is something Anna and I have to do together."
"Why?" Makara asked.
"Because..." I wasn't sure how to continue.
Thankfully, Anna picked up for me. "I have my own questions to ask the Wanderer."
"What questions?" Samuel asked.
Everyone at the council table became more alert, as if this question was more important than anything that had been discussed so far.
"I..."
Anna didn't continue. She knew as well as I did that her question was personal. It wasn't likely that Makara, Samuel, or anyone else would see it her way.
But just because it was personal didn't mean it wasn't important.
"The questions have to do with the final battle," I said. "We need specific answers that only the Wanderer can answer."
Samuel and Makara exchanged a look. Everyone waited for more of an explanation.
"I want to look the Wanderer in the eye, and ask him the reason Alex has to die," Anna said. "That's it. So far, we're just taking his word for it." She paused a moment. "There could be another way to win. We just don't know it yet."
A heavy silence followed. Makara looked at Anna, as if weighing her, and Anna merely looked back. Samuel and Julian looked down, as if in thought. Ashton stroked his chin, chewing his lower lip.
Makara at last looked at Samuel. "What do you think?"
It was a long moment before Samuel answered. "It's been a while since we've heard from the Wanderer. Maybe...maybe they're right. And Alex was right, earlier. We haven't heard from the Elekai in a while, and Augustus didn't mention anything about the Elekai being at the battle. It might be worth investigating, if they're in trouble."
"And how are they to get to the Great Blight?" Makara asked. "Perseus will be busy transporting our men."
"Askal will take us," I said.
"Even with the cold?" she asked. "And with Anna's injury? Besides, you don't even know where he is"
I knew Makara was right on all those counts. And Askal wouldn't be fast enough, anyway, for wherever we had to go – especially if the Elekai's new home was far away.
"We'd need the spaceship," I said, finally.
"This is just more time," Makara said. "Time we really don't have. And I don't like sending the person who's our only hope for victory into harm's way. What will happen if you die, Alex?"
"I won't die."
"You don't know that," Makara said sternly. "If you die, the rest of us die as well. We've had more than enough close calls. I've already put you through too much danger and I'll be damned if I do it again." She crossed her arms. "And having Anna go with you is also unsafe. She's immobilized from her injury."
The rest of the table went silent. All of her points were valid. Probably far more valid than mine. All I knew was that I had made Anna a promise, and I meant to keep it.
"Can't Augustus hold on for a few more hours?" I asked. "We have to remember why we're fighting. There are powers at work far greater than us. There's the Wanderer. And there's Askala. Only one of them can win. If there's any chance of victory, our forces will need to join. If Anna and I go find the Wanderer, we can bring back the Elekai army. We have to bring the Reapers to our side, as well. If we're not standing together, Los Angeles will fall. But we need the Elekai if we're going to win."
I looked at Makara, who looked as if she was finally being won over.
"This is something I have to do," I continued. "And Anna has to come with me. She has her own questions to ask. We're the only two who haven't had our prophecies fulfilled. We don't just need the Elekai army. We also need the Wanderer's guidance to figure out where to go next."
Makara was silent. I could tell she still didn't like this, and I didn't blame her. She would have a hard job telling Augustus that he had to wait...again. And if I died, Makara would blame herself.
"I'll make it back in one piece, and so will Anna," I said. "I promise."
Makara heaved a heavy sigh. "Fine. We'll go. We'll coordinate with Augustus and see where the best place for us to land our men. It's at least something to tide him over. After that, we can continue on to the Elekai."
Lauren was next to speak. "Who's staying here? I'm assuming me, because I'm not really much of a fighter..."
"You can stitch a wound," Makara said. "Some might say those people are more valuable than soldiers. If you're willing to offer your help, we'll sorely need it."
"My only worry is my daughter. She can't go into a war zone, and I couldn't leave her behind. If something goes wrong, I can't leave her alone."
"I..." Michael started. "I planned on staying as well."
Makara's eyes widened with surprise. Michael never missed any action.
"I made a promise to Lauren a long time ago, one I didn't keep. After seeing what almost happened to her down here – I can't leave her. Something tells me that if I go out again, I won't be coming back."
Makara looked at Michael hard. "I understand what you're saying. But we need you, Michael."
I remembered how Makara had been hesitant to let him join the New Angels at first. Now, he was an indispensable part of the team.
"Lauren and I talked about it, and I wouldn't be completely useless here," he said. "Bunker 84 will need a leader to hold the fort. I can keep order here, and my loyalty isn't in question. I promise, on my watch, nothing will go wrong here. We'll farm and be ready to help in whatever way we can."
After a while, Makara nodded. "You're right. Bunker 84 will need someone to run it, and I can't think of anyone better for the job than you."
Michael nodded his thanks, grabbing his wife's hand. Her eyes filled with unspoken relief.
Makara turned her eyes on me. "Alright, I think we're in agreement. Is there anyone else who wants to stay behind?"
"I'm going to L.A.," Julian said.
"I'm staying," Ruth said. "Hydroponics still isn't working efficiently, so hopefully I can figure out what's wrong."
"I'm staying as well," Marcus said, pointing to his cast-encased leg.
"Of course, I can pilot Perseus," Ashton said.
"And you will need a medic," Char said. "I'll go."
Several other members at the table voiced how they'd support – most wanted to go to Los Angeles, while a few wanted to stay behind to help Michael and Lauren.
"That's everyone," Makara said. "Let's regroup here in the morning. I'm going to speak with Augustus to determine our landing strategy."
The meeting adjourned, and everyone began to stand. The crowd buzzed with excitement. I helped Anna back into her chair and to the ship.
The battle for Los Angeles was about to begin.
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# Chapter 16
I sat with Makara and Samuel aboard Perseus's bridge. Anna had gone back to the cabin to rest while Ashton stood in the doorway, waiting. Michael was organizing the Bunker while Ruth made one final inspection of the hydroponics line. Julian was taking inventory of the food, determining how much should be taken and how much should be left behind. Char and fifteen of his best Raiders were also coming along.
At 20:30, Perseus lifted off. As we rose out of the hangar and into thick darkness, I looked out the windshield to see no moonlight or starlight slipping through the thick clouds. Ashton guided the ship upward, until the mist of the clouds soaked the windshield.
A minute later, we broke through, finding ourselves above a shadowed, misty plain, reflecting light from the starry heavens. The moon hung with a reddish hue, a waning crescent halfway up the eastern sky.
On the far eastern horizon came an ethereal glow.
Anna gazed toward the east.
"What is it?"
"The Great Blight," Ashton said. "Much closer, now."
"It's amazing we can see it through the clouds," I said.
The ship's fusion drive hummed and the cloud-plain passed at a crawl.
"Some of it is the celestial light from above," Ashton said. "But most of it is from the Earth. I've wondered if the Great Blight makes these clouds last longer than they should."
"What do you mean?" Anna asked.
"I can't prove anything," Ashton said, "but the clouds seem to hang extra thick in Blighted areas, don't they?"
"I thought that was everywhere," I said.
"Maybe so," Ashton said. "I've seen other places of the world, but none are as bad as this. I'd always assumed it was because Ragnarok hit here. And maybe that's the case. But as far as Blights changing weather patterns...it's a hunch I have." Ashton paused. "The more the Great Blight grows, the more quickly it can grow."
"And with more Blight comes more monsters," Anna said.
"There's no way we can fight them all," I said.
"Maybe not," Ashton said, turning to me. "But we're never going to give up. The most important time to be brave is when it's easy not to. That's something those monsters will never understand, and it's something they will never take from us."
Ashton's words rang right to me. From the moment Samuel, Makara, and I had decided to go after the Black Files, there had been an element of hopelessness. We had faced danger after danger, somehow surviving. At some point, the story had to end – for good or for bad.
But it wouldn't end just yet.
The next leg of the journey passed in silence. A little over an hour later, the LCD showed us twenty miles north of Los Angeles. Ashton began to descend.
"Look!" Anna said, pointing at the dash. It showed the time to be 22:28.
"What?" I asked.
"Look at the date."
It was December 31, 2060. Only a few hours left until the New Year.
"It snuck up on us," I said.
"It's easy to lose track of days when you're just trying to survive those days," Anna said. "This attack they're making is like clockwork."
Ashton shook his head. "I doubt they take our calendars very seriously, or even know about them. The measure of time only exists in the human imagination."
"They see time differently than we do, anyway," I said. "For the Xenominds, months, even years, pass like seconds. That's what the Wanderer said, at least."
"I can't even imagine," Anna said.
The ship lowered further. The plain of cloud rose to swallow us whole. We left the stars behind and were plunged into darkness. The bridge was lit, and we hung like a lantern in the clouds. Ashton flipped a switch, shutting the lights off. Now, the only illumination came from the glowing buttons on the dash and the LCD screen.
From the wardroom, voices murmured. Samuel came from amidships, and stood silently behind us. Makara must have still been back there. He gazed outward, waiting for the spread of the ruined city to appear before us.
At last, we broke through the clouds and it came into view. My eyes widened upon seeing the devastation. Flashes emanated from the metropolis's eastern side, lighting the tall, shadowed buildings and twisting streets with arrhythmic bursts. Streams of violent light arced from the dark towers of downtown, streaking through the sky and exploding in white-hot plumes upon the city's eastern side. Each flash revealed the silhouettes of dragons, swooping down on Augustus's position.
"They're being slaughtered," I said.
"They could be gone by morning," Makara said.
"It's almost like the Radaskim and the Reapers are working together," Anna said. "How can we throw our men into that?"
Ashton shook his head. "If they are, the Reapers' number will be up, soon. The Radaskim might be fine with a temporary truce, if only to destroy the Novans first. It's up to us to stop the war between the two sides."
"That means speaking to Black," Makara said.
Ashton switched frequencies, calling Augustus. It was a while before the Emperor's voice came online.
"Where the hell are you?"
"The first of us are here," Ashton said. "Where do you need men?"
"Where do I not need them? Those monsters are everywhere."
"What about the Reapers?" Ashton asked.
"What about them?" came an unexpected voice, low and dangerous.
Carin Black.
***
NONE OF US KNEW WHAT to say. In that lull, Black spoke again.
"We're working together now, Augustus and I," he said. "Try not to find that so surprising."
"But..." Anna said. "You're bombing the airport."
"Yeah," Black said. "And we have been since this afternoon. What's left of us are fighting behind the wall."
There was little time to contemplate how drastically things had changed. If Augustus and Carin were working together, then it had to be bad down there.
"What do you want us to do?" Ashton asked.
"You have any nukes?" Black asked. "Because that's the only way I see us getting out of this."
"No," Ashton said. "Perseus doesn't have any. And I was speaking to Augustus."
"How many men do you have?" Augustus asked.
"Twenty."
"You have to be joking," Carin said. "These are the reinforcements we've been waiting on?"
"They have the ship, too," Augustus said. "That counts for a lot."
"What's your location?" Ashton asked.
"We're sheltering behind a building, in Orion," Augustus said. "There's too many dragons out there. Every time we come out to fight, they chase us back to the towers."
"How many dragons?" Ashton asked.
"Thirty, maybe," Augustus said. "We shot down a few, but only one with the ship. The Reapers shot some others with SAM's. We're running low on rounds, though."
"I can't fight thirty dragons," Ashton said. "I don't care how long I've been piloting. That can't be done with even two ships."
"Then we have to find the Elekai," I said. "It's the only way."
Finding the Elekai in time to save the city was a tricky proposition. We didn't even know where they were. If they were relatively close, it might only take a few hours. If they were far, it could take far longer.
"Look," Augustus said. "At this rate we won't last until the morning."
"You'll have to," Samuel said. "There's nothing we can do from here. Even if we dropped off our men, what will twenty extra guns matter, and how can two spaceships fight against thirty dragons? We tried that fight already in Wyoming. We even had nukes, and still both ships went down. With the skies out of our control, Askala's ground forces can run unchecked. We have to find the Elekai dragons, or no one's surviving this night."
Both Augustus and Black went silent on the other end.
"What's our course?" Ashton asked.
Ashton's question was followed by a silence on our end. Now that it was time to make a decision, it seemed like no one wanted to make it. Trying to find the Elekai was a risk – either we found them in time, or we didn't.
But the way I saw it, we had no choice.
"There's nothing more to say," I said. "Head east. I can try to reach out for the Wanderer or Askal with my mind."
"Will that work?" Makara asked.
"I...think it might. It's always been involuntary, but maybe, if I try hard enough..."
"You don't sound too sure," Black said. "We need you here to fight those dragons. Together, we have a chance to kill them."
After a moment, Samuel spoke.
"Finding the Elekai is the only way. Go east."
Ashton turned back to the dash. "Augustus. Carin. We'll return."
"What?" Carin bellowed. "You can't leave us to die here!"
Augustus said nothing, so I wasn't sure what he thought. But no words of theirs could have stopped Perseus rising once more into the sky, above the clouds.
I exited the bridge. I needed to find a quiet place to try and communicate with the Elekai. Pray was more apt a word, because it would take nothing less than divine intervention to see us through.
In the wardroom I passed the Raiders, who looked to me for an update.
"Makara will explain," I said.
I reached an aft cabin where I could get some peace. I sat on the deck, closed my eyes, and brought up an image of the Wanderer in my mind. It came surprisingly clear. I tried to let go of all thought, and let that image guide me, until a connection was made. The picture grew firmer as the world outside faded. My mind slipped into complete calmness.
It had worked.
Then came an awful, reptilian scream. Whether that scream was in my mind, or outside the Perseus's hull, I couldn't tell.
I had gone into a vision.
***
WHEN I AWOKE, IT WASN'T on the ship. It was in the midst of battle.
Not the battle for Los Angeles, but the battle for the Elekai.
A flurry of dragons circled and fought one another in midair in the darkness above the Great Blight. Xenofungus glowed in the night; from it rose a towering Xenolith, far larger than the one outside Las Vegas.
In the storm of dragons, jaws snapped, necks thrust, tails slashed, and flying bodies pummeled and collided. Cold screams shook the air. Hundreds of white eyes glowed in an aerial dance of death.
Once again, I was present in Askal's mind, seeing the world though his eyes. I had no idea where this battle was taking place, but I needed to find out. On my right and left, other dragons flew in a V, of which I was the point.
Are you there, little human? Have you come to witness the last battle of the Elekai?
It's not the last battle, I thought. We're on our way. Where are you?
Askal flapped his large wings against the cold wind, lifting higher above the Great Blight. The other Elekai dragons followed suit. From the north, more opposing dragons flew toward the fray. The Elekai were outnumbered at least two to one.
Come east, he said. The dragons will point the way.
They will point the way? What does that mean?
We will hold them here, Askal thought. Hurry, little Elekai. And...I am sorry we could not help you with the city. It will all be in your hands, soon...
Before I could ask anything further, the vision faded, and I found myself aboard the cold deck of Perseus.
***
WHEN I CAME TO, SAMUEL, Julian, and Char stood above me. The rest of the Raiders had formed a small ring.
"His eyes..." one of the Raiders said.
"They aren't white anymore," another said.
I looked up, focusing on Samuel.
"Did you see anything?" he asked.
"I don't know where they are," I said. "Askal said the dragons would point the way. There was a battle, and they were badly outnumbered."
"We won't give up," Samuel said, grabbing my forearm and helping me up. "Not while we breathe."
We returned to the bridge, where Ashton and Anna were seated at the controls. I had no idea how much time had passed. Makara stood behind, watching out the windshield.
"Where are we going, Alex?" she asked.
I shook my head. "Askal said the dragons would point the way. Their Xenolith is growing out of some hills on a large plain. There was a huge battle of dragons in the sky. More were coming out of the north from the Radaskim."
"You sure it was from the north?" Ashton asked.
I nodded. "Yeah. I don't know how I know, but I do."
"If that's the case, then the place we're looking should be south of Ragnarok Crater, more or less. If those dragons are coming from the Crater, they'd fly on the straightest path possible to get to the Elekai."
What Ashton said made sense.
"Alright," I said. "So find a dragon, and match his trajectory?"
"The dragons will point the way," Ashton said. "Pretty simple and direct."
"Where are we now?" I asked.
"Almost over New Mexico," Ashton said.
New Mexico? There would be no way the Elekai dragons could get back in time, not even if everything went perfectly.
But before anyone could point that out, Ashton spoke.
"Two dragons, dead ahead."
I strained to see two reptilian shadows beating against the wind, distant. They didn't seem to see us.
"What's their course?" Samuel asked.
"Almost due east," Ashton said. "Assuming they're going straight to the Xenolith, we can copy their angle."
"Do it," Samuel said.
"Alright," Ashton said. "Hold on."
The ship suddenly lifted. With the change in course, one of the dragons broke, coming back to attack us. I could see its white eyes glowing in the night.
We rose higher, piercing the cloud layer and going in the direction the dragons had been pointing. Once above the clouds, Ashton allowed himself to relax. The dragon would be unlikely to follow us up here, and even if it could, we'd outpace it.
"We'll stay up here for a bit," he said. "We'll come back below when I'm sure we've passed them."
A few minutes later, Ashton lowered through the clouds. Once again, we were above the pink shimmering of the Great Blight. The land had flattened, but a line of pointy mountains rose to the northwest. I had a feeling that we were close. The clock on the dash showed 00:30.
A minute later, the ground stretched before us. A line of hills rose in the east, only visible from the glow of the fungus. As the hills neared, I saw the Xenolith rising above them.
"That's it," I said.
As we passed over the hills, there were no signs of a battle. In fact, the wide fungal plain seemed empty. The battle had concluded.
Perseus swooped by the rising Xenolith.
"Where are the Elekai?" Julian asked.
No one had an answer for that.
"I don't know," I said. "But Anna and I need to get inside the Xenolith."
"How?" Samuel asked.
"There's an entrance at its base," I said, "if it's anything like the last one."
"And what are we supposed to do in the meantime?" Ashton asked.
"Not sure," I said. "Lift off, but stay in the area."
"Radio us when you're ready to leave," Samuel said.
At last, Ashton lowered the ship, setting it down on the xenofungal bed. Samuel handed me his radio.
"The channel's on the right preset," he said. "Try to hurry."
"I'll let you know," I said.
Julian brought Anna's wheelchair around. Together, we lifted and settled her in. I took the handles and swiveled the wheelchair around.
I knew it was crazy to take Anna with me, but I also knew she would never forgive me if I left her behind. We probably had to abandon the chair at some point. Anna could walk, albeit in pain. I could only hope that the Wanderer didn't make us go too far.
Whatever the Wanderer told us, we'd hear it together.
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# Chapter 17
Anna and I walked onto the glowing xenofungus. The air was warm and humid. Ahead, the Xenolith shot hundreds of feet into the sky. It dwarfed everything else for miles. Thick branches curved from its central trunk, and these branches carried tufts of pink growth. The Xenolith's trunk radiated silvery light, the illumination intensifying in silver and purple hues upon reaching the upper branches. The low wind sifted through the limbs, making the Xenolith seem to sigh.
Perseus idled as Anna and I walked down the boarding ramp. Anna was managing, if painfully. I gave her my right arm for support.
I pointed to the base of the trunk, where the Xenolith's thick roots twisted into the fungal bed. These roots were the size of normal trees, winding around each other in complicated braids. If we were getting inside, those roots would need to untangle, somehow.
I walked forward and touched one of the roots. It reacted instantly, snapping back like a snake. I pulled my hand away as the other roots unraveled as well. They untwisted until a dark opening was revealed. Pinkish, silvery light glowed from within. We peered inside.
"What's in there?" Anna asked.
"You'll see," I said.
As soon as we passed the threshold, the roots behind closed and knotted themselves shut, leaving us in the spiraling corridor glowing pink with bioluminescence.
The tunnel circled down to the right, much like the one in the first Xenolith had, only this tunnel seemed to circle more widely, which made sense if the Xenolith was larger. As we descended, the tunnel widened.
Soon, the tunnel stopped curving, opening into a wide cavern, glittering pink and silver with alien life. A large lake of pink liquid with an almost metallic sheen on its surface lay before us. Anna's breath caught at the sight, mostly because two dragons were bathing in the lake. Only their angular heads were visible above the surface. Their eyes opened lazily upon our entrance. I felt a strange energy stir the air, now that the dragons were aware of our presence.
The dragons didn't move, merely settling back into their doze. Something seemed off about them. There was so little I knew about the Elekai, or even xenolife in general. Maybe this was how they ate – nutrients and calories might enter through the liquid itself, nourishing the dragons.
There was no island in this lake, as with the last Xenolith. And there was no sign of the Wanderer.
"Where is he?" Anna asked.
"I don't know," I said. "We have to find him."
"How?"
I stepped forward to the liquid. I didn't even know if I should be calling it that. Liquid sounded too scientific, where this fluid was the very stuff of life.
Ichor – the blood of gods. That was what it should be called.
Anna joined me on the shoreline. The surface of the lake was still as glass, except around the dragons, where tiny ripples raced outward, only to dwindle to stillness long before reaching the shoreline. The ichor looked almost solid.
"Go ahead," I said. "Step in."
Anna looked at me, as if to be sure, before placing her right boot in the lake. The ichor wrapped around, as if in embrace.
"It's warm," she said.
She stepped the other boot in, wading a bit into the ichor. I followed her in. The ichor was warm, thick, and tingled a bit as it seeped into my boots and soaked through my clothing. We waded out together until the ichor had covered us to our torsos.
There, Anna paused.
"This...feels weird," she said, after a moment.
"What's wrong?"
"It feels...tight. In my chest."
"Anna, are you..."
Suddenly, her eyes closed, and she fell backward. As if the pool had pulled her.
"Anna!"
The ichor claimed her, wrapping itself around the rest of her body until only her head remained above. Her eyes were wide with fear. She managed to scream the first syllable of my name before being completely submerged.
I rushed forward, but the ichor pushed me back, locking me to my spot. I could only watch, helpless, as Anna floated beneath the translucent ichor, eyes opened. Her body suddenly jerked in an unnatural way. I groaned as I pushed against the ichor – but it was useless. Anna's body jerked twice more, in quick succession. Her eyes were closed, her body still.
I waited, my heart pounding, for at least a full minute. Still, Anna remained under the surface.
I looked toward the dragons. "What are you doing? Can't you see she needs help?"
They stared back with fathomless white eyes, seemingly unconcerned.
"Anna..." I said.
But then, her body rose out of the liquid. I was no longer restrained, so I waded forward to pull her out. Her head appeared above, and I pulled her into my embrace.
"Anna! Anna, can you..."
She started hacking, and some of the ichor shot against my shoulder. She sucked in a sharp breath, coughing more liquid, before clinging to me tightly. Her body shook, and it was only after a long moment that she calmed.
"Are you alright?" I asked.
She gave a weak nod. "Yeah. I think so."
"We need to get out of here," I said. "It's not safe."
She felt along her ribs, her eyes widening further. "How..." She touched her rib cage, feeling each bone individually. "My ribs are fine!"
The lake hadn't tried to kill Anna. It had healed her.
I didn't see how it was possible, but I had seen a lot of things that should have been impossible in the past few months.
"I was scared to death," I said. "You'd think they'd give us a heads up."
"There isn't time to worry about it," Anna said. She faced the dragons, who watched serenely. Seeing those dragons there made me wonder where Askal was. I saw, looking into the corners of the cavern, that the lake appeared to branch off into other areas.
"Come on," I said. "We have to keep looking for him."
We waded out into the cavern. As the lake deepened, our feet could no longer touch the bottom. I'd had experience swimming in Bunker 108's pool, but Anna was struggling. Her eyes widened in fear.
"This place won't let you go under," I said. "If you want, hold onto me."
This was something that would have never worked in a normal body of water, but Anna grabbed on. As soon as her weight pressed down, the liquid pushed up from beneath, keeping us afloat. I swam forward, but wasn't making much progress on my own. A current began to push us between the dozing dragons.
Looking through the clear pink ichor of the lake, I could see deep gashes lining their forms. Wounds from the battle. I understood why they were here; the pool was healing them. The extent of their injuries must have been severe; otherwise they would have healed quickly, like Anna.
"They've already fought," I said. "That's why they haven't come to help us. They were the ones who needed help."
Anna watched the wounds along the back of the largest Elekai dragon I'd ever seen. The pink liquid of the pool was stained purple from the many deep wounds marring its body. Its wings were settled on the surface of the lake, around which wrapped the healing liquid.
The current pushed us toward the back of the lake. As it curved onward, another cavern opened, much larger than the first, where three dragons bathed in the ichorous substance. On the far side of the lake, a shore of silvery sand rose, meeting a line of xenofungus. From the fungus grew twisted, alien trees, the boughs of which glittering silvery with long, wavy fronds. There were so many trees that it was hard to discern the treetops, and it all seemed to be the same organism. Maybe the trees were the same organism. It was an entire underground forest.
"I wonder where this is," Anna said. "This cavern had to have formed naturally."
"I don't know," I said. "I'm not even sure where we're supposed to be going."
The current pushed us past a dragon whose right wing was almost ripped in two. His eyes were closed, though his chest rose and fell in short, ragged, breaths. Looking around the cavern, I didn't see any sign of Askal. Either he was in another part of the lake, or...
I didn't want to think of the other possibility.
As we neared the shoreline, I made myself stand. My feet found the lakebed, and I pushed myself up and walked onshore. We stood on the sand, the pink liquid rushing from our clothes and skin to rejoin the lake. Anna felt along her ribcage, still unbelieving of what had happened. My own body felt reinvigorated.
I reached for my radio to call everyone else above. But when I pressed the talk button, the line was completely dead. The radio had been so drenched in liquid, and we were so deep underground, that it wasn't surprising that it didn't work.
Anna stared into the trees. The land sloped downward into a narrow valley with sheer sides, in which the trees grew more thickly. The forested valley opened about one hundred feet from the shoreline. I looked upward at the sparkling ceiling of the cavern, which cast bioluminescent light. It was hard to believe this place existed; it was more magnificent than even the first Elekai home.
Somehow, I had a feeling that we'd find the Wanderer in the forest ahead. I thought it was strange that he wouldn't have come out to meet us, and it gave me a bad feeling. My Beretta was still strapped to my belt; hopefully, it still functioned, even though it was soaked. If nothing else, Anna had brought her katana – now healed, she would have no problem in using it.
We started walking. As we did, Anna placed a hand on the hilt of her katana, sheathed on her back. We entered the first of the trees, finding ourselves on a narrow path. Pink limbs overhung the path, forming a glowing tunnel. Alien blossoms spiraled from above, their scent sweet and sharp in the air. I could feel the trees' collective sentience bearing down on us as we walked, as if they were watching us. Anna's shoulders hunched. She could feel that weight, too.
The air was heavy with...something. I wanted to speak, but found my voice caught in my throat. The path twisted and turned, delving deeper into the cavern, deeper into the trees. The thickness of the xenoforest blocked any view of how deep this valley actually went.
The path snaked back and forth, plunging further and further down. The vegetation became more alien, more magnificent. Among the pink branches were now blazes of orange, of deep crimson, of yellow, of silver. The colors were incredibly bright, almost unreal in hue. Thin tree trunks rose in perfect spirals high toward the ceiling, forming arches before descending somewhere far distant in the xenoforest. Some tree trunks even fused together, two or three at a time, before forming an enormous, multicolored treetop.
Just when I was wondered if we should turn back, the path made a final bend, going from there in a straight line. Before us a massive tree grew from the fungus in a wide meadow, hemmed in by trees. The meadow was a near-perfect circle, and the trees on its rim leaned forward, as if bowing to the central tree. This tree had silver bark, and was probably larger than any tree that had ever existed on Earth. Its colossal trunk shot upward in a straight line – even at my distance, I had to crane my neck to see its height. Its treetop billowed near the cavern's ceiling, casting silvery shade onto the meadow below. Sparkles lit the air, blossoms falling thick as rain.
Though beautiful, something about the scene just felt wrong. Again, the Wanderer wouldn't have made us walk all this way just to see him.
And in the shadow of the tree, right against the bark, a small shape sat hunched. Anna drew her katana, its metallic ring violent in the still, blossom-strewn air. Time stood still as the trees waited, like a whip about to snap.
The form turned from the tree, slowly, revealing a monstrous face, long spindly limbs, and a drooping chin. I didn't know what to call it, but I knew it was nothing like the Elekai. It wasn't a crawler either, but I knew only Askala could have made a monster so hideous.
The monster's mouth opened; from it emanated a long, high wail, which pierced the air, my skin, my bones. It was the most horrible sound I'd ever heard, paralyzing me to my spot. The thing advanced, its hideous call never abating. Its six scythed legs clawed into the xenofungus, piercing it like knives. From its long mouth shot twin fangs, dripping silvery liquid. Perhaps the sap from the tree. Or poison.
As if emerging from a dream, Anna changed her stance, holding her blade aloft and bracing herself with her back foot while the left leaned forward. I took out my gun.
The creature paused and ceased its wail.
Then it shot forward.
It sailed through the air, its horrible white eyes widening and its front legs extending, sharp as blades. As the monster came down, a strange pulse of energy sizzled through the air. The creature's limbs went limp as it tumbled toward the ground. Anna and I stepped to either side, and the creature crashed between us, rolling on the fungus. Its body lay immobile.
Anna didn't wait. She plunged her blade deep into the creature's long, gray neck, where its chitin-like armor didn't protect it. As the blade entered, the thing gave a sharp hiss. Anna twisted the blade and withdrew it, the metal covered in a purple slime so dark that it could almost be called black. The creature gave a final quiver before settling into stillness.
I turned back to the tree. Its side had been gnawed on, and from the puncture silvery sap spilled out, covering the fungus below.
"Where did that thing come from?" Anna asked.
I shook my head. The answers to that question were too horrifying to consider. Somehow, the Radaskim had penetrated the Elekai home. If there was one of those things, there could be more.
"We need to find the Wanderer," I said.
We came close to the tree and circled around its wide trunk. On the other side lay a cliff, falling into darkness. We had come to the end of the forest. If the Wanderer wasn't here...
Anna pointed. "There!"
A human body, covered in a light brown cloak, lay not far off, still. I ran forward, suppressing my desire to scream.
"No..."
I knelt beside the Wanderer, touching his shoulder.
Slowly, he stirred. He wasn't dead. Not yet.
His eyes opened, completely white. It was hard to read them, but I could tell he was in pain. From the way he held his torso, it wasn't hard to see why. Blood stained the front of his robe, soaking between his fingers.
"Alex..." he said, voice raspy. "They came. We fought them above, but they..."
The Wanderer coughed – that action must have sent him into horrible pain.
"Don't speak," I said. "We'll get you to the lake."
The Wander gave a slow, weak nod.
"Come on," I said to Anna. "We need to carry him."
She sheathed her katana, and helped me lift the Wanderer. He wasn't too heavy. He closed his eyes, and though he continued to hold his torso, the red still leaked out. Such a wound would take hours to bleed out, but the poison might work faster. If we had arrived a little later, it might have been too late.
We all but ran. We found the path and ran up the twisting incline. The Wanderer's eyes were shut, and his body had stiffened.
Our breaths came out heavy, and my heart pounded at the exertion. I pushed myself beyond what I thought possible. I kept my eye out for additional threats, but the forest was silent and empty. If there were more monsters, they weren't here.
Five minutes later, we crested the final incline, finding the pink ichor of the lake glittering ahead. The dragons still lay there, bathing. Anna and I ran forward, the ichor accepting as we plunged into it.
The Wanderer fell face first, and immediately sunk below the liquid's surface, as if the ichor itself recognized the need for haste. The Wanderer sunk further into the depths, falling downward and away. He was getting quite distant, now, veering toward the center of the pool. At last, he did reach the center, near the leg of a slumbering dragon. There, his movement ceased. His form showed wavy from the liquid in between.
Anna and I had watched in silence for about thirty seconds, when the liquid began to hollow out in the center, forming a depression in the surface of the pool. The depression deepened and widened, and an emanating wave pushed Anna and me back to the shoreline. The ichor continued to rise, even as the center of the pool emptied. Anna and I scrambled onshore. The Wanderer was down there. The center of the lake had emptied of all ichor, and it whirled, pushed back by some unseen force.
I could see the Wanderer standing there, his face shrouded within his cowl. He lifted one arm, as if in command. A line formed in the water between us and the Wanderer, and from that line, the water parted, forming a corridor.
The dragon nearby came out of his sleep, looking down at his master with white eyes. The dragon's positioning had not changed; the ichor had remained around him, not wanting to disturb his rest.
The path from us to the Wanderer had been fully carved from the lake. Walls of pink ichor rose on either side, as if frozen in time. At the end of this narrow path, the Wanderer beckoned us to come forward with a single hand.
"Does he really want us to...?" Anna asked.
"Come on."
I grabbed her hand, and we walked into the canyon. The lakebed sloped down to where the Wanderer waited, his face masked in shadow.
I felt as if I were approaching a god. Maybe I was.
As we neared, the shadow no longer covered his face.
At last, we stood before him. He wore a small smile, and there was no evidence of his wound from before.
Anna and I would have our meeting with the Wanderer.
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# Chapter 18
No one said anything for a long while. The Wanderer's eyes glowed white within his cowl.
He was the first to speak.
"There is little time to speak, Alex. We both know the end is near. As you might have guessed, the Elekai were attacked. This morning, we tried to fly to Los Angeles. But Askala guessed our intent. She caught us on the open fields, and many Elekai fell. Almost all the Elekai rest in the pool."
"What about Askal?" I asked.
"Yes, he is here, too," the Wanderer said. "I suppose in one of the further caverns."
"How did that...thing...get in here?" Anna asked.
"It was not only from above that we were attacked," the Wanderer said. "Several burrowers found their way into the caverns. That one disturbed my meditation in the Sacred Grove."
I thought "disturbed" was too light a word for what the Wanderer had suffered.
"How long will it take for the dragons to heal?" I asked.
"For some, not long at all," the Wanderer said. "As for the rest...they will be ready in time for the final battle. That is, if there is a final battle to be fought. Los Angeles must be saved first, but I dare not let the Elekai fly there. They will not be ready."
"Then how are we going to save the city?" Anna asked. "That's why we came."
The Wanderer looked at her with solemn eyes. "I will come with you. With luck, I may be able to turn the tide."
"What?" Anna asked. "How?"
"You're riding on our spaceship?" I asked.
Somehow, it was hard to imagine the Wanderer doing that.
"Yes," he said. "I'd rather fly on a dragon, but there isn't time for that."
The Wanderer was probably the only person who could honestly say that.
"But...how are you going to stop the Radaskim?" Anna asked. "It's just you, and there are thousands of them."
"There is only one way, for which I must pay the ultimate price. I must release the hidden power of the Elekai. It is the only chance, I think, to save the city."
Hidden power? The ultimate price?
"What do you mean?" I asked. "Are you..."
The Wanderer's eyes seemed to answer the question that I left unasked.
Whatever the Wanderer had planned, it would cost him his life.
"The Releasing...it involves my giving up my power. It will allow me to directly control the Radaskim dragons. I'm unsure how long it will last, or even if it will work. It's the only chance I see."
"And it will kill you?"
"Yes." The Wanderer's eyes seemed to focus on something far away. "I can do this, because there is now another, to take my place..."
It was then that I realized he was talking about me.
"Wait...you want me to become..." I didn't even know what to call it.
"My time to lead has passed. Askala has grown too powerful, and she has forced our hand..." The Wanderer gave a bitter smile. "It is the same, on every world. She forces our hand before we're ready to play it."
I had no idea what the Wanderer was talking about. All I knew was that he wanted me to fill his shoes, and that was something I could never do.
"You mean – I have to lead the Elekai when you're gone?"
The Wanderer nodded. "Yes."
"Why not one of the dragons? Why does it have to be...?"
I shook my head. There was no way I was up to this.
"This is your world," the Wanderer said. "This is your fight."
The Wanderer gestured upward – above the rim of frothing ichor, the nearest dragon's head was visible.
The Wanderer continued. "They were the first to lose their world, millions of years ago. On each world, we make a home like this for them. We've saved the genetic blueprints for every species located on our home world, Askalon. You see the trees, the plants, the life of Askalon within these caverns." The Wanderer turned back to me. "The Askala have already fought their war, but through the xenovirus and the Elekai, they are preserved, for as long as we exist in the universe. But the time of humanity has come, as it has come for every race discovered by the Radaskim."
I guessed I understood what the Wanderer was saying. This was our war. Our fight. And if the Wanderer was going to die, I was the only one left who could take control.
"If it's the only way..." I said. "I'll do it."
"No," Anna said.
The Wanderer turned to her, his face questioning.
"I don't understand why Alex has to die. Isn't there another way?"
The Wanderer looked down at us. He said nothing at first. After what seemed an eternity, he spoke.
"Alex was chosen by the Elekai," the Wanderer said. "He was chosen by me. And Alex accepted the call."
"That doesn't answer anything!" Anna said, frustrated. It was all she could do to keep herself in control. "Why him? And why does he have to die?"
"Anna..."
She didn't seem to hear me, instead focusing on the Wanderer, demanding an answer from him that he couldn't give.
The Wanderer waved us up the path. We looked at each other before following him. As we walked, the ichor closed behind us. It wasn't long before we stood on the white, crystalline shoreline. The dragons continued to slumber in their healing trance.
"You ask me why he is to die," the Wanderer said. "The process of converting Askala requires a giving up of the spirit, similar to what I am going to do. When you ask me why he in particular has to die, you are forgetting something important: in the end, we will all die, sooner rather than later."
Anna didn't respond. The Wanderer had a good point: it seemed unimportant to focus on the death of a single person when, most likely, we were all going to have to face that death.
All that the Wanderer had told me before came back: on hundreds of worlds over millions of years, the Radaskim had invaded and the Elekai had resisted. The Elekai always chose a champion to fight the Radaskim. That champion, on each of these worlds, had always failed, in the end. The Radaskim were always too powerful to be resisted, crushing all who stood in their way. Earth was just a tiny planet in a vast universe. What was one death compared to all that?
"There are many things I do not know, Anna," the Wanderer said. "Many things I will never know. I do know that Alex must face Askala, that he has been chosen by the Elekai to destroy her. He can only do so by infecting her. And yes, it involves dying."
"But, why the sacrifice?" Anna asked. "Can't he just infect her and get away. Or..."
"The...transformation," the Wanderer said. "It is hard to imagine how such a thing might be accomplished without death."
"Wait..." Anna said. "Transformation? What transformation? You mean Askala, or..."
"Askala will become Elekai. I don't know what that will do to Alex, but it certainly involves his death."
"So, is there a chance that it won't?"
"There have been...times...where we thought we won," the Wanderer said. "Just on a few worlds. Always, the champion had died, the Radaskim faded...but in time, they came back, more powerful than ever..."
The Wanderer went quiet, thinking. Anna waited for him to continue.
"I see that you care for him very much. And Alex will need that. Love is the only thing that makes the darkness of the world worth enduring, and you humans have a lot of it." The Wanderer paused. "No one knows what happens in the heart of darkness, between the Champion and Askala. Only the Champions know, and they are all dead."
As the Wanderer spoke, Anna's hope died in her eyes. Watching that was more painful than anything I could ever remember
"But none of us must give up. Despite the costs we all have to bear – there is always hope. Always."
The Wanderer looked across the lake, toward the entrance of the cavern.
"We must not linger here. Take me to your ship."
***
WE SWAM THROUGH THE lake and reached the far shore, leaving the glittering cavern behind. The Wanderer led us up the twisting tunnel until we reached the roots covering the opening. The Wanderer raised his right hand. The roots unknotted and retracted into the floor and walls, revealing the glowing night. For miles upon miles, the fields emitted a pale, pink radiance, and in the distance, shining hills rose.
The wind blew warm as we stepped out of the Xenolith. The roots once more enclosed the opening.
I reached for my radio.
"Makara? Ashton? Do you read me? We've left the Xenolith."
"Coming down," Ashton responded. "We'll be there in a minute."
Within a few seconds, Perseus appeared in the sky, its landing lights gleaming. A short time later, the ship lowered to the xenofungal bed.
The Wanderer walked toward the ship, Anna and I walking on his either side. Perseus's blast door slid open, revealing Samuel and Makara standing at the top of the ramp.
When we stepped inside, we found everyone waiting, including the Raiders. Everyone looked at the Wanderer in surprise.
As the door shut behind us, we stood in silence. Everyone seemed to be shocked that the Wanderer was with us – this mysterious, prophetic man who was the leader of all the Elekai resistance against Askala and the Radaskim.
"The Elekai were attacked first," I said.
"There's no army?" Makara asked.
"Their dragons are being healed," I said. "Very few got out unharmed, if any. The Wanderer knows a way we might save the city."
Suddenly, everyone noticed what had changed about Anna.
"Anna..." Makara said. "What happened in there?"
"They have this huge lake made of...something, I don't even know. I was under for a minute, then I came out, healed."
"Completely?" Julian asked.
Anna nodded. "I don't understand how, but I'm fine. I've never felt better, actually."
Everyone turned to look at the Wanderer, who stood silent in his cloak and cowl. Some of the Raiders tried to back away upon seeing those whitened eyes. The Wanderer made no movement or sound, until the wardroom had gone completely quiet.
"There is a way to save Los Angeles," the Wanderer said. "I can gain control over the Radaskim horde's mind, if only for a few minutes. Maybe longer." He paused. "Let's hope longer."
"That's great," Makara said.
"I fear this battle serves two purposes," the Wanderer said. "Askala wants to destroy as much of humanity as she can in one stroke. That much is clear. But I also think she hopes this attack will draw me there. She knows I'm the only one with the ability to stop it – and she also knows this will cause me to die."
This news was greeted with startled silence.
"What happens to the Elekai, if you die?" Ashton asked.
"There will be no Xenomind to lead," the Wanderer said. "Our efforts would be directionless for a while, until a new leader rises to the mantle."
One by one, everyone's eyes turned on me. My face went white and my stomach twisted. I tried to speak, but no sound came. For some reason, the Wanderer's saying it out loud and everyone's looking at me made it worse.
But I had agreed. It was the only choice I had.
"I had hoped to better prepare you for this, Alex," the Wanderer said. "But time is not on our side."
"This...Releasing," Samuel said. "How long would it take? What's involved?"
"Land me on a tall building," the Wanderer said. "Somewhere I can project the wave-thought. Alex, also, must be present. Keep us safe for a few minutes. That's it."
"Wouldn't you two standing on top of a building make you targets for every dragon in L.A.?" Makara asked.
"Maybe with both spaceships, we can protect them long enough," Samuel said.
We all knew it was desperate. We all knew there was little hope for victory. But we had to fight and do whatever we could to save Los Angeles. And if we had to go down fighting, against all odds, then that was what we had to do.
"Let's go back," Samuel said.
Everyone looked at the Wanderer for final words.
"Good luck," he said at last. "Let us not make this our final stand."
***
IT WAS AN HOUR AND a half before the city came into view. Ashton piloted Perseus high in order to escape the Radaskim dragons' range. When he descended, we saw a large part of the already ruined city was in flames.
"Augustus," Ashton said. "Do you have a copy?"
It was some time before the Emperor answered.
"Where is your army?"
"We don't have time to explain. What's the tallest building in the city under your control?"
"What? Why?" Augustus asked.
"I said there was no time!" Ashton said. "We have a weapon that could destroy the Radaskim army. We need the tallest building we can to work with."
"That would be the Reaper HQ," Augustus said. "Carin Black and his men are making their final stand there."
"And your legions?"
"What's left of them are retreating to the sea."
"And you are in the Orion?"
"Yes," Augustus said. "Along with my Praetorians. Black is in Reaper HQ."
"Listen," Ashton said. "We'll need your help holding off the dragons. It'll take time for the weapon to be set up, but it will neutralize all the Radaskim invaders."
"And what is this weapon, Dr. Ashton?"
"Once again," Ashton said, "there's no time. You just have to trust me."
Augustus paused a bit before answering.
"Fine. We are very desperate, Ashton. I will let Carin know what you're doing so you don't get shot by his SAMs."
"Where is Reaper HQ?" Samuel asked.
"It's the U.S. Bank Tower, the tallest building in the city. There's a helipad."
"Make the arrangements with Black," Ashton said. "We'll be there in two minutes. And get your ass in the air. We'll need your help."
Augustus had no time to protest Ashton's gruff treatment. Ashton cut off and lowered the ship toward downtown. In the west, the sea was dark and empty, though fires and lights lit its shoreline – the remnants of the Novan legions. The streets below were dark and cluttered with rubble and flame. Swarms of movement plunged into the city.
The tall white tower approached. As we neared, human forms became discernable on its top. Reptilian screeches sounded from outside the ship.
I realized this could very much be our final battle. If we didn't win here, Askala and her hellish legions would win all of Earth.
In moments, the extinction of the human race would be decided.
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# Chapter 19
When Perseus landed on top of the tower, the Wanderer pressed the exit button. The door slid open and we headed out into the night.
After we made it to the tarmac, the boarding ramp retracted and the ship lifted off. I looked around the helipad, finding it empty. The ship turned to the north, where several dragons circled and dove. Explosions lit the night to the east. The screams of both man and monster carried on the wind.
The Wanderer made his way to the corner of the helipad. He climbed down a set of stairs, and stood above the building's precipice overlooking the southern and western portions of the city.
As the battle raged, I watched the dark ruins, the twisting highways, the distant hills to the west. It was hard to believe, standing there, that the Wanderer was about to give up his life. Before I could ask him if it was really true, he spoke.
"What I'm about to do will be my end, as we both know it. My spirit will return to the Elekai."
"So that's it?" I said. "You're going to die?"
The Wanderer gave his trademark mysterious smile – small, and a little sad. But his white eyes were bright with joy.
"I have died a thousand times, Alex. In the end, death is just the final journey."
"What do you mean?"
"Death is like waking up from a dream. Each time I die, I wake up on a new world, where the Elekai need me again."
"Well," I said, after a short pause, "that's not really how it works for us. When we die, we're gone. There's no waking up for us."
The Wanderer turned to me. In the distance, a dragon screamed and a few more took up its call. It reminded me of where I was, and that now was probably not the best time for a philosophical conversation. But the dragons were far, and I'd never see the Wanderer again. His eyes told me that what he was about to say was important.
"I think you will be surprised, when the time comes, as it must for all. Even the longest-lived Elekai must one day die. And when all goes silent, at the very end, we can be joyful that we had ever lived at all. Death isn't the end. It is an end as much as a beginning."
"What do you mean?"
"Some mysteries aren't meant to be spoken," the Wanderer said. "It ruins the magic."
The Wanderer closed his eyes, as if preparing himself. I felt a sudden panic rise in my chest. It was all going to end, and I would not have anyone to guide me on my final mission.
Before I could say anything more, the Wanderer's body stiffened. His white eyes glowed brightly, his arms outstretched, and a deep, resounding call emanated from his lungs, like a low, musical note that should have been impossible for his human vocal cords to produce. It made the very building beneath me shake.
Instantly, every dragon in the vicinity angled toward the tower. At first I counted ten, then twenty. As more materialized from the darkness, flapping their wings to fly toward the building, all the Blighters in the streets below quieted. A few gunshots sounded, but soon they, too, dimmed to nothing.
And still, the call went out. The Wanderer did not pause for breath. The dragons bore down on us, their white eyes fiery with violence. More rose from the streets below until at least fifty dragons were headed our way.
The forerunner of the dragons was only seconds away, coming at us full speed.
The Wanderer's call ceased. The dragons hovered in midair, waiting for...something. The Wanderer's form was completely still, as if he had turned to stone.
Then, in a multitude of screeches, the dragons turned away, diving once again for the Los Angeles streets.
I watched over the edge of the building as they swooped between buildings. High-pitched shrieks sounded as the dragons began ripping into the Blighters that advanced past the long wall. Both Perseus and Orion moved among the dragons, not firing on them, recognizing that the Wanderer had done his work and that the dragons were now on our side – at least for the moment.
The Wanderer fell to one knee. I grabbed him by the shoulders. His body was hunched and pained.
"Wanderer," I said. I felt weird calling him that; it seemed as if I should know his proper name by now. For all I knew, Wanderer was his proper name.
He looked at me. His eyes were fading – their power had been drained.
"It is done," he managed. "You know what you must do. There is strength inside that you do not even know...your own hidden power."
He fell forward as I watched with horror. He didn't move, and I couldn't believe.
"No..."
As the dragons above the city continued to scream and attack Askala's swarm, and as Perseus and Orion lent their aid from the air, I knew that the Wanderer was really gone. The entirety of his spirit had taken control of the dragons.
I reached for my radio.
"This is Alex," I said. "It's...done. The Wanderer's dead."
It was a moment before Ashton answered.
"Copy that. The dragons are turning on the horde right now. We're coming to pick you up."
"Got that. I'll...I'll be here."
I stood there, the cold wind whipping at my face. I looked down at the Wanderer's lifeless body. He stared upward, his eyes dark and no longer white. The only movement was the wind ruffling his worn, brown robe.
I was so drained of emotion that it was hard to feel anything. I knelt beside the Wanderer's body, touching his shoulder. I gave him a slight shake, in the vain hope that he might still be alive.
But it wasn't to be. It was then that the reality of the Wanderer's death hit me in full force. My throat tightened as hopelessness welled up within. I hated this feeling; it reminded me too much of losing my father, losing Khloe. I didn't know what I was going to do without the Wanderer. He was the only one who knew how to fight the war against the Radaskim.
Now, there was only me, and I definitely wasn't cut out for this.
The Wanderer had told me that I was supposed to lead the Elekai, and that was too much for a sixteen-year-old kid who didn't know what the hell he was doing. The Wanderer had the wisdom of the Elekai collected over eons, where I had only sixteen years, most of which was spent living a sheltered existence underground. If there was anyone who couldn't do this, it was me.
The last thing he said was to go back to the Elekai. That meant returning to the Xenolith. I didn't know what was going to happen there, but I wasn't really looking forward to it.
I couldn't be plain, boring Alex anymore. I thought of who the Wanderer was. He was a vast reservoir of knowledge with access to all the memories of an entire alien race. What would that do to me? Would I still be the same person?
Landing lights flashed above, but I lay sprawled on the rooftop right next to the Wanderer. He had saved the city, but the price was now clear: it was all on me, now. If I couldn't do it, then no one could.
I probably looked as dead as the Wanderer by the time Perseus touched down. The building vibrated upon the ship's landing. The engine gave out its deep hum, and the wind blew almost violently. The screams of dragons carried in the air.
The ship's drive idled. I waited a moment longer, not ready to get up and face reality.
I felt a hand on my shoulder. "Alex!"
I turned and saw Anna's face. I shielded my eyes because the landing lights were so bright and pointing right at me. She held me in her arms, pulling me into her embrace.
"I thought you were dead," she said.
She pulled back, looking me in the eyes. "What's wrong?"
I couldn't look at her. Not directly. "I don't know what to do, with him gone..."
Anna's eyes went over to the Wanderer. By now, Makara, Samuel, and Julian stood nearby.
"We need to return him to the Elekai," I said. "As soon as possible."
"Alright," Makara said. "I can let Augustus know. He can hold Carin in check until we get back."
"What about those dragons, though?" Julian asked. "What if they...turn?"
Makara sighed. "I don't know. But Alex is right. We need to get the Wanderer back home as soon as possible. Maybe...maybe he really isn't dead. Maybe if that stuff could heal Anna, it can heal him as well."
I wasn't holding out much hope for any miracle. The Wanderer had been pretty clear this was the end for him. Some things you just couldn't come back from.
"We need to get moving," I said. "The Wanderer wanted to make me leader of the Elekai. I don't know what's involved in that...but it's something we should do sooner rather than later. I need to get back to the Xenolith."
"Let's get on board, then," Samuel said.
I stood. Everyone's focus went to the Wanderer. His form was still. A truth came to me that must have haunted many people throughout the ages: no matter how much the world changed, death and its finality would not.
Everyone helped me lift his body. It was light with all five of us bearing it. We walked to the ship, as if in a dream. I didn't know what I was doing, and I still didn't know how long I had left to be me, Alex Keener, or how long I had left to be alive.
The clock was counting down on both.
***
WE STOOD IN Perseus's wardroom. The clinic's bed had been placed in the center of the hold, and the Wanderer was laid on the bed. His eyes, now brown, stared at the ceiling, lifeless. His hood had fallen, and his long, gray hair spread out, some of it tumbling over the bedside. His weathered face spoke the story of a man who had seen much. His form was thin – very thin. If I hadn't had known better, I would have said this man had died from his long years, not battle.
Several of the hardened Raiders wiped tears from their eyes. Both Makara's and Anna's eyes were wet with the tears, and Samuel's face was grave.
It was up to me to say something about the Wanderer and his final moments.
I cleared my throat, and everyone looked at me. There was nothing I could say to give justice to what he had done.
"No one really knew who the Wanderer was," I said. "I mean, before he became the Wanderer. We first met him in a cave, on our way to the Great Blight for the first time. This was about three months ago. There, he gave us prophecies of what was to come – things we were supposed to keep hidden until the end."
I paused a moment to collect my thoughts.
"There were five of us, then. Lisa – who I'm sure you know – was the first to have hers fulfilled. The Wanderer told her that she would have to give it her all when she thought it was over. And she did. In Bunker One, she saved our lives, even when she was dying.
"Samuel's came next. We had just met with Emperor Augustus for the first time, and Samuel struggled over whether we should join up with him, or go our own way. He was reminded of what the Wanderer told him: that he had to stay true to himself if we were to succeed. Samuel decided not to join Augustus at that time, and it was that decision that led Makara to form the New Angels."
Makara looked at me, knowing her part came next.
"With Samuel's decision, Makara became leader of the New Angels. She knew what her first action would be, because of what the Wanderer had told her. She was to seek those who were lost, which meant finding the Exiles. This led to the formation of the New Angels. We were a fractured group, only held together because of a strong leader."
Here, everyone was nodding their agreement. Makara's face turned slightly red; she didn't really like to have attention like this, but I felt like I owed it to her for all she had done.
"There are still two prophecies left," I said. "And they will be fulfilled, in time. The Wanderer told me it all hinged on me. It was only later that I learned what my part was.
"I guess what I'm trying to say is, that prophecy doesn't apply just to me. We've all played a big part in this struggle to save our planet. We've saved each other's lives, and every good deed has counted. I can't count how many times I would have died, if it hadn't been for Makara, Anna, Samuel, Julian, and countless others. We're all in this together, and the Wanderer told me that we had to unite, if we were to survive."
Again, everyone nodded in agreement. Many looked at the Wanderer as they contemplated his words.
"The Wanderer died to give us a chance to make it to the final battle. After much bloodshed, a lot of which was needless, we're finally at a point where we're all on the same page. The best way to honor the Wanderer is to stand together. I believe we are going to win this war. I can't explain how I know that, but I do."
I went quiet, not really sure what else to say. After a moment, Makara nodded.
"Thank you, Alex. In an hour or so, we'll be touching down in the Great Blight. We're going to bring the Wanderer to the Xenolith. And Alex is right – everything is coming to a head. Just be ready. In the next day or two, we'll start planning the final attack."
When Makara was finished speaking, Ashton went back to the bridge. I looked at the Wanderer one last time before following Ashton.
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# Chapter 20
An hour later, we had landed at the Xenolith. It was three in the morning – after the constant activity of Bunker 84 and the battle of Los Angeles, we were, somehow, still going.
When we disembarked, everyone came, including the Raiders. We walked across the fungus to the opening of the Xenolith. Makara, Samuel, Anna, and I all carried the Wanderer's body. I realized, as we crossed the threshold and into the spiraling tunnel, that the four people carrying him were the four about which he had prophesied.
We made our way down the glowing corridor, and it wasn't long before we stood before the glittering pink ichor. As Ashton looked around, awestruck, I could tell he could probably live down here just to unlock the Elekai's secrets. The ichor was empty of dragons – they had gone elsewhere.
I had no idea what came next. For some reason, I thought I would know what to do as soon as we entered. But the place seemed empty, so all I had to go on was my own intuition.
"Let's set him by the shore," I said.
We advanced toward the edge of the lake, softly setting the Wanderer down, face-up, where the xenofungus sloped downward to meet the ichor. We backed away a few steps.
"Do we just leave him here?" Makara asked.
Her voice seemed to trigger a reaction from the lake. The surface near the Wanderer's feet seemed to reach for him, stretching away from the rest of the ichor. It covered first his feet, and then the rest of this body.
"Back," Samuel said.
As we stepped back, the ichor covered the entirety of the Wanderer's form, pulling him into its main body. He still faced upward, just below the surface of the pink, translucent liquid. His long, gray hair fanned out, floating.
And then...the Wanderer's form faded. His image became less clear as the surrounding liquid clouded. Anna stepped forward, as if to do something about it. I held her back.
"He's returning to the Elekai," I said. "I should have seen what he meant."
As the ichor continued to cloud, I explained.
"The Wanderer wanted to return to the Elekai – to literally return to them. His memories will become part of the Elekai consciousness."
The nature of the pool became altogether clear to me. It was the wellspring from which all Elekai life sprung, whether it was xenofungus or xenodragon. Anything created by the Elekai had to come from here, just as anything that passed had to return.
Over time, the liquid became less murky. The sediment had either drifted to the bottom of the pool or dispersed throughout. Over the next minute, the lake's surface and depths gleamed once more, clear and smooth as glass.
I realized what I had to do. I turned to face everyone.
"I have to go in there," I said. "Whatever the Wanderer wanted me to do...it starts in that lake."
"Alex...?" Anna said.
I looked at her, taking her hands and looking her in the eyes. I didn't know what was going to happen in there, and I didn't know who I would be when I came out...Alex Keener, or the new Wanderer.
"Be careful," Anna said.
"Whatever happens," I said. "I love you."
I reached out and touched her face. Her eyes held back tears. I wished I didn't have to do this, but the Wanderer had told me it was what I had to do. Still, it was hard to turn from Anna and walk forward.
Everyone was quiet as my first boot was covered by the sticky ichor, then my second. I kept walking forward. Once the ichor was up to my shins, it seeped inside my boots. The warmth rushed around my socks, soaking through. Still, I strode forward, until the ichor was up to my torso, until I stood in the spot the Wanderer had been mere minutes before. It felt strange knowing that his body, his memories, were now part of the pool I stood in.
I turned to see Anna, Samuel, Makara, and Julian standing on the shoreline, shoulder-to-shoulder. Ashton stood a few feet off, and the Raiders stared, probably unsure what to think. I turned back to the pool and swam out further.
That was when the ichor pulled me under.
Panic rose in my chest, and still, I was pushed to the very bottom by a current. The pressure increased, and the thick warmth of the ichor was suffocating.
My panic subsided a bit when a familiar face appeared before me. The Wanderer stood on the lake bottom, as if he were standing in the free air and did not have thirty feet of liquid above him. I realized that this was a hallucination, from my lack of oxygen.
Breathe.
I had been holding my breath for well over a minute now. There was no way I was coming out of this pool unless I did as the Wanderer said.
But I needed air to breathe, not this liquid substitute. My lungs burned with the need for oxygen – my only hope was that there was enough in the liquid to help me survive.
So, I breathed.
The liquid seemed to squeeze into my lungs. My body immediately rejected the liquid, forcing it back out. But it sucked it in again, greedily – only to force it out once more. My body was caught between the need to breathe and the panic of having something that wasn't air in my lungs. Slowly, I became aware that my head had cleared, and the burning in my lungs had ebbed. My panic, slowly, settled into calm as my heart beat madly.
I accepted, instinctually, that I would have to breathe this liquid, at least for a little while.
I focused my attention before me. The Wanderer still stood just ten feet ahead, garbed in his usual brown robe. His white eyes shone brightly in the thick, warm liquid.
I couldn't speak, for obvious reasons. I breathed the liquid in and out, wondering whether the man who stood before me was flesh and bone, or merely a vision.
Are you real?
The thought crossed over through the liquid. The Wanderer stood, his form wavy.
Yes.
If he was alive, then it meant that he hadn't really died. He could still lead the Elekai.
Seeming to guess my thought, though, the Wanderer shook his head.
My time to walk this world has passed. The sacrifice, once made, cannot be undone.
How are you still here, then?
I will always be here, the Wanderer thought. I am the Elekai, now. And I will always be with you, wherever you go.
Do I have to lead, now?
The Wanderer nodded. The prophecy has come to pass. This is your time to shape the fate of the world.
It's...too much.
It always is, Alex. Even I didn't know what I was doing.
That's...comforting.
Isn't it? The Wanderer's thought entered my head, and the implication was that it should be comforting. It has always been a fool's quest. But we should be glad that the battle in Los Angeles was won. Askala is thwarted for the meantime, but in the coming days, she will have another army ready. We must use the time given to us to prepare the final attack, before she strikes again.
All I could think, as the Wanderer "spoke" to me, was why it had to be me, and why it couldn't have been someone far more qualified.
Why did you pick me? I asked. I'm not saying I won't try. I just have to wonder. Was there something different about me that made you decide that I was the one to do it?
There is one quality I have looked for in each Champion, the Wanderer said.
I was about to ask him what that quality was, when he spoke again.
You must think lightly of yourself, and deeply of the world.
What?
It was a saying, on a faraway world. Its name cannot be pronounced, nor understood, by any human. It was the first world attacked by the Radaskim, millions of years ago. And it was the first to fall. When the Radaskim first came there, they were greeted as gods. It wasn't long before that notion was dashed.
If all of those worlds have been conquered, then what hope do we have?
I thought I caught the Wanderer smiling. I had no idea how he could smile at something like that.
There never was much hope, Alex. Know this: it is much better to hope than to despair. The greatest things are accomplished by the tired and the discouraged, by the ones who don't give up. If they have no song to sing, they find one.
I couldn't speak, because the Wanderer had so silenced me. It had always been easy for me to get discouraged. It was hard not to, watching your world being swallowed before your eyes. Something Makara said a long time ago came back: you fought, even where there was no point in the fight. You never knew what tomorrow would bring.
Alright, I said. I see what you're saying.
The Wanderer raised an arm, as if in benediction.
The last battle comes. And remember: do not lose hope. You never know who's counting on you, or who's looking up to you. You have changed much, Alex. The Elekai are yours to command. And I believe that their powers are beyond even your imagining.
With that, the image of the Wanderer faded into the ichor.
Remember...though I have passed from this world, my words will always be with you. Never give up, even in the darkest moments. When you find yourself in a hole...look to the stars.
I stood there for a moment, my throat clenched, realizing that the Wanderer really was gone. His image had now passed; all that was left was the clear pink fluid. I almost wondered if he had been there at all, if all this had only taken place in my head.
Whatever the case, it was time to turn back. I swam to the shoreline, nearing the surface. When my head broke through, the air felt cool and light in comparison to the heavy liquid. I tried to breathe, but my lungs were full of the ichor. I coughed, spewing the liquid out of my lungs. It landed on the surface of the pool, quickly rejoining it. I sucked in a mouthful of air.
As I continued to hack, Anna strode into the pool. Makara shouted my name, while everyone else watched me worriedly.
My feet found the bottom, and I stood with the ichor up to my torso. Finally, my lungs seemed to be empty of it. I stood there, panting, feeling the liquid crawl off me and rejoin the surface of the lake.
Anna grabbed me, pulling me into a fierce hug.
"You were under there for at least five minutes," she said. "What the hell?"
"I breathed it," I said. "Whatever that stuff is, it works as good as air."
"Come on," she said.
Anna guided me back to shore. Once on dry ground, the liquid that clung to my skin and clothes flowed downward. I collapsed to the xenofungal bed, exhausted.
"Your eyes are white," Makara said.
"Huh," I said. That didn't really bother me anymore. "I talked to him. The Wanderer."
The murmuring of the Raiders quieted at this statement.
"He's alive down there?" Makara asked.
"No," I said. "I mean...I guess I really didn't talk to him, but maybe to his memory. He wants...he wants us to not give up hope, even now. I know it's easy to give up, but we can't. We have to keep fighting."
"Alex, you need to worry about getting some rest," Makara said. "Are we good to head back to the army?"
I shook my head. "No. There's one thing I have to do here."
"What's that?" Anna asked.
I stared at the pool, toward the inlets leading deeper into the cavern.
"I have to visit my army."
***
THE DRAGONS WERE GATHERED in one of the farthest caverns from the entrance. It was a lot of swimming to get there. While the rest remained onshore, everyone who could swim decided to come with me. This included Makara, Samuel, Anna, and Julian, along with several of the Raiders. Even Ashton had decided to come; he said he wanted to investigate the cavern more fully, along with experiencing the lake's vaunted regenerative properties. He said something about a Fountain of Youth.
The final cavern was the largest of all. From side to side it stretched for hundreds of feet. Perhaps it had been carved out by the Elekai. The walls and ceiling glowed silvery pink, filling the cavern with dim bioluminescence. The surface of the lake glowed, and on the far end where liquid met shoreline, the dragons waited – some dozen of them. Another dozen or so were in the lake itself, only their long necks and angular heads visible above the surface.
The dragons watched us enter their sanctum, their white eyes blazing fiery light. Their scales were various shades of red, dark crimson, all the way to a pink so light that it could almost be white. The dragons, of course, were large, though not as large as the typical Radaskim. The Elekai dragons' forms were sleek compared to their counterparts – the Radaskim were angular and brutal, as if they had been carved from stone, hewn only for war.
One of the dragons in the center gave a resounding call that echoed throughout the cavern. The call was taken up by the others, until a melodious song filled the entirety of the space. For the amount of creatures singing, it was not very loud; the fungus-lined walls probably had something to do with dampening the volume. For some reason, the song made me sad. I realized that the dragons were mourning the death of the Wanderer.
When we reached the shoreline, walking out of the lake, the song continued. I stood and watched the dragons as they paid tribute to their fallen master. I realized then that I had never seen the dragons outside the context of battle. At heart, they were peaceful creatures, and it was for the promise of peace that they fought.
When the song ended, only one of the dragons continued, letting his voice resound a moment longer than the others. It was Askal. He paused a moment, waiting for the sound to dissipate, before turning to me. Something about his face, and those eyes, seemed morose – there was no outward difference in appearance, but it was something I felt coming from him.
The Elekai did not express emotions the same way humans did. They could directly empathize with one another and literally feel one another's thoughts and feelings. As a human, I could only feel a bit of that. Being human had its limitations, but if I could have felt every bit of these creatures' souls, it probably would have been too much to bear.
Finally, Askal seemed to relax. I walked past several of the dragons while everyone stayed behind. I stood in front of Askal, craning my neck to look at his face, about thirty feet above. Askal backed away before lowering his head to bring it level with mine.
We meet again, little human.
I wasn't touching Askal, so I hadn't expected any thought to pass from him. The thought must have carried through the xenofungus. I understood little about how the Elekai communicated. At times, touch was necessary. Sometimes, the xenofungus was sufficient to bridge the gap. But then, there were the visions from the Wanderer, and my being able to dream and experience the world through Askal's eyes. Those were the most baffling, because neither touch nor xenofungus served as a medium. That would probably remain a mystery to me, but it seemed as if in those times that the communication was carried by the air itself.
We meet again, I said.
We have lost a dear friend today, Askal said. The Elekai will mourn the Father's loss for many cycles, until he is born again.
I didn't know what a cycle was, but I supposed it was a very long time.
"I know," I said. "I'll miss him, too."
I switched to speech; that way everyone who was gathering behind me could listen, too. Askal did not seem confused by the transition.
We have much to do, little human, Askal thought. Though I suppose you are not so little anymore.
"What are we going to do, Askal? Half the dragons are still hurt."
We must wait, a little while longer. We have to heal in both heart and flesh. It will be a few days before the Elekai fly again.
A few days. I didn't even know if we had that long. But there was no way we could win the battle without the Elekai's support. A few days would give us time to make sure things were taken care of in Los Angeles. It was time enough to come up with a strategy for the final battle.
"Will you guys be okay here for the next few days?"
I cannot guarantee our safety, Askal thought. But all the entrances will be closed until you return. If we are attacked, you will be the first to know, Lord of the Elekai.
Don't call me that, I thought.
Something like amusement came from Askal.
You and our Father are the same. He did not like being called that, either. But that didn't make it any less true.
Still... I thought. Just call me Alex.
Askal nodded his assent – a surprisingly human gesture.
"We probably need to get back to L.A.," I said, turning to the others. "The Elekai will be okay until we get back."
"Are you sure?" Makara asked.
"They can't leave their wounded behind. Askal says they'll be ready in a few days."
"Let's get moving, then," Ashton said.
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# Chapter 21
We went on Perseus and just slept. Ashton took us as far as Los Angeles and put the ship on hover mode. There was no way we could keep on going without getting some much-needed sleep.
I slept so deeply that it might as well have been a coma. I was sleep-deprived, not just from last night, but from every night before where I'd only gotten a few hours here and there. It wasn't until it was night of the next day – a full fourteen hours later – that I woke up, finding that I could sleep even more if I wanted to.
So I did. I kept sleeping until 03:30. No one in the ship was moving. I had no idea if Makara or the others were corresponding with Augustus and Black at this time. Probably. But sleep was hard to come by in this world, so I was going to take advantage while I could.
I finally got up and went to the wardroom, where various Raiders were sprawled on the floor, still sleeping. After almost a full day of doing nothing, it was hard to believe I was the only one up. My head pounded, and my eyes were heavy. I continued on to the wardroom and went to the faucet, where I filled my canteen with water. I drank the entire thing down.
After using the lav, I went to the bridge, finding the lights on. Ashton was awake. He turned upon hearing my approach. He held a mug of coffee in his right hand, from the top of which steam curled up.
"Morning," he said. "Would you like a cup?"
I nodded, sitting in the co-pilot's seat. Ashton took a thermos sitting on top of the dash. If that thing were to spill, it could do some serious damage to the electronics on the dash. Ashton didn't seem to be worried, though.
"You've got a mug?" Ashton asked.
I didn't really want to get back up, and was about to reach for my canteen, when my eyes came across a mug sitting on the dash. Anna must have left it there earlier.
"Right here," I said, grabbing it.
The mug was a relic of the old world, made of thick white ceramic.
Ashton poured the hot, black coffee into the mug. I could feel its heat radiate into my skin, and its rich aroma met my nostrils.
"Careful," he said. "It's hot."
I took a sip, and still the liquid scalded my tongue. It'd been a while since I had coffee.
Ashton and I just sat there, looking out the windshield. Clouds spread low and dark, covering the city. Stars blanketed the sky. I could never tire of watching them. The time showed that it was 03:36.
"Quite the show, isn't it?"
I nodded my agreement, taking another sip. My favorite part of watching the night sky was seeing the Milky Way – that thick band of purple, bluish stars, the entire arm of our galaxy. It amazed me to know that there was life out there. How many other worlds were there, like ours? The Wanderer had said there were over a thousand that had been conquered by the Radaskim. How many others were out there?
"How is everyone still asleep?" I asked. "It's been almost twenty-four hours since the Xenolith."
Ashton smiled. "If they had stayed awake any longer, they would not be asleep, but dead. The human body has its limits."
"Are you feeling any effects from the Fountain of Youth?"
Ashton chuckled. "Well, my bones didn't hurt so much this morning, getting out of bed. Might have been my imagination, though."
I laughed. "What do you think is in that stuff, anyway?"
Ashton shrugged. "You'd have a better idea than me."
"I think it's where all Elekai life is made," I said.
Ashton grunted, taking another sip of coffee. "Hard to imagine them all coming from there. Are you sure they don't reproduce on their own?"
I shrugged. "Don't know. I thought with the Wanderer gone, I'd somehow know a bit more than I do now. I guess not."
"Learning...really learning...is never free. Did you think you'd become Superman or something?"
"Who's Superman?"
Ashton looked at me blankly. "The world truly has ended."
I laughed. "I'm joking. Of course I know who Superman was."
"You scared me for a minute. But there will come a day kids don't know who Superman was."
"Maybe they'll have some new thing," I said. "Maybe Dragon Man."
"That's just...no."
"Alright, whatever. On that learning thing, though...maybe a lesson isn't learned until you accept it."
"Hm." Ashton took a long drink of coffee before refilling his mug from the thermos. Once done, he set the thermos back on the dash. "And have you accepted it? Your new job, I mean."
"I don't think I'll ever be ready to turn into an alien."
"Is that what you think the Wanderer was?"
Maybe the Wanderer had a human form, but he was definitely not human.
"I think he was a god of some sort," I said. "A Xenomind. Who can comprehend what that really is? I just want a few more days to be me. That's it. And that's not going to change as long as I can help it."
Ashton just stared out the dark windshield, at the stars.
"I don't see why he chose a young buck like you, instead of an old fart like me," he finally said. "What do I have to live for? I've done my living. I've had a long life, mostly full of sorrow. But some good moments, too. The Rock changed everything, kid. Those Radaskim ruined billions of lives at a single stroke. The lucky ones died. God, ain't that the truth. I must have wished a hundred times to have been one of them."
Ashton took another sip of coffee before continuing.
"But I had twenty years of the good life. I was young, like you, even though I'm the only one to remember it. Sometimes I doubt it, though. Was that really me? That kid, riding the bike, going camping, sneaking into movies? If no one else believes it happened, why should I?"
I smiled. "You're rambling, old man."
"Bah. You don't know enough to become a god."
"You do, though?"
Ashton chuckled. "Hell, no. Nor would I ever want to. But if someone has to do it...why not the old man?"
I shrugged. "I don't think I'd wish it on anyone."
"That's because you're too noble. What about the girl?"
"Anna? Why would I wish it on her?"
"No, not that. That's what being young is about. You're supposed to fall in love and do crazy, stupid things. Things that don't make any logical sense. As an old man, it's my job to shake my head at the stupid crap kids do. You don't do that stuff, though. Please, can you just do something stupid, so I can shake my fist at you?"
I stared at Ashton, blankly, not really sure what he was talking about.
"Bah. You don't know anything. All I'm trying to say is...when you're doing what the Wanderer told you to do, you're not just giving up your life. You're giving up your right to be young."
"I feel like being young was never a right," I said. "It was a privilege. And it ended when the door of Bunker 108 opened."
Ashton grunted. "Maybe so, Alex. Just because the world is unjust doesn't mean we have to sit back and accept it. You know...maybe Anna's right. Maybe there is another way we don't know about. Maybe..."
First Anna, now Ashton? I wondered who would be next.
"I don't think there is another way, Ashton. I wish there was, but people have a history of believing what they prefer to be true."
Ashton shook his head. "Maybe you aren't as young as I thought."
Ashton sounded kind of sad when he said that. His eyes were full of remembrance, as if he was thinking about when he was young, back when kids had the luxury of being kids.
"You'll never know," Ashton said. "You'll never know what it was like to go to school, to ask a girl to a dance, to ride your bike in neighborhood streets, to explore forests like they were some lost world that only you knew. To drive your car for the first time. College."
"Is that what kids did, back then?"
"It's what I did. I sometimes wonder if it was all just a dream."
"It wasn't," I said. "I've seen movies where that sort of stuff happened, so it had to have happened. Right?"
Ashton nodded. "Yeah. It did happen. You're still a kid, you know? But you don't act like it. None of you do. You are as much adults as anyone I knew. That's what scares me. It's not normal and never will be, in my mind."
"It's normal now," I said.
"What about Anna, though? Don't you love her? Would you go to the end and die, even if she didn't want it?"
That made me go quiet. Ashton had pinpointed something that had been troubling me greatly. He waited for my answer, and I had no idea what to say.
"I don't know what's going to happen," I said. "I don't want anything to happen. I...I do love her. Admitting that is hard, because I know what it means. I'm convinced that I'm the only one who can stop this. The Wanderer said that much. I know Anna will never be okay with that. I don't know if I am. But I see no choice. The other choice is the world ending, just as every world the Radaskim have invaded has ended. I can't let that happen. I mean, what would you do?"
Ashton sighed. "No matter how old you get, kid...there's always going to be questions that can never be answered. Sometimes, age just makes the questions all the more perplexing. I worked my ass off, raised my family, all to make the questions go away. I buried myself in work. And they always remained, those questions, haunting me like ghosts. Most people ignore them and just go on with their lives. That's what I tried to do. Your question, about what you should do...I don't have the answer. You don't have the answer. Even the goddamn Wanderer didn't have an answer. Didn't you ask him that, when you and Anna went to go see him? What did he say?"
I smiled. "A whole lot of nothing."
"See? If a god doesn't know, doesn't that tell you something? It all rests on you to figure out the rest of the riddle. Whatever happens, I wouldn't blame you either way. When you go to the heart of darkness, when you are facing down Askala, it's just going to be you and her. And you're not going to know anything more than you know now."
"Thanks for reminding me."
"Some answers just don't come, kid, no matter how much you ask them, no matter how much you think about them. Take it from an old man." He held up the thermos. "More coffee?"
Ashton refilled my cup, and we stared out into the starry night. After a while, he turned off the lights, so that we could see the stars more fully. The stars were bright before, but now they were magnificent.
"The Wanderer said something," I said. "He said you were supposed to look to the stars when you were in a hole. That's all I feel I can do. I have to hope, even if there's no reason for it. People need hope like the lungs need air."
Ashton slurped on his coffee. "Aye, you're right about that. What are you hoping for, Alex?"
I took a sip of coffee, and thought for a moment. "I'm just hoping I don't mess anything up. I hope, when the end comes, I can figure out how to save my scrawny hide. I hope I can find a way to make Anna happy, and take away all her worry and pain."
Ashton said nothing – he only listened.
"I know," I said. "There are some things you can mess up, and some things you just can't. There are the doors you can't come back from."
I looked at the clock on the dash. It was 04:00.
I stood to leave the bridge. "Thanks for the coffee."
"Where you off to?"
"Breakfast."
I just couldn't talk about what I had to do anymore. I'd have plenty of time to think about that in the coming days.
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# Chapter 22
We landed in the streets of downtown Los Angeles. Morning had just dawned, casting the eastern clouds a dull red.
We left the ship, stepping out into the cool, dry air, finding the streets a chaotic mess of rubble, ash, and purple blood. Corpses of dead crawlers lined the streets, some still twitching. Severed limbs and spindly legs were scattered among the refuse. Behind our landing site, the once-white U.S. Bank Tower rose into the morning air.
"Augustus is in there with the Reapers now," Makara said.
"Is that where we're headed?" I asked.
She nodded. "Yeah."
Everyone walked toward the tower, picking our way through rubble and monster remains. Radaskim dragons swooped the sky above, from time to time, still placid. It seemed as if the creatures had turned to our side for good from the Wanderer's sacrifice – it wasn't just temporary. Or at least, that was what I hoped. They could switch back at any moment. If they did, there wasn't much we could do about it.
We entered the wide glass doors of the building, finding ourselves in a dim lobby. There, Augustus and Black stood face-to-face, deliberating in the center of the lobby floor. Praetorians lined up behind Augustus, while Black's elite guards backed him up. The Reapers, however, were without weapons. It looked as if Augustus was having his way with Los Angeles after all.
Char stood next to me and spoke in his low voice.
"Come on, kid. These talks will be long."
He led me out from the lobby of the building, stepping through the open door and into the red-lit street outside. I wondered what he wanted to talk about.
The formed-up legions parted for us as we headed east down rubble-strewn streets. Parts of the tall buildings of downtown had crumbled, forming piles of debris. Many buildings' sides were wasted and black, the result of a fire long ago. Even the sides of the white tower had been charred, though it seemed to have mostly escaped the damage. Sand covered much of the street. In places, asphalt and faded yellow paint showed through the dirt.
Random streets and intersections had been blocked off – either in preparation for the Radaskim's attack, or from some earlier time. Beneath a pile of rubble lay some large bones from a creature I couldn't identify –they'd been there for a while. Toward the east, brown mountains walled in the Los Angeles Basin. Their tops were lost to thick red clouds.
A few blocks to the east, an enormous dragon wheeled above some buildings, settling on the precipice of a crumbled freeway. Another dragon screeched in the air, settling nearby the first. The large one – almost as big as Chaos had been – was dark gray, almost black. The smaller dragon was crimson in hue. Their long, serpent-like necks turned toward us.
"What are we going to do about these dragons?"
That was something we had to figure out. "We might want to get closer."
"It's like they're waiting for something," Char said.
"Or someone," I said. "Maybe they're looking for the Wanderer."
"The Wanderer's gone. They're looking for you, kid."
"I know," I said. "Somehow, this is working differently than we expected. The dragons have stayed on our side."
"They could turn any second," Char said. "That's what the Wanderer said, isn't it?"
Yeah, that was true.
"Alex!"
The sound of footsteps came from behind. I turned to see Anna sliding to a stop, kicking up a cloud of dust.
"Couldn't wait for me, could you?" she asked.
I shrugged. "You snooze, you lose."
She forced a frown. "I don't think that saying really works in this situation."
"You know what I mean."
Char just watched both of us, shaking his head.
"Whatever," Anna said. Her eyes gazed ahead to the dragons. "I want to see one of those things up close. Plus, you need your bodyguard."
"I don't need a bodyguard," I said.
"I wasn't talking to you."
Before I could respond, Char cut in.
"Shall we?"
We turned back to the dragons perched on the edge of the freeway. They continued to watch.
We walked forward once more. The dragons swarmed the sky, circling above the eastern side of the decayed cityscape. They seemed to be gathering above the large, dark dragon.
"I wonder what they're doing," Anna asked.
Char watched as well, narrowing his eyes to see into the distance. "It sure is something."
"It's like..." Anna paused mid-sentence. "It's like this world isn't ours anymore, you know?"
Anna was right. Even if we somehow defeated Askala and the Radaskim were stopped, against all odds, I had no idea what the future would look like. The Elekai would live among us, I supposed. The Old World would crumble completely to dust, leaving humanity to live among an alien race. We'd probably never be able to rebuild – not to the level of the Old World. I imagined that all traces of that world would be buried and gone, in a few centuries. I'd be long dead by then, and so would everyone else. But it was the future we fought for – the only future we knew.
***
WE REACHED THE OVERPASS. The dragons remained still, so still that they could have been confused for statues. Up close, however, there was no way these creatures were anything but real. Sharp talons enclosed the concrete railing on the freeway's side, which was riddled with cracks and caked with dust. The dragons' scales shimmered in the dull red sunlight, crimson for the smaller one, and dark gray, almost black, for the large one. From those scales rose sharp spikes from the creatures' backs and tails.
These creatures of conquest and brutality were under the command of the Elekai. Which meant that I could control them.
Or, so the theory went.
As Char and I stood, staring, I definitely had my doubts. It seemed impossible that I could control these beasts – especially the big one. Those whitened eyes stared at me with such hatred – especially from the dark one.
I guessed there was nothing to do but try.
"I need to get up close," I said.
"There's an on-ramp that way," Char said, pointing to our left.
We walked in that direction as the dragons looked on.
"This is crazy," Char said.
"I know," I said. "Crazy times call for crazy measures."
Char grunted in response.
Anna walked on my other side. She seemed tense, and it was probably all she could do to not draw her katana. A blade wouldn't do much good against a dragon.
We reached the on-ramp. It was a short climb to the battered highway above. The charred shell of a sedan blocked the on-ramp horizontally. Flecks of turquoise paint clung to its chassis.
We went around the car and reached the top of the highway, turning to where the dragons roosted. Their heads swiveled upon stalk-like necks toward our position. Even with dusty distance, their white eyes glowed. Though these creatures were on our side, just seeing them stare at me sent a chill down my spine. There was something different about these dragons. I could feel the violence boiling within them. It was hard to imagine these dragons getting along with the Elekai ones, especially when their allegiance might only be temporary.
Anna, at last, unsheathed her blade. The dragons gave no reaction. More dragons wheeled above, a menacing foreground against the dilapidated towers of downtown Los Angeles. A chilly wind gusted, stinging with dry cold. About twelve dragons flew above our heads like vultures. I wondered if this was some sort of trap; were they pretending to be friendly, only to deliver a death blow? The possibility, or perhaps, the probability, didn't escape my mind.
But we had come too far. Char drew his handgun, keeping it pointed at the pavement. The dragons continued to circle, some of them dropping lower.
At last, we stood in front of the two dragons, so much larger now that we were close. They towered above, the dark gray one twice as tall as any Elekai dragon. Scars and crevices crisscrossed scales that appeared to be as hard as stone. Their long tails, fitted with cruel spikes, shifted back and forth, stirring dust from the surface of the freeway. A large, especially deadly spike jutted from the tip of each of those tails, sharper than any human blade. The white eyes of the dark dragon narrowed, as if in challenge. I focused on those eyes. I knew this was a test of dominance and will; I'd lose them if I looked away first. Those eyes were white, abyssal depths. Haunting emptiness.
Unlike the Elekai, these were creatures without souls. They were organic robots obeying the will of their true master: Askala.
For now, that link had been severed by the Wanderer's sacrifice. If it hadn't, these dragons would have killed me a long time ago.
I saw the Wanderer in my mind's eye, and calmness permeated my entire being. Wild, tumultuous power radiated from these dragons. They had hearts of chaos.
Tame them, the Wanderer's voice intoned.
I focused a single thought to push against that madness. That thought came in the form of two words.
Your names.
The thought pushed against the anarchic, powerful stream of consciousness that blared from the Radaskim dragons' minds. They regarded me with cold, reptilian eyes.
The smaller, crimson one linked to me first, begrudgingly giving his name.
Mordium.
The dark dragon waited a long time, before the force of my will linked him to me.
Quietus.
A chill passed over me. Feeling the emotions of the Radaskim was so much different than with the Elekai. Sharing thoughts with Askal seemed to give energy. This drained it. They would obey me, yes – but only because they were forced. Whether they could find a way to override the Wanderer's Releasing, I couldn't say.
Mordium, I thought. Quietus. You are both Elekai, now.
Immediately, twin hisses escaped from their serpentine mouths. Anna and Char jumped back from either side of me. I remained still, attempting to affirm my control over these creatures. The dragons above circled lower. Something like a cruel smile came to Quietus's thin, reptile lips – but really, it was more a baring of teeth. Those teeth were cracked, decaying, sharp.
Who are you to command us, puny mortal? Quietus thought. We are the Radaskim, hearts of chaos. Eternity lies ahead, and behind – we have drunk full of its emptiness. The Void calls us home, in the end. The Radaskim have known this, from the dawn of time until the dusk of all things. Like twin mirrors, the light ever reflects, unending, unyielding. But what can you know of these things, you of sixteen years, when my spirit has endured more than sixteen million?
Even as he spoke such terrible things, I knew them to be true.
I am Quietus, bane of a thousand worlds. The screams of races reside within my soul, of which your puny kind are no match. You have killed the Lord Chaos, no small feat – but you will find that many of Askala's children are his equal, and that some are far more terrible. You challenge me, mortal, and seek to rule that which is eternal?
I seek only the victory of the Elekai, I thought. And you will help me secure it.
Something like glee danced in those white eyes.
Will I, now? Curious that the old fool gave up his spirit for this slight boon. For in so doing, he sealed his doom and that of the cursed Elekai. Never again shall that trickster arise. Though there be thousands of worlds yet to smite, never shall he walk them again – should you fail, little one. That he placed trust in you, least of all races, dirtiest, foulest, most reviled! Had we not come to destroy you, you would have destroyed yourselves.
I stood stunned at Quietus's revelation. Mordium looked on in silence, his eyes as empty as the vacuum of space. On the sides of the highways, more dragons settled, watching the proceedings.
The Elekai were first, I thought in response. And the Elekai will be last.
The Elekai are weak! The Elekai plant the gardens; the Radaskim reap them. Such is the order of the Universe. In order to create, you must first destroy. This is the heart of truth: that existence is a lie, and we no more than a thought, the destruction of universes infinite. All things shall pass, all courses have been set from the beginning for ruin. There is no use in fighting this inevitability. The only control we have is in sweet destruction. Destruction of all, destruction of the lie that is existence! In the end, nothing shall bloom, no star shall shine, and darkness deep shall stretch eternal. No laughter, no tears, nor even memories shall be preserved. That is the fate of all things, little Elekai, if the Radaskim are not permitted to destroy. We are the gods, and Ragnarok our judgment. And like all others, humanity shall enter our thrall.
The cold wind blew, as if in time to Quietus's dark thoughts. I stood still, not wanting to think about them. No thought crossed from Quietus for a long while.
Anna and Char watched the dragon from my either side, ready to run or fight at any moment.
Is this how you attacked every world? I thought. With a meteor?
Quietus answered. The method is different, for each world. Different diseases necessitate different cures. And the pestilence of your race runs deep.
What makes you hate us so?
Death is my name. Quietus. The End of all Things. Yes. I was created by my Mother to hate, and that hatred I have nursed for uncounted eons, mostly in darkness. You know not the darkness of space, being trapped with your thoughts for thousands of years, with nothing but the fell whispers of the Dark Mother – such imprisonment I have endured between Reapings. Death brings succor, and I drink deeply of its draft – though I am never permitted to embrace it in full. How I long for that death, little Elekai. But she will not let me have it, not until all fades to silence.
You can be free, I said. It has been said, in order to know hate, one must first know love.
I have debated this vile thought for years untold, Quietus answered. And there is only nothing. You seek, and you do not find. Everything is a lie. There is only...nothing...the ultimate truth.
Something like a sigh escaped the dragon – a strange sound to come from a creature so large, so brutal. It sounded...pained. Tired.
What is truth, Elekai, but your conception of it? I have searched the depths of your heart and know that you hold no conviction. A being of no conviction is but a leaf in the wind, having no control over its course. It goes where the wind wills. What do you believe, young Elekai? I see you hope that I might be changed. Hope is but a dream, a dream best abandoned before it blinds with false brightness.
Hope does not always lead to a false end.
There is no fighting the Radaskim, Quietus thought. On countless worlds they brood, and this world is just one among thousands. Four hundred years hence, they shall return. That was the duration of our last stasis – four hundred long years. From the world of Tar'Sha'Lak we sailed the Void, four hundred years ago. The Dark Mother may be killed, but never shall the Radaskim fall. We are too many. A Second Reaping shall come, far more terrible than the first...
Four hundred years, little Elekai. Are you ready?
I sensed no lie in Quietus's words. I only felt a numb, cold shock overtake me. What Quietus said was true. Even if we won, in four hundred years, they would return from the nearest Radaskim world.
For a long time I stood there, until Anna touched my arm.
"What is it? What's wrong?"
It was a while before I could bring myself to answer.
"I know when Xenofall is coming."
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# Chapter 23
A long silence passed. I explained nothing, and Quietus said nothing further, feeling his work complete. His eyes, once again, were empty of thought or emotion. He had said his piece, and now only waited for my reaction.
"I thought..." Anna began. "Wasn't this Xenofall? Wasn't Xenofall us fighting Askala and winning the war?"
I had to explain everything to Anna and Char. It wasn't easy; most of what Quietus told me was thoroughly depressing. I stressed that Quietus had not lied to me. Though the dragons railed against it with all their spirit, I was in control and they couldn't disobey a direct command – at least until Askala had found a way to reclaim them.
Then again, the truth they revealed could be misleading, and the outlook Quietus had painted was incredibly bleak.
"So if we defeat Askala," Anna said, "They'll just come back again. In four hundred years?"
I nodded.
"That means," Char said. "If we win, it's not really a victory. It just delays the inevitable."
"That's right," I said. "Even with four hundred years, it's hard to imagine how we could prepare for something like another invasion. Even if we could, would it really be enough?"
"So they attack," Anna said, "over and over, until they finally win."
Char shook his head. His blue eyes seemed angry, more than anything.
"Would've been nice if the old man mentioned something about that," he said. "He must have known."
"I don't know if he did," I said. "Or maybe he didn't want us to think the fight was hopeless. Maybe, even with Xenofall...there's a chance we can still win."
They both looked at me, as if I had that answer.
"I'm not saying I know what it is," I said. "But four hundred years is a lot of time to search for an answer."
Anna shook her head. "All of this is just so hard to imagine. So, the Radaskim are on that planet, and it's apparently four hundred years' travel time away?"
I nodded. "That's the gist. He said that world was called Tar'Sha'Lak."
"Right," Anna said, wincing at the weird name. "This makes the battle so much bigger. So much more...impossible. I'm starting to realize how tiny we are."
"The Wanderer had faith in us, that we would find the answer," I said. "Maybe...maybe we're a lot closer than we think."
Once again, Char and Anna looked at me.
"What would that answer be, Alex?" Anna asked.
I shook my head, and sighed. "I don't know. I'm still thinking."
"Maybe he knows," Anna said, pointing at Quietus. "You said that dragon will tell you anything, and that he can't disobey a direct order. Get him talking."
I nodded. "It's worth a shot. But...it's draining."
Anna grabbed my hand. "I'll be right here. If you need to break off the connection...do what you have to do."
I nodded. I turned back to Quietus, who met me with those chilling white eyes.
It was time to find out more.
***
CAN XENOFALL BE STOPPED?
Something like amusement came from the dragon.
Stopped? Do not waste your thoughts on this, little human. You will not even reach Askala, who dwells in the depths of Ragnarok Crater.
There has to be a way, I thought. The Wanderer would not have died if there wasn't a way.
You understand nothing, Quietus said. We are but thoughts that take mortal shape. We never truly die, until our thoughts die. Your kind has not learned this. If you are a thought, you can live forever.
You mean with the xenofungus, I said. Your memory, your being, returns there.
Yes, Quietus thought. You begin to understand now. You humans thought to do this with machines, but life is so much more complicated than machinery – infinitely more complex, beautiful and meticulous. Life gives endless possibilities for growth, for nurturing, for evolution – machines are hard lines, angles, inflexibility.
Your life is like a machine, I thought.
It is, in its essence, Quietus thought. All life is. To destroy us, you would have to destroy that which could never be destroyed: a thought. Even if all the xenofungus on this world were destroyed, the information is saved on countless other worlds. Thought-waves emanate outward, communicating, storing our beings for eternity.
Maybe you have no hope, I thought. But I do. I don't believe the Wanderer would die for nothing. That's why you're going to help me.
And what is your command?
You and your dragons are to come with me. Immediately.
Quietus cocked his head. You mean...to ride me?
Yes, I thought. Is that a problem?
Yes. The problem is that I should be killing you. Only, I cannot.
I shook my head. You're going to meet the Elekai.
Quietus hissed. I continued to stare, until at last, Quietus relented.
It shall be as you say.
I turned from Quietus. Anna and Char awaited the report.
"We're going to the Great Blight," I said.
"Now?" Anna asked.
I nodded. "You and me. Char, you can go back to Makara and the others and let them know what we've learned. When they're done here, tell them to meet me at the Xenolith."
"What are we going to do in the Great Blight?" Anna asked.
"Two things," I said. "One, the Radaskim will hopefully be fully converted to our side by the Elekai dragons. And two...I need answers."
"What answers? Do you think there's a way to stop this?"
"I don't know," I said. "But I have to think beyond the Wanderer. Maybe the Elekai know something."
"You mean the dragons?"
"The Wanderer spoke of songs crossing between the stars, a while ago. Perhaps there is something in them – some secret that will help me understand what I'm supposed to do."
Anna touched my arm. "Alex...I don't understand any of that, but if you think there might be answers...I'll come with you."
"I don't know anything either, Anna. Maybe Quietus is right. Maybe it's hopeless. But it's not hopeless until we give up. The Wanderer believed there was hope, and that's all I need."
"Alright," Char said. "I'm not sure about this, Alex. Just...don't go doing anything stupid. Though I'm afraid you already are."
"This is what I have to do," I said.
"Maybe so," he said. "But did you have to drag her into it?"
"We'll be fine," Anna said.
Char hugged Anna before backing away. It was time for us to depart.
I turned to Quietus. At a wordless command, the large Radaskim knelt to the ground to be mounted. Anna and I carefully climbed Quietus's right haunch and settled on his back. His back was a little rougher than Askal's, but not unbearably so. We were squeezed pretty tight between two spikes. Quietus raised his head, letting out a horrible roar. We were so high off the tarmac; Quietus was much bigger than any of the Elekai dragons.
"Do we have to do this?" Anna asked.
The other dragons watched, awaiting my command.
"We need to," I said. "I have to get these dragons to the Elekai. I don't trust them to make it on their own. My power might lessen with distance."
"Maybe," Anna said.
She wrapped her arms around me, and I gave the order.
Go.
Almost violently, Quietus shot into the sky. He gave two mighty flaps, and we were flying upward. I held on tightly so as not to fall, and Anna held tightly to me. Askal would never have lifted with such power, threatening to shake us off. It was as if Quietus was finding every opportunity to defy me.
As the ground dropped away, the other dragons lifted, forming a V behind Quietus. I realized that Quietus was the leader of the Radaskim dragons, now that Chaos was gone. If I controlled Quietus, I could control those who followed him.
Or at least, that was what I hoped.
***
THE WASTED, BLIGHTED landscape passed below over the next few hours. The air was frigid until we were over the xenofungus, where warm updrafts took the edge off the cold. Even so, without pause, it wasn't until nightfall that the Elekai Xenolith came into view.
Anna still held onto me, her head resting on my shoulder. She had been sleeping, or tying to sleep, for the past two hours now.
There it is, I said. Land nearby.
You do not want us to enter by the Dragons' Way?
The Dragons' Way?
I will show you. You have much to learn.
The Xenolith grew in size as we neared. Quietus flew around the Xenolith until we were on its opposite side.
Command it, human.
I thought for a moment. He wanted me to open the way into the Xenolith.
Open.
Below, the xenofungal floor began to open. Narrow at first, just a sliver of darkness. But the gap widened, until the entire surface of the fungus had pulled back to reveal a pink, iridescent glow, which I knew to be the surface of the lake.
Immediately, Quietus dipped below, stretching his wings as he descended. His dragons copied his movements almost exactly. The entrance rose to meet us. We sunk below the surface of the Great Blight; the air became muggy and thick from the humidity of the ichorous lake.
Quietus glided above the surface, settling on a far, empty shoreline. The other dragons, with flaps of wings, settled nearby, forming a line. They faced the pool itself. Above, the opening closed.
Anna and I hopped down from Quietus, stretching our stiff legs. All I could think about was how hungry I was and how much I had to pee. We'd been riding that thing for almost twelve hours straight.
"I don't even care if this is holy ground or whatever," Anna said. "Or if there are twenty dragons watching me."
Toward our left, the shoreline bent around and was hidden by the wall of the cavern.
"You might get some privacy over there," I said.
Anna went off while I waited with the Radaskim.
This place is vile, Quietus thought.
It's your new home, I said. You better get used to it.
Quietus gave his trademark hiss, but thought nothing more.
Mordium, the crimson dragon, just stared with baleful, empty eyes. The rest of the dragons gazed across the surface of the lake, their thoughts unreadable.
Their stench fills this place, Quietus thought, his lower lip curling.
Anna returned from around the bend, and I decided it was my turn to go. I told her to stay where she was; I didn't want her too close to the dragons while they were out of my sight. She stopped, and I went around the bend and relieved myself after twelve hours.
When I came back, the dragons were pretty much as I'd left them. The cavern was empty of the Elekai. They had to know that we were here, so either they would come, or we would have to go find them.
I stepped into the liquid, and closed my eyes.
Askal.
From somewhere in the cavern, I felt Askal's consciousness respond.
You bring great evil into our home.
They will become Elekai, I said. The Wanderer purchased them with his death.
Hmm. Even those as dark as these?
Yes, I said. They have not come to realize it.
Their minds are dark, Askal thought. As is their intent. If we appear, it will go badly.
They will not disobey a direct command from me, I said. Is there a way we can make them turn faster?
They must decide for themselves, Askal said. The Wanderer has opened the path to their redemption. It is up to them to continue down that path.
Then they'll never turn, I said. They truly hate everything. They have no hope and find joy only in killing.
Many things that seem impossible actually aren't, Askal thought.
Can't I just order them into the pool?
Willing obedience is better than forced obedience, Askal though. Do you think we began as Elekai?
The Wanderer changed you?
He helped us...remember.
Remember what?
Who we truly were.
That really didn't help me. I certainly didn't know anything more than the Wanderer, and I was expected to just step up and do his job.
It's hard to make them remember when I don't even know what it is they're supposed to remember.
You came here to listen, did you not? The Eternal Song still resounds in the stars.
How do I listen?
You must quiet yourself, until there is nothing. Then a Voice will fill the silence...many voices. When you hear them, you will know. And make sure they also hear. The Song is the core of our being. Hearing it will help them remember.
I didn't really understand exactly what Askal meant, but I decided to give it a try.
"What were you doing?" Anna asked.
"Talking to Askal. He says that I have to listen to the song of the Elekai."
"And then you'll know what to do?"
I paused. "I really don't know. I have to start there, though."
I closed my eyes, and listened. All I heard were the sounds around me – a dragon padding his foot on the fungus, the drip of liquid in the cavern. It became so quiet that I thought I could hear even the fungus growing. Something about that fungus drew me – I felt its consciousness fill my mind. The feeling was...overwhelming. I realized that this – the collective consciousness of the fungus – was the beating heart of the Elekai. Here were stored all the memories, thoughts, and instincts that guided a race. All the dead who had passed. All the spirits that awaited rebirth. Information continued to be added in a never-ending stream, from the stars themselves...
I realized that this was the Eternal Song of the Elekai.
Now in tune with the fungus, I turned my attention skyward – and I heard it. I could never interpret this information myself, or translate it into words. The fungus was the intermediary, and only through its vast intelligence could I begin to comprehend the soul of the Elekai beaming down from the heavens. There were Elekai, not just on this world, but on many others. There were also worlds that should have been, and worlds now dead. But still, every world had its own Song – all blended with perfect majesty of sadness and joy. It was a Song, never-ending, that I could never tire of hearing.
And I understood, in that moment, why the Radaskim could not win.
I conveyed the song to the Radaskim dragons. It was as if some ripple of energy coursed through them, paralyzing them. They realized the truth, just as I did, that the Radaskim's quest for the Secrets of Creation could never come to pass. They sought to acquire the consciousness of every race in the universe, and even if successful, they would still come up short.
Because the Elekai held the Secrets. And the Elekai could never be conquered.
The truth was in the xenovirus itself. The dominant gene was Elekai, the recessive was Radaskim. A recessive gene could never countermand the dominant. The Radaskim could never conquer the Elekai, so long as the Elekai version of the xenovirus was in existence – and if it wasn't in existence, that meant the Elekai were gone from the universe, along with the Secrets.
What these Secrets were, I could never comprehend – something older and deeper than the universe itself, woven into the very fabric of space-time. And, somehow, the Elekai were the key to unlocking it – a key the Radaskim could never acquire.
All of this information passed in a single moment from me to the Radaskim. Their entire philosophy – their religion – existed so that they could find those Secrets to recreate the universe. And here those secrets were, all along, living within themselves, the only place they could never find them.
Unless they became Elekai.
We will do as you say, Alex, Quietus answered at last. The Song speaks a truth higher than any I have ever known. This day, we become Elekai!
With that pronouncement, Quietus gave a terrifying roar, echoed by the rest of the dragons. He stepped forward toward the pool, setting one foot inside at first. The ichor surrounded it, but Quietus did not hesitate, only pressing on until the liquid reached the tips of his wings and rose up to his breast. Then, his long neck descended and plunged into the surface. The liquid frothed and churned about him, becoming cloudy. I realized it was cleansing him, converting the Radaskim virus that formed the essence of his being.
Next, other dragons stepped forward, entering the pool.
"What's happening?" Anna asked.
"They are becoming Elekai," I said.
"It worked!"
"They realized the truth."
Anna turned to me. "What's that?"
"The Secrets of Creation – the Secrets that can destroy and recreate the universe – they are known only by the Elekai. And the Elekai can never be conquered."
"How can they not be conquered?" Anna asked. "Haven't they been conquered on every world so far?"
"I mean, yes, they can be physically destroyed. I was referring more to the xenovirus itself. The Elekai version can't be conquered by the Radaskim version."
"Because it's dominant, right?"
I nodded. "The only way a Radaskim can learn the Secrets is if they become Elekai. But doing so defeats the purpose of being Radaskim. If the Elekai ever die out – the Secrets go with them. That is why they can't be conquered."
"And what are those Secrets?"
I shook my head. "You kidding me? I don't know. I don't think anyone does. It's so deep that it can only be understood by the entire collective consciousness of the Elekai, and maybe not even then. Maybe that's the Elekai's purpose: they only know the Secrets because they are its guardians. They plant the gardens, that they might understand them more. But maybe with all the gardens in the universe, it wouldn't be enough."
Anna was quiet, thoughtful. "So, what you're telling me is that there is no way the Radaskim can win?"
I nodded. "That's the way I understand it. I guess...this is how I turn Askala. She must listen to the Song and learn the truth as well."
"They can still kill us," Anna said. "Maybe the Elekai will win, in the end, but we're still just caught in the middle of their war. It doesn't stop us from dying."
"That's true," I said. "And it doesn't stop Xenofall from coming."
"I guess all we can do," Anna said, "is stop Askala and try to figure out the rest later."
Knowing that the Secrets were safe with the Elekai didn't change anything about what I had to do. Like the Wanderer, I had to take control of Askala, probably by giving up my life, just as the Wanderer had. But like the Radaskim dragons, would that only open the door to her redemption, or would it save her entirely? That answer remained unclear.
Quietus emerged from the pool, walking onto the shoreline. His form dripped pink ichor. He didn't look any different, besides the eyes. They were still white, but not so empty.
I am...Elekai.
The other dragons emerged from the pool. As they lined up along the shoreline, Askal's voice resounded in my head.
Well done.
I don't feel like I did much.
We will show our new sisters our home, and they will learn all there is to know about being Elekai.
Wait. Sisters?
Yes – did you not know? They are the daughters of Askala. We are the sons of Elekim. But the dragons are all of one kind.
All this time, I'd thought of the Radaskim as males.
Elekim? He is...
Yes. He was the Wanderer. But now, you are Elekim, little human.
I didn't know how I felt about that. There was no way I could fill the Wanderer's shoes. I did not know as much, and to know as much as he did would change who I was. I didn't want to change, but it was inevitable.
Your friends are coming, Askal thought. Enter the pool, and it will carry you to meet them.
"It's time to go," I said to Anna. "The dragons have found their home."
She nodded. "We're done then?"
"Yeah. We've done all we could."
The current bore us away from the dragons, taking us through the Elekai caverns. We twisted through tunnels and openings, until at last we reached the familiar shoreline, the one with the tunnel that would lead to the exit. We left the ichorous lake behind and strode up the spiral tunnel, until we reached the door, enclosed with the roots of the Xenolith. The roots pulled back upon my touch, revealing the glowing alien fields that shone in the night.
We stepped into the warm air, to find Perseus parked about a hundred feet away. A moment after our exit from the Xenolith, people streamed off board to meet us. I couldn't discern who they were in the darkness.
At last, we met halfway there. It was Ashton, Makara, Samuel, Julian, and Char.
"The dragons?" Makara asked.
"They are Elekai, now."
"Bunker 84?" Anna asked.
"Michael, Lauren, and Ruth are keeping it safe," Makara said. "Everything seems to have actually...stabilized."
This had never been the case. We had been running like rats in the maze, just trying to reach the end. We had arrived at a place of relative peace, and the only thing left to be done was...
"It's almost time, isn't it?"
Ashton nodded. "We will have the final council on Skyhome. As soon as we are ready...the attack will begin."
It wasn't long, now. In a few days, it would all be over. Our last gamble. Finally, we had everyone on our side – Reapers, the Empire, and the New Angels. We even had the strongest of the Radaskim dragons converted to the Elekai cause. All that remained was the final battle, and that could only be decided at Ragnarok Crater.
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# Chapter 24
Perseus lifted off a few minutes later. Our course was Skyhome, which we hadn't seen since we left for the Empire what seemed ages ago.
Over the next few days, every leader of the Wasteland resistance would be gathered here. Meanwhile, the people in Bunker 84 would be relocated to Los Angeles.
I would be out of action for a few days – Anna and I were to wait here as the others worked to transport Bunker 84's residents to Los Angeles. Makara said I needed the rest, but really, who didn't? All this was just to keep me out of action, safe for a short while. When I pointed this out to Anna, she actually agreed with Makara.
It was only for a couple of days, so I agreed. Besides, the break would be nice.
By the second day, though, I was starting to get restless. I ran around the Outer Ring, at distances and speeds far greater than a few months ago. I felt I could run forever, if I wanted to. Anna worked out with me as well. We did pretty much everything together. Anna was basically a glorified babysitter, making sure I didn't get too bored or restless before everyone else arrived.
I'd often stand in the observation deck in the Outer Ring. Surrounded by the manifold plants and crops that grew there, I'd watch Earth pass by. Each time I saw North America, it was the same story. The Great Blight had grown to dominate almost all of the Continental United States. Before, it had just been the central U.S. – but now that pink fungus stretched from sea to shining sea. Just a thin sliver of coastline remained free of Blight, on the eastern and western seaboards, and also along the Gulf Coast. I wondered if there was something about the sea that stopped the xenofungus.
"There you are."
I turned to see Anna approach. She had just showered, and her long black hair was damp.
"Looking for me?"
"Maybe."
She smiled and came to stand by me. We watched the Earth pass in silence. The cloud cover was too thick for me to tell where it was.
"Is this where you go to contemplate the end of the world?" she asked.
"I thought the world already ended."
"You know what I mean."
I grabbed her hands, pulling her close. There was a bench, facing the windows for Earth-viewing. I led her there, and we sat. While I sat, she leaned against me, resting her head on my shoulder.
This was the first real chance we had to be together without worrying about survival. We had these two days, at least, before...
"Don't talk about it," Anna said.
"How did you know I was...?"
"I felt your body tense up," she said. "Just...don't talk about it. If I could, we'd go down there."
She pointed, where the port gave a view of a dazzling, azure sea, pockmarked with green islands.
"The tropics?" I asked.
"Somewhere that's not so cold," she said. "That was my dream; after all this ended, we could get everyone to move somewhere far away from the Wasteland. Somewhere safe, somewhere beautiful, so we could forget the ugliness we grew up with. Maybe, with time, we could all heal." She sighed. "Everything changed when that Wanderer came around."
"You really don't like him, do you?"
"I know it's not his fault. He just had the misfortune of giving bad news." She sighed. "Here we are, talking about it."
"It's alright," I said.
She leaned further into me. I wished there was a way for me to take her worry and put it on myself. I didn't have that power, though.
I caressed her shoulder, and she closed her eyes. We sat like this, for a while, long enough to watch the Earth pass three times. When the third rotation was over, I touched Anna on the shoulder.
"Hey."
"Hmm?"
I waited a moment, to make sure she was really listening.
"I know we said not to talk about it. But...you'll still be there with me, at the end?"
She turned to look at me. "What are you thinking? Of course I will. Even if I die."
I felt a chill at those words. Yes, this was our final battle. People would die, most likely. But Anna...I couldn't let her die. I would do anything to keep her alive.
"Just...I don't want you to die," I said. "There's no reason for you to."
"Alex, I'm going with you, and that's final. If you don't stay alive until the end, what's the point of any of this?"
It was a good question, one which I couldn't answer. Finally, I nodded.
"Alright. We'll both go."
"And anyone else who wants to," Anna said. "We'll need everyone down there if we're to make it."
I pulled Anna upward, until she was sitting in my lap. She faced me, wrapping her legs around my torso, keeping her eyes on me.
"We have a few days, Alex," she said. "Let's enjoy life, while it's here to be enjoyed."
She kissed me, deeply and passionately. I held her close.
"Anna..." I said.
She pulled back, looking at me. "What?"
The Outer Ring turned, and once more, the Earth came into view. There were no clouds. Only a shining blue sea.
"I..."
I was about to ask her a question that I'd been thinking about for a while – something that had only been in my dreams, since it could never be reality. But if I only had a few days left of life, then I needed to ask her.
Maybe it could be reality. If I could live in a dream, let it be for a few more days.
"Anna...this is going to sound crazy, but..."
I paused. I couldn't stop now. I couldn't deny what I felt, despite everything we had to go through.
"I want you to marry me."
Her jaw dropped open. She looked at me, in disbelief of what I had just asked. My heart pounded; I was unbelieving myself. Had I really just opened myself up to look like a fool like this?
I waited for her to say no, that it was the stupidest thing that anyone had ever asked her.
"I..." she started. She still looked into my eyes. "Are you serious?"
"Anna," I said. "I need you by my side. I need you as my wife."
"Alex, how is that supposed to work? We'd be married for less than a week!"
"I can only tell you what I feel. I love you – deeply. I...want to be married to you. I want to be close to you, as close as I can ever be."
Anna just looked at me, still in shock.
"I would say yes, but...how could we be married for just a few days? Doesn't that just make it more painful?"
"I understand if you say no. It's...asking a lot."
Anna grew quiet. She didn't remove herself. She laid her head on my shoulder, and held me.
"I love you, Alex. Even...even if this doesn't happen, I just want you to know that. You've sort of dropped a bomb on me. You can't expect me to answer without thinking about it."
"I know," I said.
She pulled back, looking into my eyes. I knew she would say no, in that moment. She was distancing herself from me. She would still help me, in the end, but why would she add to that pain of it by saying yes?
"Alex...let's do it."
I looked up at her, almost in disbelief. I wasn't sure if I'd heard her right.
"What? Really?"
"Love is enough, isn't it?" she asked. "Who cares about anything else? Maybe...maybe this could have a happy ending. I know it's stupid to think that, but if I'm in love, I have a good excuse to be stupid."
"It's not stupid. I think."
"So, we're really going through with this?"
I nodded. "I'm serious."
She smiled. "Okay. When?"
"I'll let everyone know when they get up here. We'll set up something small."
She laughed. "They'll think we're crazy."
"Except that we are crazy. I'm serious, though."
Anna smiled. "I know. I am, too."
"Good."
"Well...now that we're engaged, what should we do?"
I shrugged. "Don't know. I'm kind of hungry."
"Ugh. That's an Alex answer."
"Aren't those the best kind?"
"Maybe."
"Are you saying you're not hungry?"
Her brow furrowed, as if in deep thought. "Yeah. I guess I am."
Anna didn't move. She held on to me tightly, as if pinning me to the bench.
"But give me a kiss, first."
I pulled her close, pecking her on the lips.
"Like you mean it. Like you want to marry me."
I kissed her deeply, strongly.
"That's more like it," Anna said. "I guess we can go eat, now."
"After that, we'll let everyone know."
***
WE LET THE CREW KNOW about our decision later that day when they updated us on how things were going below.
"You what?" Makara asked.
"It's true," Anna said. "We want to get married."
Shocked silence was the only answer from the other end. I had no idea what they were thinking.
"When?" Samuel asked.
"We were thinking as soon as you guys got up here," I said. "It wouldn't have to be anything major, so that it wouldn't distract from the mission..."
"We'll do it," Makara said. "It would be good for morale, and a great way for everyone to let off a little steam."
Anna and I looked at each other. She was holding back laughter from the way Makara put it.
"One word," Michael said. "Party."
"Michael!" I said. "Didn't you realize you were there."
"Now you do, I guess," Michael said. "Congrats, buddy! How'd you pop the question?"
I was thrilled that everyone was so enthusiastic about it. I thought they'd think I was crazy.
"It's been something I've been thinking about for a while," I said. "We were just watching the world pass by and the time seemed right."
"Where'd you find a ring?" Julian asked.
"Er...what?"
"Tell me you got her a ring, Alex."
Was I missing something? Getting a ring seemed a little old-fashioned. The only rings you could get these days were off the fingers of dead people, of whom Jade was a case in point. I definitely wasn't going to give Anna one of that greaseball's rings.
"Don't listen to them." It was Ruth. "It's the commitment that counts."
"Yeah, I don't really care, either," Anna said. "It didn't even cross my mind."
"We'll whip something up for you, buddy," Michael said. "Don't worry about a thing."
I was honored, and humbled, that everyone was already celebrating. I didn't think it would be that big a deal, but apparently, it was. Everyone seemed truly happy about it, which was more than I could wish for.
"I just have to say that you are an inspiration, Alex," Julian said.
One person who hadn't yet spoken was Ashton; neither had Char. Maybe they weren't present. Voices buzzed in the background, and I couldn't really discern what they were saying through the speakers.
"How much longer until you guys get here?" I asked.
"We should be another day," Makara said. "And don't worry. We'll take care of everything."
I almost felt tears come to my eyes at their level of commitment.
"Thanks. I can't say that enough."
"Don't worry about it," Michael said. "We'll do this first thing. Like Makara said, we need something to celebrate. We've been working way too hard!"
"We can find some beer in Los Angeles," Samuel said. "Or maybe Augustus has some wine we can use."
This was becoming real. It was a bit overwhelming.
"Where are we having it?" Julian asked. "I know by New America, the beach is not so far away."
"Yuck," Makara said. "I've seen that water, and it looks like a giant Behemoth took a dump in it."
"This is my homeland you speak of," Julian said.
"Just telling you how it is," Makara said.
"And where would you suggest, Makara?"
"Easy," Makara said. "Why not Baja California? It's beautiful, with that beach and those waves, with water so blue that you wouldn't even believe it's real. Besides, it's warm, and not far from Nova Roma. Augustus might be willing to part with some food and beverages. For the sake of morale."
Makara was right. The one time we did go there, to refill Odin, Baja was very beautiful. It was also where I'd kissed Anna for the first time, and told her how I felt.
"That's perfect," I said.
"I feel the same way," Anna said. "I thought we were just going to do it here."
"Skyhome?" Makara asked. "No way. There's nothing romantic about that place."
"I don't know," Samuel said. "You're sort of among the stars, and all that greenery in the Outer Ring? It's like something out of a dream."
"I hope you're kidding."
"They could get married in front of the viewing deck," Samuel went on, "and kiss right when the Earth passes!"
"Well," Makara said, "it's a good thing you're not planning this wedding, brother."
"I like Baja," Michael said. "It should be warm enough if we go far enough south, like Makara said. The beach is a classic choice."
Everyone murmured, apparently agreeing with Michael's decision. I didn't know how he nominated himself for the wedding planner position. It seemed like it should be more of a female role, but I trusted Michael to know what he was doing.
"Gus might be able to help out with flowers and stuff as well," Makara said.
"Let's keep it simple," I said. "Let's just see what we can throw together in a couple of days and go from there."
"Alex, my lad," Michael said. "It takes many weeks to plan a wedding."
"Really? How long did yours take?"
Michael went quiet. "I don't know. Five minutes, I guess. This is different, though. Everyone's eye will be on you, and it has to be perfect."
"Michael," Anna said, suppressing a laugh, "I've never seen this side of you before. I think you've missed your calling."
"We'll see what we can cook up," Michael said. "We've got the location down, and that's half the battle."
The others talked amongst themselves for a while, hammering out details. I heard Julian say something about getting Augustus on the line.
"Look," I said. "We have to go."
"Wait, we have questions to ask," Michael said. "Who's invited?"
"Anyone who wants to come!" I said. "This is your baby, now."
"We need music," Michael said. "And food."
"New America can help with both!" Julian said. "My sister plays the guitar and sings. And she has friends that play guitar as well."
"I'm glad," I said. "Let us know what you guys come up with."
Anna took me by the arm. "Alex, I know this isn't your thing, but we're going to need your input. Okay?"
I knew Anna was right. I nodded. "Alright."
For the next hour or two, we planned. Food. Music. Invitees. We could use the ship's galleys to cook, as well as make fire pits for roasted pork. If Augustus agreed, we could get some wine, food, and flowers from him. Maybe even a dress for Anna.
The only thing that wasn't mentioned was that this marriage could only last a few days – and from the way they were throwing together this celebration, it was as if they planned on its lasting a lifetime.
When most everything had been settled, Michael spoke.
"Alright. All we have to do is get Augustus on the line and confirm that we can use his resources. We'll have to get everything to the wedding site. And there are cooks to find, people to raise the pavilion, chairs, tables..."
"It sounds like you'll be busy," Anna said.
"All of us will," Julian said. "There's so much to do."
I was grateful to have friends like these, who not only supported my decision, but were going to do everything they possibly could to make it unforgettable. The most important part, though, was that I was going to marry the girl I loved. Even if it wasn't going to be forever, we were showing the world that it mattered. That we mattered, even if our mission demanded that we make the ultimate sacrifice. It would be the last thing people expected, to defy the tragedy of our situation.
The fact that we would all die made the living all the more important. Life was sacred because it was so easily extinguished; it was also sacred because of its uncanny ability to endure, no matter what. We had the power to be happy, even in the midst of pain. We could celebrate, even if there was no real reason.
We celebrated because we were alive, because that was what we did.
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# Chapter 25
The arrangements with Augustus were made, and he agreed to provide all the requested items, on one condition: that he, his wife, and his children could be invited to the wedding. I would have wanted him there, anyway; he had proved a friend, even after our original distrust.
However one broke it down, I was glad Augustus was on our side. Maybe he still wanted to control everything, but in the end, Augustus was Augustus, and the Wasteland would probably be better off in his hands, anyway.
I was in the process of getting ready. There was less than an hour left, and I was getting nervous. Michael had trimmed my hair aboard the Perseus; Anna was getting ready on Orion. Thankfully, Michael left some length so that I wouldn't be totally unrecognizable.
I looked in the mirror in the lavatory one last time. I was shaven, my hair trimmed, and clean. Even a healthy tan remained from my time in Nova Roma. I had to admit, Michael had made me as handsome as I'd probably ever be. There was a maturity to my face that had been lacking three months ago.
"Not bad," Michael said. "You clean up good."
"Thanks."
I sat on my stool, staring at the brown clumps of hair that had fallen on the deck. Michael began to sweep them up, throwing them in a nearby garbage can. Once he was done, he brushed his hands together.
"Let's get ready," he said. "Ceremony's getting close."
I went to my cabin, dressing myself in the uniform laid out for me. I had protested at first, because I really wasn't a soldier, and didn't feel like I'd earned the right to dress like one. Michael told me that I was more soldier than a lot of soldiers he knew, and that put an end to the argument.
In fact, all my groomsmen – Michael, Samuel, and Julian – were wearing the same uniform as I was: we found the military clothing in the dresser aboard Perseus. Michael was my best man. I would have been happy with any of them, really, but Michael had done so much to help out that he deserved the spot.
The uniforms were dusty and wrinkled from lack of use, but Augustus put some of his party staff on the job, and had them crisp and clean in a matter of minutes. The uniforms were Air Force blue, both the pants and the three-buttoned jacket, and with a matching dark blue tie. Ashton said the Air Force uniforms were fitting, given how much we had flown around, though he also said real airmen wouldn't have lost over sixty percent of the United States Space Fleet.
Michael and I got dressed, and I felt nothing but uncomfortable in the uniform. I figured, though, if Anna was wearing a dress, I'd look a little off if I didn't dress up as well. Apparently, Augustus had given her an amazing white dress, so people probably wouldn't be looking at me much, anyway.
Finally, Michael handed me some shiny, black shoes. I slipped them on.
"Alright, stand up straight," Michael said.
I stood off my stool. Michael held out a small mirror, where I looked at my reflection again.
"You've come a long way, kid."
I turned to see Char standing in the cabin's doorway.
"Everyone's made it?" I asked.
Char nodded. "Yeah. People are getting seated right now. Me..." He smiled. "I get the honor of escorting the bride."
"Have you seen her?"
Char chuckled. "That's bad luck, kid. Did no one tell you that?"
"Yeah," I said. "Think I may have heard that somewhere. What about Samuel? Julian?"
Samuel and Julian appeared behind Char, dressed and looking snazzy.
"So, we ready?" Michael asked.
Samuel and Julian nodded, but they were watching me.
"You have the ring?" I asked.
Michael nodded. "Yeah. Still in my pocket."
"Let me see it."
I didn't know why, but I wanted to make sure it was really there. The ring was also a gift from Augustus, which he got from one of the best jewelers in Nova Roma. Augustus had gone through so much trouble and had given so much that I had no idea how to thank the man.
The ring itself was probably the fanciest and most beautiful piece of jewelry I'd ever seen. It was a product of the Old World. It was hard to imagine anyone having the ability to craft such a magnificent piece, even in the Empire. The band was platinum, twisting and swirling intricately, and the top had a large diamond, which must have cost a fortune. Smaller diamonds were set in the platinum swirls.
"It looks incredible," Julian said. "So shiny."
I felt like the wedding expenditures had gotten out of proportion, but it had all been others' idea. Anna and I loved each other, and that was what mattered. If all of our friends wanted to celebrate in style, who were we to stop them?
"Alright," I said, handing the ring back to Michael. "I'm ready."
***
I WAITED ON THE CONSTRUCTED wooden stage. Behind me, the waves crashed against the shoreline. We were set up in the exact location where we had rested all those months ago. Down the beach, at water's edge, a huge cliff rose, against which waves crashed. Two sections of seating faced toward me, a lane of sand running in-between them. Ashton and I were the only ones standing here above. He would be conducting the ceremony, since he was the closest thing we had to a preacher. Which was a scary thought, if you thought about it.
Familiar faces filled the crowd: there were Marcus, Grudge, and several other Raiders and Exiles I recognized. Augustus was sitting up front, as we had promised, along with his wife, the Empress, and his youngest son, who was probably ten years old. The Empress was a beautiful woman, tall, regal, with long, dark hair, and dressed in a flowing, green gown.
The fact that such a diversity of people had gathered was a symbol: we were no longer divided. We were all on the same side, and we were going to celebrate together, fight together, and even die together.
The sky was a perfect blue, like out of a storybook, and puffs of white clouds trailed above. The air was slightly cool, but perfect. I wasn't too hot in my uniform, which had been my fear.
My heart began to race when, at long last, a stream of people exited a large pavilion set up on the green hill rising above the beach. That line flowed downward, including all my friends. First came Callie, with a basket of flowers, followed by the groomsmen and bridesmaids, set in two lines, after which came a beautiful girl in a white dress that dazzled in the sunlight, still too distant to make out the details. Next to Anna walked a man who must have been Char, also in an airman's uniform.
The line advanced slowly toward the seating. My eyes kept trying to find Anna, but she was mostly hidden behind the others. Julian was escorting Makara, Samuel was escorting Ruth, while Michael escorted Lauren. Makara, of course, was Anna's maid-of-honor. Their first encounter, three months ago, had consisted of Anna's placing a blade at Makara's throat. It was funny how things had worked out.
Standing in front of everyone, with a basket filled with white and yellow rosebuds, was Callie. When the two guitar players from New America began to play, one strumming and one picking, she walked forward, flowers streaming and falling in the wind. When she reached the end of the lane, she was grabbed by Ada before she could cause any trouble, as six-year-olds were wont to do. Ada pulled Callie close, sitting next to Deborah and some of the other former Community women.
As the music continued, Julian and Makara walked forward, arm-in-arm. Makara wore a light blue dress that shimmered like deep sapphire. Augustus had taken pains to provide all the women in the wedding these beautiful blue dresses, chosen to reflect the color of the Pacific Ocean. Makara was striking in her dress, and until this day, I could have never imagined her wearing one. She looked radiant.
As Makara and Julian fanned out, each taking their place on either side of the stage, Samuel and Ruth walked up next. Samuel looked strong and handsome in his uniform. Ruth, also, was striking in her blue dress, and her wavy blonde hair tumbled downward and her blue eyes shone brightly.
Last of all came Michael, escorting Lauren. They both smiled up at me. As they parted, Michael winked before taking his place beside me.
Everyone in the wedding party now stood on the stage – everyone but Anna. With Michael and Lauren now on stage, Anna was revealed. She was so beautiful that my heart nearly stopped. Her dress shimmered, almost silvery in the sunlight. Everyone drew their breath at how lovely she was. The dress itself was bedecked with colorful sequins that gave contrast to the brilliant white. Augustus, clearly, had spared no expense, and this wedding dress would have been fitting for an Empress.
But the true beauty was Anna herself, and everyone knew it. Her green eyes found mine, and she gave a small smile. There was nothing in the world but that face, those eyes looking into mine.
The guitar players transitioned into a new song – one which I could not name, but was instantly familiar to me. I'd heard it in movies I'd watched back in Bunker 108. I only knew it was the wedding song.
As the music started, Anna strode forward. My eyes were so focused on her that I didn't even notice Char escorting her. His blue eyes dazzled as I shifted my focus to him. As with Michael, he gave a sly wink.
Before he let Anna go, he said something to her. She nodded before he helped her up the steps.
Char retreated to his chair, sitting next to Augustus on the front row.
Anna stood facing me. Looking at her face, I could almost forget how nervous I was. We held hands and looked into each other's eyes.
Ashton stepped forward to begin the proceedings. I had no idea what he had prepared.
"Dearly beloved," he said, loud enough to be heard by everyone, "we're gathered here today to see two young people join together in matrimony, and..." He paused a moment, shaking his head. "I feel like these things are in danger of going on too long, especially when there's food to eat and booze to drink. But more importantly than that, there's love to celebrate."
Ashton motioned to Michael. Michael moved forward, handing me the ring. I held the ring, turning to face Anna.
"In accordance with the happy couple's wishes, I'm going to keep this short. So, Alex...repeat after me."
He said the words, which I repeated: "I, Alexander Lorin Keener, take you, Anna Bliss, to be my wife, to love and to cherish, in good times and in bad, in sickness and in health, in life and in death, from this day forward. I give to you this ring, as a symbol of my constancy and devotion, from now until the end of my days. With this ring, I thee wed."
I slipped the ring on her left ring finger.
"And you, Anna," Ashton said, voice thick. "Repeat after me."
Anna said the same words, and as she did so, a tear came down her face. I knew what she was thinking. We would only have a few days to enjoy this. But at least the world would know that we had loved each other.
Anna slipped my ring on: a thick gold band that was unadorned, but classic. We continued holding hands as Ashton continued.
"You have said your vows. You have exchanged your rings. There is nothing left to separate you from becoming man and wife."
Ashton turned to me. "Alex. Do you take Anna to be your wife, to hold and to cherish, to love and to protect, for the rest of your days?"
"I do."
Ashton waited a moment before turning to Anna.
"Anna...do you take Alex to be your husband, to love and protect, to hold and cherish, for the rest of your days?"
She smiled. "I do."
Ashton waited a moment, before breaking into a smile. "That's it. You're husband and wife. Alex, you may kiss the bride."
I smiled, lifting Anna's veil. She smiled as I leaned forward to kiss her.
The crowd applauded, and I probably kissed her a little longer than I was supposed to, but as far as mistakes go, I guessed that one was okay to make.
I held out my arm, which she took, and escorted her down the lane toward the green hill.
We waited at the base of the hill, where all of our friends came to congratulate us.
I looked at Anna.
"Well. We're married."
"Yeah," she said. "It's weird, huh?"
"I guess so."
From the sea, thunder boomed. I turned to see dark gray clouds on the horizon.
"That's not a good sign," I said.
"Quit being superstitious," Anna said.
"We should get to the pavilion," I said. "They'll have the food ready soon."
Anna smiled. "Always thinking about food."
"I know you are, too," I said. "Don't lie."
"Come on," Michael said. "Let's get up there before we get drenched!"
The storm was fast advancing, and already, a few fat drops were falling and the wind had picked up. We vacated the beach, where the waves had increased in size, and made our way up the hill.
Hopefully, during the meal, the pavilion would stay standing.
***
BY THE TIME THE STORM came, everyone had gathered inside the pavilion. The wind and rain buffeted against its westward side, but not so hard that the pavilion was in any danger of collapsing. In fact, the pavilion's interior was cozy, filled with a rosy, red light from hanging lamps. The food had been cooked and prepared beforehand, overseen by none other than Mayor Herbert Shaw of New America. He and several of his residents had roasted pork and vegetables, as well as baked some flatbread. From the farms of Skyhome had come fruits and salads, and from the Empire itself Augustus had a cake prepared, which I was now seeing for the first time. It was three-tiered and covered with thick, white frosting.
The cake, along with the food, was set on a long banquet table in the center of the pavilion. Tables and chairs were set up around various tables inside the tent. No less than two vats of wine had been brought along with the cake, from Augustus's own cellar. With the rings, the dresses, the food, and the wine, there was no way we could ever repay him. Still, I knew all this stuff had barely made a dent in his vast wealth.
I had the opportunity to meet Augustus' wife, the Empress of Nova Roma, along with the Royal Heir, whose name was Hadrian. She congratulated me on my wedding and said that she and her husband had been looking forward to it.
When everyone was seated, Ashton stood and whistled, getting everyone's attention.
"We'll go by table, starting with the bride and groom, and work our way around," he said. "Let's eat!"
There was general cheering and applause. We filled our plates with food and our cups with wine, and dug into our meal. The food was delicious. Even as it rained and thundered outside, we laughed and celebrated for what seemed the first time in my life. I was so happy, maybe aided a bit by the wine (which Anna made me stop drinking by my second cup). There wasn't enough happiness in the world, and it was great that there was at least this one day to celebrate it.
The pavilion was filled with a lively hum as everyone ate. At our large table sat all of our closest friends – most of the inner crew, along with Marcus, Augustus, and his wife and son. From time to time, I would catch glimpses of other people I had met throughout my journey. Jonas Sparks sat a few tables away, talking intently to Deborah – who seemed to be vaguely interested in what he had to say – while Marcus sat on Deborah's other side, staring daggers at Sparks. Mayor Shaw sat with Julian's mother and sister the next table over. Julian was among them, laughing – and standing next to Julian, holding his hand, was none other than Makara herself.
Grudge sat quietly with some of his Suns in a corner, sipping a cup of wine – the gang was no more, but the friendship among their members had continued. Meanwhile, the guitarists from New America began to play music, and from time to time, Julian's sister would get up to sing. I never really knew the songs. There hadn't been much time for music in my life. Still, it was lively, and seemed good to dance to.
"Come on," Anna said, grabbing me by the hand.
Some space had been cleared in one of the pavilion's corners, near where the band was set up. I noticed that Anna and I were the only ones dancing.
"I never really learned how to dance," I said.
Anna smiled. "Me neither. Now's a good time to learn then, huh?"
I didn't feel too nervous about it. I was only happy, and happiness made it easy to dance.
"Just do this," Anna said, placing my right hand in the small of her back. "And hold my left hand, over here..."
She moved her feet, slowly to the tempo of the music. Everyone quieted as all eyes turned on us. I felt a bit self-conscious. I just tried to time my steps with Anna, which was harder than it seemed. After a moment, though, I got the hang of it. We danced and turned with ease, almost becoming one.
"Not bad," Anna said.
"You're really graceful," I said.
Her face reddened a bit. "Well...I kind of lied. I had a bit of practice. My dad, when he was alive...he showed me how to dance when I was little."
I nodded. That must have been a long time ago.
We continued dancing until the song was over. Anna suddenly stopped, but I didn't want her to. But everyone around us was clapping, and the band struck up another tune, with a faster beat.
I then realized that we weren't done dancing, only that everyone else was coming to dance with us.
An hour went by in what seemed minutes. We were only stopped when Ashton interrupted us, pointing to the cake.
"Not to stop the party," Ashton said, "but we non-dancers are getting a bit impatient staring at that cake."
"Then have some!" I said.
Ashton laughed, and shook his head. "Bride and groom get the first bite. And make sure you feed her the first bite."
"Why?" I asked.
"It's just a thing," he said. "Trust me."
We stood around the cake. Ashton did the honors of cutting a rather large piece. We didn't have cake in Bunker 108. I'd heard about it, seen it in movies, but no one ever actually made it. Most of the desserts we had were fruit-based, and were actually pretty good. But the way this cake even smelled was heavenly; the frosting was wavy and looked sticky. I noticed the cake itself was brown.
"It's chocolate," Anna said, with wonder.
I'd never eaten chocolate before – it wasn't really native to the Wasteland, but apparently, they grew it in the Empire. Ashton put the cake on my plate, looking almost sad to let it go.
"I haven't eaten chocolate in almost twenty years," he said. "This alone is almost enough for me to forgive the old bastard."
I took the cake and a fork. I partitioned off a small piece, and lifted it to Anna's mouth. She took a bite that was both eager and delicate at the same time. I didn't see how she managed that.
After she took her bite, she did the same for me. When the cake entered my mouth, an explosion of rich sweetness radiated from my taste buds. Words could not describe how good it tasted. If it weren't for all the people there, I probably would have started stuffing my face at that moment.
Now that we had our cake, everyone else started to get theirs. The dancing was suspended for the moment while everyone devoured the sweet dessert.
We ate and danced late into the night. By the time people filtered out of the pavilion, the rain had ceased. It was a strange thing, having a wedding all the way out here. For the first time, I realized that there was nowhere for people to sleep. That was when I realized that people would be sleeping aboard both of the spaceships.
The air was warm, even after the rain. I followed the procession of people into the night.
"What now?" Anna asked.
"I don't know," I said. "I'm really tired."
"You're not sleeping, yet!" Michael said, stepping outside the pavilion. "There's still the honeymoon."
"And where will this honeymoon take place?" Anna asked. "Everyone is sleeping on the spaceships!"
"We have something set up for you guys on the beach," Michael said, giving a wink. "Come on."
Michael turned to the crowd, telling them that we were going away for a while. Everyone laughed and cheered; Augustus and Ashton raised their cups in tandem. Alcohol, it seemed, had made them forget their differences for the moment.
Michael led us down the hill and away from the crowd streaming in to the two spaceships for the night. They waved as we walked away from them. We arrived at the ocean, and Michael led us across the sand.
"Where are you taking us?" I asked.
"Almost there," he said. "Just another minute."
We rounded the promontory that jutted out into the sea – the rock formation arched over the sand, leading to a beach on the other side. Set up under a stand of palm trees was another pavilion – smaller than the one we had used.
"This is it," Michael said. "It's all yours. You have all day tomorrow to enjoy it."
"All day?" I said. "What will we do?"
Michael chuckled. "You're about to find out."
My cheeks burned as he turned and jogged away.
"Not much for subtlety, that one," Anna said, kicking off her shoes and leaving them on the sand.
"Well..." I said. "Yeah."
Anna smiled, facing the sea. She still wore her white dress, of course, but the veil had been lost somewhere on the way.
"They didn't really plan this well," she said. "Having me wear this beautiful dress on this beach." She pulled up the train, to show me how the sand had worked its way between the fine fabric and sequins. "See?"
"I see that."
We both turned to watch the waves, calmer now that the storm was over. The stars had come out in full, painting the sky with crystalline light.
"I'm going to change out of this," she said. "Let's sit by the water for a bit."
She went into the pavilion. I kicked off my shoes and socks – the shoes were practically ruined from walking over the wet grass and sand. I took off my jacket and white dress shirt, following Anna into the pavilion.
The inside of the pavilion was covered thickly with fine carpets. A full-size bed filled with plush pillows and soft, luxurious covers dominated the space, all lit by an overhanging lamp.
"Where'd you find your clothes?" I asked.
"They left our stuff in here," she said, pointing to a duffel bag filled with her clothes. "Yours is on the other side."
I dug out a more comfortable pair of pants, and changed, leaving my dress clothes behind.
Anna looked at me before heading back outside. I followed her to the shoreline, to the edge of where the surf stopped and retreated to the vast sea.
We sat there, Anna sitting in front and leaning into me, our legs outstretched. The surf touched our feet, cold and dark.
We'd been sitting in silence for ten minutes, when she spoke.
"What do you think about, looking at those waves?"
I took a moment to think of an answer.
"I'm...just at peace, right now. I'm not really thinking of anything." I squeezed her arm. "What about you?"
"I don't know. It's amazing, though. The sea. Each wave is like a generation. The ones that come after build on the ones that came before. Even as the greatest wave recedes, another rises to take its place."
We sat there a while longer, watching the moon make its ascent. With it the tides rose, forcing us to get up and move. I took Anna's hand, making her face me.
We kissed under the stars and high moon, as the waves passed up the shoreline. When we broke, we walked back to the pavilion.
We undressed and lay on the bed together, and looked into each other's eyes. I held her in embrace. I kissed her again, and we became one.
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# Chapter 26
Even after morning dawned, we continued to sleep. By midmorning, I finally forced myself up. I exited the pavilion to watch waves crash on the shoreline.
After ten minutes, Anna came out to join me, wearing a white tee shirt and shorts. Her hair was strewn by the strong wind.
"Good morning, husband," she said.
"Good morning, wife."
"I had to get that out of the way." She paused. "I'm starving. Did they leave us out here without nothing?"
We went back to the pavilion to find an ice chest set against the leeward side. We must have missed it last night, in the darkness. I opened it, finding a jug of wine, next to some of the leftovers from last night's meal.
"Guess that answers that," I said.
We ate and spent the rest of the morning and afternoon together. Anna and I talked, about anything we could think of. From time to time, reality would cross my mind, like a dark cloud – what I had to do, or how this couldn't last. Anna seemed to know when this happened, because she'd go quiet.
It was hard not to find myself thinking her way. Maybe she was right. Maybe there was a way for me to survive. I only had to do my job, and in the end, I might find a way to do it without dying.
I didn't tell Anna what I was thinking. What was important was enjoying our one day together.
***
DUSK CAME AT LAST. We lay on the beach, watching the stars appear one by one – just a few, and then hundreds.
"I wish I knew what they were called," I said.
"There's still time to learn."
"Do you know them?"
"No. When you grow up without the stars, you don't really think about them."
The remains of the sunset were just a purple glow on the west. The stars were now full, bright, and spectacular. I felt like I could watch them for hours.
Anna pointed. "Look."
It was a bright star, moving across the sky at a steady pace.
"Skyhome," Anna said. "It's so bright!"
"Wish we had a telescope," I said. "It'd be great to see it up close."
We watched Skyhome for another minute. I turned my attention to other stars.
"It's changing," Anna said. "Look!"
Skyhome brightened, dimmed, brightened again.
"That's weird," I said.
We continued watching Skyhome – over time, its brightness dimmed.
"What's happening to it?" Anna asked.
"A trick of the light, maybe."
Even though I said that, I suddenly got a very bad feeling.
"Come on. We need to find the others."
We grabbed all our things and ran along the beach, back to the shoreline on the other side of the promontory. As we ran, Skyhome brightened, soon followed by falling streaks of light, streaming downward from above.
"What's happening?" Anna asked, horrified. "It's gone, isn't it? Skyhome is gone!"
I looked up at the sky again – where once there had been one bright point of light, there were now two.
We reached the hill – the pavilion was no longer there, but Perseus was. Michael ran down the hill, as if coming out to meet us.
"Michael!" I said. "What's going on?"
He stopped at the edge of the beach, staring upward as if he didn't believe his eyes.
"Skyhome's gone," he said.
Dozens of lights streaked from the sky – pieces of the former sky city burning as they entered the atmosphere. I thought of all those people up there, how I had been there just two days before.
"We need to get everyone inside Perseus," I said. "Is everyone still here? Did someone let Ashton know?"
Michael didn't answer. From his silence, I knew something was wrong.
Ashton had been up there.
***
AT FIRST, WE DIDN'T know how Skyhome fell, whether it was a chance collision with a stray piece of debris or rock, or something more nefarious. We knew it had to be something big, because the space city had torn in half.
Ashton had been up there, managing some business related to the sky city, when it was struck. Augustus had allowed him to use Orion to get up there.
Makara decided to go into space to survey the damage. Such a suggestion was dangerous – the amount of debris generated by Skyhome ripping apart would be huge. Makara thought there might be some clue about what had happened to Skyhome, and there was the possibility that Ashton had survived aboard the Orion.
If Ashton was gone, I didn't know what we'd do.
Perseus broke through the upper atmosphere, and when we achieved orbit gravity loosened its hold. Everyone sat positioned in the bridge; Anna and I had to share a seat in order to give everyone else enough space to sit.
"Orion," Makara began, for at least the sixth time. "Do you have a copy?"
She repeated this for the next five minutes as we orbited around Earth. At long last, the shattered remains of Skyhome came into view above the blue curve of the planet. Most of the colony was gone, burned in the atmosphere below. Only broken chunks were left behind
"Orion." Makara's voice had thickened. "Do you have a copy? This is Makara. This is Perseus. If anyone is alive out there...please respond."
Silence was our only answer. Shredded metal and plastic swirled in a deadly, zero-g dance. Small remnants of the former city burned upon reentry above a blue-shining ocean.
Everyone was dead, destroyed in a single instant. It could have been an impact, but I knew the truth.
It was Askala.
"How did they get up here?" Anna asked.
No one answered, because no one knew. Maybe they had shot something. Was that beyond their capabilities? There were too many questions, too few answers.
"Can we track Orion?" Samuel asked. "Maybe they got out..."
"Orion had no tracker," Michael said. "Sparks removed it."
That was that, then. The only sliver of hope we had was that Ashton had escaped on Orion. If he had done so, he would have contacted us by now. We had no way to track him, so we could only assume the worst.
"Doesn't Skyhome have escape pods?" Anna asked.
"Yes," Makara said. "They would be on the surface by now. We'd have to search out their radio signal, and I have no idea what that signal would be."
"Only Ashton would know that," Anna said.
In the space of two days, my world had turned upside down. The Wanderer, dead. Now, Ashton was gone, too.
"He was a good man," Samuel said. "Wise. Knowledgeable. He knew so much, not just about our mission and the xenovirus, but about life."
To my side, Anna's eyes watered with tears. Makara clenched a fist, no longer controlling the spacecraft.
"What are we going to do?" I asked.
The bridge was quiet for a whole minute. There was no sound, no feeling but the shock. The idea of Ashton's being dead was unimaginable.
"Sometimes, you lose someone so important," Makara said, "so vital, that you wonder how you're ever going to go on." She wiped her eyes, and took up the controls. "That's when you have to go on." She nodded to Anna. "Get Augustus on the line. He needs to know."
With shaking fingers, Anna opened the frequency.
"Augustus. You there?"
The Emperor responded shortly.
"If this is about the old man, he's here in Nova Roma at the hospital."
Everyone in the bridge started.
"What?" Makara yelled. "Why didn't you call, you stupid..."
"There is no time for banalities," Augustus said. "Please see that you come to the Imperial City within the next hour. His wounds are grievous. My surgeons are doing all they can, but he may not last the night."
"Alright," Makara said. "We'll be there soon."
"Park in Central Square," he said. "And hurry."
We left the shattered space city behind and ducked into the atmosphere.
***
WE MADE LANDING IN Central Square around midnight. Also parked in the square was the Orion, surrounded by the curious citizens of the Empire's capital. Imperial soldiers kept the peace by holding the crowds back, also clearing space for our landing.
When Perseus touched down, everyone rushed off-ship, Makara taking the lead. We entered the warm night and were greeted by the pressing crowd. I didn't understand why the crowd was so agitated, until I noticed that some of the buildings were in ruins.
Nova Roma, of course, had not recovered from the attack of dragons and crawlers it had suffered a few weeks prior. The people were obviously scared and in need of a leader – but Augustus was preoccupied with Ashton – Ashton, who could very well die, if he wasn't already dead. We'd have no idea how bad it was until we got there.
"Come on," Makara said.
She broke through the crowd, and people shouted questions at us in Spanish. Eventually, the crowd made way, and several of the soldiers pointed in the direction of the hospital. We knew the way from our time here before. Julian, along with Makara, led the rest of us onward.
We ran through the streets. Within a few minutes and various twists and turns, we found ourselves on the lawn in front of the hospital. The building looked rougher than the last time we'd seen it; several of the first-floor windows were shattered, their curtains billowing in the breeze. An entire corner in the northern section had crumbled, creating a cascade of rubble that buried the drive leading from the front doors. Perhaps a dragon had pummeled into it. Despite this, a few windows were lit on the second floor.
"There," Makara said.
We ran inside the building. As we burst through the automatic doors, stuck open, a short woman behind the entry desk regarded us with widened eyes. We ignored her and found a stairwell down the right-hand hallway. Black and purple blood was still caked on the walls from our fight with the crawlers. There hadn't been time for anyone to clean up the mess we'd made.
We ran up the stairwell, bursting onto the second landing. A few rooms down was an open door, flanked by two Praetorians. As we ran toward it, Augustus stepped into the hallway. He nodded toward the door, stepping back into the room.
We entered, finding Ashton lying, bloodied and bruised, in a hospital bed. He was wrapped head to toe in bandages. His eyes were closed, his breathing shallow. There was so little life in him – his pallor was deathly, and if he ever came awake, it definitely wouldn't be as the same man.
He came alone aboard Orion," Augustus said. "From what little I could get out of him, he had only survived because he was well on his way to the hangar. Other than that, he's said nothing."
Makara stepped up to the bedside, taking Ashton's hand.
"Ashton. You there?"
There was no response. If it weren't for his breathing, I wouldn't have thought he was alive. It was unsettling to see Ashton, usually so full of life and spirit, without movement. This man, who had given me so much advice and wisdom, who had married me to Anna...
He was slipping away, and there was nothing I could do about it.
Suddenly, Ashton's hand tightened on Makara's. Without opening his eyes, he rasped:
"It came from space," he said. "Not Earth."
"What came from space?" Makara asked.
I had no idea what Ashton was talking about. If he was talking about the xenovirus, about Ragnarok, then that much was obvious. From everyone's faces, it seemed like they were thinking the same thing as I was.
Until I realized what he was talking about.
"He's talking about Skyhome," I said. "Whatever ended it came from space, not Earth. So it couldn't have been the Radaskim..."
"No."
It must have taken almost all of Ashton's strength to make that emphatic answer. He was silent for a long, terrible moment, before he spoke again.
"They came from space. They...they attacked Skyhome."
Ashton's eyes opened, so thinly that it was like looking into slits.
"See..." he said, lips now trembling. "See for yourself. Look...look to the stars..."
With that, Ashton breathed his last, and went completely still. Every muscle went slack, and I felt desperation clench my chest. He couldn't be dead. An emptiness permeated my soul – an emptiness that could only be caused when someone close to you was taken away, forever.
It was an emptiness I had grown to know too well.
"No..." Anna said.
This time, I knew Ashton really was dead. There was no denying that fact.
Makara squeezed the doctor's hands, her arm shaking. She placed two fingers at the base of Ashton's neck, as if to be sure. She waited for a long time. Then, her face softened, and she shook her head.
"He's gone," she said.
We all stood still, not knowing what to do next. After all the fighting, this was our first loss. It was amazing that everyone had made it this far. And it had come from something over which we had no control, something...
"What do you think was up there?" I asked.
Everyone looked at me, but didn't answer.
"Let's...let's bury Ashton first, alright?" Makara asked, looking up. "Then we can talk about it."
A tear had coursed down her face. She cleared her throat.
"Who will help me carry him?"
Everyone stepped forward, Augustus included, bearing Ashton's weight from the room. As we walked down the steps of the stairwell, out into the lobby, and into the night-covered streets, I had no idea where we were going. We were carrying him back to the ship, but to go where?
After ten minutes of silence, disrupted only by the Praetorians marching around us, we made it back to Central Square. We carried Ashton on board the ship. Augustus followed us up, taking with him several of the Praetorians, while telling Maxillo to stay behind.
When the blast door shut, we just stood there. Julian went to get the gurney, which of late had been used more to carry the dead than the living. Once it was wheeled into the wardroom, we laid Ashton there.
"Ashton wanted to learn more about the Elekai Xenolith," I said. "Maybe...maybe we should bring him there."
"His memory could join the Elekai," Anna said. "Maybe he can discover their secrets after all."
Another thought entered my head: maybe he wasn't dead. Maybe the ichor of the lake could heal him.
I didn't share that with anyone else, and I had no idea if anyone else was thinking it. It was a small hope, and probably pointless. Even if there was no way for that to happen, Ashton would be buried somewhere that had filled him with wonder.
It was the best we could do for him.
***
ONCE WE LANDED, WE made our way down to the ichor lake. Askal and a few of his dragons were there to greet us. We stopped along the shoreline, holding Ashton's limp form, and it seemed Askal read our intent.
Can we save him?
No, Elekim, Askal thought. But his memory can join the Elekai. Bear him to the pool.
I sighed. "Askal says he can't be saved. But his memory can join the Elekai, and he can live on that way."
Maybe he would say hello to the Wanderer for me.
We walked forward until we stood in the pink ichor. Slowly, we lowered Ashton's frail body into the lake.
We stepped away, watching him float toward the center of the cavern. In due time, he sank beneath the surface, his face placid and serene.
A weight lifted from my shoulders. I knew we had done the right thing.
"He said to look to the stars," Samuel said. "He might have meant in the literal sense, to see what had ended Skyhome. But perhaps he was being metaphorical as well."
"I just want to know what happened," Anna said. "What destroyed Skyhome?"
I looked at Askal; the intensity of his eyes told me he had something to say.
What's wrong, Askal?
They are here.
Who?
The second comers. Unforeseen. But I feel their hatred from above. They mean to end us before we can end them.
Who has come?
Askal's mind went silent, for a long time, before he answered.
The Radaskim. Xenofall is here.
A chill passed over me. Quietus had said Xenofall wasn't to come for another four hundred years. We still had time.
Didn't we?
How have they come so soon? I asked, dreading the answer.
After Ragnarok, she kept some in reserve, dormant. Now, they sense their master's need. Askala has called the rest of her children to Earth.
Everyone watched me, knowing from my whitened eyes that I was communicating with Askal.
"What is it?" Makara asked. "What's he saying?"
"He's saying the Radaskim ended Skyhome," I said. "Not from Earth, but from space."
"What do you mean?" Samuel asked.
"Askala...she kept some monsters somewhere in space, dormant. Close by, so when she was ready, she could call them down to deal the final blow."
"So – what?" Makara asked. "We're going to be fighting monsters from space, now?"
Apparently.
I turned back to Askal. How long?
Askal's eyes burned into me, and from that gaze, I knew the answer.
They were already here.
We need you, Elekim, Askal said. Are you ready to accept your fate?
Until now, I wasn't. I'd wanted to be me, at least for a little while longer. Perhaps I was Elekim in name, but I needed to fully accept my responsibility if the Elekai were to have a chance.
"I'm going into the pool," I said.
No one said anything, understanding my intent. My mind reeled from all the changes in the past few days. The Wanderer's death. My marriage to Anna. And now, Ashton's death.
The Elekai needed a fully-fledged leader.
I turned to face Anna.
"Your eyes," she said.
I brushed a strand of hair from her face. "I know. Soon, they'll be that way forever."
She looked at me a long moment before answering. "I knew what this meant, to marry you. I did, anyway. Because I love you, Alex."
"I'll always love you, Anna. Nothing will change about that, no matter what happens in there. I promise."
She nodded, but I couldn't tell what she was really thinking. It was time to accept my role in full. The Elekai needed a strong leader to survive the coming storm.
I kissed her, let go of her hand, and faced the pool. I waded in, the ichor wrapping around in embrace. I continued to walk, facing almost no resistance, even when the ichor reached my chest.
Soon, my head was submerged. I opened my eyes, waiting.
A Voice then spoke, not quite his, but deeply familiar.
Are you ready?
I am.
Then step forward.
I continued to walk, my vision fading to black. I walked until there was nothing but stillness – no senses – only my consciousness entering a dark void.
***
TIME WAS ABSENT AS I swam through vivid images, absorbing the history of the Elekai. Dreams of fallen worlds and the stories of a thousand races flowed through my mind. I didn't speak; I could only stand, mesmerized, as a thousand histories became my own.
And then, at the end of it all, came the prophecy from the Voice, the one the Wanderer had said was as deep as energy.
Though a thousand worlds will fall, one will remain.
But would it be ours?
In time, the images ceased, and all that was left was the void. Stars appeared, studding the cosmos, thicker than seemed possible.
The Universe is in your hands, Elekim, the Voice said. The Secrets of Creation lie within, Guardian of the Stars. The spirits of a thousand worlds fly with you. You are never alone, because even in darkness, the stars still shine.
The starry black void slowly reverted to the pink shimmering of the ichorous pool.
I paused a moment, collecting my thoughts. I felt no different than before. I still had no idea what came next.
One thing was certain; when I returned to that shoreline, I wouldn't be my former self. While that part of me remained, there was also this new part, given to me by the Voice and its visions.
When I returned, I would do so as Elekim.
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# About the Author
Kyle West is the author of The Wasteland Chronicles. From a young age, he has been a voracious reader of sci-fi and fantasy. He graduated from the University of Oklahoma with a degree in Professional Writing. He writes full-time and resides in the bustling metropolis of Oklahoma City.
Find out immediately when his next book is released by signing up for The Wasteland Chronicles Mailing List, found at eepurl.com/A1-8D. Be sure to follow him on Facebook for updates, book giveaways, and general shenanigans. Xenofall, the final installment of The Wasteland Chronicles, will be released this summer.
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# Contact
Facebook
Twitter
Goodreads
Blog
kylewestwriter[at]gmail[dot]com
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# Glossary
10,000, The: This refers to the 10,000 citizens who were selected in 2029 to enter Bunker One. This group included the best America had to offer, people who were masters in the fields of science, engineering, medicine, and security. President Garland and all the U.S. Congress, as well as essential staff and their families, were chosen.
Alpha: "Alpha" is the title given to the recognized head of the Raiders. In the beginning, it was merely a titular role that only had as much power as the Alpha was able to enforce. But as Raider Bluff grew in size and complexity, the Alpha took on a more meaningful role. Typically, Alphas do not remain so for long – they are assassinated by rivals who rise to take their place. In some years, there can be as many as four Alphas – though powerful Alphas, like Char, can reign for many years.
Askala: Askala has two meanings – one is the name of the Radaskim Xenomind dwelling in Ragnarok Crater, while the other form refers to the dragons the xenovirus spawns. While Askala and xenodragon (or just dragon) are interchangeable, the Askala themselves (at least the Elekai Askala) refer to themselves as Askala.
Batts: Batts, or batteries, are the currency of the Wasteland and the Empire. They are accepted anywhere that the Empire's caravans reach. It is unknown how batteries were first seen as currency, but it is rumored that Augustus himself instigated the policy. Using them as currency makes sense: batteries are small, portable, and durable, and have the intrinsic quality of being useful. Rechargeable batteries (called "chargers") are even more prized, and solar batteries (called "solars," or "sols") are the most useful and prized of all.
Behemoth: The Behemoth is a great monstrosity in the Wasteland – a giant creature, either humanoid or reptilian, or sometimes a mixture of the two, that can reach heights of ten feet or greater. They are bipedal, powerful, and can keep pace with a moving vehicle. All but the most powerful of guns are useless against the Behemoth's armored hide.
Black Reapers, The: The Black Reapers are a powerful, violent gang, based in Los Angeles. They are led by Warlord Carin Black. They keep thousands of slaves, using them to serve their post-apocalyptic empire. They usurped the Lost Angels in 2055, and have been ruling there ever since.
Black Files, The: The Black Files are the mysterious collected research on the xenovirus, located in Bunker One. They were authored principally by Dr. Cornelius Ashton, Chief Scientist of Bunker One.
Blights: Blights are infestations of xenofungus and the xenolife they support. They are typically small, but the bigger ones can cover large tracts of land. As a general rule of thumb, the larger the Blight, the more complicated and dangerous the ecosystem it maintains. The largest known Blight is the Great Blight – which covers a large portion of the central United States. Its center is Ragnarok Crater.
Boundless, The: The Boundless is an incredibly dry part of the Wasteland, ravaged by canyons and dust storms, situated in what used to be Arizona and New Mexico. Very little can survive in the Boundless, and no one is known to have ever crossed it.
Bunker 40: Bunker 40 is located on the outer fringes of the Great Blight in Arizona. It is hidden beneath a top secret research facility, a vestige of the Old World. Many aircraft were stationed at Bunker 40 before it fell, sometime in the late 2050s.
Bunker 108: Bunker 108 is located in the San Bernardino Mountains about one hundred miles east of Los Angeles. It is the birthplace of Alex Keener.
Bunker 114: Bunker 114 is a medical research installation built about fifty miles northwest of Bunker 108. Built beneath Cold Mountain, Bunker 114 is small. After the fall of Bunker One, Bunker 114, like Bunker 108 to the southeast, became a main center of xenoviral research. An outbreak of the human strain of the xenovirus caused the Bunker to fall in 2060. Bunker 108's fall followed soon thereafter.
Bunker One: Bunker One was the main headquarters of the Post-Ragnarok United States government. It fell in 2048 to a swarm of crawlers that overran its defenses. Bunker One had berths for ten thousand people, making it many times over the most populous Bunker. Its inhabitants included President Garland, the U.S. Senate and House of Representatives, essential government staff, and security forces, along with the skilled people needed to maintain it. Also, dozens of brilliant scientists and specialists lived and worked there, including engineers, doctors, and technicians. The very wealthy were also allowed berths for helping to finance the Bunker Program. Bunker One is the location of the Black Files, authored by Dr. Cornelius Ashton.
Bunker Six: Bunker Six is a large installation located north of Bunker One, within driving distance. It houses the S-Class spaceships constructed during the Dark Decade – including Gilgamesh, the capital ship, and three smaller cruisers – Odin, Perseus, and Orion. While Gilgamesh and Odin are under Cornelius Ashton's care, Perseus and Orion are still locked inside the fallen Bunker.
Bunker Program, The: The United States and Canadian governments pooled resources to establish 144 Bunkers in Twelve Sectors throughout their territory. The Bunkers were the backup in case the Guardian Missions failed. When the Guardian Missions did fail, the Bunker Program kicked into full gear. The Bunkers were designed to save all critical government personnel and citizenry, along with anyone who could provide the finances to construct them. The Bunkers were designed to last indefinitely, using hydroponics to grow food. The Bunkers ran on fusion power, which had been made efficient by the early 2020s. The plan was that, when the dust settled, Bunker residents could reemerge and rebuild. Most Bunkers fell, however, for various reasons – including critical systems failures, mutinies, and attacks by outsiders (see Wastelanders). By the year 2060, only four Bunkers were left.
Chaos Years, The: The Chaos Years refer to the ten years following the impact of Ragnarok. These dark years signified the great die-off of most forms of life, including humans. Most deaths occurred due to starvation. With mass global cooling, crops could not grow in climates too far from the tropics. What crops would grow produced a yield far too small to feed the population that existed. This led to a period of violence unknown in all of human history. The Chaos Years signify the complete breakdown of the Old World's remaining infrastructures – including food production, economies, power grids, and the industrial complex – all of which led to the deaths of billions of people.
Coleseo Imperio: El Coleseo Imperio, translated as the Imperial Coliseum, is a circular, three-tiered stone arena rising from the center of the city of Nova Roma, the capital of the Nova Roman Empire. It is used to host gladiatorial games in the tradition of ancient Rome, and serves as the chief sport of the Empire. Slaves and convicts are forced to fight in death matches, which serves the dual purpose of entertaining the masses while getting rid of prisoners and slaves who would otherwise be, in the Empire's eyes, liabilities. Ritual sacrifices routinely take place on the arena floor.
Crawlers: Crawlers are dangerous, highly mobile monsters spawned by Ragnarok. Their origin is unclear, but they share many characteristics of Earth animals – mostly those reptilian in nature, although other forms are more similar to insects. Crawlers are sleek and fast, and can leap through the air at very high speeds. Typically, crawlers attack in groups, and behave as if of one mind. One crawler will, without hesitation, sacrifice itself in order to reach its prey. Crawlers are especially dangerous when gathered in high numbers – at which point there is not much one can do but run. Crawlers can be killed, their weak points being their belly and their three eyes.
Dark Decade, The: The Dark Decade lasted from 2020-2030, from the time of the first discovery of Ragnarok, to the time of its impact. It is not called the Dark Decade because the world descended into madness immediately upon the discovery of Ragnarok by astronomer Neil Weinstein – that only happened in 2028, with the failure of Messiah, the third and last of the Guardian Missions. In the United States and other industrialized nations, life proceeded in an almost normal fashion. There were plenty of good reasons to believe that Ragnarok could be stopped, especially when given ten years. But as the Guardian Missions failed, one by one, the order of the world quickly disintegrated.
With the failure of the Guardian Mission Archangel in 2024, a series of wars engulfed the world. As what some were calling World War III embroiled the planet, the U.S. and several of its European allies, and Canada, continued to work on stopping Ragnarok. When the second Guardian Mission, Reckoning, failed, an economic depression swept the world. But none of this compared to the madness that followed upon the failure of the third and final Guardian, Messiah, in 2028. As societies broke down, martial law was enforced. President Garland was appointed dictator of the United States with absolute authority. By 2029, several states had broken off from the Union.
In the last quarter of 2030, an odd silence hung over the world, as if it had grown weary of living. The President, all essential governmental staff and military, the Senate and House of Representatives, along with scientists, engineers, and the talented and the wealthy, entered the 144 Bunkers established by the Bunker Program. Outraged, the tens of millions of people who did not get an invitation found the Bunker locations, demanding to be let in. The military took action when necessary.
Then, on December 3, 2030, Ragnarok fell, crashing into the border of Wyoming and Nebraska, forming a crater one hundred miles wide. The world left the Dark Decade, and entered the Chaos Years.
Elekai: The Elekai are the peaceful counterpart to the Radaskim. They seek the harmony and growth of all life, even if that means that the universe will eventually one day end. The Radaskim take the opposite viewpoint – that the destruction of all life is a fair price in exchange for controlling the destruction and reconstruction of the universe, called the Universal Cycle:
Eternal War, The: The Eternal War has spanned hundreds of thousands of years across the cosmos. The Radaskim seek to infect and conquer every life-bearing world. The Elekai are always with them and fight the Radaskim at every turn. However, the Radaskim are more numerous and have always won on every world. The Wanderer and the Elekai hope to reverse this on planet Earth.
Exiles, The: The Exiles are led by a man named Marcus, brother of Alpha Char. The Exiles were once raiders, but were exiled from Raider Bluff in 2048. Raider Bluff faced a rival city, known as Rivertown, on the Colorado River. A faction led by Char wanted to destroy Rivertown by blowing up Hoover Dam far to the north. Marcus and his faction opposed this. The two brothers fought, and in his rage, Marcus threw Char into a nearby fireplace, giving him the severe burns on his face that Char would live with for the rest of his life. For this attack, the Alpha at the time exiled Marcus – but in solidarity, many Raiders left to join him. For the next twelve years, the exiled Raiders wandered the Boundless, barred from ever returning farther west than Raider Bluff. The Exiles at first sought to found a new city somewhere in the eastern United States, but the Great Blight barred their path. Over the next several years, they hired themselves as mercenaries to the growing Nova Roman Empire. Now, they wander the Wastes, Marcus awaiting the day when his brother calls upon him for help – which he is sure Char will do.
Flyers: Flyers are birds infected with the xenovirus. They fly in large swarms of a hundred or more. They are only common around large Blights, or within the Great Blight itself. The high metabolism of flyers means they cannot venture far from xenofungus, their main source of food. They are highly dangerous, and cannot be fought easily, because they fly in such large numbers.
Gilgamesh: Gilgamesh is an S-Class Capital Spaceship constructed by the United States during the Dark Decade. It holds room for twelve crewmen, thirteen counting the captain. Its fuselage is mostly made of carbon nanotubes – incredibly lightweight, and many, many times stronger than steel. It is powered by a prototypical miniature fusion reactor, using deuterium and tritium as fuel. Its design is described as insect-like in appearance, for invisibility to radar. The ship contains a bridge, armory, conference room, kitchen, galley, two lavatories, a clinic, and twelve bunks for crew in two separate dorms. A modest captain's quarters can be reached from the galley, complete with its own lavatory. Within the galley is access to a spacious cargo bay, where supplies, and even a vehicle as large as a Recon, can be stored. The Recon can be driven off the ship's wide boarding ramp when grounded (this capability is the main difference between Odin and Gilgamesh...in addition to the cargo bay boarding ramp, Gilgamesh also contains a passenger's boarding ramp on the side, that also leads into the galley). The porthole has a retractable rope ladder that is good for up to five hundred feet. Gilgamesh has a short wingspan, but receives most of its lift from the four thrusters mounted in back, thrusters that have a wide arc of rotation that allows the ship to fly in almost any direction. The ship can go weeks without needing to refuel. As far as combat capabilities, Gilgamesh was primarily constructed as a reconnaissance and transport vessel. That said, it has twin machine gun turrets that open from beneath the ship. When grounded, it is supported by three struts, one in front, two in back.
Great Blight, The: The Great Blight is the largest xenofungal infestation in the world, its point of origin being Ragnarok Crater on the Great Plains in eastern Wyoming and western Nebraska. Unlike other Blights, the Great Blight is massive. From 2040-2060, it began to rapidly expand outside Ragnarok Crater at an alarming rate, moving as much as a quarter mile each day (meaning the stretching of the xenofungus could actually be discerned with the naked eye). Any and all life was conquered, killed, or acquired into the Great Blight's xenoparasitic network. Here, the first monsters were created. Animals would become ensnared in sticky pools of purple goo, and their DNA absorbed and preserved. The Great Blights, obeying some sort of consciousness, would then mix and match the DNA of varying species, tweaking and mutating the genes until, from the same pools it had acquired the DNA, it would give birth to new life forms, designed only to spread the Blight and kill whoever, or whatever, opposed that spreading. As time went on and the Xeno invasion became more sophisticated, the Great Blight's capabilities became advanced enough to direct the evolution of xenolife itself, leading to the creation of the xenovirus, meaning it could infect species far outside of the Blight – including, eventually, humans.
Guardian Missions: The Guardian Missions were humanity's attempts to intercept and alter the course of Ragnarok during the Dark Decade. There were three, and in the order they were launched, they were called Archangel, Reckoning, and Messiah (all three of which were also the names of the ships launched). Each mission had a reason for failing. Archangel is reported to have crashed into Ragnarok, in 2024. In 2026, Reckoning somehow got off-course, losing contact with Earth in the process. In 2028 Messiah successfully landed and attached its payload of rockets to the surface of Ragnarok in order to alter its course from Earth. However, the rockets failed before they had time to do their work. The failure of the Guardian Missions kicked the Bunker Program into overdrive.
Howlers: Howlers are the newest known threat posed by the xenovirus. They are human xenolife, and they behave very much like zombies. They attack with sheer numbers, using their bodies as weapons. A bite from a Howler is enough to infect the victim with the human strain of the xenovirus. Post-infection, it takes anywhere from a few minutes to a few hours for a corpse to reanimate into the dreaded howler. Worse, upon death, Howlers somehow explode, raining purple goo on anyone within range. Even if a little bit of goo enters the victim's bloodstream, he or she is as good as dead, cursed to become a Howler within a matter of minutes or hours. How the explosion occurs, no one knows – it is surmised that the xenovirus itself creates some sort of agent that reacts violently with water or some other fluid present within the Howlers. There is also reason to believe that certain Howlers become Behemoths, as was the case with Kari in Bunker 114.
Hydra: A powerful spawning of the xenovirus, the Hydra has only been seen deep in the heart of Bunker One. It contains three heads mounted on three stalk-like necks. It is covered in thick scales that serve as armor. It has a powerful tail that it can swing, from the end of which juts a long, cruel spike. It is likely an evolved, more deadly form of the crawler.
Ice Lands, The: Frozen in a perpetual blanket of ice and snow, the northern and southern latitudes of the planet are completely unlivable. In the Wasteland, at least, they are referred to as the Ice Lands. Under a blanket of meteor fallout, extreme global cooling was instigated in 2030. While the glaciers are only now experiencing rapid regrowth, they will advance for centuries to come until the fallout has dissipated enough to produce a warmer climate. In the Wasteland, 45 degrees north marks the beginning of what is considered the Ice Lands.
L.A. Gangland: L.A. Gangland means a much different thing than it did Pre-Ragnarok. In the ruins of Los Angeles, there are dozens of gangs vying for control, but by 2060, the most powerful is the Black Reapers, who usurped that title from the Lost Angels.
Lost Angels, The: The Lost Angels were post-apocalyptic L.A.'s first super gang. From the year 2050 until 2055, they reigned supreme in the city, led by a charismatic figure named Dark Raine. The Angels were different from other gangs – they valued individual freedom and abhorred slavery. Under the Angels' rule, Los Angeles prospered. The Angels were eventually usurped in 2056 by a gang called the Black Reapers, led by a man named Carin Black.
Nova Roma: Nova Roma is the capital of the Nova Roman Empire. It existed Pre-Ragnarok as a small town situated in an idyllic valley, flanked on three sides by green mountains. This town was also home to Augustus's palatial mansion – and it was around this mansion that the city that would one day rule the Empire had its beginnings. Over thirty years, as the Empire gained wealth and power under Augustus's rule, Nova Roma grew from a small village into a mighty city with a population numbering in the tens of thousands. Using knowledge of ancient construction techniques found in American Bunkers, Augustus employed talented engineers and thousands of slaves to build the city from the ground up. Inspired by the architecture of ancient Rome, some of the most notable construction projects in Nova Roma include the Coleseo Imperio, the Senate House, the Grand Forum, and Central Square. An aqueduct carries water over the city walls from the Sierra Madre Mountains north of the city. The city grows larger each passing year, so much so that shantytowns have overflowed its walls, attracted by the city's vast wealth.
Nova Roman Empire, The: The Nova Roman Empire (also known as the "Empire") is a collection of allied city-states that are ruled from Nova Roma, its capital in what was formerly the Mexican state of Guerrero. The Empire began as the territory of a Mexican drug cartel named the Legion. Through the use of brutal force, they kept security within their borders even as other governments fell.
Following the impact of Ragnarok, many millions of Americans fled south to escape the cold, dry climate that permeated northern latitudes. Mexico still remained warm, especially southern Mexico, and new global wind currents caused by Ragnarok kept Mexico clearer of meteor fallout than other areas of the world. At the close of the Chaos Years, Mexico was far more populous than the United States. Many city-states formed in the former republic, but most developed west of the Sierra Madre Mountains. Language clashes between native Mexicans and migrant Americans produced new dialects of both Spanish and English. Though racial tensions exist in the Empire, as Americans' descendants are the minority within it, Americans and their descendants are protected under law and are entitled to the same rights – at least in theory. The reality is, most refugees that entered Imperial territory were American – and most refugees ended up as slaves.
Of the hundreds of city-states that formed in Mexico, one was called Nova Roma, located inland in a temperate valley not too far east of Acapulco. Under the direction of the man styling himself as Augustus Imperator, formerly known as Miguel Santos, lord of the Legion drug cartel, the city of Nova Roma allied with neighboring city-states. Incorporating both Ancient Roman governmental values and Aztec mythology, the Empire expanded through either the conquest or annexation of rival city-states. By 2060, the Empire had hundreds of cities in its thrall, stretching from Oaxaca in the southeast all the way to Jalisco in the northwest. The Empire had also formed colonies as far north as Sonora, even founding a city called Colossus at the mouth of the Colorado River, intended to be the provincial capital from which the Empire hoped to rule California and the Mojave.
Because of its size and power, the Empire is difficult to control. Except for its center, ruled out of Nova Roma, most of the city-states are autonomous and are only required to pay tribute and soldiers when called for during the Empire's wars. In the wake of the Empire's rapid conquests, Augustus developed the Imperial Road System in order to facilitate trade and communication, mostly done by horse. In an effort to create a unifying culture for the Empire, Emperor Augustus instigated a representative government, where all of Nova Roma's provinces have representation in the Imperial Senate. Augustus encouraged a universal religion based on Aztec mythology, whose gods are placed alongside the saints of Catholicism in the Imperial Pantheon. Augustus also instigated gladiatorial games, ordering that arenas be built in every major settlement of his Empire. This included the construction of dozens of arenas, including El Coleseo Imperio in Nova Roma itself, a large arena which, while not as splendid as the original Coliseum in Old Rome, is still quite impressive. The Coleseo can seat ten thousand people. By 2060, Augustus had accomplished what might have taken a century to establish otherwise.
Oasis: Oasis is a settlement located in the Wasteland, about halfway between Los Angeles and Raider Bluff. It has a population of one thousand, and is built around the banks of the oasis for which it is named. The oasis did not exist Pre-Ragnarok, but was formed by tapping an underground aquifer. Elder Ohlan rules Oasis with a strong hand. He is the brother of Dark Raine, and it is whispered that he might have had a hand in his death.
Odin: Odin is an S-Class Cruiser Spaceship built by the U.S. during the Dark Decade. It is one of four, the other being Gilgamesh, the capital ship, and the other two being Perseus and Orion, cruisers with the same specs as Odin. Though Odin's capabilities are not as impressive as Gilgamesh's, Odin is still very functional. It contains berths for eight crew, nine counting the captain. It has a cockpit, armory, kitchen, galley, two dorms, one lavatory, and the fusion drive in the aft. A cargo bay can be reached from either outside the ship or within the galley. Unlike Gilgamesh, it is not spacious enough to store a Recon. It contains a single machine gun turret that can open up from the ship's bottom. Odin, in addition to being faster than Gilgamesh, also gets better fuel efficiency. It can go months without needing to refuel.
Praetorians, The: The Praetorians are the most elite of the Empire's soldiers. There are one hundred total, and they are the personal bodyguard of Emperor Augustus. They carry a long spear, tower shield, and gladius. They wear a long, purple cape, steel armor, and a white jaguar headdress, complete with purple plume. They are also trained in the use of guns.
Radaskim: The Radaskim seek to conquer all life in the universe in their quest to discover the Secrets of Creations, Secrets which the Radaskim believe are the key to recreating the universe. Whether this is even possible is unknown, but all Radaskim motives are based on this idea. They have conquered hundreds of worlds and are in the process of conquering Earth. The Radaskim on Earth are led by the Xenomind, Askala.
Raider Bluff: Raider Bluff is the only known settlement of the raiders. It is built northeast of what used to be Needles, California, on top of a three-tiered mesa. Though the raiders are a mobile group, even they need a place to rest during the harsh Wasteland winter. Merchants, women, and servants followed the Raider men, setting up shop on the mesa, giving birth to Raider Bluff sometime in the early 2040s. From the top of the Bluff rules the Alpha, the strongest recognized leader of the Raiders. A new Alpha rises only when he is able to wrest control from the old one.
Ragnarok: Ragnarok was the name given the meteor that crashed into Earth on December 3, 2030. It was about three miles long, and two miles wide. It was discovered by astronomer Neil Weinstein, in 2019. It is not known what caused Ragnarok to come hurtling toward Earth, or how it eluded detection for so long – but that answer was revealed when the Black Files came to light. Ragnarok was the first phase of the invasion planned by the Xenos, the race of aliens attempting to conquer Earth. Implanted within Ragnarok was the xenovirus – the seed for all alien genetic life that was to destroy, acquire, and replace Earth life. The day the Xenos arrive, according to the Black Files, is called "Xenofall." The time of their eventual arrival is completely unknown.
Ragnarok Crater: Ragnarok Crater is the site of impact of the meteor Ragnarok. It is located on the border of Wyoming and Nebraska, and is about one hundred miles wide with walls eight miles tall. It's the center of the Great Blight, and it is also the origin of the Voice, the consciousness that directs the behavior of all xenolife.
Recon: A Recon is an all-terrain rover that is powered by hydrogen. It is designed for speedy recon missions across the Wastes, and was developed by the United States military during the Dark Decade. It is composed of a cab in front, and a large cargo bay in the back. Mounted on top of the cargo bay is a turret with 360-degree rotation, accessible by a ladder and a porthole. The turret can be manned and fired while the Recon is on the go.
Secrets of Creation, The: The Secrets of Creation are the name given to the knowledge the Radaskim must have in order to destroy the universe and remake it in the exact same way it had been created – allowing the universe to exist indefinitely.
Skyhome: Skyhome is a three-ringed, self-sufficient space station constructed by the United States during the Dark Decade, designed to house two hundred and fifty people. Like the Bunkers, it contains its own power, hydroponics, and water reclamation system designed to keep the station going as long as possible. Skyhome was never actually occupied until 2048, after the falls of both Bunker One and Bunker Six. Cornelius Ashton assumed control of the station, along with survivors from both Bunkers, in order to continue his research on the xenovirus which had destroyed his entire life.
Universal Cycle, The: The Universal Cycle is a Radaskim prophecy stating that the universe has been destroyed and reborn an infinite number of times, only because the Radaskim discovered the "Secrets of Creation" in every manifestation of the universe in time to recreate it. The rebirth of the universe depends on the Radaskim discovering these secrets, and it involves acquiring all known life in order to discover the knowledge that might unlock the Secrets – whatever they are.
Voice, The: The Voice is the name given to the collective consciousness of all xenolife. It exists in Ragnarok Crater – whether or not it has a corporeal form is unknown. However, it is agreed by Dr. Ashton and Samuel that the Voice controls xenolife using sound waves and vibrations within xenofungus. The Voice also sends sound waves that can be detected by xenolife while off the xenofungus. The Voice gives the entire Xeno invasion sentience, and is a piece of evidence pointing to an advanced alien race that is trying to conquer Earth.
Wanderer, The: A blind prophet who wanders the Wasteland. He is also the Xenomind who leads the Elekai, the alien faction that wants to stop the Radaskim from conquering all life.
Wastelanders: Wastelanders are surface dwellers, specifically ones that live in the southwestern United States. The term is broad – it can be as specific as to mean only someone who is forced to wander, scavenge, or raid for sustenance, or Wastelander can mean anyone who lives on the surface Post-Ragnarok, regardless of location or circumstances. Wastelanders are feared by Bunker dwellers, as they have been the number one reason for Bunkers failing.
Wasteland, The: The Wasteland is a large tract of land comprised of Southern California and the adjacent areas of the Western United States. It extends from the San Bernardino Mountains in the west, to the Rockies in the east (and in later years, the Great Blight), and from the northern border of Nova Roma on the south, to the Ice Lands to the north (which is about the same latitude as Sacramento, California). The Wasteland is characterized by a cold, extremely dry climate. Rainfall each year is little to none, two to four inches being about average. Little can survive the Wasteland, meaning that all life has clung to limited water supplies. Major population centers include Raider Bluff, along the Colorado River; Oasis, supplied by a body of water of the same name; and Last Town, a trading post that sprung up along I-10 between Los Angeles and the Mojave. Whenever the Wasteland is referred to, it is generally not referred to in its entire scope. It is mainly used to reference what was once the Mojave Desert.
Xenodragon: The xenodragon is the newest manifestation of the xenovirus. It is very much like a dragon – reptilian, lightweight, with colossal wings that provide it with both lift and speed. There are different kinds of xenodragons, but the differences are little known, other than whether they are large or small. A particularly large xenodragon makes its roost on Raider Bluff.
Xenofall: Xenofall is the day of reckoning – when the Xenos finally arrive on Earth to claim it as their own. No one knows when that day is – whether it is in one year, ten years, or a thousand. It is feared that, when Xenofall does come, humans and all resistance will have been long gone.
Xenofungus: Xenofungus is a slimy, sticky fungus that is colored pink, orange, or purple (and sometimes all three), that infests large tracts of land and serves as the chief food source of all xenolife. It forms the basis of the Blights, and without xenofungus, xenolife could not exist. The fungus, while hostile to Earth life, facilitates the growth, development, and expansion of xenolife. It is nutrient-rich, and contains complicated compounds and proteins that are poison to Earth life, but ambrosia for xenolife. It is tough, resilient, resistant to fire, dryness, and cold – and if it isn't somehow stopped, one day xenofungus will cover the entire world.
Xenolife: Any form of life that is infected with the xenovirus.
Xenomind: A Xenomind is an ancient sentient being, evolved over the eons by the xenovirus and xenofungus. They are split into two factions – the Radaskim and the Elekai. The Radaskim are warlike and want to conquer all life in the universe – a seemingly impossible aim. The Elekai want to stop the Radaskim from achieving this. So far, on every world the Eternal War has been fought, the Elekai have lost.
Xenovirus: The xenovirus is an agent that acquires genes, adding them to its vast collection. It then mixes and matches the genes under its control to create something completely new, whether a plant, animal, bacteria, etc. There are thousands of strains of the xenovirus, maybe even millions, but most are completely undocumented. While the underlying core of each strain is the same, the strains are specific to each species it infects. Failed strains completely drop out of existence, but the successful ones live on. The xenovirus was first noted by Dr. Cornelius Ashton of Bunker One. His collected research on the xenovirus was compiled in the Black Files, which were lost in the fall of Bunker One in 2048.
Also by Kyle West
The Wasteland Chronicles
Apocalypse
Origins
Evolution
Revelation
Darkness
Extinction
Watch for more at Kyle West's site.
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\section{Introduction}
\label{sec:intro}
Collisionless $N$-body simulations are standard tools to study the
evolution of stellar systems such as galaxies and clusters of
galaxies, with typical applications ranging from stability analysis,
to the study of galaxy interactions and mergers, tidal stripping of
satellites, and dynamical friction. When the phenomenon studied with
the $N$-body simulation involves composite collisionless stellar
systems, to set up the initial conditions it is often necessary to
build $N$-body realizations of stationary multi-component models.
Here we present effective $N$-body models that allow one to study
efficiently the evolution of such composite systems.
The approach presented here can be used in several studies
of galactic dynamics, provided the studied galaxies are gas-poor, so
that they can be modelled as multi-component stellar systems, with
dark matter (DM) halos and one or more stellar components,
representing different stellar populations. Examples of potential
applications are $N$-body simulations of tidal stripping aimed at
reproducing the observed properties of satellite dwarf galaxies
\citep[e.g.][]{Bat15,Ura15,San18,Ior19} or those of tidal streams
\citep[e.g.][]{Lok10,Die17,Lap18,Vas20} in the Milky Way. But, more
generally, the effective $N$-body models presented here can be used in
$N$-body simulations of dissipationless galaxy mergers
\citep[e.g.][]{Nip03a,Boy06,Fri17} or of the dynamical evolution of
galaxies in clusters of galaxies \citep[e.g.][]{Nip03b,Lap13}.
The method proposed in this paper builds on and bears resemblance with
other techniques previously proposed in the literature. The key of the
effective $N$-body models considered here is to design composite
stellar system starting from the total distribution function (DF) and
then obtain its component by subtraction. In the literature, there
are a few other studies in which composite stellar systems are built
starting from the total DF or mass density
distribution. \citet{Eva93,Eva94} built axisymmetric composite stellar
systems with total logarithmic or power-law gravitational potential,
starting from the analytic DF of the total distribution. Other authors
\citep{Hio94,Cio09,Cio18,Cio19} used instead the total mass density
distribution as starting point to build multi-component anisotropic
spherical stellar systems. \citet{Whi80} and \citet{Cio95} used a
technique similar to the one used in this work to build equilibrium
models of isotropic or radially anisotropic spherical stellar systems
with metallicity gradients (see also \citealt{Nip03b} and
\citealt{Nip20}). Within this framework, here for the first time we
exploit the idea of building different components by subtraction from
the total DF to envisage a very effective and general method for
$N$-body modelling. This method allows us to use $N$-body simulations
involving only one-component systems to model the dynamical evolution
of entire families of composite stellar systems, with stars and DM.
The paper is organized as follows. In \Sect\ref{sec:stationary} we
review the properties of stationary composite collisionless stellar
systems and introduce the concept of their effective $N$-body
modelling. In \Sect\ref{sec:dynevo} we extend our view to the
dynamical evolution of such systems, when they are not isolated.
\Sect\ref{sec:isotwocomp} treats in more detail the case of
two-component spherical isotropic systems. In \Sect\ref{sec:app} we
present the application of our method to an $N$-body simulation of
tidal stripping. \Sect\ref{sec:concl} concludes.
\section{Stationary composite collisionless stellar systems}
\label{sec:stationary}
\subsection{Distribution functions and portion functions}
\label{sec:stat_df}
Let us consider a stationary composite stellar system with $\Ncomp$
components, in which the $k$-th component has DF $f_k$. The total DF is
$\ftot=\sum_{k=1}^{\Ncomp} f_k$. The total gravitational potential
$\Phitot$ generated by these components satisfies the Poisson equation
\begin{equation}
\nabla^2\Phitot(\xv)=4\pi G\rhotot(\xv),
\label{eq:poisson}
\end{equation}
where
\begin{equation}
\rhotot(\xv)=\int \ftot \d^3\vv
\end{equation}
is the total mass density distribution, and $\xv$ and $\vv$ are,
respectively, the position and velocity vectors. We know from Jeans'
theorem \citep[e.g.][]{Bin08} that the DFs of stationary collisionless
stellar systems depend on the phase-space coordinates $(\xv,\vv)$
through $n\leq 3$ integrals of motion $\Iv=I_1,...,I_n$, which are
functions of $(\xv,\vv)$ that are conserved along the orbits. If we
extract from $\ftot=\ftot(\Iv)$ an orbit with integrals of motion
$\Iv$, the probability that a particle on such orbit belongs to the
$k$-th component is
\begin{equation}
\Pk(\Iv)=\frac{\fk(\Iv)}{\ftot(\Iv)}.
\label{eq:pk}
\end{equation}
In this paper we will refer to the function $\Pk(\Iv)$ as the {\em
portion function} of the $k$-th component.
\subsection{$N$-body realizations}
\label{sec:nbody}
\subsubsection{Standard multi-component $N$-body model}
\label{sec:standard_nbody}
The standard approach to build an $N$-body realization of a stationary
multi-component stellar system is to represent the $k$-th component
with $N_k$ particles with phase-space coordinates extracted from the
DF $\fk(\Iv)$, with $\Iv=\Iv(\xv,\vv)$: the $j$-th particle
($j=1,...,N_k$) has mass $m_j$, phase-space coordinates
$(\xv_j,\vv_j)$ and integrals of motion $\Iv_j=\Iv(\xv_j,\vv_j$). The
total number of particles is $\Ntot=\sum_kN_k$. Jeans' theorem
guarantees that each component of this $N$-body realization is
stationary, because its particles are extracted from a DF depending
only on integrals of motions. In this approach, in the $N$-body
realization we assign to each particle a given ``kind'', for instance
``DM particle'' or ``stellar particle'' if it belongs to,
respectively, the DM halo or the stellar component. In a purely
collisionless $N$-body system the orbits of particles are determined
only by gravity and are thus independent of the particle kind and on
the particle mass. This suggests to explore different $N$-body
realizations in which the particles are not labelled as being of a
given kind or belonging to a given component. In the following we
introduce such an alternative approach.
\subsubsection{Effective multi-component $N$-body model}
\label{sec:stat_eff}
Instead of extracting a set of particles for each component, as in the
standard method described above, we can construct an $N$-body
realization of a stationary composite stellar system by extracting
$\Ntot$ particles from the total DF $\ftot(\Iv)$, obtaining for the
$i$-th particle a set of phase-space coordinates $(\xv_i,\vv_i)$ and
corresponding integrals of motion $\Iv_i=\Iv(\xv_i,\vv_i)$
($i=1,...,\Ntot$). In this way, we do not assign a given particle to
one of the components, but we can nevertheless interpret our system as
multi-component as follows. Given that $\P_k(\Iv_i)$ is the
probability that the $i$-th particle belongs to the $k$-th component
(\Eq\ref{eq:pk}), the mass contribution of the $i$-th particle to the
$k$-th component is $\xiki m_i$, where $m_i$ is the mass of the $i$-th
particle and $\xiki\equiv \P_k(\Iv_i)$ is the mass fraction of the
$i$-th particle that belongs to the $k$-th component. For instance in
a two-component system with a stellar component (with DF $\fstar$) and
DM component (with DF $\fDM\equiv \ftot-\fstar$), the $i$-th particle
has stellar mass $\xistari m_i$ and DM mass $\xiDMi m_i$, where
$\xistari=\Pstar(\Iv_i)$ and $\xiDMi=1-\xistari$ are, respectively,
its stellar and DM mass fractions, and
$\Pstar(\Iv)\equiv\fstar(\Iv)/\ftot(\Iv)$ is the portion function
(\Eq\ref{eq:pk}) of the stellar component. For any choice of
$\P_k(\Iv)$ the $k$-th component is univocally defined. For instance,
the total mass of the $k$-th component is $M_k=\sum_i\xiki m_i$, and
similarly one can compute the mass density and velocity distributions
of the $k$-th component simply by weighting the contribution of the
$i$-th particle by $\xiki m_i$. If such an $N$-body system is evolved
in isolation, the properties (e.g. density and velocity distributions)
of all its $\Ncomp$ components are time-independent in the
limit\footnote{Of course this is not true, strictly speaking, for
finite $\Ntot$ because of discreteness effects.} $\Ntot\to\infty$,
because $\ftot(\Iv)$ is the DF of a stationary system and $\Pk(\Iv)$
is a function of the integrals of motions. The main advantage of this
method with respect to the standard method
(\Sect\ref{sec:standard_nbody}) is that $\Pk(\Iv)$ must not be
specified {\em a priori}, so each simulation can be interpreted in
infinite different ways by assuming $\Pk(\Iv)$ {\em a posteriori}. Of
course, the aim of $N$-body simulations is to study systems whose
physical properties evolve in time: in the next section we move to
discuss such a case.
\section{Dynamical evolution of composite collisionless stellar systems}
\label{sec:dynevo}
$N$-body simulations are often used to study the dynamical evolution,
in the presence of an external perturbation, of stellar systems that
are initially close to equilibrium. Examples are simulations of the
evolution of satellite stellar systems orbiting within a host stellar
system (for instance satellite galaxies orbiting within a host galaxy)
or simulations of galaxy mergers. In order to illustrate our approach,
let us focus on the case of satellites and consider, for instance, the
simulation of a satellite dwarf galaxy made of stars and DM orbiting
in a host galaxy. As often done in this kind of simulations, we
assume that the host galaxy is represented simply as a static
gravitational potential, while the satellite is represented with
particles as a two-component $N$-body system (with a stellar component
and a DM halo) that would be in equilibrium if isolated
\citep[e.g.][]{Bat15}.
\subsection{Standard multi-component $N$-body models}
\label{sec:evo_standard}
In the standard method the satellite is set up as a two-component
stationary stellar system with $\Nstar$ stellar particles extracted
from a DF $\fstar$ and $\NDM$ DM particles extracted from a DF $\fDM$,
both in equilibrium in the total gravitational potential of the
satellite $\Phitot=\Phistar+\PhiDM$. The total density distribution of
the satellite is $\rhotot=\rhostar+\rhoDM$, where $\rhostar$ is the
density of the stellar component and $\rhoDM$ is the density of the DM
component. At the initial time of the simulation the phase-space
coordinates of the centre of mass of the satellite are assigned so
that the satellite is in orbit in the fixed external gravitational
potential of the host galaxy. Due to the tidal interaction with the
gravitational field of the host galaxy, the satellite evolves
modifying the distributions of its components, for instance producing
tidal tails, and losing stellar and DM particles via tidal
stripping. The relative distribution of the dark and stellar
components of the satellite are fixed in the initial conditions, so
the outcome of the simulation is univocal. To explore the evolution
of a satellite on the same orbit, with the same total distribution
function $\ftot=\fstar+\fDM$, but with different dark and stellar DFs,
a new $N$-body simulation is necessary in this standard approach.
\subsection{Effective multi-component $N$-body models}
\label{sec:dyn_eff}
When the effective multi-component $N$-body modelling is used, the
satellite is set up as a one-component stellar system with $\Ntot$
particles extracted from a DF $\ftot(\Iv)$, with total density
distribution $\rhotot$. As in the standard approach
(\Sect\ref{sec:evo_standard}), at the initial time of the simulation
the satellite is put in orbit in the fixed external gravitational
potential of the host galaxy, and the evolution of all the particles
is followed for the time spanned by the simulation. The simulation is
then interpreted, {\em a posteriori}, by assigning to each particle a
stellar mass and a DM mass, by choosing a stellar portion function
$\Pstar(\Iv)$, where $\Iv$ are the integrals of motion of the particle
{\em when the satellite is set up in equilibrium and isolated}. In
practice, if the $i$-th particle has mass $m_i$, its stellar mass is
$\mstari=\Pstar(\Iv_i)m_i$ and its DM mass is
$\mDMi=m_i-\mstari=[1-\Pstar(\Iv_i)]m_i$, where $\Iv_i$ are the values
of the integrals of motion of the $i$-th particle in the isolated
satellite. For given $\Pstar$, from the simulation we can infer the
evolution of the stellar and DM components of the satellite,
separately, for instance measuring the stellar and DM mass loss due to
tidal stripping. The same simulation can be reinterpreted in infinite
ways by choosing different $\Pstar$.
\begin{figure*}
\centerline{\psfig{file=multi_pstar.eps,width=\hsize}}
\caption{Density $\rhotilde\equiv\rho/(\Mtot a^{-3})$ (bottom row of
panels) as a function of radius and DF $\ftilde \equiv
f/(G^3M_{\rm tot}a^3)^{-1/2}$ (middle row of panels) as a function
of specific relative energy $\Etilde\equiv\E/(G\Mtot a^{-1})$ for
models with the same total distribution (solid curves), but
stellar distributions (dotted, dashed and dot-dashed curves)
obtained with different generalized Schechter stellar portion
functions $\Pstar$ (top row of panels). When not specified
otherwise, the parameters of the portion function
(\Eq\ref{eq:por}) are $\alpha=2$, $\beta=4$, $A=0.3$ and
$\Etildezero\equiv\Ezero/(G\Mtot a^{-1})=0.7$, which are the
values adopted for the model represented by the dotted curves. In
each column, the values of the parameters reported in the top
panel apply also to the middle and bottom panels. $\Mtot$ and $a$
are, respectively, the total mass and scale radius of the total
density profile, which is a Hernquist sphere
(\Eq\ref{eq:den_her}).}
\label{fig:multi_pstar}
\end{figure*}
\section{A simple case: two-component isotropic spherical systems}
\label{sec:isotwocomp}
Here we present an application of the effective $N$-body models
introduced above to spherical two-component collisionless stellar
systems with isotropic velocity distributions.
\subsection{Two-component spherical stellar systems with ergodic distribution functions}
The simplest family of multi-component collisionless stellar systems
generated by DFs is the family of two-component spherical stellar
systems with isotropic velocity distribution. In this case the DFs of
both components are ergodic, i.e. they are functions only of the
energy per unit mass $E$. For the sake of clarity, we specialize to
the case in which one of the component is the stellar component, with
DF $\fstar(\E)$, and the other is the DM halo, with DF $\fDM(\E)$,
where $\E=-E$ is the relative energy per unit mass. The total
distribution function is $\ftot(\E)=\fstar(\E)+\fDM(\E)$. As
explained in \Sects\ref{sec:stat_eff} and \ref{sec:dyn_eff}, when
building an effective $N$-body model of such a system, we consider a
single component with DF $\ftot(\E)$. The stellar and DM components
are defined by choosing a stellar portion function
$0\leq\Pstar(\E)\leq1$, so $0\leq \fstar(\E)\leq \ftot(\E)$
$\forall\,\E$. The portion function of the DM component is
$\PDM(\E)=1-\Pstar(\E)$, so $0\leq \fDM(\E)\leq \ftot(\E)$
$\forall\,\E$. One-component systems with the same $\ftot(\E)$ can be
interpreted as different two-component systems, depending on the
choice of $\Pstar(\E)$. For instance, for an isolated spherical
isotropic system with DF $\ftot(\E)$, the stellar density profile is
\begin{equation}
\rhostar(r)=4\pi\int \Pstar(\E)\ftot(\E) v^2\d v,
\end{equation}
where $\E(r,v)=\Psitot(r)-\frac{1}{2}v^2$ and $\Psitot(r)=-\Phitot(r)$
is the relative total potential (here $r$ is the spherical radial
coordinate and $v$ the magnitude of the velocity vector). The DM
density distribution is
\begin{equation}
\rhoDM(r)=4\pi\int [1-\Pstar(\E)]\ftot(\E) v^2\d v.
\end{equation}
\subsection{An analytic expression of the portion function}
Our aim is to have an analytic expression of $\Pstar(\E)$, depending
on a handful of parameters, flexible enough to represent realistic
stellar components of spheroids. In this work we adopt as analytic
expression of the portion function for spherical isotropic systems the
four-parameter function
\begin{equation}
\Pstar(\E)=A\left(\frac{\E}{\E_0}\right)^\alpha\exp{\left[-\left(\frac{\E}{\E_0}\right)^\beta\right]},
\label{eq:por}
\end{equation}
where $\alpha$, $\beta$ and $A$ are dimensionless parameters, and
$\E_0$ is a characteristic relative energy. In the following we will
refer to this analytic function as generalized Schechter function,
because when $\beta=1$ it reduces to the well known \citet{Sch76}
function, widely used in a different context to model the galaxy
luminosity function. In \Sect\ref{sec:hernq} we show a representative
case in which the generalized Schechter $\Pstar(\E)$ performs well in
producing stellar components with realistic density profiles.
However, we stress that the method proposed in this paper can be
applied with $\Pstar(\E)$ with functional forms different from
\Eq(\ref{eq:por}), for instance with more free parameters if an even
more flexible function is required.
\subsection{A case study: a system with total Hernquist density profile}
\label{sec:hernq}
Let us focus on the case of a self-gravitating system in which the
total density distribution follows a \citet{Her90} profile:
\begin{equation}
\rhotot(r)=\frac{\Mtot}{2\pi a^3}\frac{1}{(r/a)[1+(r/a)]^3},
\label{eq:den_her}
\end{equation}
where $a$ is the scale radius and $\Mtot$ the total mass. This total
density distribution is shown in the bottom row of panels of
\Fig\ref{fig:multi_pstar} as a solid curve. The total gravitational
potential of the system, related to $\rhotot$ by
\Eq(\ref{eq:poisson}), is
\begin{equation}
\Phitot(r)=-\frac{G\Mtot}{r+a}.
\end{equation}
The ergodic DF $\ftot(\E)$ generating a self-gravitating system with
mass density distribution (\ref{eq:den_her}) is know analytically
\citep{Her90} and is shown in the middle row of panels of
\Fig\ref{fig:multi_pstar} as a solid curve.
Such a spherical system with Hernquist total density profile can be
split in a stellar component and a DM component by assuming a stellar
portion function $\Pstar(\E)$. In particular, adopting as $\Pstar$
the generalized Schechter function (\Eq\ref{eq:por}), we can build
stellar components with double power law density profile, whose
detailed properties depend on the values of the parameters $\alpha$,
$\beta$, $A$ and $\Etildezero\equiv\Ezero/(G\Mtot a^{-1})$. For
instance, for $\alpha=2$, $\beta=4$, $A=0.3$ and $\Etildezero=0.7$ we
obtain the stellar portion function, DF and mass density distribution
represented by the dotted curves in \Fig\ref{fig:multi_pstar}: the
bottom row of panels shows that the resulting density profile is a
double power law with logarithmic slope $\gammastar\equiv
\d\ln\rhostar/\d\ln r\simeq -0.5$ in the centre and
$\gammastar\simeq-5.5$ in the outskirts. Different slopes can be
obtained by changing the values of the parameters. The parameter
$\alpha$ determines the probability of having weakly bound stars
(i.e.\ with low relative energy $\E$): in particular the lower
$\alpha$ the shallower the outer stellar density profile (see the
leftmost column of panels in \Fig\ref{fig:multi_pstar}). The parameter
$\beta$ determines the probability of having strongly bound stars
(i.e.\ with high $\E$), in the sense that large values of $\beta$
penalize the most bound orbits, thus the higher $\beta$ the shallower
the inner stellar density profile (see the second column of panels in
\Fig\ref{fig:multi_pstar}): in this case a core of constant density is
obtained for $\beta=12$, while for $\beta=1$ $\rhostar\propto r^{-1}$
in the centre. The parameter $A$, which is the normalization of
$\Pstar$, does not affect the shape of the stellar density profile
but, by shifting vertically $\fstar(\E)$, it determines the fractional
mass contribution of the stellar component, in the sense that the
stars contribute more for higher values of $A$ (see the third column
of panels in \Fig\ref{fig:multi_pstar}). Finally, the parameter
$\Ezero$ tunes the energy $\Epeak$ at which $\Pstar$ peaks, which for
the generalized Schechter function is
$\Epeak=\Ezero\left(\alpha/\beta\right)^{1/\beta}$. Thus, the value of
$\Ezero$ influences mainly the position of the knee of the stellar
density distribution, which is at larger radii for lower $\Ezero$ (see
the rightmost column of panels in \Fig\ref{fig:multi_pstar}). Note,
however, that also the logarithmic slope $\gammastar$ at radii smaller
than the position of the knee changes with $\Ezero$, because the
stellar DF $\fstar$ (shown in the second row of panels in
\Fig\ref{fig:multi_pstar}) depends not only on $\Pstar$, but also on
the shape of $\ftot$. The portion function, DF, and density profile
of the DM component, not shown in \Fig\ref{fig:multi_pstar}, can be
obtained simply by subtraction: $\PDM=1-\Pstar$, $\fDM=\ftot-\fstar$
and $\rhoDM=\rhotot-\rhostar$. All these quantities are guaranteed to
be everywhere positive because $\Pstar<1$ $\forall \E$.
\begin{figure}
\centerline{\psfig{file=density_profile.eps,width=\hsize}}
\caption{Angle-averaged initial ($t=0$, solid curve) and final
($t=12\Gyr$, dashed curve) total (DM plus stars) density profiles
of the satellite in the $N$-body simulation.}
\label{fig:rho}
\end{figure}
\begin{figure*}
\centerline{
\includegraphics[trim=40 20 75 10,clip,height=0.3\textwidth]{map04t.eps}
\includegraphics[trim=70 20 75 10,clip,height=0.3\textwidth]{map04c.eps}
\includegraphics[trim=25 20 25 10,clip,height=0.3\textwidth]{map04e.eps}
}
\centerline{
\includegraphics[trim=40 10 75 20,clip,height=0.3\textwidth]{map08t.eps}
\includegraphics[trim=70 10 75 20,clip,height=0.3\textwidth]{map08c.eps}
\includegraphics[trim=25 10 25 20,clip,height=0.3\textwidth]{map08e.eps}
}
\centerline{
\includegraphics[trim=40 0 75 30,clip,height=0.3\textwidth]{map12t.eps}
\includegraphics[trim=70 0 75 30,clip,height=0.3\textwidth]{map12c.eps}
\includegraphics[trim=25 0 25 30,clip,height=0.3\textwidth]{map12e.eps}
}
\caption{{\em Left column of panels.} Total (DM plus stars) mass
surface density distribution of the satellite in the $N$-body
simulation at $t=4\Gyr$ (top panel), $t=8\Gyr$ (middle panel) and
$t=12\Gyr$ (bottom panel), for a line of sight along the $y$ axis,
in the adopted Cartesian coordinate system, centred in the Galactic
centre, in which the $z$ axis is orthogonal to the Galactic
equatorial plane $xy$. {\em Middle column of panels.} Same as left
column of panels, but showing the stellar mass surface density
distribution of the satellite according to model C. {\em Right
column of panels.} Same as middle column of panels, but for model
E. In each panel the inset represents a zoomed-in surface density
map of $3\kpc\times3\kpc$ centred in the peak of the density
distribution of the satellite.}
\label{fig:snap}
\end{figure*}
\begin{figure}
\centerline{\psfig{file=mass_loss.eps,width=\hsize}}
\caption{Evolution of the total (DM plus stars) mass (black curve)
and of the stellar mass (red curve for model C and green curve for
model E) of the satellite in the $N$-body simulation. Here
$\Mthreekpc$ is the mass within $3\kpc$ from the satellite's
centre and $\Mthreekpci$ is the initial value of $\Mthreekpc$.}
\label{fig:massloss}
\end{figure}
\begin{figure*}
\centerline{\psfig{file=Evoluzione_J95_B04_star_DM.eps,width=\hsize}}
\caption{{\em Upper panels.} Angle-averaged density profiles of the
satellite in the $N$-body simulation at different times, indicated
in the legend, for models C (left panels) and E (right
panels). The red and green curves represent the stellar density,
while the blue curves represent the DM density. {\em Lower
panels.} Stellar and DM mass profiles for the same times and
models as in the corresponding upper panels.}
\label{fig:evo}
\end{figure*}
\section{Application to an $N$-body simulation of tidal stripping}
\label{sec:app}
Here, we apply the effective multi-component method described above to
an $N$-body simulation that follows the evolution of a satellite
galaxy in the gravitational potential of the Milky Way.
\subsection{Set-up of the $N$-body simulation}
\label{sec:setup}
The initial conditions of the $N$-body realization of the satellite
have been produced using the Python module
\texttt{OpOpGadget}\footnote{https://github.com/iogiul/OpOpGadget}
developed by G.\ Iorio. The $N$-body system is realized as a
one-component spherical isotropic stellar system with density profile
\begin{equation}
\rhotot(r)=\frac{\rhozero}{(r/a)[1+(r/a)]^3}\exp{\left[-\left(\frac{r}{\rt}\right)^2\right]},
\label{eq:den_her_trunc}
\end{equation}
representing the total (DM plus stellar) distribution of the
satellite, which is a Hernquist profile (\Eq\ref{eq:den_her})
exponentially truncated at $\rt$. In particular, we adopt $a=0.9\kpc$,
$\rt=17\kpc$ and central density $\rhozero$ such that the total mass
of the system is $\Mtot\equiv4\pi\int_0^\infty \rhotot(r) r^2\d
r=4.5\times10^7\Msun$. The satellite's initial total density
distribution in physical units is shown in \Fig\ref{fig:rho} as a
black solid line. The number of particles is $\Ntot=10^5$, and all
particles have the same mass $m=\Mtot/\Ntot=450\Msun$. The positions
and velocities of the $\Ntot$ particles are assigned in Cartesian
coordinates (relative to the satellite's centre of mass) as in
\citet{Ior19}, using the ergodic DF $\ftot(\E)$ obtained numerically
via Eddington's inversion formula \citep{Edd16}. The $N$-body system
is in equilibrium if isolated, as we verified by running a simulation
with the same initial conditions as that presented in this work, but
with the satellite in isolation, i.e.\ without the Milky Way external
potential.
The simulation was run using the collisionless code \texttt{FVFPS}
\citep{Lon03,Nip03a} with the addition of the axisymmetric Milky Way
model of \citet{Joh95} as external static gravitational potential
\citep[see][]{Bat15}. We adopted $\theta_{\rm min}=0.5$ as the
minimum value of the opening parameter, softening length
$\epsilon=0.02\kpc$ and constant time step $\Delta t= 0.01 \tdyn$,
where $\tdyn = 1/\sqrt{G \bar{\rho}_{\rm h}}$ is the initial dynamical
time of the satellite and $\bar{\rho}_{\rm h}$ is its initial average
density within the stellar half-mass radius $r_{\rm h}$. For the
adopted initial conditions $\tdyn\simeq 3.5\times 10^8\yr$.
As orbit of the satellite we assume the orbit dubbed P07ecc in
\citet{Bat15}, which is almost polar with eccentricity $\simeq0.4$ and
pericentric radius $\simeq 61\kpc$. At the initial time of the
simulation the phase-space coordinates of the centre of mass of the
satellite are $(x,y,z)=(35.814,\ 0,\ 137.389) \ \kpc$ and
$(v_x,v_y,v_z)=(-94.875, -77.81, 2.901 ) \ \kms$, in a Cartesian
coordinate system, centred in the Galactic centre, in which $xy$ is
the Galactic equatorial plane. The simulation is evolved for
$12\Gyr$. For each snapshot of the simulation we measure the
angle-averaged density distribution $\rhotot(r)$ and integrated total
mass distribution $M(r)$, by binning the particles in concentric
spherical shells. Here $r$ is the distance from the satellite's
centre, which is defined as the position of the peak of the density
distribution of the satellite, computed as in \citet{Ior19}. In a
similar way, for given stellar portion function $\Pstar$, we can
measure for each snapshot the angle-averaged stellar density
distribution $\rhostar(r)$ and stellar mass profile $\Mstar(r)$, by
weighting the particles' masses as described in
\Sect\ref{sec:stat_eff}. The DM density and mass distributions are
obtained using as portion function $\PDM=1-\Pstar$.
\subsection{Results}
\subsubsection{Evolution of the total mass distribution}
The projected total (DM plus stars) density distribution of the
satellite at different times in the simulation is shown in
\Fig\ref{fig:snap} (left column of panels), for a line of sight
parallel to the equatorial plane of the Milky Way. As expected, the
initially spherical density distribution of the satellite is distorted
by the interaction with the tidal force field of the Milky Way, which
produces two significant tidal tails, one leading and one trailing,
departing from the main body of the disrupting satellite. However, as
illustrated by the zoomed-in surface density maps in the insets in
\Fig\ref{fig:snap}, the central regions remain close to spherical
symmetry. While the central total density profile hardly evolves, at
larger radii the total density profile changes drastically with time,
and at $t=12\Gyr$ (black dashed curve in \Fig\ref{fig:rho}) it is
heavily truncated at $r\approx 1\kpc$ and characterized by a shallow
tail at $r\approx 10\kpc$ produced by the stripped particles. To
quantify the mass loss we take as reference mass at each time the mass
$\Mthreekpc$ of all the particles within a sphere of radius $r=3\kpc$
from the centre of the satellite. The choice of $3\kpc$ as reference
radius (which is about twice the initial half-mass radius) is somewhat
arbitrary, but is empirically motivated by the requirement to include
most of the stellar mass at $t=0$ (see \Sect\ref{sec:evostellardm})
and to exclude most of the stellar tidal tails in the subsequent
snapshots (see insets in \Fig\ref{fig:snap}). We note that
$\Mthreekpc\simeq 0.69\Mtot$ at $t=0$. The black curve in
\Fig\ref{fig:massloss}, which plots $\Mthreekpc$ as a function of
time, shows that, within $3\kpc$, the satellite loses almost $70\%$ of
its initial mass over 12 Gyr of evolution.
\subsubsection{Evolution of the stellar and dark matter mass distributions}
\label{sec:evostellardm}
The simulation is interpreted {\em a posteriori} in different ways by
choosing different portion functions $\Pstar(\E)$, where $\E$ is the
{\em initial} particle relative energy, computed for the isolated
satellite. Here we consider two models: {\em model C}, in which the
initial stellar distribution is more {\em compact}, and {\em model E},
in which the initial stellar distribution is more {\em extended}. Both
models are obtained assuming as functional form of $\Pstar$ the
generalized Schechter function (\Eq\ref{eq:por}). The values of the
parameters of $\Pstar$ are $\alpha=3$, $\beta=1$, $A=1.35$, and
$\Etildezero=0.8$ for model C, and $\alpha=0.5$, $\beta=12$, $A=0.5$
and $\Etildezero=0.7$ for model E. The initial stellar density
profile of model C (red solid curve in upper left panel of
\Fig\ref{fig:evo}) has a central cusp ($\rhostar\propto r^{-1}$) and
declines steeply in the outer parts, while the stellar density profile
of model E (green solid curve in upper right panel of
\Fig\ref{fig:evo}) has a central core ($\rhostar\propto {\rm constant}$)
and is shallower in the outskirts. The position of the knee of the
stellar density profile (i.e.\ the radius of transition between inner
and outer slope) occurs at larger radius for model E than for model C.
The stellar and DM density and mass profiles at different times in the
simulation are shown in \Fig\ref{fig:evo} for model C in the left
column of panels and for model E in the right column of panels. In
model E the initial DM density is higher than the initial stellar
density at all radii. In model C the initial stellar density is higher
than the DM density in the centre ($r\lesssim 200\pc$), while the dark
halo dominates at larger radii. In both cases the evolution of the DM
density profile resembles that of the total mass distribution, with
substantial losses at large radii. The evolution of the stellar
component is instead very different in the two cases: the stellar
distribution of model C remains almost unaltered for 12 Gyr, while it
is heavily stripped in model E. The fractional stellar mass loss for
the two models is quantified in \Fig\ref{fig:massloss} using as
reference the stellar mass within a sphere of radius $3\kpc$ from the
satellite's centre: over 12 Gyr in model C the satellite loses about
$30\%$ of its stellar mass, while in model E more than $70\%$ of the
stellar mass is tidally stripped along the orbit. We note that the
reference radius $r=3\kpc$ encloses the large majority of the stellar
mass at $t=0$ ($98\%$ in model C and $78\%$ in model E), but is small
enough to exclude most of the tidal tails during the orbital
evolution.
The extent, density and morphology of the stellar tidal tails can be
assessed by looking at \Fig\ref{fig:snap}, showing, for models C
(middle column of panels) and E (right column of panels), the
projected stellar density distribution of the satellite at different
times in the simulation for a line of sight parallel to the Galactic
equatorial plane. The stellar streams are extremely tenuous in model
C, while are much more pronounced in model E.
\subsubsection{Stellar kinematics}
\label{sec:stellarkin}
{Here we study the stellar kinematics of the satellite and of the
streams, focusing in particular on the line-of-sight stellar
velocity dispersion $\sigmalos$. As an illustrative case, we take as
line of sight the direction of the $y$ axis in our reference
Galactic Cartesian coordinate system and we take as fiducial
boundary between the main body of the satellite and the tidal tails
$R=3\kpc$, where $R=\sqrt{(x-x_0)^2+(z-z_0)^2}$ is the projected
distance from the satellite's centre $(x_0,z_0)$ in the $xz$
plane. For the main body of the satellite we compute $\sigmalos$
from sets of particles belonging to circular annuli: the
line-of-sight velocity dispersion $\sigmalosj$ of the $j$-th radial
bin $R_j<R<R_{j+1}$ is given by
\begin{equation}
\sigmalosjsquared=
\frac{\sum_i\mstari \left(\vyi-\av{\vy}\right)^2}
{\sum_i\mstari},
\end{equation}
where $\vyi$ is the $y$ component of the velocity of the $i$-th
particle,
$\av{\vy}=\left(\sum_i\mstari\vyi\right)/\left(\sum_i\mstari\right)$,
and the sums are over all particles with $R_j<R_i<R_{j+1}$. Here
$\mstari=\Pstar(\E_i)m_i$, where $m_i$ is the mass of the $i$-th
particle and $\E_i$ is its initial energy in the isolated satellite.
\Fig\ref{fig:sigmalossat} shows the satellite's initial and final
profiles of $\sigmalos$ for models C and E. In the initial conditions
$\sigmalos$ is higher for model E, which has a flatter stellar density
profile, than for model C, which has a steeper stellar density
profile (see upper panels of \Fig\ref{fig:evo}). This just reflects
the fact that, for a given gravitational potential, a higher velocity
dispersion is needed to maintain in equilibrium a more extended
stellar component. For both models the final $\sigmalos$ profile has
a shape similar to the corresponding initial profile, but lower
normalization: $\sigmalos$ decreases with time mainly because the
potential well becomes shallower, owing to substantial mass loss (see
\Fig\ref{fig:evo}, lower panels).
\begin{figure}
\centerline{\psfig{file=velocity_dispersion_3kpc.eps,width=\hsize}}
\caption{{Line-of-sight (along the $y$ axis) stellar velocity
dispersion profile for the satellite at the beginning and
at the end of the simulation for models C and E.}}
\label{fig:sigmalossat}
\end{figure}
It is also interesting to assess how the kinematics of the stellar
tidal tails depends on the initial stellar density distribution. For
this purpose we consider the $t=12\Gyr$ snapshot and, taking again the
$y$ axis as line of sight, we distinguish the leading tail (lying
above and to the right of the satellite in the bottom panels of
\Fig\ref{fig:snap}) and the trailing tail (lying to the left of the
satellite in the bottom panels of \Fig\ref{fig:snap}). Specifically,
we assign to the leading tail all particles with $R>3\kpc$ and
$z>z_0-1.8(x-x_0)$, and to the trailing tail all particles with
$R>3\kpc$ and $z<z_0 -1.8(x-x_0)$, with $x_0=31.9\kpc$ and
$z_0=-124.8\kpc$.
$\sigmalos$ as a function of $R$ is shown in
\Fig\ref{fig:sigmalostail} for the leading and trailing tails of
models C and E. For given model, the two tails have similar
$\sigmalos$ profiles out to $R\approx 40\kpc$: at larger distances
form the satellite the leading tail tends to have higher stellar
velocity dispersion than the trailing tail. For given tail (leading
or trailing), the $\sigmalos$ profile is systematically higher for
model E than for model C, which reflects the higher velocity
dispersion of the stellar component of model E in the initial
conditions (\Fig\ref{fig:sigmalossat}). To quantify the overall
velocity dispersion of each tail, we compute the quantity
$\sigmalostail$, defined by
\begin{equation}
\sigmalostail^2=
\frac{\sum_{j=1}^{\Nbin}\sigmalosjsquared\Sigmastarj}{\sum_{j=1}^{\Nbin}\Sigmastarj},
\end{equation}
where $\sigmalosj$ and $\Sigmastarj$ are, respectively, the
line-of-sight stellar velocity dispersion and stellar surface density
of the $j$-th radial bin of the tail (we used $\Nbin=24$ bins
uniformly spaced in $R$ between $R\simeq3\kpc$ and $R\simeq100\kpc$).
The leading tail has $\sigmalostail\simeq 2.8\kms$ for model C and
$\sigmalostail\simeq 3.6\kms$ for model E; the trailing tail has
$\sigmalostail\simeq 2.8\kms$ for model C and $\sigmalostail\simeq
3.5\kms$ for model E.
\begin{figure}
\centerline{\psfig{file=velocity_dispersion_tail.eps,width=\hsize}}
\caption{Line-of-sight (along the $y$ axis) stellar velocity
dispersion profile of the leading and trailing tidal tails at the
end of the simulation for models C and E.}
\label{fig:sigmalostail}
\end{figure}
\subsubsection{A family of models with smoothly varying $\Pstar$}
\label{sec:deponpstar}
So far we have applied to our simulation two models (C and E), that is
two choices of $\Pstar$. However, the power of the presented method
lies in the fact that infinite models can be explored by varying
continuously the values of the parameters of $\Pstar$. Thus we
illustrate here how some properties of the satellite and of the tails
vary in entire family of $n$ models whose extremes are models C and
E. The $i$-th member of this family of models (for $i=1,...,n$)
has $\Pstar(\E)$ given by \Eq(\ref{eq:por}) with parameters
\begin{equation}
\alpha=\alphaC+\frac{i-1}{n-1}(\alphaE-\alphaC),
\end{equation}
\begin{equation}
\beta=\betaC+\frac{i-1}{n-1}(\betaE-\betaC),
\end{equation}
\begin{equation}
A=\AC+\frac{i-1}{n-1}(\AE-\AC),
\end{equation}
and
\begin{equation}
\Ezero=\EzeroC+\frac{i-1}{n-1}(\EzeroE-\EzeroC),
\end{equation}
where ($\alphaC$, $\betaC$, $\AC$, $\EzeroC$) and ($\alphaE$,
$\betaE$, $\AE$, $\EzeroE$) are the sets of values of parameters
of models C and E, respectively (see
\Sect\ref{sec:evostellardm}). With this definition we get model C for
$i=1$ and model E for $i=n$; for $1<i<n$ we get models with $\Pstar>0$
that are intermediate between models C and E: the stellar component is
more embedded in the DM halo for lower values of $i$. Each member of
this family of models can be conveniently labelled with the value of
its initial ($t=0$) stellar half mass radius $\rhalfstar$ (that is the
radius of the sphere containing half of the stellar mass), which
increases monotonically with $i$. The initial stellar density profile
of the simulated satellite is shown in \Fig\ref{fig:rhoembed} for
models C and E, and for three representative intermediate models,
labelled with their values of $\rhalfstar$.
\Fig\ref{fig:rhalf} shows the dependence on $\rhalfstar$ of some
global properties of the stellar component of the simulated satellite
for the family of models defined above. The upper panel of
\Fig\ref{fig:rhalf} plots the fraction of stellar mass lost (defined
as the stellar mass in particles more distant than 3 kpc from the
satellite's centre) as a function of $\rhalfstar$ after $5\Gyr$ and
$12\Gyr$ of evolution. The fraction of stellar mass lost increases
smoothly from more embedded (smaller $\rhalfstar$) to less embedded
(larger $\rhalfstar$) models. The lower panel of \Fig\ref{fig:rhalf}
plots the line-of-sight stellar velocity dispersion $\sigmalostail$
(see \Sect\ref{sec:stellarkin}) of the leading and trailing tails as a
function of $\rhalfstar$. $\sigmalostail$, which is similar for the
two tails for a given model, increases smoothly with $\rhalfstar$: the
less embedded the initial stellar component, the higher the velocity
dispersion of the stellar streams.
\begin{figure}
\centerline{\psfig{file=rhoembed.eps,width=\hsize}}
\caption{Initial stellar density profile of the simulated
satellite for models C and E, and for three intermediate models,
labelled with the value of their stellar half-mass radius.}
\label{fig:rhoembed}
\end{figure}
\begin{figure}
\centerline{\psfig{file=sigma_and_f_lost.eps,width=\hsize}}
\caption{{\em Upper panel.} Fraction of stellar mass lost by the
satellite in the simulation after $5$ and $12\Gyr$ of evolution as
a function of the initial stellar half-mass radius for a family of
models with smoothly varying initial stellar density distribution,
ranging from the compact model C to the extended model E. Here
$\Mstarlost$ is the stellar mass in particles more distant than
$3\kpc$ from the satellite's centre and $\Mstari$ is the initial
stellar mass. {\em Lower panel.} Final line-of-sight stellar
velocity dispersion of the leading and trailing tails in the
simulation for the same family of models as in the upper panel. }
\label{fig:rhalf}
\end{figure}
}
\section{Discussion and conclusions}
\label{sec:concl}
We have presented a new approach to $N$-body modelling of composite
collisionless stellar systems. The method, which we refer to as
effective multi-component $N$-body modelling, allows one to build a
one-component system, and interpret it {\em a posteriori} in infinite
ways as a multi-component system using functions of the integrals of
motion, dubbed portion functions. In an $N$-body simulation the
construction of the different components can be done in post
processing, thus greatly extending the applicability of the
simulation. As an example of application, we presented the results of
an $N$-body simulation of a satellite orbiting in the tidal field of
the Milky Way, which is interpreted {\em a posteriori} as a
two-component (stars plus DM) system. {This example nicely
illustrates the potential of the presented method, by showing the
dependence of the structure and kinematics of the final satellite
and stellar streams on the choice of the portion function.}
For simplicity, we have presented as an application only the case in
which the parent one-component stellar system is spherical and
isotropic, and the portion function depends only on the initial
particle energy. But the very same method can be applied to
anisotropic spherical system as well as to non spherically symmetric
systems, provided their DF is known analytically or numerically. For
instance, one could build anisotropic multi-component spherical
systems with total DF $\ftot=\ftot(\E,L)$, where $L$ is the magnitude
of the angular momentum \citep[see][]{Bin08}, by using portion
functions $\Pk(\E,L)$. {A straightforward case is that of
Osipkov-Merritt anisotropic spherical models \citep{Osi79,Mer85}, in
which the DF is a function of a single variable $Q$, which is a
combination of $\E$ and $L$, so $\Pk=\Pk(Q)$.} Moreover, the method
is not limited to spherical systems, and can be also applied to
axisymmetric systems with total DF $\ftot=\ftot(\E,\Lz)$, where $\Lz$
is the component of the angular momentum along the symmetry axis
\citep[see][]{Bin08}, using portion functions $\Pk(\E,\Lz)$, as well
as to both spherical and flattened models with total distribution
function $\ftot(\Jv)$ depending on the action integrals $\Jv$
\citep[e.g.][]{Bin14,Vas19}, and portion functions $\Pk(\Jv)$.
Of course, the presented effective $N$-body modelling method has its
own limitations. A necessary condition to use the effective modelling,
and thus to obtain the components' DFs by subtraction from the total
DF, is to know, numerically or analytically, the total DF, which can
be straightforward only in systems in which the total distribution is
simple, for instance because one of the components (typically the DM
halo) is dominant. Moreover, the construction of the portion functions
is relatively easy when the shapes of the system's components are
simple and similar among each other, but can be unfeasible in very
complex configurations. However, as it is well known, the build-up of
a complex composite stellar system (for instance an equilibrium galaxy
model with disc, bulge and non-dominant dark halo) is a hard task also
in standard approaches based on the DFs of the system's components.
The main power of the effective $N$-body modelling is that the
components of a composite simulated stellar system can be assigned in
post-processing. This is especially useful when a simulation aims to
reproduce an observed distribution of stars, as it is often the
case. A typical case is that in which the composite system consists of
a stellar component and a DM halo. For a given simulation, one can
{\em a posteriori} explore the space of the free parameters of the
stellar portion function (for instance the four-parameter space
$\alpha$, $\beta$, $A$ and $\Ezero$, when $\Pstar$ is in the form of
\Eq\ref{eq:por}) to find the set of parameters (and thus the initial
stellar and DM distributions) such that the final stellar distribution
represents best the observed data. In the near future we are going to
apply this approach to try to reproduce with $N$-body simulations the
observed properties of satellite dwarf spheroidal galaxies and
reconstruct their dynamical evolution and stellar mass loss history.
\section*{Acknowledgements}
FC acknowledges support from grant PRIN MIUR 20173ML3WW00 and from the INAF main-stream (1.05.01.86.31).
\section*{Data Availability}
The data underlying this article will be shared on reasonable request
to the corresponding author.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,472
|
\section{Introduction}
Black holes and string theory have a long and intimate relationship, see \cite{Bowick:1985af,Wiltshire:1988uq,Callan:1988hs,tHooft:1990fkf,Witten:1991yr,Dijkgraaf:1991ba,Horowitz:1993jc,Susskind:1993ki,Susskind:1993ws,Sen:1995in,Strominger:1996sh,Maldacena:1996ky,Solodukhin:1997yy,Larsen:1997ge,Cvetic:1997uw,Youm:1997hw,Maldacena:1998bw,Peet:2000hn,Lunin:2002qf,Ooguri:2004zv,Arkani-Hamed:2006emk,Konoplya:2011qq,Chen:2021dsw} and Refs.~therein. An emblematic result by Strominger and Vafa is the microscopic origin of the Bekenstein--Hawking (BH) entropy
\eq{
S_{\textrm{\tiny BH}} = \frac{\textrm{Area}}{4G}
}{eq:null1}
of certain extremal black holes \cite{Strominger:1996sh}. The BH-law \eqref{eq:null1} is a template for falsification in quantum gravity, given the paucity of experimental data (see e.g.~Sec.~10.2 in \cite{Carlip:2001wq}). That is why the {\tt hep-th} and {\tt gr-qc} communities spent a lot of resources deriving the BH-law microscopically for more general black holes \cite{Horowitz:1996fn,Maldacena:1996gb,Gubser:1996de,Ferrara:1996um,Klebanov:1996un,David:2002wn,Dabholkar:2004yr,Mathur:2005zp,Sen:2007qy,Denef:2007vg,Bena:2007kg,Skenderis:2008qn} (see also \cite{Ashtekar:1997yu} for an alternative proposal).
Despite these efforts, it is unclear how to construct stringy microstates for non-extremal black holes, which is an obstacle to further progress in string theory and black holes since extremal black holes are zero-temperature states that do not occur in Nature. Another slightly puzzling aspect of \eqref{eq:null1} is its universality: For large black holes, the entropy depends only on the horizon area and not on any details of the microstates.
A complementary approach to deriving the BH-law \eqref{eq:null1} is to focus on near horizon-symmetries, see e.g.~\cite{Strominger:1997eq,Carlip:1998wz,Cvetic:1998xh,Birmingham:1998jt,Cadoni:1998sg,Carlip:2002be,Padmanabhan:2003gd,Guica:2008mu,Bagchi:2012xr,Barnich:2012xq,Donnay:2015abr,Hawking:2016msc,Afshar:2016uax,Afshar:2016wfy,Afshar:2016kjj,Afshar:2017okz,Carlip:2017xne,Haco:2018ske,Grumiller:2019fmp}. In these derivations, the universality of the BH-law \eqref{eq:null1} and its applicability to non-extremal black holes becomes more transparent, but the precise identification of the microstates typically remains obscure or requires ad-hoc input: Even when succeeding in counting the microstates from some horizon-symmetry arguments it remains unclear what these microstates are.
The goal of our Letter is to combine both approaches and to construct, based on horizon-symmetry considerations, stringy microstates of non-extremal black holes. For technical reasons, we restrict ourselves to three-dimensional (3d) black holes (BTZ black holes) \cite{Banados:1992wn}.
We postulate that the string worldsheet relevant for the microscopic description of the BTZ black hole coincides with the black hole horizon, a null hypersurface generated by some null vector $\ell$. The symmetries preserving such a null hypersurface are a direct sum of diffeomorphisms of the null hypersurface and scalings of the null vector $\ell$ \cite{Adami:2020ugu}. In 3d, these are precisely the symmetries of the worldsheet of a null string theory \cite{Schild:1976vq,Isberg:1993av,Bagchi:2013bga,Bagchi:2015nca, Bagchi:2020fpr,Bagchi:2019cay, Bagchi:2020ats}, thus motivating our postulate.
Our main result derived from this postulate (and additional assumptions that we shall spell out) is an explicit list of BTZ microstates in terms of null string excitations (explained below) and proof that their number accounts for the correct BH-law \eqref{eq:null1} as well as the subleading semiclassical corrections thereof (see \cite{Sen:2012dw} and Refs.~therein).
\section{Horizon strings}
We start with a classical analysis of null strings on the BTZ black hole horizon. A theory of null strings is consistent only if the string tension vanishes, i.e., null strings are tensionless \cite{Schild:1976vq, Isberg:1993av, Bagchi:2013bga,Bagchi:2015nca}. Tensionless null strings on a target space with coordinates $X^\mu$ and metric $G_{\mu\nu}$ are governed by the action \cite{Isberg:1993av} ($a=1,2$ and $\sigma\sim\sigma+2\pi$ for closed strings~\footnote{See the supplemental material for a brief review of the near horizon and tensionless limits.})
\eq{
{\cal S}=\frac{\kappa}{2}\,\int \extdm \!\tau\extdm \!\sigma\ \big(V^a\partial_a X^\mu\big)\,\big(V^b\partial_b X^\nu\big) \,G_{\mu\nu}(X)\,.
}{eq:null2}
We fix the coupling constant $\kappa$ below in Eq.~\eqref{eq:null28}. The weight-$\frac12$ vector density $V^a$ is gauge-fixed to $V=(1,0)$ exploiting worldsheet diffeomorphisms \cite{Isberg:1993av, Bagchi:2015nca}. We pick for the target space the BTZ metric evaluated in a suitable gauge and a co-rotating frame zooming onto the horizon (with radius $R_h$)
\eq{
G_{\mu\nu}\extdm \! X^\mu\extdm \! X^\nu= -2 \extdm \! x^+ \extdm \! x^- + R_h^2 \,\extdm \!\phi^2 = -2 \extdm \! x^+ \extdm \! x^- + \extdm \!\varphi^2\,.
}{eq:null3}
The lightcone coordinates $x^\pm$ take arbitrary real values,
and the angular coordinate is periodic, $\phi\sim\phi+2\pi$. The gauge-fixed version of the null string action \eqref{eq:null2} for the target space metric \eqref{eq:null3} simplifies to
\eq{
{\cal S}_{\textrm{\tiny gf}}= \frac{\kappa}{2}\,\int \extdm \!\tau\extdm \!\sigma\ \Big(-2(\partial_\tau X^+)(\partial_\tau X^-) + (\partial_\tau X^\varphi)^2\Big)
}{eq:null4}
where $X^\varphi=R_h\,X^\phi$. Varying the gauge-fixed action \eqref{eq:null4} yields the equation of motion
\eq{
\partial_\tau^2 X^\mu = 0
}{eq:null5}
solved by the mode expansion \cite{Bagchi:2015nca}
\eq{
X^{\mu}(\tau,\sigma)=x^\mu+A^\mu_0 \sigma+ B_0^\mu\tau+i\sum_{n\neq0}\frac{1}{n} \left(A^\mu_n-in\tau B^\mu_n \right)e^{-in\sigma}\,.
}{eq:null6}
We allow closed strings to wind around the compact $\varphi$-direction,
\eq{
X^{\varphi}(\sigma + 2\pi,\tau) = X^{\varphi}(\sigma,\tau) + 2 \pi\, R_h\, \omega\qquad\omega\in\mathbb{Z}
}{eq:null7}
and have trivial identifications otherwise, $X^\pm(\sigma + 2\pi,\tau) = X^\pm(\sigma,\tau)$, implying $A_0^\varphi= R_h \omega$ and $A_0^\pm= 0$. As usual in string theory \cite{Zwiebach:2004tj,Polchinski:1998rq}, the momentum along the circle is quantized in units of one over its radius.
\eq{
p^\varphi=\kappa\,B_0^\varphi=\frac{n}{R_h} \qquad n\in \mathbb{Z}
}{eq:null8}
We refer to null strings on the BTZ black hole horizon as ``horizon strings''.
\section{Constraints and physical state conditions}
In addition to the equation of motion \eqref{eq:null5}, the horizon string modes \eqref{eq:null6} classically must obey the constraint equations
\eq{
G_{\mu\nu}\,\partial_\t X^{\mu} \partial_\sigma X^{\nu} = G_{\mu\nu}\,\partial_\t X^{\mu} \partial_\t X^{\nu}=0
}{eq:null9}
that follow from varying the action \eqref{eq:null2} with respect to $V^a$. In terms of the bilinears \cite{Bagchi:2015nca}
\eq{
L_m := \kappa\,\sum_n A^\mu_{n} B^\nu_{m-n} G_{\mu\nu}\quad\; M_m := \kappa\,\sum_n B^\mu_n B^\nu_{m-n} G_{\mu\nu}
}{eq:null10}
the constraints \eqref{eq:null9} are $L_m=M_m=0$. The notation chosen for the bilinears is compatible with conventions for the Bondi--van~der~Burgh--Metzner--Sachs algebra in 3d (BMS$_3$), see e.g.~\cite{Bagchi:2012yk,Bagchi:2020fpr}; it is facile verifying that $L_m, M_m$ obey the (centerless) BMS$_3$ algebra \cite{Ashtekar:1996cd,Barnich:2006av} employing the canonical commutators \eqref{eq:null12} below. The appearance of BMS-symmetries is compatible with residual worldsheet diffeomorphisms generated by vector fields $\xi=(h(\sigma)+\tau f^\prime(\sigma))\,\partial_\tau+f(\sigma)\,\partial_\sigma$, where $h$ generates so-called supertranslations and $f$ superrotations. These residual diffeomorphisms are the most general ones that leave invariant the gauge-fixed vector density $V^a=(1,0)$, which transforms as $\delta_\xi V^a = \xi^b\partial_b V^a - V^b\partial_b\xi^a + \frac12\,V^a\,\partial_b\xi^b$.
Standard canonical quantization, $[X^\mu,\,\kappa\,\partial_\tau X^\nu]=i\,G^{\mu\nu}$, yields the non-zero commutation relations
\eq{
\big[A_n^\mu,\,B_m^\nu\big] = \frac{n}{\kappa}\,G^{\mu\nu}\,\delta_{n+m,\,0} \qquad \big[x^\mu,\,B_0^\nu\big]=\frac{i}{\kappa}\,G^{\mu\nu}\,.
}{eq:null12}
Imposing the quantum version of the constraints \eqref{eq:null9} --- analogous to the Virasoro constraints for the tensile string \cite{Polchinski:1998rq} --- yields physical state conditions \cite{Bagchi:2020fpr}
\eq{
\langle\textrm{phys}^\prime| L_m |\textrm{phys}\rangle = 0 = \langle\textrm{phys}^\prime| M_m |\textrm{phys}\rangle
}{eq:null13}
where $|\textrm{phys}\rangle$ and $|\textrm{phys}^\prime\rangle$ denote generic physical states.
\section{Vacuum choice and near horizon symmetries}
Additional subtleties of the physical state conditions \eqref{eq:null13} and their relation to different vacuum choices are discussed in \cite{Bagchi:2020fpr}. One such subtlety is a normal-ordering contribution to $L_0$, which turns out to vanish for our construction. More relevantly, there are three different eligible vacuum conditions for tensionless strings. They are, respectively, defined by the conditions 1.~$B_n^\varphi|0\rangle=0$ for $n>0$, 2.~$B_n^\varphi|0\rangle=0$ for $n\neq 0$, and 3.~$B_n^\varphi|0\rangle\neq 0$ but $\langle 0|B_n^\varphi|0\rangle=0$ for $n\neq 0$. The last option is called ``oscillator vacuum''. In the lightcone gauge, only the $\varphi$-component matters because one can fix the residual diffeomorphisms such that $X^+=x^++B_0^+\tau$ and $X^-$ is determined by $X^\varphi$, see \cite{Bagchi:2021rfw}.
We choose the oscillator vacuum
\eq{
\C_n|0\rangle = 0 = \tC_n|0\rangle\qquad n\in\mathbb{Z}^+
}{eq:null15}
where the operators
\eq{
\C_n := \sqrt{\tfrac{\kappa}{2}}\,\big(A_n^\varphi+B_n^\varphi\big)\qquad
\tC_n := \sqrt{\tfrac{\kappa}{2}}\,\big(-A_{-n}^\varphi+B_{-n}^\varphi\big)
}{eq:null16}
obey an oscillator algebra (commutators not displayed vanish)
\eq{
\big[\C_n,\,\C_m\big]=n\,\delta_{n+m,\,0}=\big[\tC_n,\,\tC_m\big]\,.
}{eq:null17}
Remarkably, the oscillator algebra \eqref{eq:null17} appears as BTZ near horizon symmetries generating soft (Heisenberg) hair \cite{Afshar:2016wfy, Afshar:2016kjj}. This algebraic equivalence provides confidence that the oscillator vacuum is appropriate for BTZ black holes.
\section{Hilbert space, level-matching, and mass}
To construct the physical Hilbert space, we start with classifying states at different levels $\level\in\mathbb{N}$,
\begin{eqnarray*}
&&{\text{Vacuum:}} \; |0, p^\mu, \omega \rangle \equiv |0\rangle \\
&&{\text{Level 1:}} \; \C_{-1} |0\rangle, \tC_{-1} |0\rangle \\
&&{\text{Level 2:}} \; \C_{-2} |0\rangle, \C_{-1}^2 |0\rangle, \C_{-1}\tC_{-1} |0\rangle, \tC_{-1}^2 |0\rangle, \tC_{-2} |0\rangle \\
&& \ldots
\end{eqnarray*}
where $p^\mu=\kappa\,B^\mu_0$. The level splits into two integers associated with each set of oscillator modes, $\level=r+s$. For instance, the five states above at level $\level=2$ have, respectively, $(r,s)=(2,0)$, $(r,s)=(2,0)$, $(r,s)=(1,1)$, $(r,s)=(0,2)$ and $(r,s)=(0,2)$. The levels $r,s$ in turn can be decomposed into a collection of integers associated with individual creation operators, $r=\sum_i ir_i$ and $s=\sum_i is_i$. For example, the first two states at level $\level=2$ with $r=2$ and $s=0$ split, respectively, into $r_2=1$ and $r_1=2$ (with all other $r_i=0$ in each case).
A generic state in the above set,
\eq{
|\Psi\rangle = |p^\mu,\,\{r_i\},\,\{s_i\},\,\omega\rangle,
}{eq:null18}
is given by arbitrary combinations of the creation operators $\C_{-m},\tC_{-m}$ acting on the oscillator vacuum $|0,\,p^\mu,\,\omega\rangle$. Physical states are a subclass of these generic states subject to the constraints \eqref{eq:null13}. We are interested in physical states without momentum in the radial direction, which in our lightcone coordinates implies $p^+=p^-$, and we keep $p^\varphi$ arbitrary, see \eqref{eq:null8}.
We now follow a route analogous to the tensile string \cite{Zwiebach:2004tj,Polchinski:1998rq} and impose the physical state conditions \eqref{eq:null13} to obtain the physical Hilbert space. For the zero modes, $L_0$ and $M_0$, these give us, respectively, a level-matching condition and a formula for the mass spectrum of the theory. The remaining physical state conditions are automatically satisfied once states are level matched (for a more detailed treatment, see \cite{Bagchi:2020fpr}). By virtue of the oscillator vacuum \eqref{eq:null15}-\eqref{eq:null17}, the requirement $L_0|\Psi\rangle = 0$ establishes a level-matching condition
\eq{
s-r=\omega\,n \,.
}{eq:null20}
From the vanishing of the $M_0$-eigenvalue, $M_0\,|\Psi\rangle = 0$, we deduce the mass $m:=\sqrt{2p^+p^-}$ of the state $|\Psi\rangle$,
\eq{
m^2 =(r+s){\kappa}+\frac{n^2}{R_h^2}\,.
}{eq:null42}
Physical states of a given mass are thus labeled by the integers $r_i, s_i, \omega$, and $n$ subject to the level-matching \eqref{eq:null20} and the mass-shell condition \eqref{eq:null42}.
\section{BTZ black hole microstates}
We label BTZ black holes by the horizon string mass $m$ and define the set of BTZ black hole microstates as the collection of all physical states in the horizon string Hilbert space. Each microstate
\eq{
\btz=|\{r_i\},\,\{s_i\},\,\omega,\,n\rangle
}{eq:null22}
is labeled by a collection of mode excitation numbers $r_i,s_i$, the winding number $\omega$, and the momentum number $n$, subject to the level-matching \eqref{eq:null20} and the mass-shell condition \eqref{eq:null42}.
The remaining task is to fix the value of the mass $m$ in terms of the geometric input, the value of the horizon radius $R_h$. Since $m$ is the mass of our string states at the horizon, on dimensional grounds it is plausible to identify it (up to some factor) with the near horizon mass of the associated BTZ black hole. Due to the near horizon first law, the latter scales linearly in the horizon radius \cite{Donnay:2015abr,Afshar:2016wfy}, i.e.,
\eq{
m = \kappa R_h\,.
}{eq:null23}
While we could include some arbitrary numerical factor in the relation \eqref{eq:null23}, we absorb such a factor by the freedom to fix the coupling constant $\kappa$, which we shall do below in Eq.~\eqref{eq:null28}.
Plugging the mass \eqref{eq:null23} into the result \eqref{eq:null42},
\eq{
\kappa R_h^2 = s+r + \frac{n^2}{\kappa R_h^2} := \level + \frac{n^2}{\kappa R_h^2}\,,
}{eq:null41}
and assuming $\level \gg n$ yields
\eq{
\kappa R_h^2 = \level + \frac{n^2}{\level} + {\cal O}(n^4/\level^3) \,.
}{eq:null40}
In the other limit, $N\ll n$, we have instead $\kappa R_h^2=n+\tfrac N2+{\cal O}(N^2/n)$.
\section{Combinatorics of microstates}
There are various sectors of states, depending on the behavior of the quantum numbers. We discuss all of them and determine their respective combinatorics. In all cases, we assume $R_h\gg1/\sqrt{\kappa}$
to guarantee the validity of the semiclassical approximation.
\subsection{Soft sector}
When the string has vanishing momentum, $n=0$, we call it soft. In this case, there is a mundane infinite degeneracy from the winding modes: no amount of winding changes anything about the spectrum. We thus declare $n=0$ states equivalent to each other if they differ only by their winding numbers.
For fixed (large) mass $m$ the counting is now straightforward. The total level $\level$ must be large and splits evenly, $s=r=\tfrac N2$. The (large) numbers $s$ and $r$ can be partitioned arbitrarily into positive integers. The number of integer partitions, $\Pi(\level)$, is given by the Hardy--Ramanujan formula \cite{Hardy:1918} [OEIS: \href{https://oeis.org/A000041}{A000041}]
\eq{
\Pi(\level) \approx \frac{1}{4\sqrt{3}\,\level}\,\exp{\bigg(2\pi\,\sqrt{\frac{\level}{6}}\bigg)}\,.
}{eq:app1}
Thus, the contribution to the partition function from the soft sector (at fixed $m$) is given by
\eq{
Z_{\textrm{\tiny soft}}(\level) = \Pi^2\Big(\frac{\level}{2}\Big) \simeq \frac{1}{\level^2}\,\exp{\bigg(2\pi\,\sqrt{\frac{\level}{3}}\bigg)}\,.
}{eq:null100}
Here and in what follows, we use $\simeq$ to denote the approximation of $\ln Z$ to the leading ${\cal O}(\sqrt{N})$ and subleading ${\cal O}(\ln N)$ contributions, while dropping terms subleading to these. This simplification has the added benefit that we can assume non-negative $\omega$ and $n$ since considering all possible sign combinations would produce some overall factor of order unity in the partition function. We make this assumption from now on.
\subsection{High momentum sector}
Consider the opposite of the soft sector: The mass is dominated by high momentum, $n\gg\level$. Level-matching \eqref{eq:null20} then implies vanishing winding number $\omega$. So we get the same result as for the soft sector \eqref{eq:null100}, but $\level$, while it still can be a large number, is now much smaller than $m^2/\kappa$.
Therefore, this sector is suppressed exponentially as compared to the soft sector and, as we shall see, also as compared to the sectors below. The same logic applies to the sector $n\approx\level$. Thus, we conclude that typical microstates require large levels, $\level\gg n$, and the contributions to the partition function $Z_{n\gg\level} + Z_{n\approx\level}$ are negligible for large masses.
\subsection{Generic sector}
The higher the level, the stronger the exponential enhancement in the integer partitions \eqref{eq:app1}. Thus, typical microstates require $\level\gg n$. Generically, there are no further constraints on winding or momentum other than level-matching and mass-shell conditions. In particular, generically neither of them vanish, $n>0$, $\omega>0$. To reduce clutter, we assume $\level$ is even (none of our results change essentially for odd $\level$).
A curious aspect of the mass-shell condition \eqref{eq:null42} is that changing the momentum number $n$ alters the almost-integer number $m^2/\kappa$ slightly. So we should not consider a fixed mass in our ensemble but allow for a range, depending on the allowed range of the momentum number. For the time being, we fix the level $\level$ but permit varying the mass by changing the momentum number.
The partition function in the generic sector for fixed $\level$
\eq{
Z_{\textrm{\tiny generic}}^{\textrm{\tiny fixed}}(\level) = \sum_{l=1}^{\tfrac{\level}{2}}\Pi\Big(\tfrac {\level}{2}-l\Big)\, \Pi\Big(\tfrac{\level}{2}+l\Big)\,\tau(2l)
}{eq:null101}
involves the number of divisors $\tau(k)$ of the integer $k$. It appears due to the level-matching condition \eqref{eq:null20}, which requires the difference of the levels $s-r$ to be the product $\omega\,n$.
The combinatorial problem \eqref{eq:null101} is not trivial but solvable at large $\level$~\footnote{We thank our colleagues at the mathematics departments at TU Wien and IPM Tehran for helpful discussions and proofs, in particular Bernhard Gittenberger, Martin Rubey, Iman Eftekhari, and S.M.~Hadi Hedayatzadeh.}.
\eq{
Z_{\textrm{\tiny generic}}^{\textrm{\tiny fixed}}(\level) \simeq \frac{1}{\level^{5/4}}\,\exp{\bigg(2\pi\,\sqrt{\frac{\level}{3}}\bigg)}
}{eq:null104}
We recover the same exponential degeneracy as in the soft sector \eqref{eq:null100} but with a monomial enhancement in $\level$, plus other subleading corrections.
It can be shown that the essential part of the partition function $Z_{\textrm{\tiny generic}}^{\textrm{\tiny fixed}}(\level)$ comes from levels $r$ in the range $\level/2-{\cal O}(\level^{3/4})$ to $\level/2-{\cal O}(1)$, implying typical ranges of winding and momentum numbers between ${\cal O}(1)$ and ${\cal O}(\level^{3/4})$.
From the range of the momentum number, we deduce an interesting physical fact: Since the momentum number generically scales at most like ${\cal O}(\level^{3/4})$, the quantity $m^2/\kappa$ changes like ${\cal O}(\sqrt{N})$. We conclude that we must consider Gaussian fluctuations of the level $\level$.
We can finally be precise about the allowed mass range in our definition of BTZ microstates. We cannot insist on a fixed value of $m$ but instead must allow fluctuations of the mass, $m\to m+\Delta m$, of order unity to guarantee Gaussian fluctuations in $\level$~\footnote{\label{Gaussian-footnote}
It is reassuring that the near horizon analysis in \cite{Afshar:2016wfy,Afshar:2016kjj} leads to analog conclusions. Namely, if we want Gaussian fluctuations in the BTZ black hole mass as measured by an asymptotic observer, $\Delta M = {\cal O}(\sqrt{M})$, we need to allow order unity fluctuations of the black hole mass as measured by a near horizon observer, $\Delta m = {\cal O}(1)$. It works because $M$ and $m$ are related by a Sugawara construction, yielding $M\propto m^2$. This implies $M+\Delta M \propto (m+\Delta m)^2 = m^2 + m\, {\cal O}(\Delta m)$, from which we deduce $\Delta M \propto\sqrt{M}\,{\cal O}(\Delta m)$.}.
\eq{
\Delta m = {\cal O}(1) \qquad \leftrightarrow \qquad \Delta N = {\cal O}\big(\sqrt{N}\big)
}{eq:null102}
While it was pure combinatorics that drove us to consider Gaussian fluctuations \eqref{eq:null102}, such fluctuations may have been anticipated on physical grounds, as we are in an ensemble of fixed temperature rather than fixed energy.
The number of generic microstates subject to the fluctuations \eqref{eq:null102} is then given by
\eq{
Z_{\textrm{\tiny generic}}(\level) =\!\! \sum_{\level_0=\level}^{\level+{\cal O}(\sqrt{\level})} Z_{\textrm{\tiny generic}}^{\textrm{\tiny fixed}}(\level_0)\simeq \frac{1}{\level^{3/4}}\,\exp{\bigg(2\pi\,\sqrt{\frac{\level}{3}}\bigg)}\,.
}{eq:null103}
\subsection{Non-winding sector}
There is one remaining sector, namely $\omega=0$. In this case, the level-matching \eqref{eq:null20} implies $s=r$, exactly as in the soft sector. However, since the momentum number $n$ appears in the mass-shell condition \eqref{eq:null42}, we get a cutoff on the spectrum.
For compatibility with the generic sector, we allow the same range of mass fluctuations and hence get Gaussian fluctuations of the level $\level$. We obtain
\eq{
Z_{\omega=0}(\level) =\!\! \sum_{\level_0=\level}^{\level+{\cal O}(\sqrt{\level})}\;\sum_{n=1}^{{\cal O}(N_0^{3/4})} \Pi^2\Big(\frac{\level_0}{2}\Big) \simeq \frac{1}{\level^{3/4}}\,\exp{\bigg(2\pi\,\sqrt{\frac{\level}{3}}\bigg)}\,.
}{eq:null105}
While the generic sector contains infinitely more states than the non-winding sector, the approximate equality
$Z_{\textrm{\tiny generic}}(\level)\simeq Z_{\omega=0}(\level)$
shows that the non-winding and the generic sectors yield the same leading and subleading result for the partition function.
\subsection{Full partition function}
The final result of our combinatorial excursion is the partition function of BTZ black hole microstates
\eq{
Z_{\textrm{\tiny BTZ}} = Z_{\textrm{\tiny soft}} + Z_{n\gg\level} + Z_{n\approx\level} + Z_{\textrm{\tiny generic}} + Z_{\omega=0}\,.
}{eq:null108}
For large horizon radii, the partition function is dominated by the contribution from the generic sector
\eq{
Z_{\textrm{\tiny BTZ}}(R_h) \approx Z_{\textrm{\tiny generic}}(\kappa R_h^2) \simeq R_h^{-3/2}\,\exp{\bigg(2\pi\, R_h\,\sqrt{\frac{\kappa}{3}}\bigg)}
}{eq:null107}
where we used the relation \eqref{eq:null40} between the level $\level$ and the horizon radius $R_h$. The approximation of the partition function \eqref{eq:null107} is the main result of our counting. It is valid for large horizon radii, $R_h\gg1/\sqrt{\kappa}$, which permits comparing with semiclassical results for the black hole entropy.
\section{Bekenstein--Hawking law}
Summarizing the combinatorics, we find the entropy of our horizon string microstates is given by the logarithm of the partition function \eqref{eq:null107}, viz.,
\eq{
S = \ln\,Z_{\textrm{\tiny BTZ}} = 2\pi\,R_h\,\sqrt{\frac{\kappa}{3}} - \frac32\,\ln R_h + o(\ln R_h)\,.
}{eq:null27}
The entropy \eqref{eq:null27} contains the correct scaling with the area of the BTZ event horizon, $2\pi\,R_h$, and a well-known numerical factor $-\frac32$ in front of the logarithmic corrections (see, e.g., \cite{Sen:2012dw}). The sub-subleading terms are small as compared to $\ln R_h$ but still infinite for $R_h\to\infty$. In particular, they are not of order unity \footnote{This is qualitatively different from the precision counting e.g.~in \cite{Dabholkar:2005dt}, where the next term is of order unity.}. Comparison with the BH-law \eqref{eq:null1} fixes the coupling constant as
\eq{
\kappa = \frac{3}{16G^2}\,.
}{eq:null28}
The scaling with $1/G^2$ follows from dimensional analysis. The numerical coefficient 3/16 is a non-trivial input and cannot be derived within the setup presented here.
Since the log corrections to the entropy depend on the thermodynamical ensemble let us finally check if we are in the right ensemble. From near horizon considerations we have fixed the temperature, and the tensionless string spectrum forced us to let the asymptotic mass have a Gaussian profile, see \cite{endnote84}. The metric \eqref{eq:null3} in a co-rotating frame has fixed angular momentum. In the conventions of \cite{Sen:2012dw}, we thus have a mixed ensemble and should get a coefficient of $-\frac32$ in front of the log corrections. This is precisely what we obtained in our main result \eqref{eq:null27}.
\section{Concluding remarks}
Based on the matching of symmetries of the horizon, we formulated and implemented the idea that microstates of a 3d black hole are physical states \eqref{eq:null22} of a null string theory. Typical microstates are excited strings with nonzero winding around the angular direction. The growth of their degeneracy with the exponential of the square root of the excitation number \eqref{eq:null103} is a known characteristic feature of strings \cite{Gross:1987kza,Gross:1987ar,Atick:1988si}. This degeneracy correctly accounts for the Bekenstein--Hawking entropy and its subleading log corrections \eqref{eq:null27}.
Before addressing generalizations, we discuss similarities and key differences to some previous proposals for black hole microstates. Our construction can be viewed as an implementation of the membrane paradigm \cite{Thorne:1986iy}. Our mass spectrum \eqref{eq:null42} is quite different from naive area quantization \cite{Bekenstein:1995ju,Ashtekar:1997yu} since, for large black holes, we have a very fine (quasi-continuous) spacing between neighboring energy levels, due to the presence of the momentum term $n^2/R_h^2$. Such a fine spacing of energy levels is more in line with general lessons from statistical mechanics than a Planck-quantized area spectrum, see e.g.~\cite{Foit:2016uxn}. Our setup differs conceptually from the fuzzball proposal \cite{Mathur:2005zp,Skenderis:2008qn}, according to which black hole microstates are horizonless configurations that asymptotically look like a black hole spacetime. By contrast, our construction relied on the existence of a horizon, and the microstates are understood as all possible horizon string configurations, i.e., the microstate degeneracy is due to excitations, winding, and momentum of (tensionless) strings on the horizon. In comparison to the fluff proposal \cite{Afshar:2016uax,Afshar:2017okz}, our construction only rests on the near-horizon information, whereas the former used both near-horizon and asymptotic symmetries. Moreover, we avoid ad-hoc input such as the quantization of Newton's constant or the quantization of conical deficit angles that were an integral part of the construction in \cite{Afshar:2016uax,Afshar:2017okz}. The realization of the soft hair proposal in \cite{Haco:2018ske} exploits near horizon Virasoro symmetries and the Cardy formula to account for the black hole entropy but is not explicit about the precise spectrum of microstates.
While our proposal stands firmly on its own, it would be nice to understand how it emerges as a limit of tensile strings approaching a black hole, building on the discussion in \cite{Bagchi:2021ban}. Relatedly, the critical dimension of our horizon strings is 26 \cite{Bagchi:2021rfw}, and it could be rewarding to consider the impact of these extra dimensions on our analysis. An obvious generalization is to consider higher dimensions, specifically, four spacetime dimensions. Instead of tensionless null strings, one would have to consider tensionless null branes wrapping the horizon. All these issues are works in progress.
\acknowledgments
\section*{Acknowledgments}
It is a pleasure to thank Mangesh Mandlik for helpful correspondence.
AB wishes to acknowledge the support and hospitality of the Erwin-Schr\"odinger Institute, University of Vienna and Vienna University of Technology, Austria where the collaboration was initiated. AB was partially supported by a Swarnajayanti Fellowship (SB/SJF/2019-20/08) of the Science and Engineering Research Board (SERB), India, a visiting professorship at \'Ecole Polytechnique Paris, and by the following grants: CRG/2020/002035 (SERB), Research-in-groups grant (Erwin-Schr\"odinger Institute).
DG was supported by the Austrian Science Fund (FWF), projects P~30822, P~32581, and P~33789, and acknowledges the hospitality of the Perimeter Institute (PI) during the final stage of this work.
MMShJ is supported in part by SarAmadan grant No ISEF/M/401332, and acknowledges support from the ICTP through the Senior Associates Programme (2022-2027) and Vienna University of Technology, Austria where the collaboration was initiated.
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Home » Kentucky Court Halts Federal Contractor Vaccine Mandate in Three States
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Kentucky Court Halts Federal Contractor Vaccine Mandate in Three States
Conservative federal court extends temporary halt but lawsuit's merits await further debate
Kentucky federal judge appointed by President George Bush has paused the vaccine mandate for federal contractor employees in that state and in Ohio and Tennessee.
Debra K. Rubin
KEYWORDS Biden administration / federal court / Government Contracting / OSHA / OSHA vaccine mandate / vaccine mandate
President Joe Biden does not appear to have the authority to require vaccines for employees of federal contractors and subcontractors, a federal court judge in Lexington, Ky., ruled Dec. 1, halting implementation of the mandate in three states that filed suit against it—Kentucky, Ohio, and Tennessee—until the case merits are decided.
The states had claimed that Biden's executive order, announced in September, was unconstitutional. The order set an Oct. 15 effective date; mandated vaccine requirements be included in new contracts as of Nov. 14; and required employee vaccinations by Dec. 8. The deadline for full vaccination has since been moved to Jan. 18. The president also sought a mandate amendment to the Federal Acquisition Regulation (FAR), which governs federal procurement.
Biden administration officials have strongly argued that the federal government has the legal authority to require the mandates.
In his ruling, U.S. District Court Judge Gregory F. Van Tatenhove, appointed by President George Bush, has taken the opposite stance.
"Can the President use congressionally-delegated authority to manage the federal procurement of goods and services to impose vaccines" on the federal contractor workforce? asked Van Tatenhove. "In all likelihood, the answer to that question is no."
The judge disputes the government's contention that the states lacked standing to sue because the mandate only applied to future contracts. He said that some agencies are applying the mandate to current contracts, with contractors fearful of being "blacklisted from future contracting opportunities."
Kentucky, Ohio, and Tennessee state agencies, as well as contractors based in those states, had about $9.9 billion, $12.5 billion, and $10 billion in government contracts in 2020, respectively, according to the ruling.
The mandate exceeds contracting authority, Van Tatenhove said, noting that the Biden administration use of a 1949 procurement statute to justify it "could be used to enact virtually any measure at the president's whim under the guise of economy and efficiency. This power has its limits."
The federal contractor vaccine order was followed in early November by an emergency temporary standard from the U.S. Occupational Safety and Health Administration that covers all employers with 100 or more employees, as well as others applying to federal workers and to health care companies paid by Medicare and Medicare. All have faced legal challenges.
States and private entities, including the Associated Builders and Contractors, joined in lawsuits against the large-employer OSHA standard in a number of courts, with the U.S. Court of Appeals in New Orleans temporarily halting it on Nov. 12, and the Cincinnati appellate court set to consider the now consolidated lawsuits. That case is expected to reach the U.S. Supreme Court, observers say.
Related to the federal contractor order temporary halt, Kentucky Attorney General Daniel Cameron termed it "a significant ruling." The Justice Dept. had not issued a statement by ENR posting time.
Even so, some legal experts say the federal government may still win on the merits of its order. "It is going to be more defensible" than the OSHA standard, Stewart Schwab, employment law professor at Cornell University, told Bloomberg Law. "It plays out differently than OSHA, because no company has to take a government contract," he said.
Rob Duston, a labor and employment attorney at Saul Ewing Arnstein & Lehr, noted a long history "of broad discretion to impose requirements [in] a contract." He points to federal contractor requirements in areas such as non-discrimination that are stricter than for other private-sector companies, which have been in effect for more than 30 years.
Recent Articles by Debra Rubin
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As ENR Editor-at-Large for Management, Business and Workforce, Debra K. Rubin has a broad vantage for news, issues and trends in global engineering, architecture and construction—from corporate finance and executive management to regulation and risk, next-generation workforce and developing markets such as offshore wind energy.
Debra also launched and manages ENR's Top 200 Environmental Firms ranking, which defines a $51-billion global market; and is editor of ENR WorkforceToday e-newsletter on industry talent management; She also is a key organizer of ENR's annual Groundbreaking Women in Construction conference, a major AEC industry forum for talent management and women's career advancement.
Click here for more detail on plans in formation for the 2022 live event.
Debra's reporting for ENR on the 2001 Ground Zero attack damage, response and recovery is featured in the National September 11 Memorial & Museum in New York City.
US Appeals Court Halts OSHA Vaccine Mandate, Unclear How Long
OSHA Vaccine Mandate Set For Jan. 7 Supreme Court Hearing
OSHA Suspends Vaccine Mandate Compliance After US Court Affirms Halt
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{"url":"https:\/\/zbmath.org\/?q=an%3A1213.35274","text":"## Very singular diffusion equations: second and fourth order problems.(English)Zbl\u00a01213.35274\n\nSummary: This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an $$H ^{ - 1}$$ gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove by giving an explicit example that the solution may instantaneously develop a jump discontinuity for the fourth order total variation flow.\n\n### MSC:\n\n 35K67 Singular parabolic equations 35K92 Quasilinear parabolic equations with $$p$$-Laplacian 74N05 Crystals in solids 62H35 Image analysis in multivariate analysis 35K59 Quasilinear parabolic equations 35K25 Higher-order parabolic equations\nFull Text:","date":"2022-05-24 22:44:40","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\": 0.6990287899971008, \"perplexity\": 622.4258099660051}, \"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-2022-21\/segments\/1652662577259.70\/warc\/CC-MAIN-20220524203438-20220524233438-00662.warc.gz\"}"}
| null | null |
About MIBP
Click to Search Search
A multimedia source for news and updates from Blue Cross Blue Shield of Michigan.
CNBC's 'American Greed' features Michigan health care fraud case cracked by Blues
A dark side of health care fraud will be revealed tonight on CNBC's "American Greed" when the program reports on a Michigan doctor who bilked insurers, Medicare and patients out of at least $1.9 million.
The program profiles Dr. Robert Stokes, who ran a thriving dermatology practice in East Grand Rapids. Dr. Stokes had a dark secret. He was stealing patients' money. He reused medical supplies and endangered patients.
Blues Vice President Greg Anderson (left), a former detective sergeant with the Michigan State Police, describes to an "American Greed" reporter how the Blues work with law enforcement agencies to fight health care fraud.
The program includes an interview with Greg Anderson, vice president of Blue Cross Blue Shield of Michigan's Corporate and Financial Investigations unit, who explains how the Blues' investigative team worked with law enforcement agencies to convict Stokes of health care fraud.
Stokes, who bilked the Blues out of more than $600,000, is serving a 10½-year sentence in a federal prison.
"American Greed" is narrated by award-winning actor Stacy Keach. The program airs tonight on CNBC at 9 p.m., 10 p.m., 12 a.m. and 1 a.m.
For a preview of tonight's episode, go to "American Greed" online. Or check out a preview of the episode in this trailer below.
http://plus.cnbc.com/rssvideosearch/action/player/id/1479384195/code/cnbcplayershare
by bcbsm
Topics: Community, For You, Our News
Tags: Fraud
Answering the Call: Blue Cross' Efforts to ...
Don't Be a Victim of Health Care Fraud
CFI Celebrates Law Enforcement Cooperation
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 447
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This proposal focuses on a critical, but often neglected skill for business, communication, and engineering students namely visual literacy, or the ability to assess abstract visual representations of information and knowledge, to evaluate their advantages and disadvantages, to improve their shortcomings, to use them to create and communicate knowledge, or to devise new ways of representing insights.
The faculties of Communication Sciences and Informatics of the University of Lugano will collaborate with the University of St. Gallen and the University of Applied Sciences North-Western Switzerland to develop an online tutorial in English onvisualization principles and methods and their applications. The didactic approach consists of rooting visualization in its application contexts, giving students the necessary critical attitude, principles, tools and feedback to develop their own high-quality visualization formats for specific problems (problem based learning). The students thus learn about the commonalities of good visualization in diverse areas, but also explore the specificities of visualization in their field of specialization (through real-life case studies). They will not only learn by doing, but in doing so contribute new didactic training material for their peers to evaluate. Peer learning is thus an important didactic element of this proposal. The tutorial consists of three building blocks: basic modules, application modules, and exploration modules. The basic modules motivate students for the topic and enable them to understand visualization rules and conventions. The application modules show how these principles can be used for specific communication, engineering, or business problems. The exploration modules let students develop their own solutions and assess each others' approaches critically.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,807
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Le siège de Riga est une tentative ratée des Russes de s'emparer de Riga, la ville la plus peuplée du royaume de Suède, au cours de la guerre russo-suédoise de 1656–1658. Le siège dura du au et se termina par le repli de l'armée russe et du tsar Alexis .
Contexte
Au début de 1654, la Pologne-Lituanie n'est pas parvenue à réprimer le soulèvement de Khmelnytsky dans le sud-est du royaume. Peu après, le tsar Alexis Ier a signé avec Bohdan Khmelnytsky et les cosaques zaporogues le traité de Pereïaslav, au terme duquel l'Ukraine à l'est du Dniepr devient un protectorat du tsar de Russie. Le tsar Alexis décide ensuite de se joindre au conflit et d'apporter son appui aux Cosaques rebelles : il espère ainsi reprendre les territoires autour de Smolensk et en Ukraine qu'il a perdus à l'issue de la guerre polono-russe (1605-1618) ; mais les arrière-pensées religieuses, c'est-à-dire la reconquête du royaume originel des Varègues, n'étaient pas non plus absentes. C'est ainsi qu'éclata la guerre polono-russe (1654–1667). Jusqu'à la fin de 1655, la Russie occupa de larges portions de la Pologne-Lituanie. Après la chute de Vilnius en , le tsar Alexis n'hésita plus à se proclamer grand-duc de Lituanie. Dans le sud-ouest, l'armée russo-cosaque envahit une grande partie du pays, mais son élan se trouva brisé par l'intervention des Tatars de Crimée, alliés des Polonais, commandés par le khan Mehmed IV Giray.
La Suède, voyant ses fiefs baltes menacés, entra dans le conflit. La Pologne-Lituanie était alors au bord de l'effondrement militaire. Un nouvel assaut russe dans le territoire de la République venait de fragiliser un peu plus les positions suédoises. C'est pourquoi à l'été 1655 le roi Charles X eut l'idée de dépêcher une armée de hommes en territoire polonais pour s'opposer à de nouvelles conquêtes russes.
Les premiers succès des Suédois alarmèrent le tsar Alexis . En , il déclara finalement la guerre à la Suède. Les provinces baltes suédoises étaient alors encore sans défense et se trouvaient à la merci d'une invasion ; la Suède, en effet, ne stationnait guère plus de hommes répartis dans les différentes garnisons de ses provinces baltes. L'armée russe marcha sur l'Estonie, l'Ingrie et le Kexholm. Elle s'empara des places-fortes de Chlisselbourg et de Nyenskans en Ingrie. À l'été 1656, le tsar Alexis fit traverser la Daugava à l'armée de Polotsk, s'emparant de Daugavpils en juillet puis de la forteresse de Koknese en août. Par quelques combats d'arrière-garde, la petite armée suédoise se replia sur Riga.
Les troupes russes avaient déjà atteint Riga en 1559, mais n'avaient pas pris le risque de s'y attaquer. Au cours de la guerre de Livonie (1558–1583), la plus grande partie de l'actuelle Lettonie était passée au royaume de Pologne-Lituanie. La Confédération livonienne s'était dissoute, et Riga, après une défense de principe, rendit les armes. La ville fut de nouveau la cible d'attaques au cours de la guerre polono-suédoise et en 1621 les Suédois, emmenés par Gustave-Adolphe, s'étaient finalement emparés de la ville. Avec une population de habitants, Riga était alors trois fois plus peuplée que la capitale de Suède, Stockholm.
Le siège de Riga
Le , l'armée russe atteignit Riga. La ville était défendue par cavaliers et dragons, et un certain nombre de miliciens, soit tout au plus hommes. Le gouverneur militaire de la ville, Magnus Gabriel De la Gardie, contre l'avis de son état-major, décida malgré le déséquilibre des forces, de tenir le chemin couvert, très étendu et d'ailleurs inachevé ; mais dès la nuit suivante, les avant-postes furent détruits et les Suédois durent s'enfermer dans la citadelle. L'armée russe prit position sur la Daugava, assiégeant le château avec sept régiments, et commença les travaux de sape.
L'état major russe s'aperçut tardivement qu'il aurait mieux fait de prendre position à l'embouchure du fleuve, ce qui aurait coupé Riga des ravitaillements par voie de mer. Les travaux d'approche étaient à présent trop avancés, et de surcroît la coordination laissait à désirer : les tranchées étaient inégalement creusées, le bombardement s'effectuait de façon erratique : malgré les dégâts considérables provoqués par les tirs de canons sur les bâtiments, les remparts étaient pratiquement intacts. Mais les tirs de l'artillerie russe affectaient énormément le moral des bourgeois ; et les Russes savaient par les fuyards interceptés et les prisonniers que les bourgeois demandaient à l'état-major suédois de remettre les clefs de la ville au tsar.
Les sapeurs russes ne parvinrent jamais à passer les circonvallations, et l'on ne tenta finalement aucun assaut direct. D'ailleurs, les alliés danois des Russes s'avérèrent incapables d'instaurer un blocus efficace de l'estuaire.
Abandon du siège
Le , la garnison suédoise reçut le renfort de soldats. Sur ce, le tsar tint un conseil de guerre, où l'on examina aussi bien les chances de succès d'un assaut immédiat que l'opportunité de poursuivre le siège. La plupart des officiers exprimèrent des doutes quant à la tentative d'assaut. Quelques jours plus tard, la levée du siège commençait. Simultanément, la rumeur d'une épidémie de peste dans Riga se répandit, justifiant l'abandon du siège. Le , l'armée russe s'ébranla vers le sud. Selon les sources suédoises, elle avait alors déjà perdu hommes, ce qui toutefois paraît fort exagéré au vu du faible nombre d'assauts finalement entrepris.
Du côté russe, les raisons de la levée du siège étaient surtout de nature diplomatique. Au cours des dernières semaines du siège, la situation politique avait évolué : les buts de guerre avaient perdu de leur pertinence, dans la mesure où il n'était plus question d'une coalition polono-suédoise. La campagne du tsar dans les pays baltes prenait désormais l'apparence d'une démonstration de force, qui poussait à présent la Pologne, le Brandebourg, la Courlande et le Danemark à se concerter militairement. Dans de telles circonstances, la perte de prestige du tsar que ne manquerait pas susciter un siège désastreux parut plus regrettable qu'un repli en bon ordre. Alexis Mikhaïlovitch, en tant que chef de guerre, mit un terme aux campagnes aventureuses, préférant préserver son armée et se tourner vers d'autres méthodes.
La décision de lever le siège fut annoncée après une dernière tentative de négocier la reddition de la garnison de Riga. Les espoirs du tsar d'un appui diplomatique du duc de Courlande et de l'électeur de Brandebourg s'étaient envolés.
Malgré l'échec militaire de l'attaque sur Riga, la campagne des pays baltes de 1656 fut applaudie à la cour de Moscou comme un grand succès. Les documents montrent le retour triomphal du tsar dans les villes de Polotsk, Smolensk et Moscou. L'invasion de presque toute la vallée de la Daugava, avec les places de Daugavpils et Koknese, donnait désormais à la Russie une forte ligne défensive le long de la frontière balte.
Conséquences
Malgré le repli devant Riga, Dorpat tomba en aux mains des Russes. Mais la guerre russo-suédoise avait donné à la république des deux nations un répit suffisant pour regrouper ses forces. L'année suivante, il fallut dépêcher de nouveaux contingents russes en Livonie, puis finalement en 1658 le tsar et le roi de suède convinrent d'un armistice de trois ans.
Riga devait finalement se rendre aux troupes du tsar Pierre le Grand en 1710 ; la ville sera rattachée à la Russie jusqu'à la Première Guerre mondiale.
Bibliographie
Geijer, Carlson, Stavenow, Heeren et al.: Geschichte Schwedens: Bis zum Reichstage 1680, volume 4, Gotha 1855
William Young:International Politics and Warfare in the Age of Louis XIV and Peter the Great, Lincoln 2004
Theatrum Europæum, vol. 7, Francfort-sur-le-Main, 1685
Notes et références
Histoire de la Lettonie
Riga
Bataille impliquant l'Empire suédois
Riga 1656
Bataille de 1656
Riga
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,727
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The list of ship commissionings in 1945 includes a chronological list of ships commissioned in 1945. In cases where no official commissioning ceremony was held, the date of service entry may be used instead.
References
1945
Ship commissionings
Ship commissionings
Ship commissionings
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,347
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Abdelmalek Madani, né le , est un coureur cycliste algérien.
Biographie
Le , il est suspendu quatre ans pour dopage après avoir été contrôlé positif en 2017.
Palmarès
2008
du championnat d'Algérie du contre-la-montre
Médaillé de bronze au championnat d'Afrique sur route
2009
Champion d'Algérie du contre-la-montre
et du Tour de Tipaza
du championnat d'Algérie sur route
du Perlis Open
2010
Champion d'Algérie du contre-la-montre
du championnat d'Afrique sur route
2011
Grand Prix Souk Ahras
du Tour du Faso
du Challenge du Prince - Trophée princier
2012
Médaillé d'argent sur route au championnat arabe des clubs
du championnat d'Algérie sur route
du Challenge du Prince - Trophée princier
2013
Champion arabe du contre-la-montre par équipes (avec Azzedine Lagab, Youcef Reguigui et Fayçal Hamza)
du championnat d'Algérie sur route
Médaillé d'argent au championnat d'Afrique du contre-la-montre par équipes
Médaillé de bronze au championnat arabe du contre-la-montre Médaillé de bronze au championnat arabe sur route 2014
du championnat d'Algérie du contre-la-montre du championnat d'Algérie sur route 2015
du Tour international de Constantine
du championnat d'Algérie sur route du Tour international de Constantine''
Classements mondiaux
Notes et références
Liens externes
Coureur cycliste algérien
Naissance en février 1983
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 2,997
|
from corehq.apps.locations.models import all_locations
from corehq.apps.commtrack.models import Product, SupplyPointCase
def supply_point_ids(locations):
keys = [[loc.domain, loc._id] for loc in locations]
rows = SupplyPointCase.get_db().view(
'commtrack/supply_point_by_loc',
keys=keys,
include_docs=False,
)
return [row['id'] for row in rows]
def get_relevant_supply_point_ids(domain, active_location=None):
if active_location:
return supply_point_ids([active_location] + active_location.descendants)
else:
return supply_point_ids(all_locations(domain))
def product_ids_filtered_by_program(domain, program):
products = Product.by_program_id(domain, program, False)
return [p['_id'] for p in products]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,598
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{"url":"http:\/\/math.stackexchange.com\/questions\/255773\/can-someone-show-me-what-to-do-with-these-fund-theorem-of-calc-problems?answertab=active","text":"# Can someone show me what to do with these Fund. Theorem Of Calc problems?\n\nI have 3 problems dealing with the Fundamental Theorem of Calculus that I'm working on. The first one is relatively easy, but the second two are above my head. If anyone has the time to explain them, I would be thankful.\n\n1: $\\int_1^8 [3x^3 - \\frac{5}{x} + 2 \\sqrt[3]{x^2}$\n\n$\\int [ \\frac{3*(8)^4}{4} - 5 ln(8) + \\frac{2(8)^{3\/2}}{3\/2}] - \\int [ \\frac{3(1)^4}{4} - 5 ln(1) + \\frac{2(1)^{5\/2}}{5\/2}$\n\n2: $\\int_0^{\\pi\/6} (sin(3x) - cos(3x))$\n\nI don't have one yet, I'm unsure of how to get the anti-derivative of a trig function with a variable larger than x.\n\n3: $\\int_{ln(1)}^{ln(2)} \\frac{e^{2x}}{e^{2x} + 4}$\n\nI don't know where to begin with this one. Do I do u = $e^{2x} +4$ and du = $\\frac{e^{2x}}{2}$?\n\nEDIT\n\n1: $\\int [ \\frac{3*(8)^4}{4} - 5 ln(8) + \\frac{2(8)^{5\/3}}{5\/3}] - \\int [ \\frac{3(1)^4}{4} - 5 ln(1) + \\frac{2(1)^{5\/3}}{5\/3}$\n\n2: $\\int_0^{\\pi\/6} \\frac{1}{3}cos(3(\\frac{\\pi}{6})) + \\frac{1}{3}sin(3(\\frac{\\pi}{6}))$\n\n3:\n\n-\n\nExercise 1:\n\nThe basic idea is right, your notation is weird though. Once you've found an antiderivative and are going to evaluate it at the endpoints of the integral, you don't write the integral sign anymore. It looks like this (for example):\n\n$$\\int_1^5 \\frac{dx}{x} = \\ln 5 - \\ln 1 = \\ln 5$$\n\nAlso, there's a mistake: $\\sqrt[3]{x^2}$ is $x^{2\/3}$, so what's its antiderivative?\n\nExercise 2:\n\nHere the problem is finding the primitive of something like $\\cos (3x)$. If that was $\\cos x$, it would be easy: the primitive would just be $\\sin x$, since $(\\sin x)' = \\cos x$. But there's a $3$ there, so let's try something involving a $3$. $3 \\sin x$ is not going to work, because its derivative is $3 \\cos x$. What about $\\sin(3x)$? Its derivative is $3 \\cos (3x)$, so we're getting there! Can you think of a function similar to $\\sin(3x)$ such that its derivative is $\\cos(3x)$?\n\nExercise 3:\n\nTry a substitution $u = e^{2x}$. Then $du = 2e^{2x}\\ dx = 2u\\ dx$. Remember to change the bounds of integration accordingly. Since your substitution is $x \\mapsto e^{2x}$, $\\ln 2$ should be changed to $e^{2\\ln 2}$ (this can be simplified) and similarly for $\\ln 1$ which, by the way, equals $0$.\n\nClarification: The idea is that when you do a substitution, the bounds of the integral have to change. Let's say you have $\\int_1^4 \\frac{2x+1}{x^2+x}\\ dx$. You can use a substitution $u = x^2+x$, $du = (2x+1)\\ dx$. The integrand then becomes $\\frac{du}{u}$. But it's not right to say that $\\int_1^4 \\frac{2x+1}{x^2+x}\\ dx = \\int_1^4 \\frac{du}{u}$, because you haven't changed the limits of integration. Since your substitution is $u(x) = x^2+x$, your new limits will be $u(1)$ and $u(4)$, that is, $2$ and $20$. So $\\int_1^4 \\frac{2x+1}{x^2+x}\\ dx = \\int_2^{20} \\frac{du}{u} = \\ln u |_2^{20} = \\ln 20 - \\ln 2$.\n\n-\n#1: $x^{2\/3} -> x^{5\/3}$. #2: Divide by 3? #3: I'm not sure I understand, so after I do u substitution, I should be plugging in $e^{2ln(2)}$ and $2 ln(1)$? \u2013\u00a0 StrugglingWithMath Dec 10 '12 at 21:14\n1: Remember to divide, too: $\\int x^{2\/3}\\ dx = \\frac{x^{5\/3}}{5\/3} + C$. 2: Yes, that's right. As always, you can check: $(\\frac{1}{3}\\sin (3x))' = \\frac13 3 \\cos (3x) = \\cos(3x)$. 3: I'll edit more information. \u2013\u00a0 Javier Dec 10 '12 at 21:19\nAh, thank you for clarifying. I didn't realize I had to plug the values into u. So if u = $e^{2x}$, I plug $ln(1)$ and $ln(2)$ into it, so my new bounds are: a = 1, b = 2? If so, then I just struggle on what the equation is supposed to look like. \u2013\u00a0 StrugglingWithMath Dec 10 '12 at 21:39\nAlmost. $e^{2\\ln 2} = e^{\\ln (2^2)} = 2^2 = 4$. But otherwise that's right, yes. \u2013\u00a0 Javier Dec 10 '12 at 21:46\nThank you for your patience and kindness in helping me. Do you mind showing me what the equation should be to plug the new bounds into? \u2013\u00a0 StrugglingWithMath Dec 10 '12 at 21:48","date":"2015-08-30 02:12:05","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\": 0.9513775110244751, \"perplexity\": 220.12815093211938}, \"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-2015-35\/segments\/1440644064865.49\/warc\/CC-MAIN-20150827025424-00093-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
| null | null |
Parkeston may refer to:
Parkeston, Essex
Parkeston, Western Australia
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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The seventies produced some great songs some of which we attempted this evening with various degrees of success and no lack of enthusiasm. Lots of wistful, oh yeah, I remember that. For me it was one of those pivotal decades, marriage, a child, responsibility and Dire Straits (in my case financially - trying to make ends meet). I could see the same thoughts crossing the faces of others during the evening. This is what these nights are about.
I started things off with Albert Hammond's "It never rains in southern California", followed by Slow Hand's "Wonderful Tonight" and later, Six blade knife by Dire Straits with Terry playing lead.
Clive decided to bracket the decade with 1970 "Wandering Star" he didn't quite get the Lee Marvin growl but it came close. He followed this with Mike Batt's 1979 "Bright Eyes" ever associated with rabbits. Clive also ran the desk for us and gave us some mellow sounds.
Manus took us to the seventies folk scene with James Taylor's "Something in the way she moves" followed by John Rembourn's version of "Sweet Potato" – a great choice.
Next up was Natasha with a charming version of John Denver's "Annie's Song", I was always a John Denver fan so I particularly enjoyed that. It is time for a John Denver revival he wrote some great stuff. This was followed by Hal Ketchum's "Past the point of rescue" from the 1990's but we pretended not to notice.
Then it was time for a debut, it was the first time Lance visited the Six Bells and we hope it won't be the last. He gave us Paul Simon's "Slip sliding away" followed by "Knights in white satin" the Moody Blue (re-issue version). A couple of tricky songs to play and sing but he nailed them.
Sylvie was next with her own composition "In praise of Bobby Dylan" followed by her "Please, please will you do these little things". Not enough songs about toilet training in my humble opinion.
Pat followed Sylvie and sang unaccompanied, Dusty Springfield's "Goin' back" followed by "Leaving on a jet plane". We all remember these.
Enough guitars, time for Ella on the piano. Being a hippy chick at heart Ella was in her element, I did not catch the name of the first song but this was followed by "After the gold rush" and later, another Neil Young song "Hurricane". Good stuff.
Jason was wearing a flower powered shirt, he sang us Bob Dylan's "Only a hobo" followed by Cat Steven's "Longer boats" and later (with Lisa) Harry Nilsson's "Everybody's talking". Only the Cat Stevens song qualified 1970, the others were 60's songs (but then who's being picky – I would never point that out to him).
Time for Terry. He gave us the muscle memory version of Scott Joplin's "Maple Leaf Rag" followed by another Dire Straits song "Romeo & Juliet". This is what happens when you practice.
Lisa, another Hippy Chick had dressed in her seventies gear (including flares – I loved wearing those, long overdue for a comeback in my opinion) accompanied by Jason they sang the Beatles classic "You've got to hide your love away" and then "Wild Horses" by the Stones.
Finally we had Keith who gave us his own song "Little Strategies" and then, because he had to play it in public, Mozart's Concerto No. 21 in C major. A catchy title but totally inappropriate for a 70's evening, not that I would be in any way critical; he did at least add a funky ending. Why didn't old Amadeus think of that? Anyway, Keith got his "played in public" box ticked!
Thanks to everyone who came along, particularly those who went to the trouble of learning a 70's song for the occasion. Next time it is sure to be a packed house as Lisa is running the evening. I am looking forward to it. See you next time.
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{
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\section{Введение}
Данная работа посвящена обхождению лабиринта коллективом конечных автоматов. Эта часть теории автоматов породила довольно широкий спектр различных задач \cite{kudr_kili_ush_1, andzhans_1}, в том числе связанных с задачами теории сложности вычислений и теорией вероятностей. В этой работе показывается, что рассмотрение сложных алгебраических объектов, таких как бёрнсайдовы группы, может быть интересном в данном контексте.
Есть довольно большое число вариаций задачи, но в целом она выглядит так: коллектив конечных автоматов двигается по рёбрам некоторого (возможно бесконечного) графа, необходимо выяснить смогут ли автоматы посетить все вершины графа.
Простейшим примером такой задачи, является обход $\mathbb{Z}$. Конечный автомат, двигающийся по решётке $\mathbb{Z}$ по некоторым внутренним правилам, не сможет обойти эту прямую, т.к. в какой-то момент зациклится. Однако можно рассмотреть коллектив из одного полноценного конечного автомата и двух автоматов-камней, которые не имеют внутреннего состояния и могут передвигаться только совместно с главным автоматом. Легко показать, что подобная система может обойти $\mathbb{Z}$ (главный автомат бегает между камнями и постепенно раздвигает их). В случае решёток $\mathbb{Z}^k, k > 1$ для обхода достаточно коллектива из автомата и трёх камней, причём с меньшим количеством камней обойти не получиться. Если из плоской решётки $\mathbb{Z}^2$ разрешить выкидывать некоторые вершины, то окажется, что автомата и трёх камней не достаточно для обхода \cite{kilibarda_3, kilibarda_4} таких лабиринтов, при этом пяти камней хватит \cite{szepietowski_1} (4 - открытый вопрос).
Возникает закономерный вопрос, существуют ли лабиринты, которые не обходятся подобными системами. Ответ на этот вопрос положительный. Например, можно построить бесконечную лабиринт-ловушку на решётке $\mathbb{Z}^3$ для любой системы автоматов \cite{kilibarda_ush_1}. Но построение таких лабиринтов обычно довольно громоздко. В работе мы предлагаем другой подход для построения подобных ловушек.
В качестве лабиринтов можно рассматривать графы Кели конечно порожденных групп. Такой подход даёт довольно интересные результаты. Оказывается, графы Кели бесконечных свободных бёрнсайдовых групп, существование которых доказано Новиковым и Адаяном \cite{nov_adian_1}, нельзя обойти никакой системой конечных автоматов. Проблемы Бернсайдовского типа имеют большое значение в современной алгебре, поэтому кажется интересным, что такие серьёзные результаты находят применение в довольно удалённых от алгебры областях.
Данная работа была проведена с помощью Российского Научного Фонда Грант N 17-11-01377.
\section{Определения}
В этой секции мы введём формальные определения, которые нам понадобятся. Во-первых мы определим понятие лабиринта. В нашем случае лабиринтом будет граф Кэли некоторой конечно-порождённой группы. Отметим, что возможны и другие определения лабиринта. Во-вторых, определим коллектив конечных автоматов и поведение коллектива в лабиринте. Также введём определение лабиринта-ловушки.
Будем пользоваться основными понятиями и обозначениями теории из автоматов и графов, принятыми здесь \cite{kudr_podkol_ushmulich_1, kudr_alesh_podlol_1, harari_1}.
Пусть $G$ группа, $S = \lbrace s_1, s_2,...,s_n \rbrace$ -- множество образующих группы $G$, $S^{-1} = \lbrace s_1^{-1}, s_2^{-1}, ... , s_n^{-1} \rbrace$. Граф Кэли группы $G$ на образующих $S$ будем обозначать $\Gamma=\Gamma(G, S)$. Для простоты будем считать, что $G$ - это вершины графа, также будет использовать групповые операции над вершинами, если это потребуется. За $E(\Gamma)$ обозначим множество рёбер графа. $\Gamma$ - будет лабиринтом в нашей задаче.
Под конечным автоматом $\mathfrak{A}$ будем понимать пятёрку $(A, Q, B,\phi, \psi)$, где $A, B, Q$ -- конечные множества (алфавиты): входной, выходной и алфавит состояний. $\phi : Q \times A \rightarrow Q$, $\psi : Q \times A \rightarrow B$ -- функции переходов и выходов. Если начальное состояние $q_0 \in Q$ фиксировано, то такой автомат $\mathfrak{A}_{q_0} = (A, Q, B,\phi, \psi, q_0)$ будем называть инициальным. Далее, если факт инициальности будет важен для нас, тогда будем обозначать автомат нижним индексом. Часто вместо конечного автомата будем говорить просто автомат.
Пусть задана группа $G$ с множеством образующих $S$.
\begin{Def}
Набор $\mathcal{A} = (\mathfrak{A}_1, \mathfrak{A}_2,...,\mathfrak{A}_m)$ называется коллективом $(G, S)$-допустимых автоматов, где $\mathfrak{A}_i = (A_i, Q_i, B_i, \phi_i, \psi_i)$, для $i = 1,...m$, некоторые автоматы, такие что:
\begin{itemize}
\item $A_i = \lbrace a \in \prod\limits_{j=1}^m (\theta \cup Q_j) | \Pr_{i}(a) = \theta \rbrace $, где $\theta \notin \bigcup \limits_{i=1}^m Q_i$ -- некоторый выделенный элемент
\item $B_i = S \cup S^{-1} \cup \lbrace e \rbrace$
\end{itemize}
\end{Def}
Если у всех $m$ автоматов заданы начальные состояния $\vec{q_0} = (q_0^1, ... , q_0^m )$, то коллектив будем называть инициальным и обозначать $\mathcal{A}_{\vec{q}}$
Пусть $\Gamma(G, S)$ - граф Кэли. $\vec{v} = (v_1,...,v_m)$, $v_i \in G$ - его набор вершин; $\vec{q} = (q_1, ..., q_m)$, где $q_i \in Q_i$ -- некоторое состояние автомата $\mathfrak{A}_i$. Тогда для для любого $i = 1,...,m$ определим функции:
\[
a_i(\vec{q}, \vec{v}) = (q_{i1}', q_{i2}', ... , q_{im}')
\]
где для $1 \leq j \leq m$, $q_{ij}' = q_j$, если $v_i = v_j$ и $i \neq j$, иначе $q_{ij} = \theta$.
\begin{Def}
Поведением коллектива $(G, S)$-допустимых автоматов \linebreak $\mathcal{A} = (\mathfrak{A}_1, \mathfrak{A}_2,...,\mathfrak{A}_m)$ с начальными состояниями $\vec{q_0} = (q_0^1, ... , q_0^m )$, $q_i \in Q_i$ в графе Кэли $\Gamma(G, S)$ с набором начальных вершин $\vec{v_0} = (v_0^1, ... , v_0^m)$, назовем последовательность \linebreak $\pi(\mathcal{A}_{\vec{q_0}}, \Gamma_{\vec{v_0}}) = (\vec{q_0}, \vec{v_0}), (\vec{q_1}, \vec{v_1}), ...,$ где $\vec{q_t} = (q_t^1, ... , q_t^m )$, $\vec{v_t} = (v_t^1, ... , v_t^m )$, такую, что
\begin{itemize}
\item $q_t^i \in Q_i$, $v_t^i \in G$, $(v_t^i, v_{t+1}^i) \in E(\Gamma)$ или $v_t^i = v_{t+1}^i$
\item $\phi_i(q_t^i, a_i(\vec{v_t}, \vec{q_t})) = q_{t+1}^i$
\item $\psi_i(q_t^i, a_i(\vec{v_t}, \vec{q_t})) = g$, где $v_t^i g = v_{t+1}^i, g \in S \cup S^{-1} \cup e$
\end{itemize}
\end{Def}
То есть, у нас есть коллектив автоматов, который перемещается по лабиринту. Они могут взаимодействовать друг с другом, когда находятся в одной вершине. Формально - следующее состояние и направление хода одного автомата, зависит от состояний автоматов, которые находятся с ним в одной вершине.
Пусть,
\[
\mathrm{Int}_i (\mathcal{A}_{\vec{q_0}}, \Gamma_{\vec{v_0}}) = \bigcup\limits_{t=0}^{\infty} \lbrace v_t^i \rbrace, \forall i = 1,...,m
\]
\[
\mathrm{Int} (\mathcal{A}_{\vec{q_0}}, \Gamma_{\vec{v_0}}) = \bigcup\limits_{t=1}^{m} \mathrm{Int}_i (\mathcal{A}_{\vec{q_0}}, \Gamma_{\vec{v_0}})
\]
\[
\mathrm{Fr} (\mathcal{A}_{\vec{q_0}}, \Gamma_{\vec{v_0}}) = V \setminus \mathrm{Int} (\mathcal{A}_{\vec{q_0}}, \Gamma_{\vec{v_0}})
\]
Будем говорить, что $\mathcal{A}_{q_0}$ обходит $\Gamma_{v_0}$, если $\mathrm{Fr} (\mathfrak{A}_{q_0}, \Gamma_{\vec{v_0}}) = \emptyset$. Говорим, что $\mathcal{A}_{q_0}$ сильно обходит $\Gamma$, если он обходит его начиная из любой вершины лабиринта. $\Gamma_{v_0}$ -- ловушка для $\mathcal{A}_{q_0}$, если автомат не обходит граф; $\Gamma$ -- ловушка для $\mathcal{A}_{q_0}$, если автомат не обходит лабиринт с какой бы вершины он ни начинал.
\section{Построение сильной ловушки для любого коллектива автоматов}
В этой секции мы построим сильную ловушку для любого коллектива автоматов. Ловушка будет выглядеть как граф Кэли некоторой группы. Сама группа будет и, соответственно, лабиринт бесконечны, но любой коллектив автоматов обойдёт лишь конечную область в нём, что мы и докажем.
\subsection{Проблема Бёрнсайда и её решения}
Проблема Бёрнсайда о периодических группах фиксированного периода была поставлена Бёрнсайдом в 1902 году в следующей форме \cite{burnside_1}
\begin{Problem}
Пусть группа $G$ имеет $m$ независимых порождающих элементов \linebreak $a_1, a_2, ..., a_m$ и для любого элемента $x \in G$ выполнено соотношение $x^n=1$, где n — данное целое число. Будет ли определенная таким образом группа конечной, и если да, то каков ее порядок?
\end{Problem}
Сейчас группы, определенные $m$ порождающими и соотношением $x^n=1$, называют свободными бёрнсайдовыми группами ранга $m$ и периода $n$ (экспоненты $n$). Обычно они обозначаются как $B(m,n)$.
Понятно, что вопрос нетривиален для случая $m > 1$. Самим Бёрнсайдом показана конечность $B(m,n)$ для $n \leq 3$ и любого $m$ и $B(2,4)$ \cite{burnside_1}. Сановым \cite{sanov_1} показана конечность $B(m,n)$ для $n=4$ и любого, М.Холлом \cite{hall_1} для $n=6$ и любого.
В 1964 году Голод и Шафаревич \cite{golod_1} доказали, что существуют бесконечные $2$ - порожденные периодические группы с неограниченными периодами элементов. В 1968 году Новиков и Адян представили отрицательное решение проблемы Бёрнсайда \cite{nov_adian_1} для любого нечётного периода $n > 4381$ и любого $m > 1$. В \cite{adian_1} Адяном решение упрощено и доказано, что $B(m,n)$ бесконечны для любого нечётного $n > 665$ и любого $m > 1$. Недавно Адяном было объявлено о улучшении оценки до $n \ge 101$ \cite{Adyan1}.
\begin{Th}
$B(m, n)$ -- бесконечна для всех нечётных $n > 101$ при $m > 1$.
\end{Th}
В 1992 году Иванов \cite{ivanov_1} анонсировал отрицательное решение для больших чётных периодов.
Таким образом, мы имеет бесконечные группы, порождённые конечным количеством элементов, при этом период каждого элемента равномерно ограничен.
\subsection{Определение лабиринта}
Пусть $G$ бесконечная конечно-порождённая группа, $M$ такое, что $g^M = 1$ для любого $g \in G$. $S = \lbrace s_1, s_2,...,s_n \rbrace$ -- множество образующих группы $G$, среди которых нет повторяющихся и обратных. $\Gamma(G, S)$ - граф Кэли для этой группы и образующих.
Пусть $g = t_1 t_2 ... t_n$, где $g \in G$ и $t_i \in S \cup S^{-1} $, причем $n$ минимальное число с такими свойствами, т.е $g$ нельзя представить меньшим количеством элементов из $S \cup S^{-1}$. Тогда будем обозначать $d(g) = n$. Заметим, что $d(g) = d(g^{-1})$. Также, легко видеть, что кратчайшее расстояние в лабиринте между двумя вершинами $v_1$ и $v_2$, равно $d(v_1^{-1} v_2) = d(v_2 v_1^{-1})$.
\subsection{Единичный автомат}
Рассмотрим сначала случай с одним автоматом. Покажем, что единичный автомат сможет обойти только ограниченную часть $\Gamma$.
Пусть $\mathfrak{A}$ -- $(G, S)$-допустимый коллектив из одного автомата. Заметим, что ход автомата в конкретной вершине определяется исключительно его состоянием, т.к. других автоматов нет. Пусть этот автомат $\mathfrak{A}$ начинает движение из некоторой вершины $v_0$. Тогда, его движение будет довольно простым.
\begin{Lm}
Поведение автомата $\mathfrak{A}_{q_0}$ в $\Gamma$ c $|Q|$ состояниями будет обладать следующими свойствами:
\begin{itemize}
\item На начальной стадии автомат сделает $U < |Q|$ ходов, с неповторяющимися состояниями;
\item Далее, состояния будут повторяться с периодом $T \leq |Q|$;
\item Каждые $M$ периодов вершины посещаемые автоматом будут повторятся;
\end{itemize}
\end{Lm}
\begin{ProofLm}
Так как количество состояний конечно, какое-то состояние автомата повториться как минимум два раза. Пусть первое такое состояние $q_1$ и между первым и вторым появлением сделано $T$ ходов (очевидно $T \leq |Q|$), а перед $q_1$ было $U < |Q|$ ходов. Заметим, что текущее состояние однозначно определяет следующее, таким образом после второго появления $q_1$ последовательность состояний будет такой же, как и после первого появления. Отсюда получаем зацикленность состояний с периодом $T$. Обозначим через $v_1$ вершину лабиринта, в которой автомат впервые оказался в состоянии $q_1$.
Пусть $s_1', s_2', ... , s_T'$ -- направления движения автомата в цикле, $s_i' \in S \cup S^{-1}$. Пусть $g_T = s_1' s_2' ... s_T'$, $g_T \in G$. Тогда через $M$ циклов после посещения вершины $v_1$ автомат окажется в вершине $v_1 g_T^M = v_1$, то есть вернётся обратно. Таким образом, сначала автомат посетит не более $U$ вершин, а дальше в цикле будет посещать не более $TM$ вершин.
\end{ProofLm}
Из этой леммы легко видеть, что $\Gamma$ -- сильная ловушка для любого одиночного автомата.
\subsection{Коллектив}
Теперь рассмотрим случай коллектива автоматов. Пусть \linebreak $\mathcal{A}_{\vec{q_0}} = (\mathfrak{A}_1, \mathfrak{A}_2,...,\mathfrak{A}_m)$ -- $(G, S)$-допустимый коллектив автоматов. Основная наша цель -- доказать, что построенный нами лабиринт -- сильная ловушка для любого такого коллектива, причём коллектив сможет обойти только конечное число вершин лабиринта.
Пусть $|Q_1|, ... |Q_m|$ -- количество состояний у автоматов. Обозначим через $Q_A$ максимум из $|Q_i|$ $i = 1,...,m$.
Назовём состоянием коллектива $\mathcal{A}$ в момент времени $t$ набор $I = (q_t^1, q_t^2, ... q_t^m, F_t) $, где $q_t^i \in Q_i$ -- состояние конкретного автомата в момент $t$; $F_t = \lbrace F_t^1, F_t^2, ... , F_t^k \rbrace$, $k \leq m$, где $F_t^i$ -- множество номеров всех автоматов находящихся в какой-то вершине в данный момент. Причём $\bigcup \limits_{i=1}^k F_t^i = \lbrace 1,...,m \rbrace$ и один индекс принадлежит только одному $F_t^i$. Если все автоматы находятся в разных вершинах, то $F_t^i$ состоят их одного индекса каждый и их $m$ штук. То есть, состояние коллектива -- это состояния всех автоматов и разбивка автоматов на группы стоящих в одной вершине. Далее факт нахождения в одной вершине двух или более автоматов будем называть встречей.
Положением коллектива в лабиринте в момент времени $t$, назовём набор вершин
\linebreak $(v_t^1, v_t^2, ... ,v_t^m)$, в котором находятся автоматы. Заметим, что пара положение-состояние однозначно все определяет последующее поведение коллектива автоматов.
\begin{Th}
Поведение коллектива автоматов $\mathcal{A}_{\vec{q_0}} = (\mathfrak{A}_1, \mathfrak{A}_2,...,\mathfrak{A}_m)$ в $\Gamma$ в вектором начальных вершин $\vec{v_0}$ обладает следующим свойством: существуют такие $O_m$ и $H_m$ зависящее только от $m$, $M$ и $Q_A$, что:
\begin{itemize}
\item Состояния коллектива $\mathcal{A}$ любые в $H_m$ подряд идущих моментов времени однозначно определяют следующее состояние.
\item На начальной стадии коллектив автоматов сделает $U \leq O_m$ ходов, после которых состояния будут повторяться с периодом $T \leq O_m$;
\item Каждые $M$ периодов, положение коллектива будет повторяться, т.е. после первых $U$ ходов пара положение-состояние коллектива $\mathcal{A}$ будет повторяться с периодом $MT \leq MO_m$.
\end{itemize}
\end{Th}
\begin{ProofTh}
Будем доказывать индукции по числу автоматов. В случае с одним автоматом, состояние коллектива фактически состояние автомата и $O_1 = Q_A$, $H_1 = 1$, что было доказано ранее. Таким образом база индукции есть.
Пусть $m=l$, тогда по предположению индукции для всех $m < l$ утверждение верно. $H_l$ и $O_l$ будем выражать через $l$, $M$, $Q_A$, $H_i$ и $O_i$ для $i < l$. Поскольку $H_i$ и $O_i$ по предположению зависят только от $i$, $M$ и $Q_A$, $H_l$ и $O_l$ тоже можно будет выразить через $l$, $M$ и $Q_A$.
\begin{Lm}
Пусть существуют две группы автоматов размерами $a$ и $b$, $a + b = l$. Пусть $O_a$ и $O_b$ константы из индукции. Тогда, если никакие два автомата из этих групп (один из первой, один из второй) не встречались $h = \mathrm{max} (O_a, O_b) + M^2 O_a O_b + 1$ ходов подряд, то и в дальнейшем никакие два не встретятся.
\end{Lm}
\begin{ProofLm}
Заметим, что эти $h$ ходов, группы не влияют на движения друг друга, поэтому эти две группы мы можем в течении $h$ ходов рассматривать как два коллектива в лабиринте $L_G$ c некоторыми начальными состояниями и вектором начальных вершин. Заметим, так как $a,b<l$, то к этим коллективам применимы предположения индукции (но только на $h$ первых ходов, в дальнейшем в худшем случае автоматы из разных групп встретятся и независимость пропадёт).
Тогда, после $\mathrm{max} (O_a, O_b)$ ходов (могут и раньше), состояния автоматов в обоих группах зацикливаются, причем циклы длиной не более $O_a$ и $O_b$ и каждые $M$ таких циклов положения групп автоматов повторяются.
Заметим, что за следующие $M^2 O_a O_b + 1$ ходов первая группа будет находиться в не более $M O_a$ различных парах положение-состояние, аналогично вторая в не более $M O_b$ различных парах. Тогда существуют два момента времени $t_1, t_2$ из этих $M^2 O_a O_b + 1$ ходов, когда положение первого коллектива -- $\vec{v_a}$, состояние -- $(\vec{q_a}, F_a)$, положение второго коллектива -- $\vec{v_b}$, состояние -- $(\vec{q_b}, F_b)$. Заметим, что из эти двух пар положение-состояние однозначно получается пара положение-состояние всего коллектива. Таким образом, пара положение-состояние всего коллектива из $m$ автоматов повторилась, то есть оно и дальше будет повторяться с периодом $T_{ab} = t_2 - t_1 < M^2 O_a O_b$. Поскольку, между $t_1$ и $t_2$ группы автоматов не встречались, они и дальше не будут встречаться.
\end{ProofLm}
Докажем, следующую лемму, являющуюся подпунктом теоремы.
\begin{Lm}
Положим $H_l$ минимальным натуральным числом со следующими свойствами:
\[
H_l \geq \mathrm{max} (O_i, O_j) + M^2 O_i O_j + 1, \forall i > 0, j > 0, i + j = l
\]
\[
H_l \geq H_i, \forall i < l
\]
Тогда, по последовательности $I_1, I_2, ..., I_{H_l}$ из $H_l$ состояний коллектива можно однозначно определить состояние $I_{H_l+1}$.
\end{Lm}
\begin{ProofLm}
Заметим, что по этим $H_l$ состояниям мы можем определить, есть ли две группы автоматов, автоматы из которых не встречаются друг с другом за эти $H_l$ ходов (в состояниях записана информация о встречах автоматов).
Пусть есть две такие группы $A$ и $B$ c размерами $a$ и $b$. Тогда в силу выбора $H_l$ эти группы удовлетворяют условию предыдущей леммы. Следовательно, автоматы из этих групп не встретятся после этих $H_l$ ходов, а, следовательно, и на ходу $H_l+1$. Заметим, что состояние коллектива их $l$ камней однозначно определяется состоянием коллектива $A$ и коллектива $B$ и наоборот (если их рассматривать как коллективы автоматов в $L_G$) на этих $H_l + 1$ ходу.
Тогда, $I_{H_l+1}$ -- состояние коллектива $\mathcal{A}$ из $l$ камней, однозначно определяется состоянием коллектива $A$ и коллектива $B$ на $H_l+1$-ом ходу. В свою очередь, состояния коллективов $A$ и $B$ на этом ходу, однозначно определяются предыдущими своими $H_a$ и $H_b$ состояниями, а значит и предыдущими $H_l$ состояниями. Следовательно $I_{H_l+1}$ однозначно определяется через предыдущие $H_l$ состояний коллективов $A$ и $B$, а значит и $\mathcal{A}$, что и требовалось.
Пусть таких двух групп нет. Далее все моменты времени лежат в интервале от $1$ до $H_l$ (если не оговорено обратное). Заметим, что по состояниям $I_1, I_2, ..., I_{H_l}$ можно определить, направление движения автомата $\mathfrak{A}_i$ в момент $t$ (будем обозначать его $g_t^i \in D(S)$). Следовательно, если известная вершина $v_{t_1}$, в которой $\mathfrak{A}_i$ находился в момент $t_1$, то и известна вершина $v_{t_2}$, в которой он находился любой другой момент $t_2$, причём $v_{t_2} = v_{t_1}g$, где $g \in G$ произведение каких-то $g_j^i$, либо их обратных.
Пусть $\mathfrak{A}_1$ в момент времени $1$ находится в вершине $v_0$. Тогда в остальные моменты $2 \leq t \leq H_l$ он находится в вершинах $v_0 h_t^1$, где $h_t^1 \in G$ однозначно определяется через $I_1, I_2, ..., I_{H_l}$. В силу предположения о том, что разделённых групп не существует, есть автомат, для удобства $\mathfrak{A}_2$, с которым он встретится в момент $t_1$, в вершине $v_0 h_{t_1}^1$. Тогда, в момент $t_1$ известно положение $\mathfrak{A}_2$, а значит известны вершины в которой $\mathfrak{A}_2$ находится во все остальные моменты времени $t$, причём они имеют вид $v_0 h_{t}^2$, где $h_{t}^2$ однозначно выражается через $I_1, I_2, ..., I_{H_l}$ ($h_{t_1}^1 = h_{t_1}^2$, а остальные получаются из $h_{t_1}^2$ с помощью $g_j^2$).
Аналогично, найдем автомат $\mathfrak{A}_3$, с которым встречается хотя бы один из двух первых, для него положения вершин будут определятся аналогично. Так будем продолжать и далее. Заметим, что в силу условия о не существовании двух не встречающихся групп, мы всегда найдем новый автомат, с которым есть встречи у уже рассмотренных. Таким образом, любой $\mathfrak{A}_i$ из $m$ автоматов в момент $t$ находится в вершине $v_0 h_t^i$, причём $h_{t}^i$ однозначно выражается через $I_1, I_2, ..., I_{H_l}$.
Рассмотрим момент времени $H_l+1$. Состояния каждого из $m$ автоматов, однозначно получаются из $I_{H_l}$. Пусть в момент $H_l+1$ $\mathfrak{A}_i$ находится в вершине $v_0 h_{H_l}^i g_i$, $g_i \in G$. Заметим, что для определения, стоят ли $\mathfrak{A}_i$ и $\mathfrak{A}_j$ достаточно сравнить на равенство $h_{H_l}^i g_i$ и $h_{H_l}^j g_j$. $h_{H_l}^i$ и $h_{H_l}^j$ однозначно определяется через $I_1, I_2, ..., I_{H_l}$, $g_i$ и $g_j$ через $I_{H_l}$. Следовательно, разбиение на группы стоящие в одной вершине на шаге $H_l + 1$ однозначно определяется через $I_1, I_2, ..., I_{H_l}$, а значит и $I_{H_l+1}$ однозначно через них определяется. Что нам и требовалось. Итого пункт про $H_l$ выполнен.
\end{ProofLm}
Посчитаем, сколько различных состояний $I = (\vec{q}, F)$ может быть у коллектива $\mathcal{A}$. Различных $\vec{q}$ не более $Q_A^l$ (т.к. состояний у каждого из автоматов не более $Q_A$). Количество различных $F$ грубо оценим сверху числом $l^l$. Действительно, разложить $l$ индексов автоматов по $l$ множествам можно не более чем $l^l$ способами, и каждому такому способу соответствует только одно $F$ (разбиение автоматов по группам, стоящим в одной вершине). Итого, различных состояний коллектива не более чем $(Q_A l)^l$.
Тогда заметим, что существует не более $(Q_A l)^{lH_l}$ различных блоков из $H_l$ подряд идущих состояний коллектива. Положим $O_l = (Q_A l)^{lH_l} + H_l$, рассмотрим первые $O_l$ ходов коллектива $\mathcal{A}$ в $L_G$. В этих ходах можно выделить $(Q_A l)^{lH_l} + 1$ блок из $H_l$ подряд идущих состояний коллектива. Тогда, два блока начинающиеся в ходы $t_1$ и $t_2$ будут одинаковыми. Так как каждый такой блок однозначно определяет последующую последовательность состояний коллектива, мы получаем, что, начиная с момента $t_1$, состояния коллектива зацикливаются с периодом $t_2 - t_1$. Таким образом, $U$ из условия теоремы равно $t_1 \leq O_l$ (если первый ход считается нулевым), а $T = t_2 - t_1 \leq O_l$. Получаем, что второй пункт теоремы выполнен, нужное $O_l$ найдено
Для доказательства теоремы осталось показать, что каждые $M$ получившихся циклов, положения коллектива будут повторяться. Действительно, пусть любой $\mathfrak{A}_i$ на каком-то ходе $t \geq t_1$ находится в вершине $v_t$. Пусть за каждые $T$ ходов он сдвигается на $g_i \in G$, тогда через $M$ ходов он попадёт в вершину $v_t g_i^M = v_t$, что нам и требуется. Таким образом после первых $U$ ходов пара положение-состояние всего коллектива $\mathcal{A}$ будет повторяться с периодом $MT \leq MO_m$.
\end{ProofTh}
Таким образом, мы показали, что любой коллектив автоматов стартующий из любых вершин обойдёт только конечную часть $\Gamma$.
\section{Литература}
|
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| 6,580
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{"url":"https:\/\/ask.openstack.org\/en\/questions\/99556\/revisions\/","text":"# Revision history [back]\n\n### gnocchi setup\n\nHi,\n\nI am following this document http:\/\/docs.openstack.org\/developer\/ceilometer\/install\/custom.html to setup gnocchi, but I don't understand when it says\n\nThe value specified for archive_policy should correspond to the name of an archive_policy configured within Gnocchi.\n\nCould someone please help me to find out what I need to put on my gnocchi.conf file?\n\nI have:\n\nceilometer.conf\n\n[dispatcher_gnocchi]\n\narchive_policy = low\n\n\ngnocchi.conf\n\n[archive_policy]\n\n#\n# From gnocchi\n#\n\n# Default aggregation methods to use in created archive policies (list value)\n#default_aggregation_methods = mean,min,max,sum,std,median,count,95pct\n\n\nthank you very much","date":"2020-07-13 12:59:23","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\": 0.1838446855545044, \"perplexity\": 5574.873928149887}, \"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-2020-29\/segments\/1593657143365.88\/warc\/CC-MAIN-20200713100145-20200713130145-00216.warc.gz\"}"}
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Spencer Williams, né le à Vidalia (Louisiane) et mort le à Los Angeles (Californie), est un acteur, scénariste, réalisateur et producteur afro-américain. Il est principalement connu pour avoir réalisé le film The Blood of Jesus (1941) et pour son rôle d'Andy dans l'émission télévisée Amos 'n' Andy. Il est l'un des pionniers afro-américains du cinéma avec Oscar Micheaux.
Biographie
Spencer Williams dit Spencer Williams Jr., né le 14 juillet 1893 à Vidalia, en Louisiane, emménage à la fin de son adolescence à New York. Il commence à travailler la comédie sous la direction de Bert Williams, un comédien de vaudeville.
Les débuts
Après ses études secondaires à la Natchez-Adams School District (Mississipi), Spencer Williams s'inscrit à l'université du Minnesota, il la quitte avant son Bachelor of Arts pour s'engager dans l'armée américaine pendant et après la Première Guerre mondiale, il sera démobilisé en 1923. Il retourne à New York, Voulant s'engager dans le monde du spectacle et intéressé par le développement du cinéma il partira à Hollywood où il commence à travailler avec le scénariste Octavus Roy Cohen, puis il commence à tenir des petits rôles dans divers films comme Tenderfeet, Cadet d'eau douce /Steamboat Bill Jr.. En 1927, il est engagé par le First National Pictures, puis en 1928 par la Christie Film Company comme scénariste, gagman , ingénieur du son et acteur pour participer à la réalisation de films destinés au public afro-américain : Melancholy Dame réalisé par Arvid E. Gillstrom, Music Hath Harms réalisé par Walter Graham, The Framing of the Shrew réalisé par Arvid E. Gillstrom, Oft in the Silly Night par Arvid E. Gillstrom.
En 1939, il est le scénariste de deux films : un "black western" et une comédie d'horreur .
Le réalisateur de cinéma
En 1941, Alfred Sack, directeur de la Sack Amusement Enterprises sise à Dallas, l'engage comme réalisateur. Enfin, en 1941, malgré un petit budget de 5000$, Spencer Williams peut tourner son premier film The Blood of Jesus, ce sera un succès populaire auprès du public afro-américain. En 1949 il tourne Dirty Gerty avec en vedette Francine Everett.
La télévision, la série Amos 'n' Andy
En 1946, la CBS souhaite programmer des émissions télévisées à destination du public afro-américain, leur choix se porte sur une série radiophonique Amos 'n Andy, créée par Freeman Gosden et Charles Correll en 1928, série très populaire auprès des afro-américains, dont la CBS achète les droits à la NBC en 1939. Amos 'n Andy sera la première émission de télévision américaine à être composée d'acteurs afro-américains et qui comptera 78 épisodes diffusés de 1951 à 1953. Les deux acteurs principaux sont dans le rôle d'Amos Jones et Spencer Williams dans celui d'Andrew Hogg Brown (Andy).
La série provoque des polémiques, la NAACP appelle au boycott, traitant les acteurs de la série d'Oncle Tom.
Le retrait et la fin
Spencer Williams décède, le , des suites d'une insuffisance rénale majeure au de Los Angeles. En tant qu'ancien combattant de la Première Guerre mondiale il repose au .
Héritage
Son film The Blood of Jesus est le premier film afro-américain inscrit au National Film Registry en 1991, c'est la sortie du purgatoire.
Les films de Spencer Williams comme ceux d'Oscar Micheaux et d'autres productions afro-américaines ont souffert de la ségrégation et des préjugés racistes, bien loin de Hollywood et des circuits habituels de la production filmographique, son nom est absent des ouvrages américains de référence sur le cinéma comme l'International Dictionary of Films and Filmmakers ou le The New Biographical Dictionary of Film dans leurs éditions respectives de 2000 et 2002. Le décès de Spencer Williams ne fait l'objet que d'un entrefilet tardif dans la rubrique nécrologique du New York Times du 24 décembre 1969
En 2015, le Museum of Modern Art (MoMa) de New York organise une rétrospective des films de Spencer Williams.
Filmographie
Acteur
Tenderfeet (court métrage, 1928)
Melancholy Dame (court métrage, 1929)
Music Hath Harms (court métrage, 1929)
The Framing of the Shrew (court métrage, 1929)
Oft in the Silly Night (court métrage, 1929)
The Lady Fare (court métrage, 1929)
Brown Gravy (court métrage, 1929)
Fowl Play (court métrage, 1929)
The Widow's Bite (court métrage, 1929)
Georgia Rose (1930)
Reno (1930)
The Virginia Judge (1935)
Coronado (1935)
Harlem on the Prairie (1937)
Two-Gun Man from Harlem (1938)
The Bronze Buckaroo (1939)
Harlem Rides the Range (1939)
Bad Boy (1939)
Son of Ingagi (1940)
Toppers Take a Bow (court métrage, 1941)
The Blood of Jesus (1941)
Brother Martin: Servant of Jesus (1942)
Of One Blood (1944)
Go Down, Death! (1944)
Beale Street Mama (1946)
The Girl in Room 20 (1946)
Dirty Gertie from Harlem U.S.A. (1946)
Juke Joint (1947)
Rhapsody of Negro Life (court métrage, 1949)
Amos 'n' Andy (série TV, 78 épisodes, 1951-1955)
Bourbon Street Beat (série TV, 1 épisode 1959)
Réalisateur
The Blood of Jesus (1941)
Brother Martin: Servant of Jesus (1942)
Marching On! (1943)
Of One Blood (1944)
Go Down, Death! (1944)
Harlem Hotshots (court métrage, 1945)
Beale Street Mama (1946)
The Girl in Room 20 (1946)
Jivin' in Be-Bop (documentaire, 1946)
Dirty Gertie from Harlem U.S.A. (1946)
Juke Joint (1947)
Rhapsody of Negro Life (court métrage, 1949)
Notes et références
Liens externes
Naissance en juillet 1893
Naissance dans la paroisse de Concordia
Acteur américain du XXe siècle
Acteur afro-américain
Acteur américain du muet
Acteur américain de cinéma
Réalisateur américain
Scénariste américain de cinéma
Race film
Décès en décembre 1969
Décès à Los Angeles
Décès à 76 ans
|
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|
{"url":"http:\/\/mathhelpforum.com\/algebra\/54813-why-average-set-numbers-alway-smaller-than-largest-number-s-print.html","text":"# why is the average of a set of numbers alway smaller than the largest number in the s\n\n\u2022 October 20th 2008, 04:20 PM\nm3clubracer\nwhy is the average of a set of numbers alway smaller than the largest number in the s\nwhy is the average of a set of numbers alway smaller than the largest number in the set?\n\u2022 October 20th 2008, 05:54 PM\nterr13\nNot true, take the set 5,5,5. The average is 5, which is not greater than the largest element 5.\n\u2022 October 22nd 2008, 04:45 PM\nThePerfectHacker\nQuote:\n\nOriginally Posted by m3clubracer\nwhy is the average of a set of numbers always smaller than the largest number in the set?\n\nI assume smaller is in the weak sense rather than in a strict sense.\n\nLet us do it with three elements so you can see the idea. Say $\\{a_1\\leq a_2\\leq a_3\\}$.\n\nThe average $a_1=\\tfrac{a_1+a_1+a_1}{3}\\leq \\tfrac{a_1+a_2+a_3}{3} \\leq \\tfrac{a_3+a_3+a_3}{3} = a_3$","date":"2013-05-22 12:43:39","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\": 0, \"img_math\": 0, \"codecogs_latex\": 2, \"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\": 0.8447909355163574, \"perplexity\": 400.45982702909697}, \"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-2013-20\/segments\/1368701670866\/warc\/CC-MAIN-20130516105430-00048-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/physicstravelguide.com\/advanced_notions\/topological_defects\/vortices","text":"# Vortices\n\n## Layman\n\nAn example of a vortex is universally encountered by people taking baths or washing dishes. As the water flows down the drain, it circulates. We cannot interpolate the circulating velocity field all the way to the center of the vortex since it would have to become multi-valued. Instead the fluid density in the central region of the vortex vanishes.\n\n## Student\n\nConsider the Lagrangian: $$L = |\\partial _\\mu \\phi |^2 - {\\lambda\\over 4}(|\\phi |^2 -\\eta^2 )^2 \\label{glstring}$$ where $\\phi$ is now taken to be a complex scalar field. The Lagrangian is invariant under $$\\phi \\rightarrow \\phi ' = e^{i\\alpha} \\phi$$ and hence the model has a $U(1)$ (global) symmetry. The vacuum expectation value of $\\phi$ is $\\eta e^{i\\alpha}$ where $\\alpha$ can take any value. So the ground state of the model has continuous degeneracy. The degeneracy is labelled by the phase angle $\\alpha$ and hence the vacuum manifold is a circle.\n\nVortices are formed if we consider the model in two spatial dimensions and let $\\alpha$ be such as to wrap around the vacuum manifold. For example, we could take $\\alpha =\\theta$, the polar angle. Then, since the field is single valued everywhere, there must be at least one point at which $\\phi =0$. The field carries energy at this point since $\\phi =0$ is not on the vacuum manifold. The location of this point may be defined as the location of a vortex.\n\n## Researcher\n\nThe motto in this section is: the higher the level of abstraction, the better.\n\nExample1\nExample2:\n\n## History\n\nLord Kelvin famously tried to explain the different atoms as vortices.","date":"2021-09-29 02:53:56","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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9026405811309814, \"perplexity\": 242.57847634493223}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"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-2021-39\/segments\/1631780061350.42\/warc\/CC-MAIN-20210929004757-20210929034757-00109.warc.gz\"}"}
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\section{Relation-Changing Modal Logics}
\label{sec:basic}
In this section, we formally introduce extensions of the basic
modal logic with relation-changing operators. For more details,
we direct the reader to, e.g.,~\cite{fervari14phd}.
\begin{definition}[Syntax]\label{def:relchangsyn}
Let ${\rm \sf PROP}\xspace$ be a countable, infinite set of propositional symbols.
The set ${\rm \sf FORM}\xspace$ of formulas over ${\rm \sf PROP}\xspace$ is defined as:
$$
{\rm \sf FORM}\xspace ::= \bot \mid p \mid \neg \varphi
\mid \varphi \land \psi
\mid \lozenge \varphi
\mid \blacklozenge \varphi,
$$
where $p\in {\rm \sf PROP}\xspace$, $\blacklozenge \in \{\tup{{\sf sb}}, {\tup{\sf br}}, \dswap, \dgsab, {\tup{\sf gbr}},\dgswap\}$,
and $\varphi, \psi \in {\rm \sf FORM}\xspace$. Other operators are defined
as usual.
Let $\ensuremath{\mathcal{ML}}\xspace$ (the basic modal logic) be the logic without the $\cset{\tup{{\sf sb}}, {\tup{\sf br}}, \dswap, \dgsab, {\tup{\sf gbr}},\dgswap}$ operators,
and $\ensuremath{\mathcal{ML}}\xspace(\blacklozenge)$ the extension of $\ensuremath{\mathcal{ML}}\xspace$ allowing also
$\blacklozenge$, for $\blacklozenge \in \cset{\tup{{\sf sb}}, {\tup{\sf br}}, \dswap,
\dgsab, {\tup{\sf gbr}},\dgswap}$. In particular, $\ensuremath{\mathcal{ML}}\xspace(\tup{{\sf sb}}, \dgsab)$, $\ensuremath{\mathcal{ML}}\xspace({\tup{\sf br}}, {\tup{\sf gbr}})$, and $\ensuremath{\mathcal{ML}}\xspace(\dswap, \dgswap)$ will be called Sabotage Logic, Bridge Logic, and Swap Logic, respectively.
\end{definition}
Semantically, formulas are evaluated in standard relational models, and the meaning of the operators of the basic modal logic remains unchanged (see~\cite{blackburn01} for details). When we evaluate formulas containing relation-changing operators, we will need to keep track of the edges that have been modified. To that end, let us define precisely the models that we will use.
\begin{definition}[Models and model updates]\label{def:rcmodels}
A model $\mathcal M$ is a triple $\mathcal M = \langle W, R, V \rangle$,
where $W$ is a non-empty set whose elements are called points or states;
$R \subseteq W{\times} W$ is the accessibility relation; and $V: {\rm \sf PROP}\xspace \rightarrow \pow(W)$
is a valuation. We define the following notations:
\begin{center}
\begin{tabular}{ll}
\textbf{(sabotaging)} & $\mathcal M^-_{S} = \langle W, R^-_{S}, V \rangle$,
with $R^-_{S} = R{\setminus} S$, $ S\subseteq R$.\\
\textbf{(bridging)} & $\mathcal M^+_{S} = \langle W, R^+_{S}, V \rangle$,
with $R^+_{S} = R \cup S$, $S \subseteq (W{\times}W){\setminus}R$.\\
\textbf{(swapping)} & $\mathcal M^*_{S} = \langle W, R^*_{S}, V \rangle$,
with $R^*_{S} = (R{\setminus} S^{-1} ){\cup}S$, $S \subseteq R^{-1}$.
\end{tabular}
\end{center}
\end{definition}
Intuitively, $\mathcal M^-_{S}$ is obtained from $\mathcal M$ by deleting the edges in $S$, and similarly
$\mathcal M^+_S$ adds the edges in $S$ to the accessibility relation, and $\mathcal M^*_S$ adds the
edges in $S$ as inverses of edges previously in the accessibility relation.
Let $w$ be a state in $\mathcal M$, the pair $(\mathcal M,w)$ is called a pointed model;
we will usually drop parenthesis and write $\mathcal M,w$ instead of $(\mathcal M,w)$. In the
rest of this article, we will use $wv$ as a shorthand for $\{(w,v)\}$ or $(w,v)$;
context will always disambiguate the intended use.
\begin{definition}[Semantics]\label{def:relchangsem}
Given a pointed model $\mathcal M,w$ and a formula $\varphi$, we say that $\mathcal M,w$
satisfies $\varphi$, and write $\mathcal M,w \models \varphi$, when
$$
\begin{array}{lcl}
\mathcal M, w \models p & \iff & \ w \in V(p) \\
\mathcal M, w \models \neg \varphi & \iff & \ \mathcal M, w \not\models \varphi\\
\mathcal M, w \models \varphi \land \psi & \iff
& \ \mathcal M, w \models \varphi \mbox{ and } \mathcal M, w \models \psi\\
\mathcal M, w \models \lozenge \varphi & \iff
& \mbox{ for some } v \in W \mbox{ s.t. } (w, v) \in R,
\mathcal M, v \models \varphi\\
\mathcal M, w \models \tup{{\sf sb}} \varphi & \iff & \mbox{ for some } v \in W
\mbox{ s.t. } (w, v) \in R, \mathcal M^-_{wv}, v \models \varphi\\
\mathcal M, w \models {\tup{\sf br}} \varphi & \iff & \mbox{ for some } v \in W
\mbox{ s.t. } (w, v) \not\in R, \mathcal M^+_{wv}, v \models \varphi \\
\mathcal M, w \models \dswap \varphi & \iff & \mbox{ for some } v \in W
\mbox{ s.t. } (w, v) \in R, \mathcal M^*_{vw}, v \models \varphi\\
\mathcal M, w \models \dgsab \varphi & \iff & \mbox{ for some } v,u\in W,
\mbox{ s.t. } (v,u) \in R, \mathcal M^-_{vu}, w \models \varphi\\
\mathcal M, w \models {\tup{\sf gbr}} \varphi & \iff & \mbox{ for some } v,u\in W,
\mbox{ s.t. } (v, u) \not\in R, \mathcal M^+_{vu}, w \models \varphi\\
\mathcal M, w \models \dgswap \varphi & \iff & \mbox{ for some } v,u\in W,
\mbox{ s.t. } (v, u) \in R, \mathcal M^*_{uv}, w \models \varphi.
\end{array}
$$
We say that $\varphi$ is satisfiable if for some pointed model $\mathcal M, w$ we have
$\mathcal M, w \models \varphi$.
\end{definition}
The meaning of the relation-changing operators $\tup{{\sf sb}}$ (local sabotage), ${\tup{\sf br}}$ (local bridge),
$\dswap$ (local swap), $\dgsab$ (global sabotage), ${\tup{\sf gbr}}$ (global bridge) and $\dgswap$ (global swap) should
be clear from the semantic conditions above. The local operators alter one arrow which is adjacent to the point of
evaluation (deleting, adding and swapping it, respectively) while the global versions can change an arrow
anywhere in the model.
\section{Decidable Fragments}\label{sec:decid}
Interesting decidable fragments of hybrid logics with binders have been
found over time.
Such decidable fragments are convenient for our relation-changing logics
in the light of the (computable) translations presented in Section~\ref{sec:trans}.
First, let us consider restricting the satisfiability problem over certain classes
of models. The following logics are known to be decidable over the indicated classes:
\begin{itemize}
\item[-] $\mathcal{HL}({\sf E},{\downarrow})$ over linear frames (i.e., irreflexive, transitive, and trichotomous frames) \cite{FranceschetRS03,schneider07phd} (this includes $(\mathbb{N},<)$),
\item[-] $\mathcal{HL}({\sf E},{\downarrow})$ over models with a single, transitive tree relation \cite{schneider07phd},
\item[-] $\mathcal{HL}({\sf E},{\downarrow})$ over models with a single, $S5$, or complete relation \cite{schneider07phd},
\item[-] $\mathcal{HL}(:,{\downarrow})$ over models with a single relation of bounded finite width~\cite{tencate2005complexity};
as a corollary, also over finite models.
\end{itemize}
Since the translations preserve equivalence, we get:
\begin{corollary} \label{coro:semsat}
The satisfiability problem for all relation-changing modal logics over
linear, transitive trees, $S5$, and complete frames is decidable.
\end{corollary}
\begin{corollary}
The satisfiability problem for local sabotage and local swap logics over
models of bounded width is decidable.
\end{corollary}
Curiously, these results mean that relation-changing modal logics are
decidable over certain classes of models, even if the modifications
implied by evaluating RC formulas yield models that \emph{do not}
belong to such class. For instance, these two facts are simultaneously true:
sabotage logic is decidable on the class of $S5$ models, and deleting edges in
an $S5$ model can yield a non-$S5$ model.
Now, let us turn to syntactical definitions of decidable fragments.
We recall that local sabotage and local swap can be translated to $\mathcal{HL}(:,{\downarrow})$.
Consider formulas of $\mathcal{HL}(:,{\downarrow})$ in negation normal form.
$\mathcal{HL}(:,{\downarrow}) \setminus \Box{\downarrow}\Box$ is the fragment obtained by removing
formulas that contain a nesting of $\Box$, ${\downarrow}$ and again $\Box$.
This fragment is decidable \cite{tencate2005complexity}.
Our translations do use the ${\downarrow}$ binder in many places, but we can make
them a little more economical in that sense, at the expense of losing succinctness.
Take the following case for $\ensuremath{\mathcal{ML}}\xspace(\dswap)$:
$$
\begin{array}{rl}
(\Diamond \varphi)'_S = & {\downarrow} n . \lozenge ( \neg {\sf belongs}(n,S) \wedge (\varphi)'_S).\\
\end{array}
$$
Instead of using the down-arrow binder and later ensuring that we
did not take a deleted edge by using $\neg {\sf belongs}(n , S)$, we can
do the following. For all pairs of nominals $(x,y) \in S$, the current state
$w$ satisfies one combination of the truth values of the nominals $x$. Let $X$ be
the set of true nominals $x$ at $w$.
Then, $(\varphi)'_S$ should be true at some accessible state $v$ that should not
satisfy any of the corresponding $y$ nominals for all $x \in X$.
Then, the translation becomes:
$$
\begin{array}{rl}
(\Diamond \varphi)'_S =
& \underset{X \subseteq {\sf fst(S)}}{\bigvee}(
\underset{x \in X}{\bigwedge} x \wedge \underset{x \notin X}\bigwedge \neg x
\wedge \Diamond
(\underset{y \in {\sf snd}(S,X)}{\bigwedge} \neg y \wedge (\varphi)'_S))
\end{array}
$$
where ${\sf fst}(S) = \{ x \mid (x,y) \in S \}$
and ${\sf snd}(S,X) = \{ y \mid (x,y) \in S , x \in X \}$.
In the case of $\ensuremath{\mathcal{ML}}\xspace(\dswap)$ we can do the same. We recall that the case
introduced in Section~\ref{sec:trans} was:
$$
\begin{array}{rl}
(\Diamond \varphi)'_S = &
({\downarrow} n . \lozenge (\neg{\sf belongs}(n,S) \wedge (\varphi)'_S ))
~ ~\vee ~ ~ {\sf isSat}(S^{-1},(\varphi)'_S). \\
\end{array}
$$
Here the ${\sf isSat}(S^{-1},(\varphi)'_S)$ disjunct does not use
the ${\downarrow}$ binder, while the first disjunct is similar to the case
of local sabotage, and can be replaced accordingly:
$$
\begin{array}{rl}
(\Diamond \varphi)'_S =
& ~ ~ ~ ~ ~ \underset{X \subseteq {\sf fst(S)}}{\bigvee}(
\underset{x \in X}{\bigwedge} x \wedge \underset{x \notin X}\bigwedge \neg x
\wedge \Diamond
(\underset{y \in {\sf snd}(S,X)}{\bigwedge} \neg y \wedge (\varphi)'_S))\\
& \vee ~ ~ {\sf isSat}(S^{-1},(\varphi)'_S).\\
\end{array}
$$
Let $\blacklozenge$ be either $\tup{{\sf sb}}$ or $\dswap$
and $\blacksquare$ be either $[{\sf sb}]$ or $\bswap$. The following patterns in RC formulas
provoke the following patterns in the hybrid formula produced by the
translations:
\begin{center}
\begin{tabular}{|c |c|}
\hline
RC pattern & Produced pattern \\
\hline
$\Box$ & $\Box$ \\
\hline
$\blacklozenge$ & ${\downarrow}$ \\
\hline
$\blacksquare$ & ${\downarrow} \Box {\downarrow}$\\
\hline
\end{tabular}
\end{center}
By considering these new versions of the translations, and by taking into account the syntactic decidable fragment of $\mathcal{HL}(:,{\downarrow})$ mentioned above, we can establish the following result:
\begin{corollary} \label{th:syxsat}
The following fragments are decidable on the class of all relational models:
\begin{itemize}
\item[-] $\ensuremath{\mathcal{ML}}\xspace(\tup{{\sf sb}}) \setminus \{ \blacksquare\blacksquare,
\blacksquare\Box,
\Box\blacksquare,
\textcolor{gray}{\blacksquare}\blacklozenge\textcolor{gray}{\blacksquare} \}$
\item[-] $\ensuremath{\mathcal{ML}}\xspace(\dswap)\setminus \{ \blacksquare\blacksquare,
\blacksquare\Box,
\Box\blacksquare,
\textcolor{gray}{\blacksquare}\blacklozenge\textcolor{gray}{\blacksquare} \}$
\end{itemize}
\noindent where $\textcolor{gray}{\blacksquare}$ is either $\Box$ or $\blacksquare$.
\end{corollary}
\section{Comparing Expressive Power}
\label{sec:exppow}
We have introduced translations for the six relation-changing modal logics
from Section~\ref{sec:basic} into hybrid logic. In some cases (for the local version
of swap and sabotage), the obtained formulas fall into the fragment $\mathcal{HL}(:,{\downarrow})$.
On the other hand, for encoding the rest of the logics we need also to use the
universal modality ${\sf E}$. An interesting question is whether we can obtain translations
from hybrid to relation-changing logics, i.e., if some of the relation-changing
logics considered in this article are as expressive as some hybrid logic. Let us define
first, the expressive power comparisons we will use.
\begin{definition}[${\cal L} \leq {\cal L}'$]
\label{def:expcomp}
We say that ${\cal L}'$ is \emph{at least as expressive as} ${\cal L}$
(notation ${\cal L}\leq{\cal L}'$) if there is a function ${\sf Tr}$ between
formulas of ${\cal L}$ and ${\cal L}'$ such that for
every model $\mathcal M$ and every formula $\varphi$ of ${\cal L}$ we have that
$$\mathcal M \models_{\cal L} \varphi \mbox{ iff } \mathcal M \models_{\cal L'} {\sf Tr}(\varphi).$$
$\mathcal M$ is seen as a model of
${\cal L}$ on the left and as a model of ${\cal L'}$ on the right, and we
use in each case the appropriate semantic relation $\models_{\cal L}$ or
$\models_{\cal L'}$ as required.
$\L'$ is strictly more expressive than $\L$ ($\L<\L'$) if $\L\leq\L'$ but
not $\L'\leq\L$. Finally, we say that ${\cal L}$ and ${\cal L}'$ are
\emph{incomparable} if ${\cal L} \nleq {\cal L}'$ and ${\cal L}' \nleq {\cal L}$.
\end{definition}
In~\cite{areces12,areces14igpl,fervari14phd,AFH15} we discussed the expressive power
of relation-changing modal logics by introducing their corresponding notions
of bisimulations and using them to compare the logics among each other. We concluded that
they are all {\em incomparable in expressive power.}\footnote{Except for the
local and global swap operators, which is still open in one direction.} As a consequence, we conclude
that it is not possible that two of them capture the same fragment of hybrid logic.
In fact, we will prove that all the relation-changing logics considered here
are strictly less expressive than the corresponding hybrid logic in which they are translated.
\begin{theorem}
\label{th:rcstricthl}
Let $\blacklozenge_1\in\{\tup{{\sf sb}},\dswap\}$, we have $\ensuremath{\mathcal{ML}}\xspace(\blacklozenge_1)<\mathcal{HL}(:,{\downarrow})$.
For $\blacklozenge_2\in\{\dgsab,\dgswap,{\tup{\sf br}},{\tup{\sf gbr}}\}$, we have
$\ensuremath{\mathcal{ML}}\xspace(\blacklozenge_2)<\mathcal{HL}({\sf E},{\downarrow})$.
\end{theorem}
\begin{proof}
For any of the logics mentioned above, we have translations into the
corresponding hybrid logic. Now we need to prove that these translations
do not cover their entire target language (modulo equivalence). In order to do that,
we provide bisimilar models for relation-changing modal logics which can
be distinguished by some hybrid formula. In Figure~\ref{fig:models}, we show
two pairs of models already introduced in~\cite{AFH15} that cover all possibilities of bisimilarity.
\begin{figure}[!h]
\begin{center}
\begin{tabular}{|c |c|l|}
\hline
$\mathcal M,w$ & $\mathcal M',w'$ & Bisimilar for\\
\hline
\hline
\begin{minipage}{.25\linewidth}
\begin{tikzpicture}[>=latex]
\node (b) at (3,0) [shape=circle,draw,fill, inner sep=1pt, label=below:$ w$] {} ;
\path [ar,->] (b) edge [loop] (b);
\end{tikzpicture}
\end{minipage}
&
\begin{minipage}{.25\linewidth}
\begin{tikzpicture}[>=latex]
\node (a0) at (7,0) [shape=circle,draw,fill, inner sep=1pt,label=below:$ w'$] {} ;
\node (a1) at (9,0) [shape=circle,draw,fill, inner sep=1pt] {} ;
\path [ar,->] (a0) edge [loop] (a0);
\path [ar,->] (a1) edge [loop] (a1);
\draw [ar,->] (a0) edge [bend left] (a1);
\draw [ar,->] (a1) edge [bend left] (a0);
\end{tikzpicture}
\end{minipage}
& \begin{minipage}{2cm}
\smallskip
$\ensuremath{\mathcal{ML}}\xspace(\dswap)$
$\ensuremath{\mathcal{ML}}\xspace({\tup{\sf br}})$
$\ensuremath{\mathcal{ML}}\xspace(\dgswap)$
$\ensuremath{\mathcal{ML}}\xspace({\tup{\sf gbr}})$
\end{minipage}
\\
\hline
\begin{minipage}{.25\linewidth}
\begin{tikzpicture}[>=latex]
\node (b0) at (2,0) [shape=circle,draw,fill, inner sep=1pt, label=above:$ w$] {} ;
\node (b1) at (4,0) [shape=circle,draw,fill, inner sep=1pt, label=above:$ $] {} ;
\node (b2) at (3,-1) [shape=circle,draw,fill, inner sep=1pt, label=below:$ $] {} ;
\draw [ar,->] (b0) edge [left] (b1);
\draw [ar,->] (b1) edge [left] (b2);
\draw [ar,->] (b0) edge [left] (b2);
\end{tikzpicture}
\end{minipage}
&
\begin{minipage}{.25\linewidth}
\begin{tikzpicture}[>=latex]
\node (a0) at (6,0) [shape=circle,draw,fill, inner sep=1pt, label=above:$ w'$] {} ;
\node (a1) at (8,0) [shape=circle,draw,fill, inner sep=1pt, label=above:$ $] {} ;
\node (a2) at (8,-1) [shape=circle,draw,fill, inner sep=1pt, label=below:$ $] {} ;
\node (a3) at (6,-1) [shape=circle,draw,fill, inner sep=1pt, label=below:$ $] {} ;
\draw [ar,->] (a0) edge [left] (a1);
\draw [ar,->] (a1) edge [left] (a2);
\draw [ar,->] (a0) edge [left] (a3);
\end{tikzpicture}
\end{minipage}
& \begin{minipage}{2cm}
$\ensuremath{\mathcal{ML}}\xspace(\tup{{\sf sb}})$\\
$\ensuremath{\mathcal{ML}}\xspace(\dgsab)$
\end{minipage}
\\
\hline
\end{tabular}
\end{center}
\caption{Bisimilar models}\label{fig:models}
\end{figure}
The two models in the first row can be distinguished by
the formula ${\downarrow} n. \Box n$, which establishes that the only successor
of the evaluation point is itself. This formula is true at $\mathcal M,w$
and false at $\mathcal M',w'$. Models in the second row can be distinguished
by the formula ${\downarrow} n. \lozenge {\downarrow} m. n{:}\lozenge\lozenge m$,
which says that from the evaluation point it is possible to arrive to the same
state in one or two steps. This is true at
$\mathcal M,w$ but false at $\mathcal M',w'$.
\end{proof}
Notice that both hybrid formulas we introduced above belong
to the fragment $\mathcal{HL}(:,{\downarrow})$, i.e., it was not necessary to use the ${\sf E}$ operator.
This means that even though we use ${\sf E}$ in some of the translations (and we strongly believe
that it is essential for some encodings) there are fragments
of $\mathcal{HL}(:,{\downarrow})$ that cannot be captured by relation-changing modal logics.
\section{Final Remarks}
\label{sec:final}
In this article, we introduced equivalence-preserving translations from six logics we named
{\em relation-changing} to a very expressive hybrid logic.
We considered three kinds of modifications:
deleting, adding, and swapping edges, that can be performed both globally (anywhere in the model)
and locally (modifying adjacent edges from the evaluation point).
On the other hand, hybrid logic has operators to rename states
in a model with some particular atomic symbols named nominals. We use the down-arrow operator
${\downarrow}$ to name pairs of states that represent modified states. In this way, we keep
track of the evolution of a model.
It is known that the hybrid logic $\mathcal{HL}({\sf E},{\downarrow})$ has the same expressive power as $\mathcal{FOL}$,
and we introduced standard translations from relation-changing logics to $\mathcal{FOL}$ in~\cite{AFH15}.
However, by giving explicit translations to hybrid logic we can benefit from its decidable
fragments to find decidable fragments of relation-changing
modal logics. Also, these translations are useful to analyze expressive power. We showed that
the six logics we considered are strictly less expressive than $\mathcal{HL}({\sf E},{\downarrow})$. In fact,
despite we used the modality ${\sf E}$ in some translations, all relation-changing logics we
considered here cannot capture the full fragment $\mathcal{HL}(:,{\downarrow})$ (which is less expressive than
$\mathcal{HL}({\sf E},{\downarrow})$). In summary, we learned that relation-changing modal logics are languages
that enable to talk directly and succinctly about distinct kinds of model modifications,
but with a little effort they can be simulated by hybrid logics.
Translating to $\mathcal{HL}({\sf E}, {\downarrow})$ also enabled us to easily obtain an implementation
for relation-changing modal logics, by extending the hybrid logic theorem prover HTab.
Satisfiability checking and model building can thus be automated and were useful to
empirically verify our translations on concrete cases.
However, we did not implement the changes of Section~\ref{sec:decid}.
Indeed, although in some cases they provide a way to avoid using the down-arrow binder in
the translations (which is a source of undecidability for hybrid logic),
they also make the generated formulas impractically bigger.
We studied six relation-changing modal logics with the goal of covering a sufficiently varied sample
of alternatives. Clearly, other operators could have been included in this exploration, and actually
some alternative choices have been already investigated in the literature, e.g., the adjacent sabotage
operator discussed in~\cite{rohde06phd}, or the more generic approach investigated in~\cite{AFH15}.
Even though relation-changing modal logics have been extensively investigated~\cite{fervari14phd,AFH15},
there are still many interesting questions to be answered. The {\em hybrid perspective} we introduced in this article,
as well as in~\cite{areces13}, gives us a new way to think of the relation-changing framework.
As an example, we can use {\em hybridization techniques} (a very standard technique in modal logic~\cite{blackburn01})
to find complete axiomatizations or compute interpolants for relation-changing modal logics.
\section{Extensions of Modal Logic and Hybrid Logic}\label{sec:hybrid}
In this section, we present several extensions of the basic modal logic $\ensuremath{\mathcal{ML}}\xspace$.
The existential modality~\cite{gorausin92}, written ${\sf E} \varphi$, extends $\ensuremath{\mathcal{ML}}\xspace$ in the following way:
$$
\begin{array}{lcl}
\mathcal M, w \models {\sf E} \varphi & \iff
& \mbox{ for some } v \in W, ~ \mathcal M, v \models \varphi.
\end{array}
$$
In words, ${\sf E}\varphi$ is true at a state $w$ if $\varphi$ is true somewhere in the model.
The ${\sf E}$ operator, with its dual ${\sf A}$, has been extensively investigated in classical modal logic~\cite{Spaan93}.
Now we consider several traditional `hybrid' operators (see~\cite{arec:hybr05b} for details):
nominals, the satisfaction operator, and the down-arrow binder.
The basic hybrid logic $\mathcal{HL}$ is obtained by adding \emph{nominals} to $\ensuremath{\mathcal{ML}}\xspace$.
A nominal is a
propositional symbol that is true at exactly one state in a model. Fix the signature $\tup{{\rm \sf PROP}\xspace, {\rm \sf NOM}\xspace}$,
with ${\rm \sf NOM}\xspace \subseteq {\rm \sf PROP}\xspace$.
For $n \in {\rm \sf NOM}\xspace$, we require that its valuation is a singleton set,
i.e., there is a single state $w$ such that $V(n) = \{w\}$.
In addition to nominals, hybrid logic typically involves
the \emph{satisfaction operator}. Given a nominal $n$ and a formula
$\varphi$, the satisfaction operator is written $n:\varphi$. The intended meaning is ``$\varphi$ is true at the
state named by $n$''. Its semantics is given by the following clause:
$$
\begin{array}{rcl}
{\cal M}, w \models n:\varphi
& \mbox{iff}
& \mathcal M, v \models \varphi \mbox{ where } V(n) = \{v\}.
\end{array}
$$
Observe that if the language has the ${\sf E}$ operator and nominals, then
$n:\varphi$ is definable because $n:\varphi$ is equivalent
to ${\sf E}(n \wedge \varphi)$.
Finally, consider the \emph{down-arrow binder} operator, written $\downarrow$.
Let the valuation $V_n^w$ be defined by $V_n^w(n)= \{w\}$ and $V_n^w(m)= V(m)$,
when $n \not = m$.
The semantic condition for ${\downarrow}$ is the following:
$$
\begin{array}{rcl}
\tup{W,R,V}, w \models {\downarrow} n . \varphi
& \mbox{iff}
& \tup{W,R,V_n^w}, w \models \varphi.
\end{array}
$$
The language $\mathcal{HL}(:,{\downarrow})$ is a reduction class of first-order logic, and
is thus undecidable~\cite{blackburn95,tencate_phd}.
It remains undecidable even with a single accessibility
relation, no satisfaction operator, and only nominal propositional symbols~\cite{arecroad99}.
$\mathcal{HL}({\sf E},{\downarrow})$ is equivalent to first-order logic,
since ${\downarrow}$ can define the operators $\exists$ and
$\forall$ when combined with ${\sf E}$ and ${\sf A}$.
Contrary to relation-changing modal logics, the logic $\mathcal{HL}({\sf E}, {\downarrow})$
is not able to modify the accessibility relation of a model. However,
it can use the binder to name as many states as needed. Being able to
name states implies that we can also name \emph{specific edges} in the model.
This is what the translations introduced in the next section will exploit.
\section{Implementation and Examples}
\label{sec:imple}
We have implemented these translations as a new feature of the
tableaux-based theorem prover HTab~\cite{Hoffmann2007}. Its version
1.7.1 can be downloaded from \url{http://hub.darcs.net/gh/htab} along with
example formulas.
Instructions are provided, explaining how
to use HTab to check satisfiability of some relation-changing formula
and to generate a model from an open tableau.
HTab originally handles the hybrid logic $\mathcal{HL}(:, {\sf E}, {\downarrow})$,
and guarantees termination
of any fragment that lacks the ${\downarrow}$ binder.
As part of the work presented in this article, we added the following
feature: when passed the \verb_--translate_ flag, HTab interprets the
input formula as a relation-changing one. It first translates it to
the corresponding $\mathcal{HL}({\sf E}, {\downarrow})$ formula (or more precisely,
$\mathcal{HL}(:,{\downarrow})$ formula in the case of local sabotage and local swap),
and then runs its internal hybrid tableaux calculus on the translation.
Since the translation uses the ${\downarrow}$ binder, HTab may never terminate
on some specific relation-changing formulas. Even in the terminating cases,
the size of the translated formula (in particular for
swap logic) can make HTab run for a very long time before giving an answer.
However, there are several possible workarounds.
First, a time limit in seconds can be given with
the flag \verb_-t_. Also, it may be useful to disable the semantic branching
optimization by passing \verb_--sembranch=no_. Tweaking unit propagation
can also be useful for certain formulas, in some cases by disabling it
(with \verb_--no-unit-prop_), and in others, by making it more aggressive (with \verb_--eager_).
The flag \verb_--minimal_ makes HTab try reusing existing states in the
model instead of systematically generating new ones. This introduces
many more branches in the tableau, making it slower, but it can be crucial
to make some cases terminate. Finally, the flag \verb_--random_
uses pseudorandomness to select the next disjunct formula
to expand in the tableau, and also shuffles the order in which
the branches are explored, including those introduced
by \verb_--minimal_. The advantage is that some pseudorandom run of HTab
on a given input formula could terminate quickly by doing the right choices.
Then, this run can be reproduced by setting the pseudorandom seed of HTab
with the \verb_--seed_ flag.
More information is available by running \verb_htab --help_.
For all three translations, the implementation has the following particular case:
$$(\Diamond \varphi)'_\emptyset = \Diamond (\varphi)'_\emptyset$$
This avoids introducing unnecessary nominals and makes the translated formula
more readable. The generated hybrid formula can be seen by using the \verb_--showformula_ flag.
Since the translations are equivalence-preserving, the models built by HTab satisfy the
input RC formula. Let us see a few examples.
We present the formulas in multi-line way, each line is a conjunct of the whole
formula. Also, some new logical connectors are used in the expected way.
\bigskip
\begin{minipage}[t]{0.3\textwidth}
Input $\ensuremath{\mathcal{ML}}\xspace{\tup{{\sf sb}}}$ formula:
$$
\begin{array}{l}
\Diamond(A \wedge \neg B \wedge \Diamond\Diamond A)\\
\Diamond(B \wedge \neg A \wedge \Diamond\Diamond B)\\
[{\sf sb}](A \rightarrow \Box\Box\neg A)\\
[{\sf sb}](B \rightarrow \Box\Box\neg B)\\
\end{array}
$$
\end{minipage}
~
\begin{minipage}[t]{0.3\textwidth}
Translated hybrid formula:
\vspace{-0.4cm}
$$
\begin{array}{l}
\Diamond(A \wedge \neg B \wedge \Diamond\Diamond A)\\
\Diamond(B \wedge \neg A \wedge \Diamond\Diamond B)\\
{\downarrow} n_0 . \Box ( {\downarrow} n_1 . (\neg A \\
~ ~ ~ \vee {\downarrow} n_2 . \Box ( ( n_1 \wedge n_2: n_0)\\
~ ~ ~ ~ ~ \vee {\downarrow} n_3 . \Box ( (n_1 \wedge n_3:n_0 ) \\
~ ~ ~ ~ ~ ~ ~ \vee \neg A ) )) )\\
{\downarrow} n_0 . \Box ( {\downarrow} n_1 . (\neg B \\
~ ~ ~ \vee {\downarrow} n_2 . \Box ( ( n_1 \wedge n_2: n_0)\\
~ ~ ~ ~ ~ \vee {\downarrow} n_3 . \Box ( (n_1 \wedge n_3:n_0 ) \\
~ ~ ~ ~ ~ ~ ~ \vee \neg B ) )) )\\
\end{array}
$$
\end{minipage}
~
\begin{minipage}[t]{0.3\textwidth}
Model found by HTab:
\vspace{-0.4cm}
\begin{center}
\includegraphics[width=3cm]{mod_lsb.pdf}
\end{center}
\end{minipage}
\bigskip
\begin{minipage}[t]{0.3\textwidth}
Input $\ensuremath{\mathcal{ML}}\xspace{\dgsab}$ formula:
$$
\begin{array}{l}
\Diamond(A \wedge \neg B \wedge \Diamond\Diamond A)\\
\Diamond(B \wedge \neg A \wedge \Diamond\Diamond B)\\
\Box\Box(C \wedge \Box\neg C)\\
\dgsab\Box\Box\Box \bot \\
\end{array}
$$
\end{minipage}
~
\begin{minipage}[t]{0.3\textwidth}
Translated hybrid formula:
\vspace{-0.4cm}
$$\begin{array}{l}
\Diamond(A \wedge \neg B \wedge \Diamond\Diamond A)\\
\Diamond(B \wedge \neg A \wedge \Diamond\Diamond B)\\
\Box\Box(C \wedge \Box\neg C)\\
{\downarrow} n_0 . {\sf E} {\downarrow} n_1
\Diamond( {\downarrow} n_2 .\\
~ ~ ~ n_0:({\downarrow} n_3 . \Box (
( n_2 \wedge n_3:n_1) \\
~ ~ ~ ~ ~ \vee ~ {\downarrow} n_4 . \Box (
(n_2 \wedge n_4:n_1) \\
~ ~ ~ ~ ~ ~ ~ \vee ~ {\downarrow} n_5 . \Box (
(n_2 \wedge n_5:n_1) ) ) ) ) ) \\
\end{array}
$$
\end{minipage}
~
\begin{minipage}[t]{0.3\textwidth}
Model found by HTab:
\vspace{-0.6cm}
\begin{center}
\includegraphics[width=3cm,height=4.6cm,keepaspectratio=false]{mod_gsb.pdf}
\end{center}
\end{minipage}
\bigskip
\begin{minipage}[t]{0.3\textwidth}
Input $\ensuremath{\mathcal{ML}}\xspace{\dswap}$ formula:
\vspace{-0.5cm}
$$
\begin{array}{l}
\Diamond(A \wedge \neg B)\\
\Diamond(B \wedge \neg A)\\
\Box\Diamond \top\\
\Box\Box\Box \bot\\
\bswap\Box\bswap\Box\Box \bot\\
\Box\bswap\Box\Box \bot\\
\dswap\dswap\Diamond\Diamond\Diamond\Diamond\Diamond \top\\
\end{array}
$$
\end{minipage}
~
\begin{minipage}[t]{0.3\textwidth}
Translated hybrid formula:
\bigskip
\bigskip
\bigskip
\emph{\ldots too big to be displayed \ldots}
\end{minipage}
~
\begin{minipage}[t]{0.3\textwidth}
Model found by HTab:
\vspace{-0.4cm}
\begin{center}
\includegraphics[width=3cm]{mod_lsw.pdf}
\end{center}
\end{minipage}
More formulas from the six RC logics are available in the \verb_./rc/_
directory of the HTab source code.
This implementation is useful to check the correctness of the translations,
just by checking the satisfiable/unsatisfiable output of the prover for known
RC formulas. It is
also useful for checking that RC formulas build models in the expected way,
such as non-tree or diamond-shaped models.
\section{Introduction}
\label{sec:intro}
Modal logics~\cite{blackburn06,blackburn01} were originally conceived
as logics of necessary and possible truths. They are now viewed, more
broadly, as logics that explore a wide range of modalities, or modes
of truth: epistemic (``it is known that''), doxastic (``it is believed
that''), deontic (``it ought to be the case that''), or temporal (``it
has been the case that''), among others. From a model-theoretic
perspective, the field evolved into a discipline that deals with
languages interpreted on various kinds of relational structures or
graphs. Nowadays, modal logics are actively used in areas as diverse
as software verification, artificial intelligence, semantics and
pragmatics of natural language, law, philosophy, etc.
As we just mentioned, from an abstract point of view, modal logics can
be seen as formal languages to navigate and explore properties of a
given relational structure. If we are interested, on the other hand,
in describing how a given relational structure evolves (through time
or through the application of certain operations) then classical modal
languages seem a priori to fall short of the mark. Of course, it is
always possible to model the whole space of possible transformations
as a graph, and use modal languages at that level, but this soon
becomes unwieldy. It would be more elegant to use \emph{dynamic
modalities} that mimic the changes the structure will undergo.
There exist several dynamic modal operators that fit in this
approach. A clear example are the dynamic operators introduced in dynamic
epistemic logics (see, e.g.,~\cite{vanditmarsch07}). These operators are used to model
changes in the epistemic state of an agent by removing edges
from the graph that represents the information states the agent
considers possible. A less obvious example is given by hybrid
logics~\cite{arec:hybr05b,blackburn95}
equipped with the down arrow operator ${\downarrow}$ which is used to
`rebind' names for states to the current point of evaluation.
Finally, a classical example is Sabotage Logic introduced
by van Benthem in~\cite{vanbenthem05}. The sabotage operator deletes individual edges in a graph
and was introduced to solve the \emph{sabotage game}.
This game is played on a graph by two players, {\em Runner} and {\em Blocker}.
Runner can move on the graph from node to accessible node, starting from a designated point,
and with the goal of reaching a given final point.
Blocker, on the other hand, can delete one edge from the graph every time
it is his turn. Runner wins if he manages to move from the origin to the final point, while Blocker wins otherwise.
Van Benthem turns the sabotage game
into a modal logic, where the (global) sabotage operator $\dgsab$ models the moves of Blocker,
and is interpreted on a graph $\mathcal M$ at a point $w$ as:
$$
\mathcal M,w \models \dgsab \varphi ~ \mbox{ iff } ~ \mbox{ there is a pair $(u, v)$ of $\mathcal M$ such that $\mathcal M^-_{(u,v)}, w \models \varphi$}
$$
where $\mathcal M^-_{(u,v)}$ is identical to $\mathcal M$ except that the edge
$(u,v)$ has been removed. The moves of Runner, on the other hand, can
be modeled using the standard $\Diamond$ operator of classical modal logics.
More recently, Sabotage Logic was proposed as a formalism for reasoning about formal learning theory~\cite{GKVQ09}.
Learning can be seen as a game with two players, {\em Teacher} and {\em Learner},
where Learner changes his information state through a step-by-step process.
The process is successful if he eventually reaches an information
state describing the real state of affairs.
The information that Teacher provides
can be interpreted as feedback about Learner's conjectures about the current state of affairs, allowing him to discard inconsistent
hypotheses.
It should be clear that from this game-theoretical perspective, the interaction between Teacher and Learner
can be modeled using Sabotage Logic.
The dynamic approach seems appealing and very flexible: it is easy to come up with situations that nicely fit and
extend the examples we just mentioned. Discovering alternative routes for Runner in van Benthem's sabotage game, or possible shortcuts that
Learner can take in learning theory can be modeled by adding new edges to the graph.
Swapping an edge can be used to represent other scenarios such as changing the direction of a route,
or allowing Learner to return to a previous information state. All these primitives can also
be turned into a modal logic in the same way as Sabotage Logic,
in order to get a formal language for reasoning about the games.
Motivated by scenarios like the ones we just described, we investigate three dynamic primitives that can
change the accessibility relation
of a model: {\em sabotage} (deletes edges from the model), {\em bridge} (adds edges to a model), and {\em swap}
(turns around edges), both in a global version (performing changes anywhere in the model) and local
(changing adjacent edges from the evaluation point).
We have chosen these relation-changing operators with the intention of covering a sufficiently varied sample of alternatives, as it is done in previous works. In~\cite{areces12} we first introduced the primitives, and we investigated their expressive
power and model checking problem. We introduced tableaux methods for relation-changing modal logics in~\cite{areces13}.
In~\cite{areces14igpl} we studied local swap logic, in particular its decidability problem and its relation with first-order and hybrid logics.
In~\cite{AFH15} a general framework for representing
model updates is defined, and connections with dynamic epistemic logic were introduced in~\cite{areces14wollic,ADFS15}.
Finally, we know that the satisfiability problem for the six relation-changing logics considered
is undecidable~\cite{loding03fsttcs,fervari14phd,areces14igpl,martel15}.
In this article, we show that relation-changing logics can be seen as fragments of hybrid logics.
We consider hybrid logics because it is the best known \emph{modal} logic that can
simulate the semantics of relation-changing operators.
We introduce translations
to $\mathcal{HL}({\sf E},{\downarrow})$, the basic modal logic
extended with nominals, the down arrow binder ${\downarrow}$, and the universal modality ${\sf E}$ (in some cases the translations fall into
the less expressive hybrid logic $\mathcal{HL}(:,{\downarrow})$, i.e., with the satisfiability operator $:$ instead
of ${\sf E}$).
We also show that relation-changing logics are strictly less expressive than the hybrid logics they are translated into. Then, we discuss how
we can benefit from known decidable fragments of
$\mathcal{HL}({\sf E},{\downarrow})$ to find decidable fragments of relation-changing modal logics. Finally, we have implemented these
translations into the hybrid logic prover HTab~\cite{Hoffmann2007} so that it can now reason on relation-changing formulas.
The article is organized as follows. In Section~\ref{sec:basic} we introduce the syntax and semantics
of relation-changing modal logics. Section~\ref{sec:hybrid} introduces different hybrid
extensions of modal logic which are used in Section~\ref{sec:trans} to encode relation-changing operators.
An implementation is described in Section~\ref{sec:imple} with some examples.
Then, in Section~\ref{sec:decid} we discuss decidability results, and
in Section~\ref{sec:exppow} we compare the expressivity of relation-changing logics and hybrid logics.
Finally, we conclude with some remarks and
future work in Section~\ref{sec:final}.
\section{Translations to Hybrid Logics}
\label{sec:trans}
Relation-changing (RC) logics and hybrid logics with the binder
${\downarrow}$ are two families of logics that are dynamic in their own way.
The dynamicity of RC logics is quite obvious: they are able to modify
the accessibility relation in a model in an explicit way. On the
other hand, hybrid logics carefully move nominals around, avoiding to
touch anything else in the model. If we consider both formalisms, it
would seem that hybrid logics are the gentler and weaker of both.
However, this is not true. Hybrid logics have the advantage of
surgical precision over RC logics. Being able to name states of the
model and use these names turns out to be a crucial advantage. As we
will see now, naming can be used to manipulate \emph{edges} by naming
pairs of states using the pattern ${\downarrow} x . \Diamond {\downarrow} y
. \varphi$. We use this naming technique to simulate edge deletion,
addition, and swapping.
Our translations are parametrized over a set
of pair of nominals $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$. For a given RC
formula $\varphi$, we write its translation as a hybrid formula
$(\varphi)'_S$. When translating a formula, $S$ will originally be
empty and it will store pairs of nominals that we will use to simulate
the edges affected by the relation-changing operators we encounter
during the translation.
Intuitively, given that the hybrid operators cannot affect the accessibility relation, we have to
simulate the updates by recording possible affected edges using nominals and ${\downarrow}$.
Notice that as a result, in all the relation-changing logics we will consider, the RC formula
$\Diamond\psi$ cannot be simply translated into a hybrid formula $\Diamond(\psi)'_S$, even though
we have $\Diamond$ at our disposition in the hybrid language, because in the source language
$\Diamond$ is interpreted over the updated accessibility relation.
Instead, diamond-formulas need to be translated in a way that takes into account the edges
that should be considered deleted, added, or swapped. This is why
the translation of diamond-formulas involve the $\Diamond$ operator
mixed with specific considerations about the set of altered edges $S$.
Consider Sabotage Logic with either the local or global
operator. We use the set $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$ to represent
sabotaged edges, i.e., edges
that have been deleted in a given updated model.
\begin{definition}[Sabotage to Hybrid Logic]\label{transsab}
Let $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$ and $n \in {\rm \sf NOM}\xspace$.
We define the translation $(~ ~)'_S$ from formulas
of $\ensuremath{\mathcal{ML}}\xspace(\tup{{\sf sb}},\dgsab)$ to formulas of $\mathcal{HL}({\sf E}, {\downarrow})$ as:
$$
\begin{array}{rl}
(p)'_S = & p\\
(\neg \varphi)'_S = & \neg (\varphi)'_S\\
(\varphi \wedge \psi)'_S = & (\varphi)'_S \wedge (\psi)'_S\\
(\Diamond \varphi)'_S = & {\downarrow} n . \lozenge ( \neg {\sf belongs}(n,S) \wedge (\varphi)'_S)\\
(\tup{{\sf sb}} \varphi)'_S = & {\downarrow} n. \lozenge ( \neg {\sf belongs}(n,S) \wedge {\downarrow} m. (\varphi)'_{S \cup nm})\\
(\dgsab \varphi)'_S = & {\downarrow} k. {\sf E} {\downarrow} n. \lozenge ( \neg {\sf belongs}(n,S) \wedge {\downarrow} m. k{:}(\varphi)'_{S \cup nm})
\end{array}
$$
\noindent where $n$, $m$ and $k$ are nominals that do not appear in $S$, and:
$${\sf belongs}(n,S) = \underset{xy\in S}{\bigvee} (y ~ \wedge ~ n{:}x)$$
\end{definition}
A few comments are in order to understand the translation.
First, given some model $\mathcal M=\tup{W,R,V}$ and
some set $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$,
the formula ${\downarrow} n. \Diamond (\neg {\sf belongs}(n,S))$ is true at
some state $w\in W$ if there exists some state $v$ such that $(w,v) \in R$
and there is no pair of nominals $(x,y) \in S$ such that $(V(x),V(y)) = (w,v)$.
Then, observe that the cases for $\tup{{\sf sb}}$ and $\dgsab$ modify the set
of deleted pairs in the recursive call to the translation, in both cases by
adding an edge named $nm$. In the $\tup{{\sf sb}}$ case, $n$ names the evaluation state
of the formula, while in the $\dgsab$ case, $n$ names some state anywhere in
the model.
Finally, all nominals introduced by the translation are bounded
exactly once. Then we can define the following unequivocal notation:
let $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$,
we define $\bar{S}=\{(\bar{x},\bar{y}) \mid (x,y) \in S \}$,
where $\bar{n}$ is the state named by the nominal $n \in {\rm \sf NOM}\xspace$ under
the current valuation of a model.
When considering the translated formula $(\varphi)'_S$ and its truth
in some model $\mathcal M=\tup{W,R,V}$, one question that may arise is
what should be the initial valuation of the nominals that appear
in $(\varphi)'_S$. By definition of models for hybrid
logics, nominals must be true at some state. This is not
problematic: in $(\varphi)'_S$, nominals are immediately bounded
by the ${\downarrow}$ operator, so the truth value of $(\varphi)'_S$ does not depend
on their initial valuation. Hence, we can choose some state $w\in W$
and say that all nominals are bounded to it. This enables us to talk
about equivalence preservation of the translation: the same model $\mathcal M$
can be used for $\varphi$ and its translation $(\varphi)'_S$ modulo
the addition of the set of nominals that appear in $(\varphi)'_S$ and their
valuation to some arbitrary state. Then, we can state:
\begin{theorem}
For $\mathcal M=\tup{W,R,V}$ a model, $w\in W$, and $\varphi \in \ensuremath{\mathcal{ML}}\xspace(\tup{{\sf sb}}, \dgsab)$
we have:
$$\mathcal M,w\models\varphi ~ ~ \iff ~ ~ \mathcal M,w\models(\varphi)'_\emptyset.$$
\end{theorem}
\begin{proof}
We use structural induction on the relation-changing formula, the inductive
hypothesis being:
$$\mathcal M^-_{\bar{S}},w\models\varphi ~ ~ \iff ~ ~ \tup{W,R,V'},w\models(\varphi)'_S$$
with $S\subseteq {\rm \sf NOM}\xspace\times{\rm \sf NOM}\xspace$, and $V'$ is exactly as $V$
except that for all $(x,y)\in S$, there are $v,u\in W$ such that
$V'(x)=v$ and $V'(y)=u$.
Boolean cases
are straightforward, so we only prove the non-trivial inductive cases.
\smallskip
\noindent $\varphi=\lozenge\psi$: For the left to right direction, suppose $\mathcal M^-_{\bar{S}},w\models\lozenge\psi$.
Then there is some $v\in W$ such that $(w,v)\in R^-_{\bar{S}}$ and $\mathcal M^-_{\bar{S}},v\models\psi$.
Because $(w,v)\notin \bar{S}$, then there is no $(x,y)\in S$ such that $(\bar{x},\bar{y})=(w,v)$.
By inductive hypothesis, we have $\mathcal M,v\models(\psi)'_S$, and because we can
name $w$ with a fresh nominal $n$, we obtain $\tup{W,R,V^w_n},v\models\neg{\sf belongs}(n,S)\wedge(\psi)'_S$.
Therefore, we have $\mathcal M,w\models{\downarrow} n . \lozenge ( \neg {\sf belongs}(n,S) \wedge (\psi)'_S)$, and
as a consequence we get $\mathcal M,w\models(\psi)'_S$.
For the other direction, suppose $\mathcal M,w\models(\psi)'_S$, i.e.,
$\mathcal M,w\models{\downarrow} n . \lozenge ( \neg {\sf belongs}(n,S) \wedge (\psi)'_S)$.
Then we have $\tup{W,R,V^w_n},w\models \lozenge ( \neg {\sf belongs}(n,S) \wedge (\psi)'_S)$,
and, by definition, there is some $v\in W$ such that $(w,v)\in R$,
$\tup{W,R,V^w_n},v\models \neg {\sf belongs}(n,S)$ and $\tup{W,R,V^w_n},v\models (\psi)'_S$.
Because we have $\neg {\sf belongs}(n,S)$, there is no $(x,y)\in S$ such that $(\bar{x},\bar{y})=(w,v)$,
which implies $(w,v)\in R$ if and only if $(w,v)\in R^-_{\bar{S}}$. On the other hand, by inductive hypothesis
we have $\mathcal M^-_{\bar{S}},v\models\psi$, then we have $\mathcal M^-_{\bar{S}},w\models\lozenge\psi$.
\medskip
\noindent $\varphi=\tup{{\sf sb}}\psi$: For the left to right direction, suppose $\mathcal M^-_{\bar{S}},w\models\tup{{\sf sb}}\psi$. Then there is some $v\in W$ such that $(w,v)\in R^-_{\bar{S}}$ and $(\mathcal M^-_{\bar{S}})^-_{wv},v\models\psi$.
This is equivalent to say $\mathcal M^-_{\bar{S}\cup wv},v\models\psi$.
Because $(w,v)\notin \bar{S}$, then there is no $(x,y)\in S$ such that $(\bar{x},\bar{y})=(w,v)$ ~$(\otimes)$.
By inductive hypothesis we have $\tup{W,R,((V')^w_n)^v_m},v\models(\psi)'_{S\cup nm}$,
where $V'$ is exactly as $V$ but it binds all the nominals which appear in $S$.
By definition, we get $\tup{W,R,(V')^w_n)},v\models{\downarrow} m. (\psi)'_{S\cup nm}$,
and by $(\otimes)$ we
have $\tup{W,R,(V')^w_n)},v\models \neg{\sf belongs}(n,S) \wedge {\downarrow} m. (\psi)'_{S\cup nm}$.
Then (by definition) $\tup{W,R,V'},v\models {\downarrow} n. \lozenge(\neg{\sf belongs}(n,S) \wedge {\downarrow} m. (\psi)'_{S\cup nm})$,
and, as a consequence, we have $\tup{W,R,V'},v\models (\varphi)'_S$.
For the other direction, suppose $\tup{W,R,V'},w\models(\psi)'_S$, i.e.,
$\tup{W,R,V'},w\models{\downarrow} n . \lozenge ( \neg {\sf belongs}(n,S) \wedge {\downarrow} m. (\psi)'_{S\cup nm})$,
where $V'$ is exactly as $V$ but it binds all the nominals which appear in $S$.
Then, we have $\tup{W,R,(V')^w_n},w\models \lozenge ( \neg {\sf belongs}(n,S) \wedge {\downarrow} m. (\psi)'_{S\cup nm})$,
and, by definition, there is some $v\in W$ such that $(w,v)\in R$,
$\tup{W,R,V^w_n},v\models \neg {\sf belongs}(n,S)$ and $\tup{W,R,V^w_n},v\models {\downarrow} m. (\psi)'_{S\cup nm}$.
Then, $\tup{W,R,((V')^w_n)^v_m},v\models (\psi)'_{S\cup nm}$.
Because we have $\neg {\sf belongs}(n,S)$, there is no $(x,y)\in S$ such that $(\bar{x},\bar{y})=(w,v)$,
which implies $(w,v)\in R$ if and only if $(w,v)\in R^-_{\bar{S}}$.
On the other hand, by inductive hypothesis
we have $\mathcal M^-_{\bar{S}\cup wv},v\models\psi$, and thus we have $\mathcal M^-_{\bar{S}},w\models\tup{{\sf sb}}\psi$.
\medskip
\noindent $\varphi=\dgsab\psi$: this case is very similar to the previous one.
\end{proof}
For Bridge Logic, we use the set $B \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$
to represent the new edges.
New edges present in $B$ mean that the translation of
the modality $\Diamond$ should be able to take them.
This explains why the translation of $\Diamond$ does not
look like a $\Diamond$ with an extra condition, but like an ${\sf E}$
with two possibilities: we traverse an edge that is either in the original
model or an edge from the $B$ set.
\begin{definition}[Bridge to Hybrid Logic]\label{transbr}
Let $B \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$. We define $(~ ~)'_B$ from formulas of \linebreak $\ensuremath{\mathcal{ML}}\xspace({\tup{\sf br}},{\tup{\sf gbr}})$
to formulas of $\mathcal{HL}({\sf E}, {\downarrow})$ as:
$$
\begin{array}{rl}
(p)'_B = & p\\
(\neg \varphi)'_B = & \neg (\varphi)'_B\\
(\varphi \wedge \psi)'_B = & (\varphi)'_B \wedge (\psi)'_B\\
(\Diamond \varphi)'_B = &
{\downarrow} n . {\sf E} {\downarrow} m ( ~ (n{:}\lozenge m \vee {\sf belongs}(n,B))
~ \wedge ~ (\varphi)'_B )\\
({\tup{\sf br}} \varphi)'_B = & {\downarrow} n. {\sf E} {\downarrow} m . (
\neg n{:}\Diamond m \wedge \neg {\sf belongs}(n,B)
\wedge (\varphi)'_{B{\cup}nm})\\
({\tup{\sf gbr}} \varphi)'_B = & {\downarrow} k . {\sf E} {\downarrow} n. {\sf E} {\downarrow} m . (
\neg n{:}\Diamond m \wedge \neg {\sf belongs}(n,B)
\wedge k{:}(\varphi)'_{B{\cup}nm})\\
\end{array}
$$
\noindent where $n$, $m$ and $k$ are nominals that do not appear in $B$,
and ${\sf belongs}$ is defined as in Definition~\ref{transsab}.
\end{definition}
\begin{theorem}
For $\mathcal M=\tup{W,R,V}$ a model, $w\in W$, and $\varphi \in \ensuremath{\mathcal{ML}}\xspace({\tup{\sf br}}, {\tup{\sf gbr}})$,
we have:
$$\mathcal M,w\models\varphi ~ ~ \iff ~ ~ \mathcal M,w\models(\varphi)'_\emptyset.$$
\end{theorem}
\begin{proof}
A similar reasoning can be done with the following inductive hypothesis:
$$\mathcal M^+_{\bar{B}},w\models\varphi ~ ~ \iff ~ ~ \tup{W,R,V'},w\models(\varphi)'_B$$
with $B\subseteq {\rm \sf NOM}\xspace\times{\rm \sf NOM}\xspace$, and $V'$ is exactly as $V$
except that for all $(x,y)\in B$, there are $v,u\in W$ such that
$V'(x)=v$ and $V'(y)=u$.
\end{proof}
We finish with the case of Swap Logic.
We presented a different translation in~\cite{areces14igpl} for the local case only.
As we did for Sabotage Logic, we use $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$ to
represent the set of deleted edges, i.e., the edges
that should not be possible to traverse in a given updated model.
Indeed, swapping a non-reflexive edge of a model has the effect of
deleting it, along with adding its inverse.
This implies that $S^{-1}$ is a set of edges that we can currently
traverse. All of this requires that $S$ do not contain any reflexive edge,
since a swapped reflexive edge is not deleted. Neither can it contain
a pair of symmetric edges since that would be contradictory.
To ensure this, the translation gets more cautious when handling $\dswap$
and $\dgswap$. When swapping occurs, three possible cases are
taken into account. The first one is when a reflexive edge is swapped.
In that case, the translation continues with the set $S$ left unchanged,
but we require some reflexive
edge to be present, be it at the current state for $\dswap$ with
${\downarrow} n . \Diamond n$, or anywhere in the model for $\dgswap$
with ${\sf E}{\downarrow} n . \Diamond n$.
The second case is when we swap an irreflexive edge
that has never been swapped before.
Hence we ensure that this edge is present in the model, that it is
irreflexive, and that neither this edge nor its
inverse is in $S$. We then add the nominals that name it to $S$
before moving on with the translation.
The last case is when we traverse an already swapped edge.
That is, for some $xy \in S$, we traverse the edge referred to
by the nominals $yx$. In this case, we do not need to require the presence
of any new edge in the model. We assume to be standing at the state named
by $y$ and that the rest of the formula is satisfied at $x$, with the
modification that we remove $xy$ from $S$ and add $yx$ to it.
An attentive reader would object: why not just remove $xy$ from the set
$S$ since swapping some edge twice just makes it return to its configuration
in the original model? The answer is that there is a corner case when some edge
\emph{and} its symmetric are both present in the initial model. Then, the action
of swapping it twice is not supposed to restore its symmetric. This is what we do
by adding $yx$ to the set $S$: we ensure the former symmetric edge is no
longer present.
\begin{definition}[Swap to Hybrid Logic]\label{transswap}
Let $S \subseteq {\rm \sf NOM}\xspace \times {\rm \sf NOM}\xspace$.
We define $(~ ~)'_S$ from formulas of \linebreak$\ensuremath{\mathcal{ML}}\xspace(\dswap,\dgswap)$ to formulas of
$\mathcal{HL}({\sf E}, {\downarrow})$ as:
$$
\begin{array}{rl}
(p)'_S = & p\\
(\neg \varphi)'_S = & \neg (\varphi)'_S\\
(\varphi \wedge \psi)'_S = & (\varphi)'_S \wedge (\psi)'_S\\
(\Diamond \varphi)'_S = &
({\downarrow} n . \lozenge (\neg{\sf belongs}(n,S) \wedge (\varphi)'_S ))
~ ~\vee ~ ~ {\sf isSat}(S^{-1},(\varphi)'_S) \\
(\dswap \varphi)'_S = &
~ ~ ~ ( ~ {\downarrow} n. \Diamond n ~ ~ \wedge ~ ~ (\varphi)'_S ~)\\
& \vee ~ {\downarrow} n . \lozenge ({\neg}n
\wedge \neg{\sf belongs}(n,S)
\wedge \neg{\sf belongs}(n,S^{-1})
\wedge {\downarrow} m . (\varphi)'_{S\cup nm})\\
& \vee ~ \underset{xy \in S}{\bigvee} (y \wedge x{:}(\varphi)'_{(S\setminus xy) \cup yx}) \\
(\dgswap \varphi)'_S = &
~ ~ ~ (~ {\sf E} {\downarrow} n. \Diamond n ~ ~ \wedge ~ ~ (\varphi)'_S ~ )\\
& \vee ~ {\downarrow} k. {\sf E} {\downarrow} n . \lozenge ({\neg}n
\wedge \neg{\sf belongs}(n,S)
\wedge \neg{\sf belongs}(n,S^{-1})
\wedge {\downarrow} m. k{:}(\varphi)'_{S\cup nm}) \\
& \vee ~ \underset{xy \in S}{\bigvee} (\varphi)'_{(S\setminus xy) \cup yx} \\
\end{array}
$$
\noindent where $n$, $m$ and $k$ are nominals that do not appear in $S$,
${\sf belongs}$ is defined as in Definition~\ref{transsab}, and
$$
{\sf isSat}(S,\varphi) = \underset{xy\in S}{\bigvee} (x \wedge y{:}\varphi).
$$
\end{definition}
The formula ${\sf isSat}(S,(\varphi)'_S)$ says that the translation of $\varphi$ is
satisfiable at the end of some of the edges belonging to $S$. Note that
the translation maps formulas of $\ensuremath{\mathcal{ML}}\xspace(\dswap)$ to the less expressive
$\mathcal{HL}(:, {\downarrow})$, i.e., the ${\sf E}$ operator is not required.
\begin{theorem}
For $\mathcal M=\tup{W,R,V}$ a model, $w\in W$ and $\varphi \in \ensuremath{\mathcal{ML}}\xspace(\dswap, \dgswap)$
we have:
$$\mathcal M,w\models\varphi ~ ~ \iff ~ ~ \mathcal M,w\models(\varphi)'_\emptyset.$$
\end{theorem}
\begin{proof}
Again, a similar reasoning can be done with the following inductive hypothesis:
$$\mathcal M^*_{\bar{S^{-1}~}},w\models\varphi ~ ~ \iff ~ ~ \tup{W,R,V'},w\models(\varphi)'_S$$
with $S\subseteq {\rm \sf NOM}\xspace\times{\rm \sf NOM}\xspace$, and $V'$ is exactly as $V$
except that for all $(x,y)\in S$, there are $v,u\in W$ such that
$V'(x)=v$ and $V'(y)=u$.
\end{proof}
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HomeArticlesRio OlympicsMURRAY BACK ON THE OLYMPIC GOLD TRAIL
MURRAY BACK ON THE OLYMPIC GOLD TRAIL
IN BEATING ROGER FEDERER AT LONDON 2012, BRITISH TENNIS PLAYER ANDY MURRAY WON THE FIRST MAJOR TITLE OF HIS CAREER. HE HAS SINCE CLAIMED THREE GRAND SLAM TITLES, SO TRAVELS TO RIO WITH HIGH HOPES OF BECOMING THE FIRST PLAYER TO RETAIN AN OLYMPIC SINGLES TITLE.
Andy Murray went to London 2012 considered as one of the "Big Four" in men's tennis, alongside Roger Federer, Novak Djokovic and Rafael Nadal. Yet the Scot had still to win a Grand Slam title, having lost out to either Djokovic or Federer in every final he had contested. In fact, the Swiss ace had denied him in four sets in the final at Wimbledon just three weeks before the London Games.
The 25-year-old was not to be denied on the grass of the All England Club for a second time, however, and became the first Briton to win gold since Josiah Richie, who had prevailed on the very same courts at London 1908.
The home favourite began his charge to the Olympic title by defeating Switzerland's Stanislas Wawrinka in straight sets and then dispatching Finland's Jarko Nieminen by the same margin. The Scot was taken to a third set by Cyprus' Marcos Baghdatis in the third round before winning through, and then beat Nicolas Almagro of Spain 6-4, 6-1 to set up a semi-final with Djokovic.
TEAM GB'S GOLDEN BOY
Drawing on his considerable shot-making abilities and tactical acumen, Murray saw off the Serb 7-5 7-5 to move into the final against Federer, who had got the better of him to win the 2008 US Open, 2010 Australian Open and 2012 Wimbledon titles.
Yet there was to be no such outcome this time around, as Murray drew on GB team-mate Mo Farah's stunning 10,000m win at the Olympic Stadium the previous evening for inspiration.
"I watched the athletics last night, and it was unbelievable," recalled the tennis star after his straight-sets defeat of Federer, in the middle of which he won nine games in a row. "It was amazing to see Mo Farah run his final 400 metres in 53 seconds when I can only do it in 57 seconds when I'm fresh. That gave me the motivation to try to win the gold medal, because I wanted to be part of it if I could."
After conjuring up an ace to seal the 16th of Team GB's 17 gold medals of the Games, Murray, who draped the Union Jack around his shoulders for the medal ceremony, said: "It's number one for me – the biggest win of my life. I have had a lot of tough losses in my career and this is the best way to come back from the Wimbledon final."
Later that day, Murray teamed up with Laura Robson in the mixed-doubles final, the British pair going down 10-8 in the final set to No1 seeds Viktoria Azarenka and Max Mirnyi of Belarus. In the process, the Scot became the first British player since Richie in 1908 to win tennis gold and silver.
A PLATFORM FOR GRAND SLAM SUCCESS
Murray has never looked back since claiming Olympic gold. A little over a month later, he beat Djokovic in a five-hour five-setter to land the US Open and become the first Briton since Fred Perry in 1936 to win a Grand Slam event. He made another piece of tennis history at Wimbledon the following year, downing Djokovic in straight sets in the final to succeed Perry as the last British winner of the tournament.
Murray added to his collection of trophies by steering Great Britain to Davis Cup glory in 2015, going unbeaten in his 11 singles and doubles matches and inspiring his team to a 3-1 defeat of Belgium in the final in Ghent. It was a victory that ended his country's 79-year wait to lift the famous "salad bowl".
Victorious in Rome and at Queen's this season, the Scot lost out to Djokovic in the Australian and French Open finals before claiming his second Wimbledon title in July, courtesy of a straight-sets defeat of Canada's Milos Raonic in the final.
Looking ahead to the Games, which will be played on hard courts at Barra Olympic Park and where he will look to become the first player to retain an Olympic singles title, Murray said: "The Olympics are important to me. I loved the whole experience, both at Beijing 2008 and London 2012, where I won gold. It makes you realise just how special they are."
MURRAY BACK
OLYMPIC GOLD
BRAZILIAN BEACH VOLLEYBALL DUO TARGET COPACABANA GLORY
India vs West Indies: There's not much room to change, says Virat Kohli
Super Hot Vanessa Hanson Is One of the Most Beautiful UFC Girl
Mother Nature's Wind Gusts With Some Booty Shots
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Copyright © 2017 by Alice Alech and Cécile Le Galliard
All rights reserved.
Published by Familius LLC, www.familius.com
Familius books are available at special discounts for bulk purchases, whether for sales promotions or for family or corporate use. For more information, contact Familius Sales at 559-876-2170 or email orders@familius.com.
Reproduction of this book in any manner, in whole or in part, without written permission of the publisher is prohibited.
Library of Congress Cataloging-in-Publication Data 2016954732
Print ISBN 9781942934738
Ebook ISBN 9781944822361
Hardcover ISBN 9781944822378
Printed in the United States of America
Edited by Lindsay Sandberg and Elena Gonzalez
Cover design by David Miles
Book design by Brooke Jorden
Charts and illustrations by Adam Eastburn
10 9 8 7 6 5 4 3 2 1
First Edition
Stronger Bones, Cancer Prevention,
Higher Brain Function, and Other
Medical Miracles of the Green Nectar
ALICE ALECH AND CÉCILE LE GALLIARD
DISCLAIMER NOTICE
This book is not intended as a substitute for medical advice. Readers should consult their physician in matters relating to health, especially if the symptoms require medical attention.
The authors would like to point out that the olive oil world, in regards to both business and culture, is a changing one. New information, studies, results, and figures will not be the same in years to come. The authors made every effort to ensure the information in this book was correct at press time. Both the authors and the publisher disclaim any liability caused by error or omission.
Who We Are
We are two olive oil enthusiasts living in different regions in France. Cécile lives in Southwest France, a little less than 100 kilometers from the Spanish border, and Alice lives in a small village in Provence, around 90 kilometers inland from Nice.
We first met on the Internet a few years ago through our writing on olive oil features and stories, Alice through her writing covering olive oil news from France and Cécile on her French blog Jus d'olive.
We started sharing our knowledge and resources essentially for our writing, building a strong working relationship before we finally met. The idea of writing this book came one day when it dawned on us that even though the world is slowly recognizing the goodness of olive oil, there is still much to learn. We pooled our knowledge and our expertise, enlisted the help of experts, and embarked on preparing The 7 Wonders of Olive Oil. It led us to family farms and research institutions and allowed us to talk to researchers, nutritionists, and chefs.
We, too, had much more to learn about the nutritional secrets of this highly beneficial food.
We hope you enjoy reading this book as much as we did writing it.
CONTENTS
Introduction
Part I
Olive Oil: The Cornerstone of Mediterranean History
Understanding Fats
From Olive to Oil
Part II
Wonder 1: Olive Oil Contains Oleocanthal, a Natural Anti-Inflammatory
Wonder 2: Olive Oil Fights and Prevents Cancer
Wonder 3: Olive Oil Beautifies, Protects, and Rejuvenates Skin
Wonder 4: Olive Oil Keeps Bones Healthy
Wonder 5: Olive Oil Slows Alzheimer's Disease
Wonder 6: Olive Oil Helps Reduce the Risk of Diabetes
Wonder 7: Olive Oil Helps Prevent Strokes and Heart Attacks
Part III
Understanding the Label and Buying Extra-Virgin
Different Categories of Olive Oil
Olive Oil in the Kitchen
Olive Oil Quiz
Glossary
References
INTRODUCTION
A beautiful French woman, Jeanne Calment, lived to be 122 years old. Toward the end of her life, when asked the secret of her longevity and her relatively youthful appearance, she had two words: "olive oil." This supercentenarian French lady was alert right until the end of her life.
Olive oil, our gift from Mother Nature, has always been praised by dietitians, nutritionists, and medical researchers worldwide. They say, "Olive oil is good for your health and well-being." Today, consumers are more health conscious. More curious than ever, they want to understand the medical research behind the health benefits, they need to know how they should be taking this monounsaturated fat, and, most importantly, they want to understand why they should include extra-virgin olive oil in their daily diets. Countless studies on the different health benefits of olive oil have been discussed over the years, and scientists are constantly discovering more. We could not include all these research projects, but what we discuss in The 7 Wonders of Olive Oil is based on interviews with the researchers involved in the studies, published research carried out in laboratory studies, randomized clinical trials, and observational studies. Without being too academic, we describe health conditions and show the extraordinary work and positive results that scientists have achieved so far.
In order to fully grasp the work of these researchers, consumers need to understand olive oil. What you'll find in 7 Wonders—in addition to the astounding health benefits of olive oil—is a comprehensive guide to extra-virgin olive oil, how it's made, and the different types of olive oil available, as well as tips on storing and cooking.
Excellent olive oil comes from excellent fruit. This is one of the first things we as olive oil enthusiasts have come to appreciate more and more. We also realize that even though olive oil is one of the oldest products in history, not many people understand the basics of olive oil production, what a well-balanced extra-virgin olive oil should taste like, or, equally important, the practical side of buying, storing, and cooking with olive oil. This book answers all of these questions.
The 7 Wonders of Olive Oil has three major sections:
Part I looks at the history of the Mediterranean culture, followed by a description of olive oil processing. It covers harvesting and milling, looking at some of the choices modern-day producers have to face—questions such as when the best time to pick the olives is and whether or not producers should filter the oil. We conclude this section by discussing the complicated business of fats and how olive oil is different from other oils.
Part II is a more in-depth study of the healthful characteristics of olive oil. It describes the research into the nutritional, medicinal, and cosmetic values of olive oil. We show you through scientific discoveries that olive oil is a therapeutic agent fighting diseases such as cancer, diabetes, and Alzheimer's and why it is so important to incorporate this gift from Mother Nature into our daily diets for health and beauty.
Dr. Oreste Gualillo, one of the scientists whose findings contributed to this book, sums up the value of this research:
Natural products have been used for thousands of years for the treatment of many diseases and pathological conditions. Thus, nature is a vast source of bioactive molecules from terrestrial and marine environments. Many of these natural products have gone on to become current drug candidates. The era of modern pharmacology is the result of man experimenting by trial and error for hundreds of centuries through palatability trials or untimely deaths, searching for available foods for the treatment of diseases.
What he means by this is that Nature has been providing the necessary ingredients for healthful living since the beginning. We are just using the elements that she offers to us to find healing.
In Part III, we offer you practical advice and information to help you figure out what it is that you should look for when purchasing olive oil. Food magazines and cookbooks say you should be using olive oil in the kitchen, but the choices in the supermarkets these days are overwhelming. As consumers, you want to understand the tricky business of labeling, especially when olive oil quality and authenticity are being questioned. This final section explains why you should not reject oil just because it causes a sting at the back of your throat and how to store your oil once you buy it.
The olive oil world has changed over the years. Olive oil keeps making the news, attracting the attention of people from all walks of life—from health-conscious baby boomers to nutritionists and medical experts.
"Let food be thy medicine and medicine be thy food," said Hippocrates, fifth-century Greek physician and philosopher, considered by some the "father of medicine." How right he was! But today, we are afflicted by a plethora of noncommunicable diseases, the leading cause of death worldwide and a new challenge for all global health policies. If only we could put Hippocrates's advice into practice.
More than anything, we want this book to be inspiring, to show you that the humble olive fruit produces a most powerful natural oil that can make a significant difference to our health.
PART I
OLIVE OIL:
THE CORNERSTONE OF MEDITERRANEAN HISTORY
According to legend, Zeus, King of the Gods, settled the dispute between Athena, Goddess of Wisdom, and Poseidon, God of the Sea, by asking them to provide a gift to a new city over which they both claimed sovereignty. While Poseidon offered the city a horse laden with weapons and an invincible army of warriors, Athena drew a tree from the ground, an olive tree, to nourish and to heal wounds and ailments—a gift which would last forever. The people declared that the olive tree was more useful to humanity. Hence, the city was named Athens.
All civilizations and religions of the Mediterranean bestowed a spiritual importance upon the olive tree. There are an incredible number of references to olive oil and the olive tree in the religious scriptures of Judaism, Christianity, and Islam. It has been a symbol of peace and abundance to the Greeks and an emblem of benediction to Christians. One of the best-known Biblical references is the olive branch brought to Noah as a symbol of reconciliation between God and mankind and a sign that the flood was over. The Catholic Church uses it widely in its rituals, especially for anointing. In the rabbinic tradition, there is the miracle of the olive oil lamp that illuminated the temple for eight days following the military and spiritual victory of the Jews of Judea even though there was only enough oil to light the temple for one day. The significance of olive oil and its association with light has been perpetuated in the most important Jewish festival of Hanukkah, the Jewish Festival of Lights, by lighting the seven-branch candelabra, the Menorah. In addition, olive oil fuels the lamps of the Shabbat and other ritual celebrations. In the Koran, the olive tree is described as sacred, a symbol of Universal Man. It is compared to the bright light of Allah that illuminates the path of mankind. Even as far to the east as Japan, it is a symbol of social success.
In modern times, the olive branch continues to symbolize peace. The United Nations flag is a good example of the meaning of "holding out an olive branch." It shows a map of the earth framed by olive branches, a symbolic peace surrounding the world. Another present-day symbol is the Great Seal of the United States, which features an eagle holding in the left talon thirteen arrows and in the right one an olive branch. Although the arrows suggest war, the eagle's head is turned toward the olive branch, representing peace.
The origins of the olive tree, when and where it was first cultivated, remain uncertain. Fossilized leaves from the Paleolithic and Neolithic ages were found on the edges of the Sahara and dated as far back as 12,000 BC. However, it was probably first cultivated by man in Asia Minor or on the coast of Lebanon or Palestine between 6000 and 5000 BC. The olive tree and the Mediterranean have always been inseparable; the olive tree has linked the past civilizations, people, cultures, and religions in the region. Scientific evidence proves that olive trees had been growing in Crete as early as 3500 BC, long before spreading to mainland Greece.
As the first civilizations were emerging in the Mediterranean basin and the Middle East, cereals, vines, and olive trees were the top three staple food sources. The first vegetable oils were extracted from olives in the West. (In the East, it was extracted from sesame.) Extracting these oils required a simple mechanical force easily exerted by humans or animals. Just think: for a long time, these were the only edible oils the people at the time were familiar with.
We find that the first cultivated trees coincide with the birth of the Phoenician, Assyrian, Jewish, Egyptian, and Greek cultures as well as other less studied Mediterranean cultures. The first written documents on the olive tree are Mycenaean clay tablets dating back to the reign of King Minos (2500 BC), which shows how important olive oil was to the Cretan economy.
It is believed that the Phoenicians—the original inhabitants of Phoenicia in what is now Lebanon—were the first people to spread olive-growing culture in 4000 BC. This civilization died out between 1200 and 300 BC. Few records remain of this period, but it seems that the Phoenicians were excellent merchants, artisans, and navigators and that they were the first to disseminate olive growing through their commercial exchanges with Crete, Greece, the Aegean Islands, Italy, and the western part of the Iberian Peninsula. The remains of amphorae (storage jars for transporting food such as oil and wine on long voyages) were found in shipwrecks.
Olive-growing culture spread to Greece via Anatolia (modern day Turkey) before reaching Crete and Egypt and eventually the whole Mediterranean basin. The Greeks brought olive trees to Corsica, Sardinia, Sicily, all of Italy, and France, specifically the city of Marseille in 600 BC. Perhaps most notably, the Greeks spread olive trees to Al Andalus, the modern Andalusia in Spain, which is considered a leader in olive oil production today.
This healthy oil, sometimes called "green gold" or "liquid gold," played a fundamental role in forging the Mediterranean character—the common thread that binds all of its peoples. The culture of olive oil quickly spread to become a prized commercial product for cooking and cosmetics, a fuel for lighting the lamps of cities and temples, and an important part of religious rituals. At the time of Venetian trade, olive oil was highly prized for the manufacture of soap and the treatment of skin. The term "lampante" olive oil comes from the use of olive oils unfit for human consumption but good for lighting lamps.
It was the Roman Empire that introduced technical improvements, developing plantations and olive mills. Olive growing was exported to all the Roman colonies. As a result, the olive tree became a symbol of economic modernity. The precious oil was stored and carried in tall jars called amphorae. You can see just how important the olive oil trade was under Roman rule when you look at the Monte Testaccio in Rome—an ancient landfill site with an enormous pile of oil amphorae. Once the oil was poured out of the amphora pottery, it was sold in the markets of the city and the amphorae were carefully arranged in layers. Over time, the amphorae ended up forming this artificial mountain.
The decline of the Roman Empire in the fifth century and the arrival of the barbarians and the first Arab invasions interrupted the Roman advances in producing olive oil. It was not until the Crusades, specifically during the time that Venetian trade expanded, that production began to develop once more. In 1497, the sea route to India was discovered and olive trees were brought farther east. Similarly, the olive tree also traveled west to Argentina, Peru, Chile, Mexico, and California in the United States. With the discovery of the new world, Venice lost its monopoly and the market was shared by Italy, Greece, and Portugal. Even to this day, the trade and production of olive oil are widely confined to Europe. The European Union is the world's leading producer, accounting for 73 percent of production, and the leading consumer of olive oil, at 63 percent. Spain, Italy, and Greece together account for 97 percent of total olive oil production in the European Union. Olive oil cultivation in the Americas did not reach the same prominence until the twentieth century as public awareness of the goodness of olive oil became widespread.
In recent times and as the thirst for and success of olive oil grew, the need for setting up standards for chemical, physical, and sensory characteristics of olive oil was apparent. The International Olive Council (IOC), an intergovernmental body, was instituted by the United Nations in 1959 in Madrid. It is the world's only institution with the objective of promoting and defending the entire olive oil and table olive industries worldwide. They are involved in supplying aid and advice to growers and millers as well as funding research to improve the quality of olive oil.
It is a complex business as the IOC currently counts a total of forty-three member nations, including the European Union with its twenty-seven member states. These member countries account for 98% of the world's olive oil production. Today, the IOC faces new challenges when it comes to controlling quality. Their duties extend to coping with fraud and harmonizing trade and production as well as protecting the rights of consumers by implementing the agreement signed in 2005 by the member states. One of the main purposes of the agreement is that it standardizes processing controls and labeling by assigning specific names and definitions to the different oils.
Olive oil is one of the most regulated food industry products. The growers and producers put in a lot of effort to make sure that the olive oil that you bring home is full of the seven wonderful traits that make it so healthful. After all stages of development and before bottling, a sample gets sent to the laboratory for careful examination. This involves examining not ten or fifteen but a total of twenty-six physical and chemical parameters, including acidity and sensory tests, both vital for complete evaluation of the oil. The sensory tests are carried out by a panel of eight to twelve experts and can take place in any member country of the International Olive Oil Council. This means that olive oil in Puglia in southern Italy and olive oil in southern Algeria have had the same quality testing measures as enforced by the IOC.
Even non–IOC member countries have their own strict standards. They protect the consumer through commissions, associations, and organizations such as the California Olive Oil Council (COOC) in the United States and the North American Olive Oil Association (NAOOA). Two other notable organizations are the Australian Olive Association (AOA), promoting olive oil from Australia, and the Association Française Interprofessionnelle de l'Olive (AFIDOL) from France.
But why does olive oil need such high standards and regulation? You might wonder if it's even worth all the fuss. In 2013, the United Nations Educational Scientific and Cultural Organization (UNESCO) recognized the Mediterranean diet as part of the intangible cultural heritage of Spain, Portugal, Morocco, Italy, Greece, Cyprus, and Croatia. These countries' intangible heritage, normally associated with natural wonders, monuments, and temples, now included a different culture: an eating pattern which incorporated olive oil as the main source of fat.
This served as an important international recognition of olives and olive oil. However, this recognition came in a worldwide culture that used olive oil very little. Since the beginning of the industrial era, other refined vegetable oils were on the market, which seemed a more profitable venture in Western countries. As a result, very little olive oil is now consumed. Palm oil and soybean oil are the world's most consumed vegetable oils, at 32% and 28%, respectively, followed by rapeseed oil at 16% and sunflower oil at 8%.
International Olive Council Executive Director, M. Abdellatif Ghedira, stated:
In recent years, world olive oil production has increased considerably, but olive oil still only accounts for just over 3% of all the vegetable oils produced in the world. So, of course, there is room for expanding this percentage share. World consumption of olive oil can only go upwards: it is so healthy and so delicious that I am very optimistic about its future.
And he is correct! New producers have entered the market, including the United States, Canada, Australia, Brazil, and Japan. Today, olive trees are grown and olive oil is produced on five continents. It was only in 2008 that the olive oil industry took off in India, but olive trees are now cultivated successfully in the desert region of Rajasthan. The first fruit should soon appear, a good thing for the country, which presently imports a total of 2,617 tons of oil, 195 tons of which is virgin olive oil. The new domestic production of olive oil will certainly influence the Indian diet and alter the import of oil.
Still, the market beyond the Mediterranean basin is still in its infancy. The Chinese market, attracted to the western lifestyle, imports more and more olive oil each year; 14,000 tons were imported in 2010. Another burgeoning market is the United States, where consumption reached 248,000 tons in 2006, ranking it the world's second olive oil consumer after Europe and also the leading importer.
The last thirty years have seen a worldwide shift in consumer habits toward healthier and more natural foods. As a healthy alternative to other fats, the average annual olive oil consumption has already reached 2.8 million tons and will reach 3 million tons in 2016.
We asked Abdellatif Ghedira what the major advances have been these ten last years. This is what he said: "Agriculturally speaking, I think the biggest changes in the last decade have been—amongst other things—that cultural practices have improved and irrigated acreage has increased. On the consumption side, world imports of olive oil have expanded by 29% during the same period. This means that olive oil is no longer a local product but has become a global product that reaches every corner of the world."
Let's not forget the impact on health over the years. Almost 2,500 years ago, Hippocrates called olive oil "the great healer" and described at least sixty medical conditions that could be treated with olive oil, including skin conditions, wounds, burns, and other ailments. Today, researchers are delving more and more into the health benefits of the green nectar.
The ancient Greeks cherished olive oil; for them, it was a necessity for daily sustenance. Sadly, over time, olive oil lost its special place as the people turned to other sources of fat. However, olive oil was not the only fat rejected and misunderstood by consumers and nutritionists. Fats in general were seen as unhealthy, as causing heart attacks and leading to obesity; they were the first thing to eliminate if you wanted to go on a diet. Thankfully, since the 2000s, things have changed. New scientific studies are proving the health benefits of virgin olive oils and the importance of fatty acids in our daily diet. As we discuss the seven wonders of olive oil, we'll share with you some of the studies that prove its healing properties.
Today, olive oil has opened up a whole new world of sensory pleasures, taking cooking to new heights, which will no doubt continue to give olive oil a well-deserved and special role in our cuisine.
UNDERSTANDING FATS
Fat is often considered the prime suspect for weight gain, but the body needs a certain amount and certain kinds of fat to survive.
The human body requires energy to function correctly, energy which comes from the nutrients in the food we eat. Food gets processed by our digestive system and then goes into our bodies through our circulatory system, traveling to our cells and supplying them with proteins, carbohydrates, and lipids, or fats.
Fats are a necessity for the correct functioning of our bodies. They provide a rich source of energy. They supply the necessary calories used directly by the body for growth and general functioning. To respond to all their fundamental needs throughout the day, a healthy adult man or woman, doing regular physical exercise with a daily energy intake of 2000 kilocalories, should ideally obtain 55% of their energy from carbohydrates, 15% from proteins, and, finally, a minimum of 30% from lipids. Fats, a rich source of energy, come from certain foods, particularly butter, milk, vegetable oils, meat, and cheese.
Fatty acids are classed as the simplest lipids because of their structure. They serve as fuel for our muscles and help to build cell membranes. Certain fatty acids are termed "essential" because the body does not produce them—we have to consume them in food.
There are three main types of fatty acids: saturated, monounsaturated, and polyunsaturated. They are similar in that they have similar chemical structures and they contain a chain of carbon atoms bonded with hydrogen atoms. However, all three have different roles to play in the human body and have complex differences in form and function.
DIFFERENT FAMILIES OF FATTY ACID AND RECOMMENDED NUTRIENTS INTAKES (RNI)
How are they different?
Saturated fats are solid at room temperature because they are "saturated" with hydrogen molecules.
Monounsaturated fats—often classed as good fats—are liquid at room temperature because they contain fewer hydrogen atoms.
Polyunsaturated fats—also considered good fats and liquid at room temperature—are broken into two main types: omega-3 fatty acids and omega-6 fatty acids.
Oleic acid is one of the main types of monounsaturated fatty acids. It can be found directly in food, especially in olives and in olive oil, as well as in rapeseed oil, peanut oil, nuts, almonds, avocados, goose fat, meat, oily fish, and, to a lesser amount, processed meats. Products containing oleic acid carry the name "omega-9" on the food label. Oils containing monounsaturated fats are usually liquid at room temperature but start turning solid when chilled. Monounsaturated fatty acids protect the heart, playing a fundamental role in preventing cardiovascular diseases. They affect the cholesterol levels in the blood and are known to reduce bad cholesterol, or LDL, and to increase good cholesterol, or HDL.
Polyunsaturated fatty acids, commonly referred to as omega-3 and omega-6 acids, deserve special mention. The body does not produce these well-publicized fats; we can get them only from foods, mostly from plant food and certain seafood. That is why they are often called "essential fats." Fatty acids make up 60% of the brain and are necessary for the correct functioning of our gray matter.
Studies on the importance of fatty acids, their role, and the recommended nutritional intake for each family of fatty acids are very recent. In the 1950s, Ancel Keys, an American doctor and researcher, was one of the first to establish a link between our diet and our health.
During the 1950s in the United States, the population and medical professionals were confronted with a major public health concern: the number of deaths from heart attacks was rising continuously. Autopsies revealed the presence of cholesterol in the arteries. Those studies lead to the campaign against cholesterol. Up until then, scientists were convinced that having a diet rich in products of animal origin was the best way to stay in good health. This belief was later rejected by some studies.
It was during this period that Ancel Keys conducted his first survey in Naples, Italy. He discovered that cardiovascular diseases affected the more affluent populations; one of the main differences between the rich and the poor was that the rich ate more meat. It showed a clear connection between the consumption of fats and the risk of death from cardiovascular disease.
Following this first observation, he began, in 1952, to study seven countries. He decided to compare the diet and cardiovascular health of populations in the United States, Finland, Italy, Japan, the Netherlands, Yugoslavia, and Greece; he especially wanted to study those living on the island of Crete. This intriguing study included 13,000 volunteers and lasted twenty-five years. It showed that in the northern nations—Finland, the Netherlands, and the United States—a lot of animal fats were consumed in meat, butter, and cheese, while there was only a small amount of fruits and vegetables consumed. In the southern nations, specifically in Crete, vegetable fats and olive oil were the main fats consumed, together with a healthy intake of nuts, vegetables, fruit, and fish. The results of the study showed that the Cretans presented with the lowest rate of cardiovascular-related deaths—six times lower than the Finns—and enjoyed a lower rate of cancer and greater longevity.
Although Ancel Keys's results and the studies themselves were very heavily criticized in later years, the study of the Cretan diet marked a turning point in the close relationship between our food intake, our health, and the beneficial and protective quality of diets rich in monounsaturated and polyunsaturated fats.
Studies of these differences in types of fatty acids are very recent. Until 2010, people were told to reduce their intake of all fats and to switch to low-fat foods. Since then, however, daily energy intake and the role of fats have been reevaluated. The emphasis today is more on choosing the right fat, creating a shift from a quantitative view of fatty acids to a more qualitative one. The recommended daily intake now varies according to the properties of the fat and the individual's needs.
Dietician, nutritionist, and head of the School of Human Nutrition and Dietetics in Paris Anne-Laure Meunier says that we should consume fats according to our needs, as well as our taste, but warns that we should avoid hydrogenated fats. These fatty acids or trans-fatty acids are processed industrially with high-pressure oxygen; they are the culprits that clog up our cardiovascular system. Meunier, who specializes in obesity in adults, says that a healthier choice would be to opt for virgin oils, which have been pressed or extracted by mechanical means such as "real" butter and "real" cream. She says that we should also consume only what is necessary; although we do need to eat fat, we should aim for good fats and to have them in reasonable quantities.
Large amounts of saturated fatty acids are found in food, but they are also produced by the human body in the liver, the brain, and the adipose tissue (fat cells). Besides being vital sources of energy, saturated fats are also essential for transporting vitamins A, D, E, and K in the body. These vitamins are fat soluble, which means they are carried through the body in oils. Our bodies naturally create these fats to generate energy and transport nutrients, but we need to find the right balance, to get the right ratios of fatty acids for optimum health. Most saturated fatty acids are of animal origins, such as meat from cows, sheep, lambs, and pigs. We find them also in dairy products: butter, fresh cream, whole milk, and cheese. Where we can, limiting these animal products in our diets can help us maintain a healthy balance of saturated fats in our bodies.
Palm oil, soybeans, and coconut oil (plant-based vegetable oils) are also sources of saturated fats. Although the body needs saturated fatty acids, consuming too much only results in an accumulation of bad cholesterol. This is bad news for our bodies, because from that we become overweight and suffer from other related illnesses such as cardiovascular disease, high blood pressure, and heart attacks.
As we search for and select saturated, monounsaturated, and polyunsaturated fats in our food, the first challenge is knowing where to find them. Secondly, they must be in the right proportions. According to the French Agency for Food, Environmental, and Occupational Health and Safety (ANSES), for correctly balanced overall health, the ratio consumed must be 1 to 5: 1 portion of omega-3 acids for every 5 portions of omega-6 acids. Imbalance leads to excess fats stored in the body, causing first weight gain then cardiovascular problems. In France, for example, the ratio of omega-3 acids to omega-6 acids is 1 to 10; in the United States, it is 1 to 40.
Without realizing, we now consume far more omega-6 fatty acids than our grandparents' generation did. Back then, cows simply grazed on grass in the fields and ate hay during the winter. However, the story is different these days. With beef producers focusing more on increased production and maximum profits, cow feed today includes cereals such as corn, sunflower, meal, and soybeans. As a result, all products that we get from cows, such as milk, butter, cheese, and cream, are already naturally enriched with omega-6 acids.
At the same time, there has been an increased consumption of omega-6 type fatty acids due to a rise in direct consumption in the household; changes in production and extraction methods mean that these fatty acids are used to make cheap oils.
We find omega-6 acids in sunflower oil, safflower oil, soybean oil, corn oil, and grapeseed oil, whereas omega-3s come from linseed oil, rapeseed oil, soybean oil, and all oily fish. Some oils such as walnut oil and groundnut oil contain both in large quantities.
While much of the world has consumed increasing amounts of unhealthy fatty acids, there is one notable exception. Since 2010, the Mediterranean diet, which includes the Cretan diet, has been listed as intangible cultural heritage of humanity by UNESCO. The countries that submitted recommendations were Cyprus, Croatia, Spain, Greece, Italy, Morocco, and Portugal. There is no single model of the Mediterranean diet; instead, it is more about the eating habits of these countries bordering the Mediterranean Sea. According to the UNESCO definition, the Mediterranean diet is not only a combination of know-how, rituals, symbols, and traditions but also involves the practice of sitting down to share a meal and consume food together. The act of eating together is a fundamental part of the cultural lifestyle in the Mediterranean basin.
The countries that seem to ascribe to the Mediterranean diet enjoy certain things in common, such as food obtained from livestock, fishing, and agriculture. They also have an abundance of fruits, vegetables, cereals, and olive oil and only moderate consumption of meat, dairy products, and wine. Traditionally, moderation is one of the defining characteristics of the Mediterranean diet.
Meunier concludes: "Based on fish, olive oil, fruits, vegetables, and legumes, the Mediterranean diet has color; it is varied and comes with simple advice that helps to maintain a healthy balance between the different fatty acids."
To conclude, it is all a question of getting the right balance. Saturated, monounsaturated, and polyunsaturated fatty acids all have different functions within the body; they are all vital for the various systems of the body to function. As you come to better understand the characteristics of fats and your own need for the different kinds and quantities, we hope that you are able to enjoy a similarly colorful diet that is balanced and helps you discover the wonders that our foods can provide for us.
FROM OLIVE TO OIL
Producing oil from the humble olive is an exciting process. For growers and producers striving to get the best extra-virgin oil, it is both a business and a labor of love.
The olive is not an ordinary fruit. You cannot pick it off the tree and eat it; it is so bitter that you would spit it out immediately. But the fruit contains proven health-protective nutrients and organic matter. The challenge for producers is that the fruit also contains a lot of water, which must be extracted before they can obtain the oil.
October is a busy but exciting time for olive oil farmers and producers in Europe. Millers have to make sure their machines are spotlessly clean and in good running condition, ready to wash and process the picked fruit, while farmers lovingly examine their trees every day, checking to see if the olives are ripe enough. The right time for picking olives sparks all sorts of discussions amongst growers; only the best possible olives picked at the right time yield what they are all aiming for: good quality extra-virgin olive oil with excellent flavor, nutritional goodness, and a decent shelf life. After the flowering period, the olives first appear green then change color as they ripen from rosy to mauve. The color of the fruit on the outside corresponds to the different chemical changes taking place within. It's difficult to extract oil from green olives, and they don't give as much oil, but olives that are too ripe lack polyphenols and become less stable with time.
When nurtured appropriately, picked at the right time, and processed with precision, good-quality olive oil will have high levels of polyphenols, the phytonutrients in plants that are recognized scientifically as potent antioxidants. Olive oil contains a multitude of polyphenols; these are the health components that scientists love— they are extra-virgin olive oil's most powerful weapon for protecting the body and preventing diseases.
Some farmers pick their olives by hand, while others use a machine. In the past, European farmers would start the harvest in December and continue slowly until February the following year when the fruit was well and truly black in color and the oil easy to extract. By then, however, the intrinsic qualities of the oil had reduced considerably in the overripe fruit.
Things have changed significantly over the years, with farmers striving to improve quality rather than quantity. Following the advice of the IOC and other research centers, they now harvest olives earlier, achieving more nourishing oil with all the desired organo-leptic properties: taste, odor, color, and texture. The tendency these days is to pick olives while the skin is between green and mauve and the pulp still green. In Mediterranean countries, this usually happens between the second week in October and the second week in November. Experts at the IOC say this is the best time to get the maximum nutrients and aroma from the fruit. Most importantly, this is when the fruit is full of polyphenols and antioxidants. Once picked and collected, the olives must be thoroughly washed, dried, and weighed at the mill.
Though the methods vary with time and technology, making olive oil always involves picking good fruit, crushing the fruit, and separating out the water. With proper treatment of the olives, the resulting oil is beautiful and has incredible health benefits!
The Traditional Approach
Though technology has advanced to allow farmers to produce good, clean olive oil in a short time, we think it is important to remember the heritage of the olive press and the traditional method to crush olives. The idea of a donkey or a mule turning a great stone mill might seem a bit far-fetched, but that's how they crushed olives thousands of years ago.
The millstone was set in a trough where farmers poured the cleaned olives. As the millstone rotated, it broke, crushed, and ground the cleaned olives to a thick paste. The mixing or malaxing stage followed—a period of about an hour of continued grinding that made sure the olives were fully ground and allowed the fruit enzymes to develop.
After milling and malaxing, mill workers would spread the thick paste onto fiber discs, which look like mats placed one on top of the other. These stacks of mats then went into a vertical hydraulic press and were pressed; several hundred pounds of pressure applied squeezed everything to the maximum. Fifty-six kilograms of ground olives stacked and pressed would yield a little over seven liters of oil.
Olive oil processed this traditional way is sometimes referred to as cold-pressed olive oil, meaning no heat or chemicals have been added to alter the taste. You will, however, find labels saying "cold press," "first press," or even "first cold press" on packaging today when the oils were made using the centrifugal method.
Farmers with small quantities of oil still use this traditional system, especially in rural areas of France, Spain, Italy, Morocco, and Tunisia.
The main disadvantage of the traditional system, however, is the danger of oxidation and fermentation because the process is slow and discontinuous. Processing does not take place immediately after harvesting, and waiting during the processing cycle exposes the olive paste to oxygen and light.
The modern mill is certainly not as romantic, and gone is the charm of bygone days, but with the modern crusher, mixer, extractor, and stainless-steel vats, fewer workers are necessary to oversee the extracting process, meaning less worry for the mill owners, especially when they have vast quantities of oil to process. Traditional producers argue that the slow mixing of oil and water gives them better quality and quantity than modern mills.
Not surprisingly, the tastes using these two methods are quite different. Traditionally processed oil is fruity, less bitter, and slightly herbaceous but complex. Oils processed by the modern centrifugal system are more herbaceous, robust, bitter, and pungent with an aroma reminding us of fresh vegetables.
Clearly, production methods have become more sophisticated over the years, more challenging for both the small and large olive oil producer. With growing public awareness of olive oil today, keeping up with new techniques and new developments is essential to producing high-quality olive oil.
The modern methods of processing oil are clearly developed from the traditional methods. After picking the olives, either by hand or by machine, the whole olives are then crushed and ground to form a fine paste, a mixture of fruit and stone. For this process, millers use different types of equipment: the two main types of machines are the modern stainless steel crusher and the more traditional mechanical rollers.
The fruit then needs to be mixed for about thirty to forty minutes at a temperature no higher than 27° C (80° F). Mixing is a critical phase, as this is when the flavor develops. Mixing, sometimes referred to as malaxing, also allows the smaller droplets of oil to mix with the larger ones.
The next step is the separation of the oil from the water and solids. In the past, they called this the first press or cold press because old wooden vertical presses used to be operated with very low temperatures. Often, producers would press the paste a second time to obtain the maximum oil. Although you might still hear these terms, think of them mostly as commercial terms mainly used for marketing olive oil. Modern-day methods are very different.
Today, many mills are equipped with modern centrifuge decanter systems, which use high centrifugal force to allow the water and oil to separate easily. Some producers opt for a three-phase centrifuge system because these machines separate the oil, water, and solids separately, while others prefer a two-phase centrifuge, which separates the oil from the wet paste.
WORLD OLIVE OIL PRODUCTION AND CONSUMPTION
Top Ten Producers of OO in volume (million tons) from 2012–2014 | PRODUCTION
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2011 / 2012 | 2012 / 2013 | 2013 / 2014
SPAIN | 1615,0 | 1615,0 | 1615,0
ITALY | 399,2 | 399,2 | 399,2
GREECE | 294,6 | 294,6 | 294,6
TURKEY | 191,0 | 191,0 | 191,0
SYRIA | 198,0 | 198,0 | 198,0
MOROCCO | 120,0 | 120,0 | 120,0
TUNISIA | 182,0 | 182,0 | 182,0
PORTUGAL | 76,2 | 76,2 | 76,2
ALGERIA | 39,5 | 39,5 | 39,5
CHILE | 21,5 | 21,5 | 21,5
Top Ten Producers of OO in volume (million tons) from 2012–2014 | CONSUMPTION
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|
2011 / 2012 | 2012 / 2013 | 2013 / 2014
ITALY | 610,0 | 590,0 | 600,0
SPAIN | 574,0 | 513,0 | 580,0
U.S.A. | 300,0 | 293,0 | 294,0
GREECE | 200,0 | 200,0 | 185,0
TURKEY | 150,0 | 160,0 | 150,0
MOROCCO | 122,0 | 129,0 | 132,0
FRANCE | 112,0 | 97,2 | 99,6
SYRIA | 135,5 | 135,5 | 95,0
PORTUGAL | 78,0 | 74,0 | 74,0
BRAZIL | 68,0 | 73,0 | 73,0
Harvesting Time in Sunny Spain
In Hellín, Spain, they certainly know how to make fine extra-virgin olive oil, oil that is recognized worldwide.
Two olive growers, Dominguez and Olivares, from a region in the heart of Spain, worked hard at their orchards in 2015, aiming to produce the very best oil they could. Their labor was rewarded. For the first time, their brand Pago de Peñarrubia (made from the delicate Arbequina olive variety) won a gold award at the prestigious New York International Olive Oil Competition (NYIOC) in 2016. More international recognition followed, with silver medals from the Los Angeles International Extra Virgin Olive Oil competition and the Olive Japan International Extra Virgin Olive Oil Competition.
And at home in Spain, they were finalists in the Mario Solinas competition, recognized for the fine flavor of their ripe, fruity extra virgin category for 2016.
This is how they did it:
The region is blessed with great weather; the sun perched high all year round means temperatures of about 35° C (95° F) at the height of the summer season in August—perfect for the magnificent olive trees dotted all around the hillsides.
Their families have been in the business for over a hundred years, striving every year to produce the best-quality olive oil. In the olive oil world, their farms are classed as intensive and super-intensive plantations because of the high number of trees they contain. The intensive farm has a plantation density of between two hundred and six hundred trees per hectare, while the super intensive farm has a plantation density of between one thousand and two thousand trees per hectare. Their olive trees are well aligned, meticulously maintained, and equally spaced within and between the rows.
At their plantations, Dominguez and Olivares grow two varieties of olives: Picual, a medium- to large-size olive that is very popular in Spain; and Arbequina, a smaller-size variety. Picual olives produce rich, green, pungent oil, while Arbequina olives render sweet oil. Dominguez and Olivares aim for intense green oil with notes rather similar to freshly cut grass, the tomato plant, or the Granny Smith apple.
In a real spirit of cooperation, these two farmers work closely together, especially at harvest time. Dominguez owns the mill, and Olivares has the know-how, providing all the technical knowledge for growing olives and producing premier oil. Growing olives and producing olive oil is their livelihood; they are proud of their brand of olive oil, Pago de Peñarrubia, which is also the name of Dominguez's mill.
Like many farmers to whom olive farming is a way of life in this vast olive oil production area of Castilla, these two fifth-generation olive farmers use a machine to pick olives from their 155 hectares of olive trees. Hand-picking is not an option here, as there are too many olives on the trees and, more importantly, processing must be done as soon as the olives are picked. The spacing of about 1.5 meters between the trees and careful pruning so the trees don't grow too tall are essential: the machine must be able to maneuver easily and not damage the trees.
Picking at the two orchards takes place in late October and into early November, when the olives are firm and mauve in color. Luckily for the pickers, the temperatures can drop to 25° C (77° F) in the fall.
Ten people work hard collecting the olives and getting them to the factory, while another ten work in the mill, cleaning, preparing, and processing the oil. With everyone working at top speed, they can pick and process the olives all in about six hours per day. The two owners provide accommodation for the hired hands as a way of making sure production is both continuous and efficient. As fermentation or oxidation starts the moment the fruit is picked, on certain days, they work much longer than six hours, managing to complete all of the harvesting and processing in two weeks.
Olivares and Dominguez use the modern centrifugal method and, in keeping with regulations to improve sustainability in Spain, use the two-way phase system. With so many olive oil producers in the country, waste treatment is hard to control with the three-way system.
The resulting oil is slightly cloudy because bits of sediment accumulate at the bottom of the container. There is some discussion amongst producers as to whether the oil should be filtered or not before bottling. Some prefer to keep the bits floating in the oil, which, according to the experts, contain more polyphenols. Others prefer to filter the oil, saying that the sediment eventually produces nondesirable flavors. Dominguez and Olivares prefer to filter the oil.
Olive By-Products
You might be surprised that to hear that when making olive oil, only 20 percent of the fruit is used. Other products are generated throughout the process, however—lesser-known products that we hardly think about. Some of these may even be considered valued by-products, crucial today for environmental and economic issues for the sector.
The month of March is a relatively slow time for European growers. This is the time to take a closer look at the trees in their orchards, making sure they are ready for the spring season. The vital task for them at this time is pruning the trees. If they want the best quality and quantity, they must remove the old branches; this stimulates and gives more strength to the buds. Olive branches can be utilized in many ways: they can be processed to make excellent cattle feed (instead of hay or straw) and can be used for making compost. Industrially, these excess branches are used for making paper, furniture, and fuel.
The by-products of processing oil are also beneficial and valuable. In some mills, extraction of oils starts in October and continues until February. The machines run continuously, producing oil as well as kilos of pomace and vegetable water. Pomace is a solid residue, a mixture of core, pulp, and skin. These two by-products can be treated on site, but in most cases, the pomace is bought by a dedicated processing company.
The pomace still contains a little olive oil, so to not lose a single drop, growers return the mixture for extracting at high temperatures, adding some solvents. This juice of the second press must not be confused with first cold pressing. The oil that comes out has no organoleptic qualities; it cannot be consumed in this state. This crude pomace oil will be sent to the refinery to be cleansed of all impurities and also to improve its smell, flavor, and color. Once "cleaned," it is added to extra-virgin olive oil and sold under the commercial category of olive pomace oil: edible but cheap. Olive pomace oil is also used in the manufacture of soap.
Today, the pomace and vegetable waters can be used in many ways. Pomace mixed with other components makes excellent fodder. The stone (pits) are used in boilers, and the pulp is made into fuel pellets for heating. The residue of vegetable water is a sensitive issue, as it is considered highly toxic to the environment. In Spain, for example, oil producers have to follow precise regulations for evacuating olive mill waste. Some producers, though, have discovered ways to use it as a garden fertilizer.
Studies on the antioxidation properties of some of the molecules contained in these by-products are now making news—scientists are looking closely at the fruit, olive pits, and leaves. The cosmetics industries see olive oil as a source of beauty, and new technologies are being developed to retrieve the phenolic compounds. One company even makes exfoliating sponges from olive pits.
Food trends are also changing. You can now find tea made from olive leaves in specialist boutiques, and surprisingly, in Japan, they make products like pasta made from ground olive leaves, soda from decanting the vegetable water, and fermented pomace to feed animals. At one university in Spain, they are currently looking into producing functional foods such as yogurt and drinks with enriched phenolic compounds.
Indeed, the business of the utilization of the humble olive is booming.
CONSUMPTION PER PERSON PER YEAR
PART II
Here in Part II, we reveal the incredible discoveries of the seven wonders of olive oil. Each section focuses on uncovering the miraculous characteristics that we can use to improve our personal health by including olives and olive oil in our diets and lifestyles.
THE 7 WONDERS OF OLIVE OIL
WONDER 1
OLIVE OIL CONTAINS OLEOCANTHAL, A NATURAL ANTI-INFLAMMATORY
Scientists discovered that extra-virgin olive oil contains oleocanthal, which mimics the activity of ibuprofen. Oleocanthal, like ibuprofen, may also be able to help reduce arthritis. Natural, safe extra-virgin olive oil can easily be incorporated into a daily diet, say researchers who have set up the Oleocanthal International Society.
When the American scientist Dr. Gary Beauchamp accepted an invitation to an olive oil symposium of newly pressed olive oil in Sicily, he had no idea it would change his life and the rest of the olive oil world. The sensory chemist was employed at the time at Monell Chemical Senses Center in Philadelphia, working specifically on a project relating to the medication ibuprofen. The anti-inflammatory drug stings as it goes down the throat; his job was to investigate why and to improve the taste.
Along with other scientists, food specialists, and chefs, he followed the precise and somewhat complicated instructions of how to taste olive oil. He cradled the cup, sipped, and slurped as he was told and promptly experienced a déjà vu moment: a burn in his throat, one that he recognized instantly as similar to liquid ibuprofen. The discovery in Italy was totally unexpected. Here were two unrelated compounds showing the same effects.
Delighted with his accidental scientific revelation, Dr. Beauchamp traveled back to Philadelphia, taking some of the precious oil with him. There, he and biologist Dr. Paul Breslin began investigating the properties of the pungent oil with their team. They isolated the throat-irritating compound, researched it, and named it "oleocanthal" from the Latin oleo-, olive, canth-, sting, and -al, aldehyde. They confirmed in further studies that although the chemical composition was different, the compound oleocanthal behaved similarly to the anti-inflammatory ibuprofen. (They discovered another miraculous wonder in oleocanthal, but we'll discuss that in the next chapter!)
Oleocanthal (OC) is a powerful antioxidant which develops when olives are crushed into pulp; this is what causes that sting, the peppery sensation in the back of the throat when you sip it neat. It comes from the purest of olive oil; if the oil stings a little, it contains only a little OC, but if it stings a lot, it contains a good amount of OC.
Interestingly, olive oil is the only vegetable oil that contains OC, that vital component with such pungency that it causes us to cough once or twice when we taste the green nectar. Some of us are more sensitive than others, but that distinct sting and irritation will vary depending on the variety of olives used to make the oil and also the concentration of extra-virgin olive oil.
Oleocanthal and Arthritis
So how does the ibuprofen-and-olive-oil connection help us?
Do you know anyone with rheumatoid arthritis? The word arthritis comes from the Latin word itis, which means "inflammation," and the Greek word arthron, meaning "joint." It is a chronic disease that can cause a great deal of pain. It starts off attacking the fingers and then the major joints, causing pain and stiffness in the mornings; it comes with fatigue, and, later on, it can cause permanent disability. Sadly, it gets worse with time unless the inflammation slows down.
Arthritis is quite simply inflammation of the joints caused by two inflammatory enzymes, COX-1 and COX-2. If the inflammation of these two enzymes continues unchecked, it causes damage to the cartilage and, eventually, the bones themselves. And once the deformity sets in, it cannot be reversed.
Doctors recommend anti-inflammatory drugs, in particular ibuprofen, to relieve pain and swelling from rheumatoid arthritis. Classed as a nonsteroidal anti-inflammatory drug (NSAID), ibuprofen helps reduce pain and inflammation, but unfortunately, it does not slow down the disease. Also important to remember is that long-term pain-relieving drugs are harmful to the body; even ibuprofen, taken over long periods, can cause damage to the kidneys as well as bring on intestinal bleeding.
Oleocanthal, like ibuprofen, inhibits production of the two enzymes that cause arthritis, but it does so without causing harm to the body. Scientists say oleocanthal is a natural anti-inflammatory compound, not structurally identical to ibuprofen but similar in potency and profile. Dr. Paul Breslin (who, you'll remember, helped Dr. Beauchamp with the initial studies) said that by inhibiting the two enzymes, inflammation and an increase in pain sensitivity becomes dampened. Now, the Arthritis Foundation recommends foods that are rich in antioxidants, like the Mediterranean diet, which emphasizes olive oil.
This is good news for sufferers of a disease that affects over two million people in the United States alone, most of them women. There is no cure yet for arthritis; sufferers can only follow treatment to reduce inflammation and pain and to prevent joint damage.
More tests are definitely needed, but wouldn't it be great if daily consumption of olive oil could help provide the same benefits of ibuprofen without the health risks? Dr. Beauchamp said that although we don't have enough evidence to tell us that eating olive oil daily can combat rheumatoid arthritis, eating a Mediterranean diet, with healthy amounts of olive oil and fats rich in omegas-3 and -6, appears to be beneficial.
Oleocanthal cannot totally replace drugs like ibuprofen, but consider it an herbal food—a natural remedy. And as you are looking for this natural remedy, remember that not all olive oils are the same; they don't all contain the same amount of oleocanthal, and there's nothing on the label to tell you how much or even if it contains any oleocanthal at all. To be sure you're getting the maximum oleocanthal, it's best to buy one from early-harvest olives or, in other words, newly pressed olive oil. When olives are picked green, they tend to have more polyphenols such as oleocanthal and are rich in antioxidants.
The University of California began to study how oleocanthal levels in olive oil varied and its impact on people. At a meeting held in Greece in October 2015, one of their research scientists, Dr. Roberta Holt, gave the results of the first clinical trials in humans using extra-virgin olive oil with a high concentration of oleocanthal and without it. Nine healthy male adults took part in the study where early-harvest Greek extra-virgin olive oil with high levels of oleocanthal and oleacein (another constituent of olive oil) was used.
Dr. Holt reported at the conference that the two compounds had a similar effect on the reactivity of blood platelets to a 400 milligram dose of ibuprofen.
As research continues, scientists are getting closer to understanding oleocanthal's protective role in our health while olive oil producers aim to get the maximum oleocanthal from their oil.
Oleocanthal could well be the most important polyphenol in olive oil; hopefully, it won't be long before the Oleocanthal Society publishes even more conclusive results based on human clinical studies.
The Oleocanthal International Society
The discovery of oleocanthal made such a buzz amongst researchers and health professionals that they formed a society. It started in June 2015 when retired Spanish physician José Antonio Amerigo, saddened to see that nothing had been done from 2005—when the compound was discovered—to 2013, decided to act. He contacted Gary Beauchamp and got the ball rolling.
The Oleocanthal International Society (OIS) was created.
The OIS today comprises scientists, nutritionists, physicians, dietitians, chefs, and even communication professionals, all from different countries and different backgrounds. Set up as a nonprofit group, the principal aim of the members is to facilitate and increase research into oleocanthal. Through collaboration and organized special events, members hope to publish their findings so as to promote and raise awareness of oleocanthal. The founding members met for the first time on the island of Zakynthos in Greece in early 2015. Dr. Breslin, also a founding member, said that as a biomedical researcher, he wanted to see research funding come from people who are interested in finding cures, preventions, and treatments for major killers like cancer, Alzheimer's disease, other terminal dementias, and diseases associated with chronic inflammation.
But it's not all research! The OIS also wants to help spread the word of the wonders of olive oil, for nutrition and flavor. For Dr. Amerigo and Chef Daniel García Peinado from Malaga in Spain, the interest is a mix of medicine and food. Dr. Amerigo wants to spread awareness of this natural food worldwide, while Peinado, at thirty-six, is a passionate, dynamic, health-conscious chef. Dr. Amerigo said it wasn't easy to find a young chef motivated by food cooked in extra-virgin olive oil but that Peinado was interested right in the beginning, that "rich oleocanthal was Peinado's guiding principle."
Dr. Amerigo added:
Cooking with olive oil for me means using it for grilling, in the oven, and even frying, because it enhances all dishes and is good for our health. Not so long ago, the University of Granada demonstrated how frying with olive oil increases the transfer of phenols from extra virgin to the vegetables. Soon, there will be other examples with meat, fish, etc.
These two OIS members now work in tandem, attending conferences, congresses, and cooking exhibitions to demonstrate how cooking with extra-virgin olive oil, rich in oleocanthal, is scientifically proven to be good for one's health.
Dr. Amerigo thinks we should be adding extra-virgin olive oil to our daily diet, about 40 milliliters (1.5 ounces) of oil, rich in oleocanthal, each day. He said, "65% of the Spanish olive groves are rich in oleocanthal." On a cautious note, though, he added, "We need to involve the producers as well because, although there's much olive oil being produced, we only get oleocanthal from extra-virgin olive oil."
In charge of gastronomy, Chef Daniel García Peinado is taking his work seriously and was voted second vice president of the group at the second OIS meeting in November 2015. Peinado says his primary role is to transmit and diffuse extra-virgin olive oil in the world of gastronomy, especially since the mix of gastronomy and medicine is a relatively new cooking concept. "Up until now," he said, "when we speak about healthy cuisine, we refer to diet therapy or calories but not based on anything as functional as EVOO, rich with oleocanthal, with scientifically proven health benefits."
Because of the OIS's efforts, we have made more discoveries about olive oil and greater awareness is being spread. Thank goodness that Dr. Beauchamp attended that olive oil symposium and noticed the first sign that olive oil contained a powerful anti-inflammatory element. Now we have several more wonders to discover!
WONDER 2
OLIVE OIL FIGHTS AND PREVENTS CANCER
Studies have shown that olive oil has the ability to protect the body against certain benign and malignant tumors. In this chapter, we look at two common cancers: colon cancer and breast cancer.
You'll remember that Dr. Beauchamp and Dr. Breslin conducted the original discovery and research of oleocanthal. After recognizing the anti-inflammatory characteristics, Dr. Breslin continued investigating oleocanthal, this time looking at how and why cancer cells grow so quickly.
Focusing mainly on breast, pancreatic, and prostate cancer cells, Dr. Breslin headed a team of researchers as they applied oleocanthal to human cancer cells in the laboratory. The results were as unexpected as they were amazing! The team was surprised to see how quickly oleocanthal destroyed the cancer cells of the organs. The cells died within thirty to sixty minutes, unlike in other experiments and studies where cells programmed to die normally take between sixteen and twenty-four hours (scientists call this processed death "apoptosis"). Oleocanthal pierced the center of the cancer cell, targeting the lysosome, the part of the cell that stores waste. As it did so, it released the enzymes within the cell, causing it to die.
Dr. Breslin referred to the lysosome, which is full of aggressive enzymes, as "the garbage dump" of the cell. Lysosomes in cancer cells are larger than healthy cells, but what surprised the researchers was that oleocanthal did not damage the healthy cells in any way; it simply stopped their activity for a day. Dr. Breslin said, "It put the healthy cells to sleep for a day or so, then they resumed their activity. We now need to understand why cancerous cells are more sensitive to oleocanthal than noncancerous cells."
Could oleocanthal be the reason for reduced cancer incidence in the Mediterranean, where olive oil is consumed in high quantities?
When asked how much extra-virgin olive oil we should be consuming to reap the benefits of oleocanthal, Dr. Breslin said, "We need to separate OC from the plant and test it in toxicological studies to find out what [quantity] is safe. This has never been done, so we do not know how high a level would be tolerated well. In the meantime, I think 50 milliliters (2 ounces) of extra-virgin olive oil of a very high OC content each day appears both safe and desirable, provided people watch their total calorie intake."
The scientists say they still need a complete and thorough explanation of oleocanthal and to understand its protection mechanism in healthy cells. Once they achieve that, the next step will be to show that oleocanthal can shrink tumors in living animals.
Oleocanthal and Plasma Cells
Another study demonstrated the antitumor properties of oleocanthal, this time in a disease which affects blood plasma cells. This destructive cancer, called multiple myeloma (MM), involves the white blood cells made in the bone marrow, the soft center of the larger bones of our bodies. These plasma cells are vital because they are a part of our immune system; they produce antibodies which help fight infection. However, with MM, these abnormal cancer cells accumulate and push out the healthy blood cells.
Studies conducted in vitro have started to reveal the vast potential of the phenolic component of extra-virgin olive oil, mainly oleocanthal, in counteracting MM cell growth.
Spanish scientists from the NeuroEndocrine Interactions in Rheumatic and Inflammatory Diseases (NEIRID) laboratory described their results as "the first line of evidence that oleocanthal may be considered a novel antimyeloma drug, opening novel routes for the treatment of MM." How does olive oil do this? The group said that their experiments showed that oleocanthal eliminated the cancer cells by reducing the production of MIP-1 alpha, a key molecule which generates the myeloma cells.
Dr. Oreste Gualillo, head researcher, said, "This in vitro study suggests a therapeutic potential of oleocanthal in treating multiple myeloma."
Could oleocanthal work as a potential drug alone or with other established MM medicinal products for the treatment of MM in the future? When asked, Dr. Gualillo said, "Although the results are promising and suggest a wide spectrum of activity of certain molecules, further research on both in vitro and in vivo systems is always warranted."
So far, these anticancer properties of the compound were demonstrated in only in vitro studies; the real challenge will be, then, the follow-through with human trials. It may take a few years, but oleocanthal may well turn out to be effective anticancer treatment.
Extra-Virgin Olive Oil and Colon Cancer
Hydroxytyrosol, a component in olive oil, acts as a cancer suppressor in the intestine, protecting us from colon cancer. We should pay more attention to what we put on our plates because it eventually gets into our bowels.
Anne was speechless when the doctor told her she had bowel cancer; it came as a shock because the healthy, vivacious sixty-five-year-old had none of the listed factors that could affect her risk of colon cancer. Anne was not overweight, did not smoke, had no family members with the disease, nor was suffering from pain or discomfort. She was, however, sixty-five years old, an age at which the chances of developing colon cancer increase markedly. Anne had taken up the offer of a free stool test through a government screening program, but the report came back saying she had colon cancer. She was lucky; the doctors found her cancer in time, and even though she has to carry around a colonoscopy bag, she is responding well to treatment. Anne couldn't help but wonder: could she have avoided this disease by being more cautious with her food habits?
Today, this type of cancer is not considered life threatening, but people do not talk about it. There's nothing less glamorous than discussing changes in bowel habits and blood in the stool. Bowel cancer, also called colon cancer, is the second most common kind of cancer in the Western world. Doctors say it can be hereditary, or you can get it because of poor lifestyle habits and lack of exercise, but like hundreds of different cancers of the body, it occurs when cells of the affected area divide uncontrollably to form an abnormal mass.
In Anne's case, the disease was found in the last several inches of her colon as well as the rectum. As it started in the colon, the doctor said she had colon cancer. Sometimes, it can begin in the rectum, in which case it is diagnosed as rectal cancer, while cancers affecting either of these are called colorectal cancer. Unfortunately, the disease hardly shows any early signs and symptoms at the beginning stage and can progress long before it is detected, just as it did in Anne's case.
This is what happens to the body with colon cancer:
Colon cancer causes havoc with our endocannabinoid system—the body's central regulatory system—causing it to become unbalanced. You might not have heard of this complex process, a chemical messaging system discovered and identified only in the mid-1990s. It's not easy to explain how it all works, but what's important to know is that our bodies produce natural chemicals called endocannabinoids. These aren't stored in the body but are generated when necessary, binding to protein molecules or receptors present in cells throughout the body.
Endocannabinoids can change the way that a particular body part functions. Compare it to a lock-and-key system where the receptors are locks and cannabinoids act as keys: the two parts bind in a healthy system but are blocked in an unhealthy one.
With colon cancer, the intestinal receptors (known as CB1), although still present, get deactivated. However, CB1 is an anticancer agent recognized in the scientific world for its ability to suppress cancer growth.
Professor Mauro Maccorrone has been studying the cannabinoid's protective role. This is how he explains it:
CB1 is like a guardian angel, a cancer suppressor. When the receptor is turned off, colorectal tumor growth starts. With this decrease in CB1, cancer-promoting cells become free to roam; they continue to grow and divide, eventually contaminating the surrounding tissues. Those cells lose the memory of how to die; they keep on growing.
Could olive oil have an effect on our endocannabinoid system? This was the question Maccorrone and his team set out to research: to understand and study the therapeutic properties of the phenols contained in extra-virgin olive oil and, more importantly, to see whether they had an anticancer effect by influencing the CB1 receptor.
Performing both in vitro and in vivo studies on human tissue and rats, the team carried out experiments using different grades of olive oil on colon cancer cells. They first discovered that high-quality extra-virgin olive oil contains hydroxytyrosol, the principal component able to protect the body from colon cancer.
According to Professor Maccorrone, "Hydroxytyrosol in extra-virgin olive oil wakes up the sentinels that allow our bodies to keep colorectal cancer away. The mixture of olive oil is essential, but it is the hydroxytyrosol that pulls the trigger." The researcher further explained that hydroxytyrosol activates the CB1 receptor through epigenetic mechanism. He continued to explain that with this genetic system, "extra-virgin olive oil doesn't break up or alter the genes; it just makes them more readable."
In the second part of their research study, the researchers fed rats a healthy diet with extra-virgin olive oil for ten days and then looked for changes in their cell linings. They found a significant increase in CB1 expression; extra-virgin olive oil had reactivated their CB1.
Maccorrone concluded from these experiments that adding olive oil to an animal's diet can restore a healthy CB1 receptor level to protect cells against cancer; in other words, high-quality olive oil can defend us against colorectal cancer!
Consider colorectal cancer a food-related cancer, then think of the amount of food we ingest in a lifetime. Diet is bound to have a direct impact on colorectal cancer, because everything we eat passes through the digestive system, into the intestine, and out of the rectum. But food is customer oriented these days, where we, the consumers, often ignore the food value chain, not bothering to learn much about where the food came from and what processes were involved before it got to our plate.
Keeping the entire intestinal system healthy and manageable is important to allow the body to convert food into energy and get rid of waste. Adding olive oil to the daily diet can help with elimination and regulation by lubricating the whole system. It allows the food we ingest to have a smooth passage right through the digestive track, softening the stools in the rectum and making them easier to pass. It also works as a mild laxative, so if we use olive oil regularly, we should not be constipated. Choosing a higher quality of olive oil is crucial. A better-quality olive oil has more flavors and is much more efficient.
An interesting animal test showed promising results in the fight against colon cancer. In Barcelona, researchers wanted to see how olive oil compared with other oils, to study the effects of olive oil on the colon. The scientists fed mice with diets rich either in safflower oil, fish oil, or olive oil. The oils were mixed with a chemical to accelerate cancer in the bowel and given to half of the three groups. The study lasted five months, after which the doctors at the University Hospital Clinic of Barcelona Department of Gastroenterology noted that twice as many rats in the safflower group developed colon cancer as in the two other groups. What's interesting here is that these oils have different chemical structures: olive oil is a monounsaturated fat, a good type of fat, which significantly reduced the number of cancer lesions. The medical professionals said that the benefits could have been due to a lowering of inflammatory chemicals in the intestine.
Today, we have a mass of information at our disposal, knowledge passed on from clinicians and nutritionists telling us the importance of eating the right food. However, the statistics are still high: the Center for Disease Control and Prevention (CDC) reports that colorectal cancer is the third most common cancer and the second leading cause of cancer-related deaths in the United States. Instead of reaching for a drug to "treat" a digestive symptom, we should perhaps choose more carefully what we put into our mouths.
As Professor Maccorrone pointed out, we seem to forget the "importance of the quality of food, and that includes olive oil." The Italian specialist in Food Science and Human Nutrition thinks that more could be done by food manufacturers to show the authenticity of their products through correct labeling. Consumers can easily be seduced by the impressive Made in Italy label, but, as he says, the label might say Italian Olive Oil when the olives come from somewhere else entirely. Professor Maccorrone feels food awareness could start at school and that lawmakers should work harder at defending quality, adding that "we all know this is happening, but there's not enough reaction against it. I would like to see more healthy food and a deeper respect for genuine, natural food."
The hard truth, however, is that poor-quality oil and deception in the olive oil business is not new. Extra-virgin olive oil is very often in the spotlight when it comes to food fraud, and even some of the best-known brands and most highly reputed companies have been caught trying to pass off a virgin oil or simply olive oil as "extra-virgin" when it is not truly the same quality. With olive oil fraud, industrialists take advantage of the consumer's ignorance and a lack of controlled chemical and sensory tests to mix extra virgin with another vegetable oil such as rapeseed oil (also known as colza). Sadly, this widespread, deceitful, and damaging practice is likely to continue until food authorities enforce stricter laws. As consumers, we need to remember that high-quality olive oil is better for us and that fraudulent oil does not offer the same health benefits as extra-virgin olive oil.
One thing is clear in the fight against colon cancer: good food is a lot kinder to our body than drugs, especially extra-virgin olive oil, which is shown to help take care of our bowels.
Olive Oil Works Effectively to Fight Breast Cancer
It is very hard for women to live with breast cancer. Studies show that women who follow a Mediterranean diet can reduce their risks of getting the disease. Various components in olive oil help lower the risks of breast cancer.
Slim, attractive, forty-year-old Joanna answered with a smile when she heard her name called; it was her turn to go to the X-ray room. She had come for her yearly checkup at the busy Symptomatic Breast Clinic. Breast technologists have learned how to be discreet in their work; they know how vital it is to be sensitive to the anxiety of their patients when they come for their follow-up mammogram. It was not easy, though, to be discreet with Joanna. As she took her shirt off, you had to look, admire, and take in the stunning work of art running up the right side of her chest. Starting from her slim waist and going up to where her breast should have been was a truly magnificent tattoo: a rose shrub with vibrant green leaves, ending at the top with a magnificent pink rose. It replaced the breast Joanna once had.
She explained, "Trees are a symbol of life; this tattoo is sacred to me. After my mastectomy, I needed something to remind me that life must go on. This tattoo is what helps me not to be fearful of the future."
The young patient had been diagnosed with breast cancer three years before. No one in her family had had breast cancer, yet she could not ignore the lump in her breast; it seemed to be getting bigger every day. She eventually saw her general practitioner, who referred her to a diagnostic breast unit. She then had to see a surgeon.
After she'd had the tests, the surgeon gave her the bad news. He told her, "You have a choice: you can have either a mastectomy or a wide local excision." Then he explained the difference. A mastectomy would remove her entire breast; a wide local excision would remove the cancer and some of the normal tissue around it, "but the cancer could come back," he warned.
Joanna explained why she made the decision to have a mastectomy. "The choice sounded drastic at the time, but I did not want to leave any doubts. I could not afford to. I had to think of my family; the kids were still young." Shockingly early to be diagnosed with an aggressive form of breast cancer, Joanna admits she had to face many challenges—the most important, she confesses, was fear.
Women with breast cancer who are being treated, even those that have successfully been treated, all say the same thing—they are always haunted by fear and trepidation. They fear that the cancer will come back. The medical term for this is a "recurrence," and it can happen five, ten, or even fifteen years after the necessary therapy. The treatment itself to remove or destroy the cancer is stressful whether it is done through surgery, chemotherapy, or radiation therapy; they all leave changes to the body, as well as visible scars that make women feel uncomfortable with themselves.
There's no one single reason women get breast cancer, but it seems that several factors can contribute: genes, lifestyle, environment, and hormones are the chief risk factors, and any combination can trigger the disease.
Women over fifty are more likely to have breast cancer; that is why some countries have a breast cancer program for older women. These screening programs allow women to have mammograms— breast X-rays—either every two or three years (depending on the country) as well as free health treatment if they are diagnosed with the disease.
Some women can inherit abnormal genes, which means that their risk of developing breast cancer is much higher than someone who does not have those genes. The actress Angelina Jolie made the decision to have preventive double mastectomy surgery when she realized she carried the same genes as her deceased mother.
Cancer of the breast starts when cells begin to grow in an uncontrolled way, forming a tumor. The cancer is noninvasive if the cancer cells are only in the breast and have not spread any further. However, when cancer spreads and invades the surrounding tissue, it is called invasive breast cancer. Most invasive cancers spread to the rest of the body if they are not treated in time. They pass through the bloodstream and lymphatic system, forming secondary cancers in other parts of the body.
Joanna lives in the UK, where cancer rates are high. According to Cancer Research UK, there were 53,700 new cases of invasive breast cancer in 2013, and it was the most common cause of cancer death that year. A breast screening program in the UK provides a free service, aiming to find breast cancers early in women. A recent extension now includes women aged between forty-seven and seventy-three. It means that women who are registered with their local general practitioners receive letters of invitation every three years to attend a dedicated screening unit for a routine mammogram. If the results appear abnormal, they are then asked to participate in an assessment clinic where they might have new images, ultrasounds, and perhaps a biopsy. Breast Screening UK says that one in every twenty-five women who are called back for assessment is diagnosed with breast cancer.
The Mediterranean Diet Lowers the Risk
As women get older, their risk of getting breast cancer increases. They also have to put up with menopause, which has a definite influence on breast cancer. And life expectancies in Western countries are also increasing. This means that the proportion of women affected by breast cancer is continually growing.
The results of the following study are significant because it focuses specifically on postmenopausal women, where the need for prevention and control is more vital these days.
The data in this study was the first randomized trial to research whether a Mediterranean diet could protect women from breast cancer. It was a parallel study carried out in conjunction with studies on cardiovascular disease by PREDIMED (Prevención con Dieta Mediterránea). PREDIMED researchers were primarily concerned with nutrition and heart disease, but the trial was also designed to follow the incidence of five different types of cancer, including breast cancer in older women.
Researchers chose older subjects for this study, Spanish women between the ages of sixty and eighty, who were almost all menopausal. None of them had cardiovascular disease, but all had either type 2 diabetes or at least three of the major cardiovascular risk factors that could bring on the disease; these included smoking, hypertension, or family history of cardiovascular diseases. They all had an average body mass index of 30; in other words, they were obese. This in itself is a known risk for breast cancer.
The researchers divided the 4,282 women into three groups:
1,476 were put on a Mediterranean diet supplemented with extra-virgin olive oil. All the women were given a liter of olive oil every week to make sure they followed the specification required by the research team. These Spanish women use the oil anyway; it is what they are used to using, as Spain is the largest producer of olive oil.
1,285 were instructed to follow the Mediterranean diet supplemented with a daily intake of 30 grams of mixed nuts. They received weekly supplies of nuts during the research period.
1,392 received dietary training and were asked to follow a low-fat diet.
After a follow-up period of about five years, the researchers found that thirty-five women had developed malignant breast cancer. The results revealed that the women who followed the Mediterranean diet supplemented with olive oil showed a 68% lower risk than those who were on the control diet. Those in the second group with nut supplements showed a 40% lower risk of breast cancer than the control group.
The researchers concluded that preventive strategies represented the most sensible approach to treating cancer. They said they hoped to see more emphasis on a Mediterranean diet to reduce cancer and cardiovascular disease and improve overall health and well-being. They recognized, however, that the results could be considered limiting because all the participants were from Spain, were white, and lived where the Mediterranean diet and cooking with olive oil is a way of life. We have to remember that the Spanish volunteers already consumed significantly less saturated fats than Americans, Canadians, and the British.
Another limiting factor is that this was a secondary analysis, carried out in conjunction with the study of cardiovascular disease. It was concerned primarily with women with a high cardiovascular risk rather than a risk for breast cancer. What it does show, however, is that prevention of breast cancer might very well start in our kitchens.
The following large-scale studies also show the importance of consuming the right type of food.
The Mediterranean Diet and Greek Women
A poor and high-fat diet seems to be one of the leading causes of breast cancer, which is much more prevalent in Western countries. The Mediterranean diet, on the other hand, is rich in antioxidants and contains only a small amount of animal saturated fats. This study looks specifically at the association of breast cancer and the Mediterranean diet in a Mediterranean country.
14,807 women took part in the European Prospective Investigation into Cancer and Nutrition (EPIC), and the results were published in the highly regarded American Journal of Clinical Nutrition.
This study looked at what the women ate; it followed their eating habits for nearly ten years. The Greek women started the research by answering an in-depth questionnaire about their diet to see how they conformed to the Mediterranean diet. From that, they were given a score from 1 to 9. Those who adhered to typical dietary components got the high score of 9, while those with minimum adherence scored 1.
At the end of the study, 240 cases of breast cancer were reported. In their report, the researchers indicated that, in premenopausal women, there was "no significant association between conformity to the diet and breast cancer risk." What they did find, though, was an inverse situation with postmenopausal women. They concluded that "conformity to the traditional Mediterranean diet is associated with a lower risk of breast cancer amongst postmenopausal women." The report added that this could explain the lower incidence of breast cancer in Mediterranean countries.
Prevention Is a Better Cure
The Genesis Breast Cancer Prevention Centre in the United Kingdom understands how important it is to tackle breast cancer and to prevent breast cancer from developing. This UK charity is dedicated entirely to the prediction and prevention of breast cancer. The team believes that if women understand the factors that cause breast cancer, they will take positive steps to lower their risk and prevent the disease.
One of those factors is maintaining a healthy lifestyle. Studies show that an unhealthy lifestyle increases the risk of breast cancer; carrying excess weight, being sedentary, and consuming high amounts of alcohol are all strongly associated with breast cancer. To put it bluntly, we eat the wrong food, and too much, and don't exercise enough; this is why we carry that excess weight and increase our risk.
At Genesis, they developed an efficient eating plan for women at increased risk of breast cancer who are overweight, want to lose weight, and, once that is achieved, want to maintain a healthy weight to reduce their risk. Michelle Harvie is the only UK dietician who has performed in-depth studies on the relationship between diet and breast cancer prevention. Based on these research studies and trials at Genesis, she has developed a two-day diet approach to how we can reduce our calorie intake and have less sugar and processed food.
A daily diet can be restrictive, too hard to follow for many people, but with this eating plan, dieters do not check their calorie intake or skip meals; "they simply follow a low-carb plan for two days and eat sensibly for the rest of the week."
Dr. Harvie says: "We advise our patients to try to follow our two-day diet, which includes the benefits of two days of energy restriction and five days of a healthy Mediterranean diet." Dr. Harvie recommends a healthy diet such as the Mediterranean diet—which includes fish, fruit, vegetables, low-fat dairy, high-fiber carbohydrates, and moderate amounts of healthy fats like olive oil and nuts—for women who wish to reduce their risk of breast cancer and other diseases. She feels that this diet increases satiety and prevents overeating, which are key to reducing weight.
Studies show that incidences of breast cancer decrease with an increased intake of olive oil, highly present in the Mediterranean diet, but you may ask: what components in olive oil actually help lower the incidence of breast cancer?
Dr. José Gafario from the University of Jaén in Spain has been studying the effects of some of the minor components in olive oil, looking at compounds which could play a natural role in preventing disease. The research team carried out in vitro studies on squalene, one of the oil's minor polyphenols with strong antioxidant properties on healthy human breast tissue and cancerous human cells. You may not have heard of squalene, but squalene got its name because it is derived from shark liver oil. Although present in other edible oils, the concentration in olive oil is much higher, ranging from between 0.8 to 13 grams per kilogram, depending on the olive variety.
Dr. Gafario reported that "squalene protects the DNA of healthy mammary epithelial cells against oxidative damage, suggesting that this molecule could have the potential to prevent human breast cancer." He said that squalene appeared to have a preventive role in the healthy cells but less potential in tumor cells.
The team also studied the effect of oleanolic acid, another compound found in the olive skin and leaves of the olive tree. These tests were also carried out on healthy breast cells as well as cancerous ones. Dr. Gafario said that the results suggested a dual effect on healthy and tumor cells and that the data from the studies suggested that "oleanolic acid could be a chemo preventive agent" (a food or drug that prevents or delays cancer) in human breast cancer. He added that, at the same time, it "could have the ability to inhibit proliferation of highly invasive breast cancer cells."
In normal breast cells, on the other hand, the effects of oleanolic acid were different. He said that in normal breast cells, the effect is the opposite: it appears to decrease oxidative stress (which we'll discuss further later) and protects the DNA (our genes) against oxidative damage. A reduction of oxidation stress is good news! Our bodies are in danger of too much oxidation stress, the medical term used when our cells are not protected adequately from potential damage, perhaps because of a lack of antioxidant nutrients. Not enough antioxidant nutrients suggest that eating habits are not what they should be.
Interestingly, other scientific researchers have described other positive effects of oleanolic acid against other diseases; the effects of oleanolic acid are certainly not limited to breast cells.
There is one other element of olive oil that needs to be addressed in how it can fight or prevent cancer. Olive oil contains vitamin E. Though we will discuss it further later, it is important to recognize that vitamin E is an antioxidant that protects our cells from damage. The chemical name is "tocopherol," and olive oil is rich in vitamin E (alpha-tocopherol).
How does vitamin E work for us? Because it is a powerful antioxidant, it neutralizes the free radicals—the unstable molecules of oxygen that cause havoc to our DNA and eventually cause cancer. Vitamin E is often described as the scavenger of free radicals. Research shows that the best form of the vitamin is what we get from our food supply naturally, as our bodies do not absorb the artificial form as well.
Is It Just a Question of Fats?
A study which took place in Sweden differs from the above Mediterranean research because it did not take place in the Mediterranean area and also looked specifically at the intake of fats. The report came from the Karolinska Institutet in Stockholm; it was a major study involving 61,000 women, all between the ages of forty and seventy and born between 1914 and 1948.
The women filled in questionnaires that listed sixty-seven common foods, with further, more specific questions about the participants' fat consumption. The researchers then checked the Cancer Registry to see who in the group developed breast cancer. Between 1987 and 1993, there were 674 confirmed cases of invasive breast cancer. When researchers analyzed the data, they found that a high intake of monounsaturated fats—the fat you get from olive oil— was protective against breast cancer. They found that a high intake of polyunsaturated fats raised the risk of developing breast cancer considerably, reporting that a 5-gram increment in the daily intake of polyunsaturated fats corresponded to a 69% increase in risk. The researchers said that monounsaturated fat in the diet appeared to reduce the likelihood of breast cancer by 45%. It appeared that saturated fat—the kind found in meat and dairy products—had no effect on breast cancer one way or the other.
Interestingly, they found that the largest contributor to polyunsaturated fats came from margarine (33%), followed by bread and cereals (23%), meat (17%), and dairy products (11%).
These study results show the importance of the kinds of fats we eat. Eating the wrong fats can be dangerous.
Olive Oil versus Corn Oil
Studies have shown that we have such a high rate of breast cancer because of the kind of fat we eat. Trans fats, the sort of fat we get from packaged fats, is particularly dangerous for our health. Scientists compared the effect of different oils on rats, looking at corn oil—a source of omega-6 acids and considered a polyunsaturated fat—and olive oil, a monounsaturated fat. The two groups of rats were fed with the two oils, while the control group was given a low-fat diet. They injected rats with powerful doses of carcinogens and looked at the resulting development of breast cancer in the two groups. They found that the rats fed with corn oil developed more numerous breast cancers more quickly than those on the olive oil diet. These rats also developed more body weight. Those on olive oil, however, did not show any increase in body weight, and they showed fewer tumors, which developed much more slowly than those on the corn oil.
How do the researchers explain the result?
They say that a diet high in corn oil stimulates certain proteins that allow the cancer cells to grow uncontrollably. The diet based on olive oil, on the other hand, reduces the activity of these proteins and encourages cell destruction.
In the fight against breast cancer, there are factors such as genetics and age that we cannot control. We do have a choice, though, when it comes to lifestyle and diet. A good healthy intake of olive oil is one of those healthy choices that can make a difference.
What Oil You Should Be Using and When: A guide to choosing the right cooking oil
Which cooking oil should you be using?
All oils are not created equally; they come with different health benefits and nutrition. One thing is clear: vegetable oils enhance the taste of foods and dishes. Besides being tasty, they are necessary for our balance because they are essential to the body and ensure proper functioning of our cells. Vegetable oils have the advantage of not containing cholesterol, unlike animal fats such as butter. Some vegetable oils are essential because our bodies do not produce certain elements in them, so we need to get them from our diet. Omega-3 and -6 acids are good examples.
As oils do not all have the same composition, even though they have the same caloric value of 9 kilocalories, it is best to mix and choose them according to the specific type of cooking.
"No oil is perfect, so it is best to choose carefully, and don't hesitate to mix them," says Silvie Borrat, who is a pharmaceutical assistant and phyto-aromatherapist in Toulouse, France. "For the healthy functioning of our body, it is important to consume fat. It is recommended to reduce the intake of animal fats and to promote the intake of vegetable lipids. Each vegetable oil is made, in varying proportions, of saturated fatty acids, monounsaturated oils, or polyunsaturated oils. Mixing them gives a complete diet in different compounds for good health," she explains. To fully benefit from their contents and to create good balance, it is important to alternate, mix, and vary their use. The change also means developing the palate to appreciate new flavors; it is essential to use them reasonably to enjoy the benefits of each.
Remember: not only the brain but also the retina, skin, and bones need essential fatty acids for growth. Oils also help maintain general good health throughout our lifetime. We should choose cold-pressed oils and organic oils if possible.
Here's what you need to know:
We need fats to be healthy.
The healthiest oils must be cold pressed and organic.
Oils rich in omega-3s cannot stand the heat and are, therefore, best used cold.
We should limit our intake of omega-6 unsaturated fatty acids, which promote inflammation.
Try mixing different oils. A good example of this is vinaigrette made with olive oil and linseed oil. This mixture allows you to have both omega-3s and -9s.
To conserve our health and to safeguard our planet, we should avoid soybean oil and palm oil.
OLIVE OIL
Olive oil contains more than 60% of monounsaturated fats, omega-9s, a few omega-6s, and vitamin E. Olive oil is an essential oil in the kitchen; there is nothing better for cooking vegetables. Olive oil can easily replace butter or sunflower oils in cooking. For salad dressings, mix olive oil with a little linseed oil or walnut oil to get that healthy omega-3 content.
PEANUT OIL
Peanut oil, also called groundnut oil, is high in omega-6 (34%) and omega-9 (48%) acids. It has no omega-3s. This means it tolerates heat well. Peanut oil is a good choice for frying.
Be careful, though: allergy to peanuts is a very common sensitivity.
COLZA OIL OR RAPESEED OIL
Colza oil, rapeseed oil, or canola oil (in Canada and America) contains omega-9 (57%), omega-6 (23%), and omega-3 (10–15%) acids. It is the newest oil on the market, but scientists disagree about the effects it can have on our health. As over 90% of canola oil is genetically modified, processing of this oil can be a real concern.
NUT OIL
Nut oil has a perfect ratio of omega-6 and omega-3 acids. Its biggest asset is its superb taste. Nut oil cannot stand the heat, though, so keep it away from all heat and oven use.
COCONUT OIL
Coconut oil is rich in saturated fatty acids; recent studies show that it is not as bad for health as the saturated acids we get from animals. It contains medium-chain fatty acids that provide a quick source of energy and could even have positive health benefits. More studies are being carried out, but one thing is sure: coconut oil is becoming more and more popular in cosmetics. The smoke point is high.
SUNFLOWER OIL, GRAPESEED OIL, AND CORN OIL
These oils contain more than 50% of omega-6 unsaturated fatty acids that can increase inflammation in the body. We should limit their consumption in our diet or mix them with other oils with a large number of omega-3s.
Sunflower oil, with a high content of oleic acid, is produced from hybrid plants. It has a composition similar to olive oil and is also more stable than traditional varieties. However, it is hard to find these oils cold pressed.
SOYBEAN OIL
The composition of soybean oil is not great for our health, as it contains little omega-9 and omega-3 acids but more than 50% of omega-6 acids. We should consume soybean oil only occasionally and ensure that we buy organic.
PALM OIL
The attractive price and excellent heat resistance of palm oil make it the preferred industrial oil. It contains 50% saturated fatty acids and little omega-3 acids, not a very good composition for our health. Also, it poses a problem for the planet when not grown from sustainable palm groves. In contrast, red palm oil (available at health food stores) contains vitamin E and carotene and can replace butter.
LINSEED OIL OR FLAXSEED OIL
This oil is perfect for vegetarians because it contains more than 50% omega-3s. With an extremely high rate of omega-3s, flaxseed oil rectifies the imbalance between omega-6 and omega-3 acids that prevails today in the United States.
Because it is so rich in omega-3s, you cannot use it for cooking, so it can only be used in vinaigrettes. It hasn't got a very pleasant taste, so it is a good idea to mix it with tastier oil. The ideal pairing is one part linseed oil and two parts olive oil to mask the taste. To avoid the problem of the oil going bad, linseed oil should be refrigerated. Flaxseed oil is found only in health food stores and can be a little expensive.
COD LIVER OIL
Cod liver oil is very rich in omega-3s and efficient on the cardiovascular system. It also makes our cell membranes more fluid and boosts our morale. Cod liver oil contains vitamin D, essential for growth, and high levels of vitamin A, which is great for fighting infection. However, it has an especially bad taste. It is better to take it in capsules or eat cod liver on toast.
OTHER OILS
Argan oil, borage oil, and sweet almond oil are very popular in cosmetics.
COMPOSITION OF FATTY ACID OF THE MAIN VEGETABLES OILS
WONDER 3
OLIVE OIL BEAUTIFIES, PROTECTS, AND REJUVENATES SKIN
For centuries, olive oil has been recognized as an amazing substance, a beautiful gift with many uses. We now know more about the different ingredients that work toward healthier skin: vitamin E, vitamin K, phenols, and squalene. Although they weren't aware of the scientific knowledge we have today, the Greeks were among the first to cultivate, appreciate, and use olive oil extensively.
At the first Olympic Games in 776 BC, winners received olive tree branches cut with a gold-handled knife—an offering of peace. But the ancient Greeks also used olive oil in cosmetics. They loved beauty, considering it a blessing and a gift from the gods. At the Olympic Games, wrestlers used to rub olive oil on their skins before entering the arena. Just imagine how their torsos must have shone in the strong Mediterranean sunshine. They were probably not very comfortable, but to the crowds who came to watch them, they appeared strong, sturdy, and beautiful. The contestants believed that by anointing themselves with the oil, the goddess Athena would bestow power, strength, and wisdom as they fought. Rubbing olive oil on their body during training was standard practice at the time—it protected their bodies, elasticizing and soothing their muscles at the same time.
Later on, the Greeks and the Romans discovered the pleasures of bathhouses—a place to wash, chat, and contemplate. They would spend hours in these baths, enjoying the custom of anointing their bodies with olive oil, rubbing it well into the skin then scraping away all the accumulated dirt and sweat. They used a bronze metal tool for this oil cleansing, a kind of spatula called a strigil with a curved blade at the end which made it look dangerous; but it was just thick enough to skim the surface without cutting the skin. This was the first known exfoliating tool, used in what we can describe as the equivalent of our modern-day spa.
Olive oil became more and more popular as the people discovered its health and beauty benefits. Surprising as it may seem, the discovery of soap came well after exfoliants. The origin of soap is not quite clear, but we know that the first documented olive soaps (created in Marseille around 1370) were made using simple products combined with seawater and local olive oil. Increasing demands for Savon de Marseille at the beginning of the seventeenth century lead to overworked employees in the many factories, which in turn caused the quality of the soap to deteriorate.
Factories were springing up everywhere at the time, with production figures reaching as high as twenty thousand tons per year in France. A French law passed in 1668 changed that. The new regulations limited the use of the name "Savon de Marseille" to soaps made only in and around Marseille. It stipulated that only pure olive oil should be used, that no fat could be added, and that certain manufacturing processes had to be respected; only then could the soap carry the prestigious Marseille mark.
Today, Marseille is still the principal place for producing Savon de Marseille soap, with around ten savonnières (soap factories) in and around the city. Although production methods and contents have changed over time, many French people still recognize the olive oil–based soap as a traditional soap, one that their mothers and grandmothers used. For many, it is a simple, pure, natural, and hypoallergenic soap, ideal for people with sensitive skin.
Portrait of an Olive Oil Soap Maker
Nia Hafsia and her husband, Karim, live in a tiny village called Bourgnac, a beautiful little village with only 350 inhabitants situated in southwest France between Bordeaux and Bergerac. Their property, an old flour mill, dates from 1657 and comprises a thousand square meters where they have their home, Nia's soap workshop, and a boutique all on the same premises.
Indeed, this is the inspirational and motivating environment of artists and creators. This is where Nia lives her passion—making olive oil–based soaps, lotions, and creams. A true artist, she uses natural plant colors and clays in her soaps, ingeniously swirling and marbling the eight-kilo blocks of soap to transform them into true works of art. Her other specialty, she says, is blending the natural essential oils and absolutes (concentrated oil mixtures from plants). Mixing different natural perfumes together not only creates beautiful fragrances that smell good but also yields positive, therapeutic benefits for various skin types.
Her beautiful olive oil soaps are mild and gentle, cleansing without drying the skin. The secret is in the blending of natural fatty acids that help protect and regenerate the skin, leaving it soft and smooth. She uses between 50% and 80% extra-virgin olive oil in her base formula. Because of the quality oil, Nia said that individuals with sensitive skin can use her products, as the soap has soothing and nourishing properties.
Nia launched her natural, handmade soap–making business, Passion Savon, some thirty-five years ago in Wales in the United Kingdom. The venture started in her kitchen, where she could work and stay at home with her four young children. Today, the small cottage industry has snowballed into a business that manufactures one ton of soap every two months. Nia researches, blends, refines, and creates her products, increasing production to satisfy customers not only in France but also in China, Hong Kong, Lebanon, Dubai, Sweden, Holland, Malta, Finland, the United States, South Africa, and many other countries throughout the world.
As a qualified aromatherapist, she recognizes how important it is to protect and preserve the skin—the largest organ of the body—without compromising our environment. For her organic soap products, she uses only 100% natural raw ingredients, many of which are sourced locally from small producers. She says the great thing about olive oil is that it does not block the pores and that it makes soap that is suitable even for babies, children, and elderly people with fragile skin types. The extra-virgin olive oil that she uses is first cold pressed and food quality.
She explains, "A lot of soap manufacturers add palm oil to the soap mixture, but palm oil is a saturated fat, which can be damaging to the health and also to the environment, as its production is threatening our rainforests." Nia's company has a strict no–palm oil policy. "We feel that the product is destroying our rainforests, and many animals are losing their habitats because of the intensive industry that is slashing and burning trees, leaving a barren wasteland where there was once life."
Many soap makers choose a cheaper option for their craft, using the less expensive pomace oil, the last grade of olive oil. These are dregs that are that are washed with a chemical solvent. Nia feels that it should not even be referred to as "olive oil."
Passion Savon uses a true cold process saponification (the natural chemical process of soap making) for manufacturing their soap. And Nia mixes her soap at an ambient temperature of between 20° C and 24° C (68° F and 75° F). "It's like cooking vegetables. If we overcook them, all the precious nutrients are lost. We don't heat the ingredients included in our soap formula, thereby preserving the positive therapeutic properties in the final soap bar."
The company does not just make healthy, olive oil–based products; they are also very aware of the duty that small enterprises have to protect the environment. They manufacture solid products that do not need packaging. Nia says, "I hate the thought of those millions and zillions of plastic bottles that are cluttering, polluting, and ruining our world." At Passion Savon, they pack their skincare range, Kariti, in special dark glass containers, without water content, so there is no need for chemical preservatives, and the twelve products are concentrated and based on organic, effective, and active raw materials.
Nia says that adding olive oil to cosmetics and beauty products gives the skin wonderful moisturizing properties in the same way as the soap, but in a more profound way, because soap is a "wash off" product whereas creams and lotions are "leave on" products. Without giving any secrets away, she says that a very simple and easy moisturizing cream can be made with 30% extra-virgin olive oil, 5% vegetable emulsifying wax, 5% shea butter, 5% vegetable glycerin, and 55% rose water or aloe vera.
Even though the price of extra-virgin olive oil has increased tremendously, Nia makes no compromise in her choice of natural, high-quality, organic, and free-trade ingredients. She would not dream of changing anything and is determined to keep using this top-grade oil for her luxury, artisanal soaps. This is good news when you consider that today the market is full of detergent-based liquid and solid soaps, shower gels, shampoos, and bubble baths, mostly formulated with a base of sodium lauryl sulphate, sodium laureth sulphate, or ammonium laureth sulphate, all harsh chemicals that can be dangerous for our health.
Nia's spirit of innovation, environmental awareness, talent, and hard work have not gone unnoticed. In January 2016, she received the outstanding award of Maître Artisan d'Art in recognition for her work as an aromatherapist and master soap maker. Translated as "master of art," this is France's prestigious award, honoring her as one of a selection of the country's most gifted master artisans. She deserves this recognition for her hard work and is the first Welsh woman to receive this diploma in her field of work. The mayor of her village, Bourgnac, was there to support her, pleased to put their village "on the map," so to speak. Nia enthuses, "It means that I am recognized as a specialist and master of art in my work, and that is really thrilling."
The company is busy developing the business further to include more exciting, original, and innovative aromatherapy olive oil products. The vivacious entrepreneur says excitedly, "The future looks bright! The company continues to grow, and my creative skills seem to grow with it. I am so grateful to be able to live in such a beautiful and safe part of the world and be able also to have a job that is such a real pleasure. It's incredible to wake up each morning and be excited to go over to my workshop and see my new 'babies' that I made the day before."
Squalene and Squalane in Cosmetics
Squalene is a natural compound present in the outer layer of our skin, making up roughly 10% of our serum; it is a lipid that protects us from environmental toxins and helps to retain moisture. The richest source that people have cultivated, however, comes from sharks, followed by extra-virgin olive oil. Our natural squalene contributes to hydrating the body, but after age thirty or so, the body levels start declining, and by age fifty, the production levels drop. Squalene has another downside: it is highly unstable and so cannot be used in cosmetics or skincare products that need a long shelf life because it spoils quickly.
However, if we treat squalene with hydrogen (through a process called "hydrogenation"), we get a saturated form of squalene which is less susceptible to oxygen and, therefore, longer lasting. This hydrogenated state is called "squalane"; it is a more stable, colorless oil with exceptional moisturizing properties. Cosmetic manufacturers use it extensively for making face creams, foundation, eye and hair products, and lipsticks.
The preferred option for cosmetics would be squalane made from olives, as we cannot continue to target sharks for their squalene; slaughtering of sharks for their liver contents is a real concern for environmentalists. According to a study released in 2012 by Bloom, a nonprofit organization, 90% of cosmetics used shark liver oil, contributing to about 2.7 million deep sea shark deaths every year. The Bloom Association said that although Western cosmetic corporations are moving more toward plant-based squalene, the industry is still largely supplied by animal squalene.
The problem today, though, is that not all manufacturers state the origin on the package. Squalane can come from either source; they are chemically identical. But one thing is certain: with such a booming cosmetics industry, the demand for squalane is bound to increase. By checking the label and looking for "vegetable based," "vegetable origins," or "100% plant derived," we as consumers can make sure that we are opting for plant-based squalene instead of animal-based squalene.
A Comparative Study on Olive Oil and Burns
Burns happen when you expose your skin to heat from fire, hot liquids, or to electricity, and occur mostly at home through careless handling of home appliances. You can also be burned by excessive exposure to the sun or through chemicals.
Although you might experience some pain, first-degree burns are relatively minor; they affect mainly the outer layer of the skin, called the dermis. Second-degree burns are a little more serious; they can penetrate to the second skin layer, called the epidermis. With that comes pain and blisters, which need attention. If they're not treated correctly and immediately, second-degree burns can progress to third-degree burns. Third-degree burns are more complicated; that is when the dermis and the epidermis, muscles, tendons, and nerve endings of the area all become damaged.
Researchers conducted studies on patients suffering from second- and third-degree burns using sunflower and olive oil. They wanted to see if consumption of either of these two oils could have an effect on the healing of burn patients. They wanted to know if the healing process could take place internally when taking olive oil orally.
A report published in the journal Burns looked at how the body recovered with second-degree burns in a burn care hospital. The study took place on one hundred hospital patients, aged between thirteen and fifty-nine, admitted with burns affecting between 20% and 30% of their body surfaces. They were randomly divided into two groups: one that took extra-virgin olive oil and one that took sunflower oil. They all started taking the oils within twenty-four hours of hospital admission, both groups receiving similar hospital care treatment for their burn conditions—skin grafting, excision of wounds, etc.
Daily examination during their stay allowed the researchers to check and evaluate changes in the surface areas of the burns. They found no significant changes in the body surface of both sets of patients but differences in the duration of wound healing, which ultimately decreased the time spent in the hospital for those taking olive oil.
The authors concluded that "those taking olive oil appeared to have accelerated wound healing and decreased length of stay in the hospital." They added that a larger study involving more patients with more than 20% burn surface taking more olive oil might have more significant results.
Are you wondering how the olive oil made healing faster? The researchers felt that the positive results were due to antioxidants and the anti-inflammatory properties of olive oil.
Olive Oil Solutions
Here are some healthy, safe suggestions for using olive oil at home. We gathered them over time from friends, beauticians, and family. Have fun as you test them.
MORNING DRINK
Wake up the inner beauty in you by starting the day with a lemon, olive oil, and honey concoction. Squeeze the juice of half a lemon into a cup of hot water; add 1 teaspoon of honey and 1 tablespoon of extra-virgin olive oil. Have this before breakfast and make it a routine; it's an ideal way to wake up your liver with a good dose of vitamin C and to start your healthy daily intake of extra-virgin olive oil.
NATURAL EXFOLIANT 1
Our body naturally gets rid of dead skin, but sometimes it hangs on too long, giving us a dull, tired-looking complexion. Exfoliating removes dull, tired, dead cells and rejuvenates the skin. Use this natural scrub twice a week to help keep your complexion clear and healthy. You might need to double the ratios for a whole body and face scrub.
Make a paste using 2 teaspoons of sea salt and 3 tablespoons of olive oil. Using the tips of your fingers, rub the mixture on your face, neck, and body using circular movements, paying special attention to the thicker skin on your knees, elbows, and T-zone area (across the forehead and down the nose forms a T shape). This is also especially good for use on dry, cracked heels.
NATURAL EXFOLIANT 2
This sugar exfoliant, a mix of sugar and olive oil, is good for particularly hardened areas, like knees and elbows. Mix equal parts olive oil and sugar into a thick paste, rub it into the hardened areas, leave on for about seven minutes, then wash and dry. Moisturizing after will give you nice smooth skin.
OLIVE OIL BATH
For all over body smoothness, add a few drops of extra-virgin olive oil to your warm bath.
MAKEUP REMOVER
Taking off makeup is less fun than putting it on. If you don't remove all the foundation, mascara, and eyeliner, it clogs up your skin and causes smudges around your eyes. Olive oil, similar in structure to the oils used in makeup, removes all makeup without damaging the skin; it will even get rid of waterproof mascara. Simply dab a little olive oil on a cotton ball and apply over your face and eyes. Rub gently until all the makeup is removed. Rinsing with lukewarm water will remove any residue.
FOR SHADOWS AND DARK RINGS
We sometimes refer to these as bags under the eyes; stress, hereditary factors, and lack of sleep can cause dark circles under your eyes. You'll come across home remedies such as cucumber and tea bags, but did you know you can use a mixture of olive oil and lemon as well?
Mix 2 tablespoons of olive oil with the juice of half a lemon. Using a cotton ball, apply on the eyes, press lightly on the dark circles for about five minutes, then rinse. Lemon is known for its skin-lightening properties, while olive oil hydrates the skin.
MASSAGE OIL
Nothing beats a massage to help you unwind and feel relaxed. Here's massage oil you can use at home. Start by putting 200 millileters (7 ounces) of olive oil in your massage bowl; add the juice of 1 whole lemon and 8 drops or so of lavender oil, the essential oil recognized for its calming and relaxing properties. It is ready to use.
Olive oil is especially beneficial during the winter months when our bodies tend to get really dry. Just remember, though, that it's not the best oil to use on your face as a moisturizer unless you have particularly dry skin.
FACE MASK
We know how important it is to cleanse the face every evening; we do it to remove all the grime and dirt from the day as well as makeup. Every so often, though, we need to go deeper to pull out those impurities that are lurking beneath the top layer of the skin. Using a face mask once a week will do this. It will also unclog our pores and get rid of any pimples caused by bacteria. This healthy, nutritious homemade mask does not take long to prepare; your skin will feel rejuvenated after the treatment. And you get to relax for fifteen minutes while it does its magic.
Mash half an avocado with some olive oil and make into a paste. Smooth it onto your face and leave for about fifteen minutes or so, then rinse.
OLIVE OIL HAIR MASK
Today, the use of too many hair dyes and strong conditioners can cause hair loss. Olive oil has always been used as a natural, effective treatment for hair rejuvenation. Extra-virgin olive oil can be used for the growth, regrowth, and conditioning of hair.
When olive oil or an olive oil product is rubbed into the scalp, the stimulation together with the nutrients from extra-virgin olive oil acts on the hair follicle, penetrating all the way into the hair shaft. As well as providing health supplements, olive oil acts as a barrier to retain the moisture and protect the scalp from the damaging effects of the sun. As a conditioner, olive oil beats any expensive salon treatment.
Heat 3/4 cup of extra-virgin olive oil in a pan or warm in a microwave for a few minutes. Rub into the scalp, continuing right to the ends. Put a towel in the microwave for 2 minutes until it is very warm but not too hot. Wrap around your head for at least 30 minutes to allow the warm oil to penetrate. Wash your hair as normal, using a gentle shampoo.
A FEW MORE TRICKS
To energize a tired face: Mix some olive oil with a few drops of lavender oil and rub onto your face. Cover with a hot flannel and relax for a few minutes.
To help with wrinkles: Before going to sleep, massage your face with a mixture of oil and lemon juice a couple of times a week.
To conquer acne: A mixture of olive oil and salt can improve acne if used daily for one week and then two or three times a week after. Mix 5 tablespoons of salt with 4 tablespoons of olive oil, rub the paste onto your face, then rinse off with warm, soapy water.
To avoid mosquito bites on vacation: Buy some local olive oil, mix with lavender oil and some citronella, and rub on exposed skin.
To cure an earache and ease irritation of the outer and inner ear: olive oil soothes the ear, opens up the eustachian tubes, and reduces infection. Heat a little olive oil in the microwave for no more than 30 seconds. Apply to the affected ear using an ear dropper and cover with a cotton ball. Repeat if necessary, making sure the oil is not too hot.
To enhance nail care: Rub a few drops of olive oil into the cuticle area and around the nails for moist cuticles and shiny nails. If you soak your nails once a week in a bowl of lukewarm olive oil for about 10 minutes, they'll become stronger.
To even a false tan: For a safe, smooth tan, mix equal parts of olive oil with your favorite tanning lotion and apply evenly. Adding olive oil allows the lotion to go on more smoothly and will give you additional moisturizing protection.
To prevent stretch marks: To avoid stretch marks during pregnancy, apply olive oil two or three times a day and before going to bed.
To soften scars: Apply olive oil to the scarred areas twice a day; they will disappear in no time.
To ease diaper rash: Mix 3 teaspoons of olive oil with 1 1/2 teaspoons of water. Wiping baby's bottom with this helps with rashes.
To prevent fibromyalgia: Fibromyalgia is an arthritis-related condition which causes considerable pain in the muscles and joints. To lessen aches, stiffness, and pain, massage warm olive oil into the affected muscles and joints.
To ease digestive troubles: Your digestion can be put right with an olive-oil-and-honey combination. Take 1 tablespoon of honey mixed with 1 tablespoon of olive oil in 8 ounces of hot water a couple of times a day.
For an at-home foot lift: Rub olive oil on your feet before going to bed and then cover with a pair of socks. You'll have nice soft feet in the morning.
To protect your stomach and liver: If you're going out and planning to have a few drinks, have a tablespoon of olive oil on a slice of bread before you go. Besides protecting the stomach and liver, olive oil helps delay the passage of alcohol into the bloodstream.
To substitute for shaving cream: Reach for a bottle of olive oil instead of a bar of soap. Olive oil will glide over your face and legs, moisturizing your skin at the same time.
To heal slight sunburn: If you've been out too much in the sun, you might feel tenderness with mild sunburn. Mix equal parts of olive oil with water in a lidded container, shake well, and apply on the affected areas of the sunburn.
WONDER 4
OLIVE OIL KEEPS BONES HEALTHY
This is the story of a fifty-seven-year-old, seemingly healthy, retired home economics teacher who suffered a fracture of her spine. It began when osteoporosis caught Susan Smith completely by surprise.
"I fell over while on a walk on frosty grass one day, landed on my back, and expected to suffer somewhat, but I was fine. Four days later, I was out walking again when I tripped on a tree root and fell. The pain started down my back that evening, extreme pain. The doctor didn't seem too worried and put me on painkillers, but I started getting pins and needles in my leg, so I had to go back to the clinic. This time, my doctor ordered X-rays, which showed I had a wedge compression fracture in my spine. Further tests with a DEXA scan confirmed that I had osteoporosis in my spine and osteopenia in my hips. I was in shock! Why did this happen? How did it happen? And why me?"
Susan had to accept that, like many women her age, her bones had become weak and that the osteopenia (low bone density) in her hips was a clear warning that that too could develop into osteoporosis.
The doctor explained why the accident happened.
As her bones aged, it caused the vertebrae of the spine to become weaker; there was too much pressure on the weak spine. As she was five years into menopause, he said, she also had low levels of estrogen, the hormone which regulates the bone cells. These are called osteoclasts and are the cells responsible for building new bones. In short, the vicious cycle of postmenopausal osteoporosis provoking bone loss had started. And it happens to many women as they age—their bone turnover is much faster than men, causing them to suffer more frequently than men from osteoporosis.
Susan lives in the UK, where the incidence of osteoporosis is high. The British Orthopedic Association reports that over 300,000 patients go to hospitals each year with fragility fractures, the type of break you get with osteoporosis. They say most of it relates to hip fractures, resulting in medical and social care costs amounting to £2 billion. The figures are high worldwide. According to data from the World Health Organization (WHO), in 1990, approximately 1.7 million fractures of the femur were recorded in the world and 6.3 million are expected for 2050.
Susan said that after getting over the shock, she started her research, looking closely at what caused osteoporosis, including the importance of getting enough calcium, considered critical for healthy, strong bones. "Now I see that many factors played a part. And some of them I could do nothing about! Above all, I am a woman, I am in my fifties; I can't do anything about my genes, being white, European, with early menopause and a dairy intolerance, and not having had children! Not to mention our modern diet." To her, it seemed like every factor played a part.
But osteoporosis is not just a woman's disease. Statistics show that one in five men over the age of fifty will break a bone mainly due to osteoporosis.
However, statistics also show that people living in the Mediterranean basin following a Mediterranean diet suffered far less from the disease.
As we get old, our bones become weaker, and sadly, there's nothing we can do to stop it from happening. What we can do, though, is slow down the effects of this silent disease by taking in the right nutrients and staying active. Research shows that the antioxidants contained in olive oil have significant positive health benefits on our bones, one way to stay in control and keep osteoporosis at bay.
When we are young, our bone density increases, old bone cells break down, and new cells are constantly being formed. They build strength in our skeletal system. Things change as we get older, however. The turnover does not function as well, more demolition takes place, and we lose our bone strength and become accident prone with greater risks of fractures.
This is the dreaded disease osteoporosis. The disease creeps up on you unexpectedly, often becoming apparent in a dramatic way—usually after a minor injury after a fall. The problem with this disease is that it progresses silently without any signs until we strain ourselves or fall. We develop osteoporosis because, quite simply, our bones have become porous: they no longer have as much strength as they did when we were younger.
A large-scale study in 2013 illustrates this well. The study involving 188,795 people in eight European countries showed that people living in the Mediterranean who stuck to a Mediterranean diet had a lower risk of hip fracture. The bone health of participants—48,414 men and 139,981 women with an average age of forty-six—were followed for over nine years. The researchers in charge of the study assessed the participants' diets and measured how they adhered to the Mediterranean diet. Evaluation at the end of this large study showed only 802 hip fractures for those who adhered to a Mediterranean diet.
According to the study's findings, a Mediterranean eating pattern with high fruit, vegetable, and olive oil intake could be beneficial to improving bone health. But what part does olive oil play?
Scientists in Egypt also wanted to evaluate the protective effects of olive oil supplementation against osteoporosis. To investigate this, they set up comparable situations to female menopause; they wanted to see if olive oil supplements could protect against menopause-induced osteoporosis. In other words, could olive oil prevent bone loss when subjected to similar conditions?
The research was carried out on three groups of rats aged between twelve and fourteen months.
Group 1: These rats were surgically treated to remove their ovaries.
Group 2: These rats also received the same surgery. They were given olive oil for a total of twelve weeks: for four weeks before their ovaries were removed and then for eight weeks after the operation.
Group 3: These rats in the control group did not undergo any operations—they did not have their ovaries removed to induce menopause.
At the end of the experiment, the researchers examined the blood, bones, and livers of all the rats. This is what they found:
Rats that did not receive olive oil showed a decrease in calcium levels.
Rats that did not receive olive oil showed an increase in plasma.
On examination of the blood, the researchers found a lower level of nitrates in Group 2.
More importantly, microscopic examination of the bones of the olive oil–fed rats showed improvement in bone thickness and the ability to replace bone. On comparing the two groups, the researchers said there was a marked improvement in Group 2. They observed that the supplement prevented a decrease in cortical bone thickness as well as the trabecular thickness.
The researchers concluded that olive oil looked like a promising candidate for the treatment of postmenopausal osteoporosis.
Going deeper, we may ask: what components in olive oil were responsible for this sort of result, and can it work on humans?
Researchers at France's National Institute for Agricultural Research (INRA) found the answer. Ranked as the number-one agricultural unit in France, INRA carries out research for high-quality and healthy goods in eighteen centers in France. One of their studies evaluated oleuropein, one of the main components of olive oil.
After their two successful animal trials, the research team at INRA went on to human trials. The aim of this twelve-month study was to measure the effect of taking an olive extract on bone turnover in postmenopausal women with osteopenia, when bone mass had already decreased.
Researchers divided the sixty-four participants who enrolled into two groups. The women, aged between forty-nine and sixty-eight, were all at least twenty-four months past their last menses and all tested, measured, and examined for a number of factors such as weight and cholesterol levels. More relevant, they all had bone mineral density scans at the beginning at the base of the lumbar spine and the neck of the femur using DEXA scans, which were redone and compared at the end of the study.
They were randomly allocated equally between the treatment group and the placebo group. This was a double-blind study where neither the patient nor the investigators were aware of the allocation of the two groups. The test product was an olive extract made up of polyphenols derived from the olive leaf. The researchers were interested in two things: first, they wanted to evaluate the effect of taking an olive oil extract on bone turnover in menopausal women and second, they wanted to study the bone mineral changes and blood lipid profiles.
The study group was given 250 milligrams of olive extract a day (the equivalent of a dose of 100 milligrams of oleuropein per day) as well as 1000 milligrams of calcium. The placebo received 1000 milligrams of calcium with no olive extract.
What they noticed during the study period was an increase in osteocalcin levels in the treatment group. This is what their results showed:
On comparison of the two groups, they found that the levels of osteocalcin increased significantly with the study group as compared with the placebo group. They also found that the bone mineral density decreased in the placebo group but remained stable in the treatment group. Significant, also, was that the lipid levels of the treatment group improved; the total cholesterol, LDL, and triglycerides all decreased.
The researchers concluded from these results that "olive phenolic compounds may have a promising biological activity towards the maintenance of a balanced bone turnover process and blood lipid profile." They said that although this was a limited-scale study, further and larger human-intervention studies could now be set up.
Women and men who suffer from osteoporosis have a fear of something that the rest of us take for granted. They have a fear of crowds—anything from a football match to a crowded theater. Susan, who still battles with osteoporosis, said, "I am afraid to go where there's a crowd. Someone might knock into me and cause my shoulder to break. I am scared of going out in autumn when the leaves are soggy and in winter with ice and snow. Living with osteoporosis can be very hard."
But the results from both studies are encouraging, particularly for women everywhere. There's a factor for osteoporosis prevention and management that we can control just by including olive oil and olive extract in our diet!
WONDER 5
OLIVE OIL SLOWS ALZHEIMER'S DISEASE
Our fascinating, complex brain, unlike any other part of the body, has a tight security blood system, a sort of barrier controlling all substances going in and transporting certain molecules out. This blood-brain barrier is like a security fence ensuring proper functioning of the brain by allowing selected molecules and vital elements in; oxygen and essential nutrients can cross over easily. A no-entry mechanism, however, operates for toxins, bacteria, and certain drugs. In a healthy brain, this semipermeable barrier protects the brain, but with Alzheimer's, the circulatory system breaks down, allowing more harmful particles to get inside the brain. And once those harmful particles are in, our healthy brain cells or neurons start producing an abundance of two abnormal structures—two culprits that start disruption.
Imagine losing your memory of your friends and your loved ones and not fully understanding what's going on around you. These are the signs and symptoms of Alzheimer's disease, a disease that clogs the brain, reducing the ability to think, remember, and reason. Drugs can now stabilize the condition, but sadly, once the brain cells are damaged, there's no reversing the damage with this form of dementia. The good news, though, is that researchers have recently discovered that extra-virgin olive oil can help protect the brain from the condition.
Most people associate Alzheimer's disease with old age, but doctors say that even though sufferers may seem normal, damage to the brain can start some twenty years or so before the first symptoms appear. This is a progressive disease, because it takes place gradually over time, contaminating the brain, and as it develops, more symptoms appear, starting mildly and getting worse with time.
The gradual increase in the two abnormal structures accounts for the steady deterioration. The main buildup is caused by clusters of protein fragments called "beta-amyloid" that form outside the neurons. Scientists say these are chemically sticky and so clump up easily to form plaques. The plaques are full of clusters of abnormal tissue that destroy the contact points between the neurons, impeding their communication.
A second unhealthy accumulation also takes place, this time inside the neurons. Twisted fibers of a protein called "tau" develop inside the nerve cells, forming a thick mass known as "neurofibrillary tangles." A healthy brain depends on the correct functioning of the protein tau, but in an unhealthy brain, the tau protein is abnormal and damages the affected brain cells.
These masses of dead and dying cells disrupt the internal transport system of the brain, preventing adequate transportation of nutrients and other essential supplies. With this instability comes a chain of undesirable reactions, from shrinking of the brain to memory loss and lack of communication and reasoning.
Much of the mystery of Alzheimer's has been unraveled since it was first described in 1906. And while the disease affects around thirty million people worldwide, remarkably, people from the Mediterranean countries suffer much less.
Research Studies with Oleocanthal
Research scientist Amal Kaddoumi was raised in Jordan, where the olives are tasty and plentiful, and olive oil is a way of life. She learned of oleocanthal's protective qualities as they were discovered in the scientific world and wanted to delve further to understand why and how the component oleocanthal in olive oil reacted with Alzheimer's disease. Using the brains and cultured brain cells of mice, Dr. Kaddoumi and her team at the College of Pharmacy at the University of Louisiana at Monroe first set out to find out whether the oleocanthal in olive oil was able to reduce the buildup of beta-amyloid, the development of plaques outside the neurons.
Here's how Dr. Kaddoumi explained their findings:
Oleocanthal reduces the accumulation of beta-amyloid in the blood-brain barrier by boosting production of proteins and vital enzymes that are critical for the removal of beta-amyloid. Also, it has an anti-inflammatory effect. However, it's important to use extra-virgin olive oil, as it contains poly-phenols, among which is oleocanthal. Refined olive oil lacks these polyphenols and may not show the same effect observed with extra-virgin olive oil.
Encouraged by these results, the researchers then concentrated on a second question: could olive oil then protect us from Alzheimer's?
The second aspect of the study used two approaches: for the first group, the scientists fed mice with extra-virgin olive oil at an age before the amyloid buildup started inside the brain, while the second control group was put on an enriched extra-virgin olive oil diet after the buildup. The results showed that several mechanisms that caused the disease were reduced; most noticeable was a stronger blood-brain barrier and a considerable reduction in the amyloid levels of the first group.
These findings show a definite impact of dietary factors in health, and as Dr. Kaddoumi says, "What we see in animals, we could see in humans." She feels strongly that we can avoid Alzheimer's through healthy eating; a preventive means, she says, would be incorporating extra-virgin olive oil into our daily diet.
Oozing with enthusiasm, she said to us, "These results are exciting. They indicate that oleocanthal and EVOO prevent, or at least delay, the onset of the disease. I recommend people add extra-virgin olive oil, not refined olive oil, to their diet. Oleocanthal is not commercially available, but it is present in extra-virgin olive oil."
Dr. Kaddoumi feels while we wait for more medical evidence, we should include extra-virgin olive oil in our diets as early as possible and that a dose of 50 milliliters (about 3 tablespoons) each day would be a good start. This is the amount they used for their research and what the Greeks would consume daily.
Encouraged by these findings, the team would like to see more studies on humans but realizes the difficulties involved. The team leader explained, "These kinds of studies are costly and require funds. Also, they need to be well designed and controlled to evaluate such effect. We are indeed considering clinical studies, and we are preparing for them. Epidemiological studies also exist; while they don't talk about the mechanism, they showed a positive effect of extra-virgin olive oil on subjects' cognitive function."
Dr. Kaddoumi said she'd like to see more studies and eventual results using a combination of olive oil and other drugs, which she thinks could enhance the effect of the oleocanthal.
Experts will one day hopefully have a deepened understanding of this neurological disorder, which will allow us to nip the disease in the bud. This evidence that extra-virgin olive oil can help combat memory loss and mental decline is positive and encouraging. As life expectancy continues to increase, just think how much money could be saved worldwide every year by families and governments by combating this debilitating disease. The Alzheimer's Association in the United States reports that the illness and other dementias cost the nation $226 billion.
As Dr. Kaddoumi says, "Prevention is the key to protection against Alzheimer's, and the way to do it is to maintain a healthy lifestyle and a healthy, balanced diet. Olive oil is a safe food product accessible to everyone."
Feeding the Brain with the Right Stuff
Numerous studies have investigated the link between the Mediterranean diet and the brain. Briefly, we want to share two studies.
Study 1: We Can Improve Our Memory with Olive Oil and Nuts
A small randomized test conducted in Spain showed that adding a bit more olive oil to a Mediterranean diet might decrease our risk of developing Alzheimer's. Researchers chose older subjects for this study, suggesting that it's never too late to switch to a healthier diet. They selected 447 volunteers between fifty-five and eighty years old, slightly overweight, who had no memory problems but were at a risk of cardiovascular disease. The seniors were assigned to one of three groups: a Mediterranean diet with one liter of extra-virgin olive oil added every week, a Mediterranean diet with 30 grams of mixed nuts added per day, or a low-fat diet. The groups all followed the different diets for four years. All of the members had a total of six cognitive function tests, a series of memory and thinking tests to check their mental state and cognitive changes, before, during, and after the research.
Here are the results as reported by Reuters, the international news agency:
Based on the brain function tests done before and after the study, the group eating low-fat foods had a significant decrease in memory and cognitive function. The group following a Mediterranean diet with supplemental nuts had significant improvements in memory, while the group adding extra-virgin olive oil experienced significantly better cognitive function.
This test might be only a small trial, but it does show that monounsaturated fatty acids such as those found in olive oil and nuts present a holistic approach, especially if we want to slow down shrinkage of our brain, prevent the development of plaques, and maintain our cognitive powers. The researchers concluded in their report published in July 2015 that in an older population, a Mediterranean diet supplemented with olive oil or nuts is associated with improved cognitive function.
Study 2: A Look at the Brain Size of Older Seniors
When scientists looked into the eating habits and scanned the brain cells of some older seniors, they found an enormous difference in volume and size between participants.
The aim of the study, as reported in October 2015, was to examine the size of the brain in people who followed a Mediterranean diet "to develop an association." The study involved two groups of people and took place at the same time. The participants who took part had an average age of eighty; some followed a diet aligned more closely with a Mediterranean diet, while others did not.
The 684 older people who came from multiethnic groups in northern Manhattan first answered in-depth questions about their diet and then underwent MRI brain scans to measure the volume and thickness of their brains.
What did the scan results show? Those who followed a more Mediterranean-like diet had a larger brain volume than those who did not. The difference in average brain size was 3.11 milliliters, or the equivalent of five years of aging, the authors reported.
Forgetting where you left the car keys can be considered a mild cognitive behavior, but it could deteriorate into full-blown Alzheimer's with time. As we don't yet know how to target gungy amyloid or the tangles of tau protein—the destructive, deadly duo that comes with Alzheimer's—we can adopt preventive measures to keep them away. The take-home message seems simple: we should incorporate good-quality extra-virgin olive oil to protect ourselves from neurological disorders such as Alzheimer's.
4 Good Reasons for Switching to the Traditional Mediterranean Diet
The Mediterranean diet is the easiest, simplest, most practical diet around, yet in our search for losing weight, we turn to other food plans—the Paleo, the Dukan, and the Atkins diets . . . . The list goes on. However, it is overindulgence of the wrong foods and inactivity that creates the need to lose weight. Wouldn't it be better to incorporate a healthy lifestyle with simple eating—good eating habits that are easy to follow, nutritious, and healthy?
There's no real definition of the Mediterranean diet, no single model. It is more a lifestyle based on a southern European approach that includes food from livestock, fishing, and agriculture. Instead of an excess of meat, the people of the South consume lean sources of protein, drink red wine in moderation, and flavor their dishes with herbs and spices.
The Mediterranean Diet Is Neglected These Days
Sadly, the healthy eating styles and traditions of the Mediterranean diet are being neglected; fast food is gaining popularity, and with that comes the obvious—we are getting fatter. We are all in a hurry these days, not taking the time to choose what we eat, and we're losing the habit of sitting down and enjoying mealtimes.
Those of us who live in the countryside or next to the sea are lucky. It isn't too difficult to follow the Mediterranean diet. The choice of fruit and vegetables is endless in our traditional, open-air markets. Going to the market on Sunday and stocking up on fruit and vegetables is still, thankfully, an essential part of French life. But here's why we should take a second look at the Mediterranean diet:
MEDITERRANEAN DIET
1. NO FOOD RESTRICTIONS
The great thing about the Mediterranean food plan is that there are no strict laws on what we should not eat. Instead, the diet focuses on the variety of fresh vegetables and fruit and encourages us to experiment with locally sourced products. Snacking is not advisable unless, of course, you satisfy those hunger pangs with fresh fruit or nuts. Instead of chips and artificial dips, go for fresh fruit and yogurt.
2. HEALTHY
Originally, this was known as the "Poor Man's Diet" because it did not include much meat—only a bit of fish—and lots of seasonal fruit and vegetables. However, this is the diet that has been proven to reduce heart disease as well as number of medical conditions brought on by the Western lifestyle. This eating plan has been thoroughly researched over the years with scientifically backed evidence showing that you can achieve weight loss and cut back on your risks of certain diseases.
The Mediterranean diet is rich in vitamins, minerals, and fiber and relatively low in saturated fat. It consists mainly of plants, vegetables, fruits, cheese, cereals, and olive oil—the main source of fat. To that, some choose to add one or two glasses of red wine a day (excessive alcohol use can lead to increased risk of health problems), fish, and white meat in moderation and small quantities.
A weekly shopping list would include everything that's healthy, tasty, nonprocessed, and fresh:
Olive oil is the main fat in Mediterranean cuisine: we cook with it, we have it in salads, and some even use it on bread instead of butter.
Lots of fruit and vegetables low in fat and high in fiber; popular choices here are zucchini, eggplant, avocados, tomatoes, peppers, and juicy melons.
Fish packed with protein, vitamins, and minerals.
Nuts, not to be consumed every day, but two or three times a week.
3. LESS EXPENSIVE
By buying local and fresh, you will not be paying too much for food—especially with a list and a meal plan for the week. Choosing the right food automatically cuts out those costly junk food items. You might be tempted into thinking that it would cost more to go out and buy fresh, but look at what you can cut back on in your current spending; cutting back on those extra soft drinks and cookies is a good way to start. And what about including all the family in the experiment? Planning the meal together could turn out to be a fun activity for the whole family.
4. BALANCED LIFESTYLE
The Mediterranean diet is not only about food; it is a lifestyle in which families take the time to eat together—not in front of the television but around the table. It is this sharing which creates a deep appreciation of food.
Considering all the scientific evidence out there as to all of the health benefits the Mediterranean diet has to offer, it makes sense to make the switch. Forget about fad diets; study the Mediterranean diet pyramid, a doable, easy, and versatile plan. Redefining your lifestyle in this way should make you see food as a social interaction with a focus on healthy choices.
It's simple, fresh, wholesome, and, above all, healthy!
WONDER 6
OLIVE OIL HELPS REDUCE THE RISK OF DIABETES
Arenowned British diabetes specialist noticed that during times of war—when food shortages lead to the removal of white flour, white sugar, and excessive meat protein and fats from the typical British diet—the death rate from diabetes decreased by 50%. If the scientist were still alive, he would be surprised at the healthy eating trend today, the number of food blogs advocating clean, healthy eating, all challenging the way we choose and prepare our meals. And yet diabetes is on the increase. Type 2 diabetes can be managed successfully, however; we need only to choose the right food and exercise more.
We have become much more sedentary these days, spending more time in front of the television. We sit at work. We rely solely on the car to take us places. Many of these factors make us obese, and obesity is one of the reasons we suffer from diabetes. Obesity becomes even more of a problem if we have difficulty controlling our blood sugar levels. Type 2 diabetes has become a worldwide lifestyle disease, which starts when our bodies do not produce enough insulin. It also happens when the cells of our bodies become insensitive to insulin, a condition called "insulin resistance."
Insulin is a hormone produced by the beta cells of the pancreas, the key which unlocks the cells in our bodies to allow glucose in. Insulin is produced and released when the body needs it; if insulin cannot penetrate into all of the cells in our bodies, then the glucose (blood sugar) gets too high. In type 1 diabetes, the pancreas cannot make insulin, the glucose stays in the blood, and sufferers have to take insulin injections, whereas in type 2, the pancreas produces insulin but it cannot reach the cells.
What's vital to understand is that insulin controls the amount of blood sugar gaining access to our cells; it instructs the body's cells to absorb blood sugar for energy or storage. With type 2 diabetes, our blood glucose levels are high and our insulin sensitivity is decreased. This means that the excess sugar stays in the bloodstream, waiting for insulin activity to kick in so it can get to the cells to provide the energy we so need.
Also important to know is that diabetes has an influence on our good and bad cholesterol. It tends to lower our good cholesterol and raise our bad cholesterol. This combination is not good, because when it happens, we risk having a heart attack or stroke. If left unmanaged, diabetes can also lead to increased susceptibility to cuts and bruises, skin, kidney, and bladder infections, and problems with vision.
But why are we talking about diabetes and cholesterol in a book about olive oil? Because extra-virgin olive oil lowers blood glucose and cholesterol!
A small study was carried out at Sapienza University in Rome, involving only twenty-five participants, all without diabetes. The study looked at the influence diet had on blood sugar in a healthy metabolism. Each of the participants partook of a typical Mediterranean lunch consisting of fish, grains, vegetables, and fruit. They had this meal on two separate occasions; the first time, they added 10 grams (roughly 2 teaspoons) of extra-virgin olive oil to their meal, and the second time, they added 10 grams of corn oil.
They all had their glucose levels measured two hours before and after each meal. The results showed that the glucose levels were much lower after the extra-virgin olive oil meal. The results also showed reduced levels of LDL—the bad cholesterol—with the olive oil meal. On the other hand, the research team found that after the meals with corn oil, the participants had significantly higher LDL levels.
"Lowering blood glucose and cholesterol may be useful to reduce the adverse effects of glucose and cholesterol on the cardiovascular system," said the main researcher. He did stress, however, that he believed it was better to have the olive oil with a meal rather than on its own.
Perhaps, then, if we can lower our level of glucose and LDL, in conjunction with other healthy lifestyle choices, we can prevent the onset of diabetes.
In fact, Spanish scientists from the Institute of Fat in Seville say you can reduce your risk of type 2 diabetes by consuming olive oil. They based their studies on the polyphenol oleic acid in olive oil.
Sergio Lopez, the main researcher, explained that with diabetes, we need to consider the importance of postprandial insulin sensitivity, which he described as a measure of how effectively the body's cells use insulin to reduce elevated blood sugar levels after the ingestion of carbohydrate-containing foods. (The word postprandial means "after a meal.") He said that "appropriate postprandial β-cell function and insulin sensitivity are required to sustain good health." The amount and type of dietary fat are relevant to postprandial response. The researchers said that "oleic acid in olive oil aids to minimize spikes in blood sugar levels after meals by optimizing pancreas production of insulin and improving whole-body sugar utilization."
The researcher also explained that, unlike other dietary fatty acids, such as palmitic acid that you get from some meat and dairy products, oleic acid in olive oil may help to regulate insulin secretion and action in the postprandial state. Monounsaturated fatty acids, then, should be our best choice for the immediate effect of lowering glucose and improving insulin sensitivity in the postprandial state.
According to the World Health Organization (WHO), 90% of people with diabetes around the world have type 2 diabetes. The organization also says that the burden of diabetes can be reduced through prevention by adopting simple lifestyle measures, but not everyone is aware of how lifestyle changes can make a difference. Janet Amos faces that problem in her work as a diabetes specialist nurse.
Amos is based in an inner-city suburb of Manchester in the north of England, an increasingly multicultural population, predominantly from Pakistan and Bangladesh. She says, "My work is more challenging now than ever with an increasing amount of patients with diabetes and prediabetes." At this busy clinic, she sees, on average, between twenty and thirty diabetics each week. Janet says they are very overweight with poor diets and inactive lifestyles.
She explained that patients might come to see her because of a wound that's not healing or an infection that won't go away. They do not understand that the problem is really about shedding their weight, picking the right food, and exercising. What's important and crucial for their health is for her to explain that they are experiencing the effects of type 2 diabetes.
"It is all very challenging," Janet says, "for a health professional trying to get people to change." But it can be rewarding when you see the little changes, like getting them to check their blood regularly and following her advice on food and nutrition. "My greatest reward," she says, "is when I see them controlling their dietary intake and exercising."
Unfortunately, getting her patients to convert to olive oil is often out of the question: the women choose the cheapest oil for cooking, and olive oil is much too expensive for their circumstances.
Growing awareness of the the health benefits of the Mediterranean Diet has led to a rise in olive oil consumption in the United States, but like the UK and rest of the world, diabetes is still a huge problem in the country. We want to share this: the first study conducted outside of the Mediterranean basin and a project to see if higher doses of olive oil would lead to a lower risk of diabetes in the United States.
In the United States, researches from Harvard Medical School and Brigham Young University, working with two Spanish universities, conducted two large cohort studies of 59,930 nurses—aged thirty-seven to sixty-five years from the Nurses' Health Study Group (NHS)—and 85,157 nurses—aged twenty-six to forty-five years from the NHS II group. This was one of the largest studies ever, lasting twenty-two years.
Food questionnaires were sent out every four years. Total olive oil consumption was calculated as a combination of salad dressing olive oil and olive oil added to food or bread. After those twenty-two years, a total of 5,738 cases of diabetes were noted in the NHS group and 3,914 cases in NHS II group.
These are some of the highlights of the study:
Total olive oil intake was associated with a lower risk of developing type 2 diabetes for participants consuming more than one tablespoon of olive oil per day compared to those who never or almost never consumed olive oil.
Every additional tablespoon of olive oil was associated with a 6% reduction in type 2 diabetes.
Replacing one tablespoon of margarine, butter, or mayonnaise with one tablespoon of olive oil was associated with a predicted lower risk of type 2 diabetes. There was a 5% lower risk when replacing margarine with olive oil, an 8% lower risk when replacing butter with olive oil, and a 15% lower risk when replacing mayonnaise with olive oil.
In their report published in August 2015 in the American Journal of Clinical Nutrition, the authors concluded that "increased olive oil consumption was associated with a decreased risk of type 2 diabetes."
Olive Leaf and Diabetes
Olive leaf extracts are making the news these days. One of the latest, most interesting studies showed the two main polyphenols in olive oil: oleuropein and hydroxytyrosol. When taken as supplements, these polyphenols can help maintain proper insulin levels. In a twelve-week study that took place in New Zealand, the results showed the effect of these two antioxidants, demonstrating that the duo can improve insulin sensitivity, an exciting promise for those with diabetic risks.
The researchers at Liggins Institute chose forty-six men for the study, aged between forty-one and fifty-one, who were not diabetic but were overweight. Divided into two groups, one group took four capsules of olive leaf extract containing 51.1 milligrams of oleuro-pein and 9.7 milligrams of hydroxytyrosol daily as a single dose. The other group received the placebo, or a false supplement. Both groups were given capsules, so the identity of what they were taking was completely masked.
After a six-week washout period where both groups did not take any capsules, the researchers switched groups so that those taking the placebo went on the olive leaf supplement while the others also switched to the opposite treatment.
This is what the results showed:
The olive leaf extract improved the insulin action by 15%.
The olive leaf extract increased insulin secretion by 28%.
The study concluded that supplementation with olive leaf phenols for twelve weeks significantly improved insulin sensitivity and pancreatic beta cell capacity in middle-aged men, particularly those at risk of developing the group of risk factors, such as elevated blood pressure, which can lead to diabetes.
What's different about olive leaf extract? According to the researchers, olive leaf extract not only increases insulin production but decreases insulin resistance. They reported that olive leaf extract "is a natural product that has an appreciable benefit in improving insulin action in adult overweight males. It worked as well as metformin, the most commonly used diabetes medication in overweight males."
Dr. Martin de Bock, the first author of the paper who conducted the study, answered our question about how oleuropein and hydroxytyrosol—elements in olive leaf extract—can be used as supplements for diabetes:
There is continuing growth in the nutraceutical industry. It is known that in type 2 diabetes, patients are often more inclined to try alternative medical therapy rather than traditional pharmaceuticals. Therefore, while the incidence of type 2 diabetes continues to increase globally, there is likely to be continued demand for nutraceuticals such as oleuropein and hydroxytyrosol.
Genetics might contribute to type 2 diabetes, but lifestyle, exercise, and diet are things we can change. And we do have that choice today.
Bad Butter
Butter is bad for our health, according to a study funded by the butter industry itself. Amazing but true, this study backfired, showing butter is not as healthy as olive oil.
It is not common that a study commissioned by an industry goes against what they are hoping to promote. Generally, a study such as this one is an excellent marketing tool to improve the image of a product. The results of a study of butter, however, did not give it a good name. The Danish Research Foundation for the dairy operation funded a study on butter in the hope of promoting the image of this lipid.
Instead, it showed that even a moderate intake of butter increases the total cholesterol and LDL in the blood.
The results suggested that hypercholesterolemic individuals— that is, those who have high cholesterol—should keep their butter consumption to a minimum, while the rest of the population could have it as usual, in small quantities, in moderation, raw; in other words, a little butter on toast in the morning is not dangerous.
The study results, published in July 2015 in the American Journal of Clinical Nutrition, found that even when consumed in moderation, butter leads to higher cholesterol and LDL in the blood, much more than consumption of alternatives such as olive oil.
What We Should Know about Cholesterol
This waxy substance we sometimes call fat comes from our body. It is made by the liver and circulated through the blood via proteins known as lipoproteins. We also get cholesterol from consuming animal sources like meat and full-fat dairy products. With a diet high in saturated and trans fats, the liver has to produce more cholesterol.
There are two types of cholesterol: the bad cholesterol, LDL (low-density lipoprotein), and the good cholesterol, HDL (high-density lipoprotein). Research shows that when rates are normal and the balance is right, cholesterol protects our arteries by giving them flexibility and strength. Studies also show that HDL recovers the excess cholesterol and carries it back to the liver to be processed before it can be eliminated. LDL carries cholesterol from the liver to all cells. When there is a disruption, LDL cholesterols cause buildup on the walls of the arteries. The buildup accumulates and may contribute to the formation of plaques, which gradually clog the arteries.
Of all cholesterol in the blood, 60–80% is associated with LDL. An increase in bad cholesterol and a decrease in good cholesterol are risk factors which lead to plaque atheroma, causing cardiovascular disease, impotence, and even strokes.
The Composition of Butter
Butter is rich in vitamin A. It also contains vitamin D and vitamin E. With over 730 kilocalories per 100 grams, butter contains 63% saturated fatty acids, 26% monounsaturated fatty acids, and 3.7% polyunsaturated fatty acids. Butter has a low smoke point, which we will talk about in the next chapter, but what's important to know is that this low smoke point of about 177° C (350° F) means that whatever you're cooking will brown and burn quickly, which is bad for your health. All told, butter is probably best taken in moderation—and even better when we only use it raw and for light cooking.
WONDER 7
OLIVE OIL HELPS PREVENT STROKES AND HEART ATTACKS
The celebrities in their stunning red dresses parade elegantly down the runway. They were there to present the American Heart Association's "Go Red for Women" Red Dress collection, a colorful event which took place at the department store Macy's in February 2016. The gathering of top models, designers, and celebrities wanted to show their support and to help raise awareness of cardiovascular disease, an illness that affects 44 million women in the United States.
Wearing red for this illness is the obvious choice because red stands out and is the color associated with our hearts. Women suffer more than men from diseases of the heart, and the only way to bring down these high figures is by education. The American Heart Association asks women everywhere to wear red every first Friday in February; it is a ritual they say is beginning to bear fruit, as 90% of women have made at least one healthy behavior change. In the UK, the statistics are also high; the country observes National Heart Month in February as well. With more heart awareness taking place worldwide, women are now checking their cholesterol, increasing their exercise, and making lifestyle changes to prevent cardiac events from happening.
We'd all like our blood to be freely flowing through our bodies, but blood vessels can sometimes get obstructed, and when they do, they can wreak havoc. If blood flow to the brain is not progressing correctly, the body suffers a stroke—a brain attack that disrupts the abilities related to the affected part of the brain; it can come on suddenly, causing numbness, sudden confusion, and dizziness. Strokes occur when a clot (a fatty deposit) blocks the artery in the brain. Equally serious is a heart attack, which causes damage to the heart's muscles when vital oxygenated blood cannot get to those tissues. This, too, is mainly due to a blood clot, blocking the coronary artery supplying blood to the heart. We are familiar with blood clots; we can see the sticky blood cells appear whenever we injure ourselves and form a scab over the cut. That is how the body responds to injury. However, what we do not want are those clots in our blood vessels. The risk factors are the same for strokes and heart attacks: smoking, diabetes, high cholesterol, and high blood pressure, all of which provoke changes in our blood supply system.
In the last chapter, we addressed some of the factors of cholesterol as related to diabetes, but it is perhaps an even more significant risk when considering cardiovascular health. The process that causes the narrowing of the arteries is called "atherosclerosis," said to be caused mostly by the bad cholesterol LDL. All conditions involving the heart and circulation are called "cardiovascular diseases" (CVD), which is one of the leading causes of death in our world today.
You might not be surprised to know that there are fewer incidences of cardiovascular diseases in Mediterranean countries. The American Ancel Keys was the first to study diets, lifestyle, and the incidence of heart disease. On comparing lifestyle and food in seven countries, he found that people living in Greece and Italy who consumed decent amounts of fruit, vegetables, olive oil, grains, and fish were less prone to heart disease than those living in America and Finland, where they ate considerably higher amounts of saturated fats.
You may say the Greeks are noted for adding olive oil to everything and that olive oil is fat. You are right; olive oil is a fat, but you know by now that olive oil is a healthy fat. And we must remember, though, that although all types of olive oil are good sources of monounsaturated fats, extra-virgin olive oil offers higher levels of health protection because it is less processed.
Vitamin E, Olive Oil, and Heart Disease
Interesting in the fight against cardiovascular disease were the results of a study carried out on women in Naples, Italy. It showed that women who consumed more vitamin E were less likely to show signs of heart disease. The women taking part in the study did not take any additional supplements of vitamin E nor did they have any special dietary recommendations for the study. They simply followed their normal Mediterranean way of life; in other words, they continued having their usual regimen of legumes and vegetables, together with olive oil. On examination with ultrasound, these women showed little trace of thickening in the carotid arteries of the neck, an accepted medical marker for vitamin E in the body.
Three hundred and ten women, aged between thirty and sixty-nine years old, took part in the study. All the women had ultrasound tests to see whether they showed early signs of atherosclerosis in their carotid arteries. They had additional blood tests as part of the study to check their levels of vitamin E, other antioxidants, and cholesterol.
In their results, published in the American Journal of Clinical Nutrition, the researchers said that the "more vitamin E the women consumed, the less likely they were to show signs of atherosclerosis." They also found that additional vitamin E only helps those who start off with low amounts and that those with sufficient amounts of vitamin E would not benefit from increasing their amount. It also seems that the other antioxidants present in olive oil—vitamins A and C—did not have an effect on carotid thickening.
Why You Should Look at Vitamin E More Closely
We've mentioned vitamin E briefly in several sections because this little-known vitamin is so beneficial to the body. Why is it so important?
Vitamin E was first discovered in Berkeley, California, by two scientists who were exploring infertility in female rats. The scientists could not understand why the pups were dying in their mothers' wombs; something was obviously missing from the diet they were being fed. The scientists eventually discovered that a diet of lettuce and wheat germ resolved the problem; it seemed that a compound contained in both substances was the answer. With time, they were able to isolate vitamin E successfully.
Here are some of the most significant things to know about vitamin E:
Vitamin E contains eight different compounds that make up the vitamin E family; each of these is fat soluble and can be found in food substances. They are all similar except for a simple difference of molecular structure.
Olive oil contains alpha-tocopherol, the most common and the most efficient of the eight compounds.
Vitamin E is the best antioxidant for fighting free radicals. Our brain cells, in particular, are more susceptible to damage from free radicals; it is vital that they are protected.
The heart needs vitamin E. The vitamin acts on LDL, the bad cholesterol, limiting oxidation. LDL oxidation accelerates the building up of plaques within our arteries (see page 100).
Although the heart and the brain do need more protection, all of the cells of our bodies also need constant protection; the cell membranes and the tissues of the skin, eyes, breast, liver, and testes all benefit from the antioxidant properties of vitamin E.
Besides olive oil, you can get vitamin E from nuts, green vegetables, and vegetable oils.
According to the IOC, it is important to maintain correct levels of vitamin E during breastfeeding. Women need more vitamin E during pregnancy, as the vitamin is concentrated in the breast. When they breastfeed, mothers supply essential nutrients, vitamin E in particular, to the newborn baby.
Oxidative Stress and Antioxidants
If you leave a coin out in the open for some time, the rain, the sun, and other environmental factors will cause it to go green and rust, a process called oxidation. This interaction between oxygen molecules and outside substances is a normal process. A very similar process takes place in the cells of our bodies; the oxidative process can also cause destruction and havoc to our bodies.
Much of the damage is caused by free radicals. Often labeled as harmful products, they invade our immune system. They are called free radicals because these molecules roaming around our bodies are unstable; unlike other normal molecules, which exist in pairs, they lack one electron. In their search to become stable, they seek to react with other molecules looking for stability. Free radicals will continue to circulate in the system as long as the body continues to function and age. Scientists say not all free radicals are harmful, but when there are lots of free radicals in the body, it becomes dangerous for our health. Our bodies can only take on so many roaming radicals. When there is too much buildup, the body suffers.
What causes this buildup? Some of the most common causes are smoking, pollution, drugs, too much exposure to sunlight, radiation, and stress. The body suffers from oxidative stress when too many free radicals are produced, causing damage to the cells of our bodies by the oxidative process. Scientists say oxidative stress can cause a plethora of undesired human pathologies, such as Parkinson's disease, diabetes, rheumatoid arthritis, cancers, and heart diseases.
Thank goodness for antioxidants, man-made or natural substances that donate electrons that these free radicals need, such as vitamin E. Antioxidants fight and neutralize the reaction, allowing the body to function normally—in other words, by intervening, they stop the oxidation process by oxidizing themselves.
Although the body produces some antioxidants, with the increase in pollution and lifestyle demands, we need other, outside sources of antioxidants, healthy foods that will provide a healthy balance of oxidants and antioxidants. Olive oil contains some key polyphenols recognized as natural antioxidants, particularly vitamin E.
Research into Olive Oil and Heart Conditions
Today, there are numerous case-controlled studies of the Mediterranean diet showing the protective benefits of olive oil. Researchers are also continuously investigating the influence of olive oil on cardiovascular diseases; dedicated scientists from different parts of the world all use various methods, but with one aim—discovering how to improve our heart health.
Although anyone of any age can suffer a heart or brain attack, most sufferers are over sixty-five years of age. One of the first studies to look at the effect of olive oil and the risk of stroke took place in France. It lasted six years and involved 7,625 participants aged over sixty-five. Led by Cécilia Samieri from the University of Bordeaux and the National Institute of Health and Medical Research (INSERM), the primary aim of the large-scale study was to investigate the relationship between olive oil consumption and the risk of stroke in older subjects. The seniors taking part came from different regions in France: Montpellier in the south, Bordeaux in the southwest, and Dijon in central France.
They were all interviewed at the start of the study to make sure they did not have any existing heart condition and also to determine how much olive oil they consumed in general. The researchers then formed three groups based on the three levels of olive oil consumers:
Group 1 (23%): those who never used olive oil
Group 2 (40%): those who used it moderately for cooking, salad dressings, and with bread
Group 3 (37%): those who used it intensively for cooking, salad dressings, and with bread
More interviews and questionnaires followed two, four, and finally six years after the baseline visit.
The final results showed the following:
A total of 148 suffered strokes.
Participants in Group 3, who used olive oil regularly for cooking and salad dressings, had a 41% lower risk of stroke compared with those who used none.
The overall stroke rate was 1.5% for olive oil users compared to 2.6% for the participants in Group 1.
At the conclusion of the study, Samieri said that since stroke was so common in older people, olive oil would be an inexpensive and easy way to help prevent it.
This was a well-designed cohort study (a study over time); however, it was an observational study, not a randomized study where some of the subjects took olive oil and others didn't. It was based on responses from the participants making their own assessments. And, as it took place throughout different regions of France, the other food could have varied considerably. Another factor which could have influenced the result is that for whatever reason, people sometimes change their eating habits. We do not know what type of olive oil the participants used—be it expensive extra virgin or a cheaper option—but their eating habits relating to olive oil could have changed during the study.
Study in Spain
A second and more rigorously controlled study which took place in Spain showed more conclusive and clear results. In the PREDIMED study (Prevention con Dieta Mediterránea), participants changed their diet and were subjected to stricter methods of control than the French study.
The main aim of the Spanish study was to show the effects of the Mediterranean diet as a primary prevention of cardiovascular disease. It involved people with high risk factors but who had not yet developed the disease. The study wanted conclusive evidence as to whether or not the Mediterranean diet supplemented with nuts or extra-virgin olive oil could prevent strokes or cardiovascular disease–related deaths in comparison with a low-fat diet.
This was also a large-scale study; a total of 7,447 participants took part in the five-year study. Although none of the men (aged between fifty-five and eighty years old) and women (aged between sixty and eighty years old) had a history of cardiovascular disease at the time of enrollment, they were all at high risk of heart attacks and strokes. To be selected for the study, they had to have either type 2 diabetes or at least three risk factors such as smoking, hypertension, low good cholesterol (HDL), high bad cholesterol (LDL), being overweight, or having a strong family history of heart disease.
Group 1 stuck to their normal Mediterranean diet, which encourages red wine but limits red meat, soft drinks, and commercially baked products. They received extra-virgin olive oil weekly and were instructed to have as much olive oil as they could—at least 1/4 cup per day.
Group 2 continued their normal Mediterranean diet but with an increase of nut intake. They, too, were given weekly supplies and were told to eat at least 1/4 cup of unsalted mixed nuts daily: 15 grams of walnuts, 7.5 grams of hazelnuts, and 7.5 grams of almonds.
Group 3 participants were advised to follow a low-fat diet and were told to limit their intake of olive oil and nuts.
After nearly five years, of the 7,447 participants, 288 suffered one of three endpoint events—heart attack, stroke, or death from a cardiovascular cause. The results showed that those assigned to the Mediterranean diet plus olive oil had a 30% reduced risk of a combined endpoint of heart attack, stroke, and cardiovascular death. (Interestingly, the study also showed that those who followed the Mediterranean diet displayed a noticeable reduction in belly fat.)
One of the major limitations of the study was that it focused only on Spanish people, who have a common culture. Professor Salas Salvado explained: "It's hard to extrapolate our results to other populations. This is a limitation of inherence to all clinical trials. First, we need to clarify in the future if adherence to the Mediterranean diet is achievable in other non-Mediterranean countries."
He explained that some ingredients (particularly olive oil) are difficult to find in some countries, or some (like wine) are forbidden for religious reasons. He said that some cultural differences in cooking might make it difficult to adopt a traditional Mediterranean food pattern.
However, despite the limitations of the study, Professor Salvado said: "For the first time, a scientific study of large scale could demonstrate the efficiency of the Mediterranean diet in primary prevention cardiovascular pathologies."
The researchers concluded: "In summary, we found that greater consumption of total olive oil, especially extra-virgin olive oil, was associated with reduced cardiovascular disease and mortality risk in an elderly Mediterranean population at high cardiovascular risk. Our findings underscore olive oil consumption as one of the key components of the Mediterranean diet for cardiovascular disease prevention."
The PREDIMED researchers said that 50 grams (about 3.5 tablespoons) of extra-virgin olive oil is the best dose for a person at high risk of cardiovascular disease and that the incidence of the disease can be reduced by 10% for every 2 teaspoons.
Study in Scotland
The two studies above demonstrate the link between diet and health in older subjects, but both long-term studies took a long time to gather enough data for analysis. However, a study conducted in 2014 in Glasgow, Scotland, showed a new way to measure subtle changes in the heart. The good thing about their method is that it takes only a few weeks to see results. The research study involves proteomics, a science which looks at protein patterns in the urine.
Unlike the PREDIMED study involving three different groups, including the control one, all of the volunteers in Glasgow took olive oil (some with high phenolics, while others took low-phenolic olive oil). Researcher Dr. William Mullen of the Institute of Cardiovascular and Medical Sciences said this was the first time proteomics was used from a nutritional perspective and that one of the aims of the research was to show which food was truly responsible for health benefits. The conclusions of the study would influence more accurate labeling, he said, and allow informed consumer choice. Dr. Mullen stated that the main target for their studies was to provide a means to test whether certain health foods—in this case, olive oil—do in fact have a beneficial effect on heart disease. At present, he said there were no standardized tests to assess foods for heart health benefits.
How does the test work? Our urine contains not only waste products but also medical evidence that can tell us what is happening inside our bodies. These are proteins which allow scientists or doctors to work out a "protein fingerprint" for different diseases. The study uses urinary biomarkers (measured biological evaluation) as a sort of reporter for the detection of illnesses before the damage starts appearing. Dr. Mullen said that other sorts of research are limited; for example, we can measure cholesterol to get an idea of your risk of heart disease, but it does not tell you if you have the disease. In other words, we might be pleased to see our printed laboratory results indicating normal cholesterol levels, but it cannot show an existing heart disease that's totally unrelated to our cholesterol levels.
For the study, researchers chose sixty-three healthy volunteers from Glasgow, participants who did not take olive oil regularly (the Scottish are not big consumers of olive oil). Aged between eighteen and seventy-five years old, the participants were medically screened before being accepted. None of them suffered from chronic diseases of the gastrointestinal tract, were on any form of medication, or had taken any recent antibiotics. The group included smokers as well as nonsmokers but excluded pregnant and lactating women.
This was a double-blind study where neither the subjects nor the research team knew which oil the participants were taking. Participants were randomly divided into two groups: one group that took extra-virgin olive oil with high phenols and another group that took normal olive oil with lower phenols. During the study time, they all had 20 milliliters of olive oil from Portugal as a daily supplement, but the oils were not heated or used in cooking. There were no dietary restrictions; they simply took the oil at any time during the day at a single intake.
The researchers used a scoring system, with all the healthy participants beginning with an equal score. Urine samples were taken at the beginning of the study, analyzed, and given a "below-the-disease threshold" baseline score of 0.14. At the end of the first three weeks, each urine sample was again analyzed and given a score for its protein characteristic followed by a calculation of the average heart disease score. This was repeated at the end of six weeks.
The results revealed that the average measure of coronary artery disease for both groups decreased. The researchers said that after three weeks, those on low-phenolic oil showed a decline in the score of 0.3 while the group on the high-phenolic oil showed a score of 0.2. Interestingly, urine analyses at the end of the study did not reveal any significant differences between the two groups; in other words, any olive oil, low or high in phenolics, seemed to be beneficial.
The researchers concluded that although there was an improvement in scores for coronary artery disease, there was no significant contribution of olive oil phenolics. They said that in the population studied, "any olive oil high or low in phenolics seems to be beneficial, and . . . the fatty acids were probably the main contributors to the observed effect."
The beauty of this test is that it can be used to quickly confirm what foods are good for health so that the public can make informed choices about the food they include in their diet. Dr. Mullen said that identifying the early signatures of disease before they become a problem would considerably reduce medical intervention. The results, he added, could have a real, positive impact on the health of the general public, and if people in the UK reduced part of their fat intake with olive oil, it could have an even greater effect on reducing the risk of heart disease.
When asked about what the results might mean in the future, the researcher was optimistic: "We were very excited about the results of this study, as it provided evidence that, using a relatively small sample size of volunteers over a relatively short intervention period, we could provide statistically significant proof of a benefit in heart health in the general population."
If proteomics could be used to measure the health effects of certain foods, the conclusions would indeed be an incentive for people to change their diet because they would be able to see the results. In the long run, that would reduce the high healthcare costs, estimated at up to £11 billion in the UK. The economic burden of CVD, including indirect costs from premature death and disability, is estimated to be over £15 billion each year in the country.
Can Canola Replace Olive Oil for a Healthy Heart?
Making the cooking news these days is the vegetable oil canola, also called "rapeseed oil." Suddenly, canola oil is big business, and some say that it can replace olive oil. These two oils are, however, quite different. Olive oil comes from a long history. It is steeped in tradition, while canola oil is relatively new.
Rapeseed oil is made by crushing the seeds of the rapeseed plant. For this, they use high temperatures, and during manufacture, solvents and chemicals are added. Olive oil, on the other hand, comes straight from the olive fruit; it is pure juice, usually cold pressed without using heat and solvents during extraction.
What's important to consider for health reasons is their stability when we use them for cooking; in other words, how resistant are they to oxygen? This is vital, as you do not want too many chemical changes taking place while cooking; you want to avoid oxidation, a reaction with the air. Consider the smoke point of the oil, the temperature at which the cooking oil or fat breaks down. When heated, olive oil is one of the most stable, principally because it is rich in monounsaturated fats. The smoke point of canola varies according to the variety and growing conditions.
The International Olive Council says the following:
Olive oil in proper temperature conditions, without overheating, undergoes no substantial structural change and keeps its nutritional value better than other oils.
When heated, olive oil is the most stable fat, which means it stands up well to high frying temperatures. Its high smoke point (410° F or 210° C) is well above the ideal temperature for frying food (356° F or 180° C). The digestibility of olive oil is not affected when it is heated, even when it is reused several times for frying.
(This statement refers to extra-virgin olive oil; the cheaper brands have lower smoke points.)
The British Journal of Nutrition, after studying the health benefits of both oils, said the evidence for the protection afforded by extra-virgin olive oil is "convincing" but that evidence for canola oil was short term and that it is prone to oxidation when frying.
Extra-Virgin Olive Oil Reduces Blood Clots
We've addressed that clots block the arteries carrying blood and that people in the Mediterranean seem to have fewer attacks. But is this because olive oil affects the blood's ability to form clots? If we could control clotting in the blood, then perhaps we could reduce the vast number of heart attacks.
In Denmark, at the Royal Veterinary and Agricultural University, Frederiksberg scientists looked at how blood clots react with different oils. Researchers wanted to compare the effects of extra-virgin olive oil, rapeseed oil, and sunflower oil on the blood coagulation factor VII. They recruited healthy subjects for this study: eighteen fit young men who ate diets with high contents of olive oil, rapeseed oil, or sunflower oil. After three weeks, when they checked the levels of factor VII of each group, they found that the readings were considerably lower for those on the diet rich in olive oil as compared to those on the sunflower and rapeseed oils.
It does not take a long time for a clot to form in an artery and stop the flow of nourishment from getting to the brain and heart. Risk factors like age and heredity we cannot control, but we can choose to replace saturated fats with monounsaturated fat in our diet. Choosing a healthy diet such as the Mediterranean diet coupled with regular exercise helps keep the interior walls of our blood vessels free to pump blood to our hearts, the way it should be.
PART III
UNDERSTANDING THE LABEL AND BUYING EXTRA-VIRGIN
Once you understand the label, buying extra-virgin olive oil becomes easy.
How often have you stood in front of a row of extra-virgin olive oils, bewildered by the choices and confused by the labels? Don't worry; you are not alone. The wording that manufacturers use is not always easy to follow, and the bottles come in various sizes and different colors, some with impressive-sounding labels. Mandatory information such as provenance, quantity, and company brands are clear, but how can we be sure of the quality of the contents? What does "light olive oil" mean?
As consumers, we scrutinize the label because we want to buy the best olive oil, especially now that we are more aware of the health benefits and the organoleptic aspects of extra-virgin olive oil. Buying the right olive oil, however, is not easy these days because of complicated phrases and confusing terminology, and besides, we are put off by the alarming increase of fraud in the olive oil world.
Phrases like "packed in Italy" or "bottled in Italy" do not mean that the oil was made in Italy or that it was made from Italian olives. Often, too, a label claiming that the oil is extra virgin is nothing but cheaper oil sold at the same price as good-quality extra-virgin olive oil or, even worse, cheaper, low-grade oil blended with refined olive oil and passed off as fresh olive juice.
To safeguard against these fraudulent practices and to protect the consumer, The International Olive Council has enforced strict standards for their member countries, which account for 98% of the world's olive oil supply. Most non-IOC members also have established norms and practices to ensure that the contents live up to what the label says, particularly regarding chemical and organo-leptic standards.
Things were different a few years ago when olive oil was mostly a Mediterranean product and a Mediterranean way of life, but extra-virgin olive oil has become a global product present in 150 countries. Because of the considerable increase in olive oil consumption with new markets and new producers all over the world, it is important to protect consumers with common regulatory statements: compulsory declarations relating to bottling and selling all types of olive oil everywhere.
If you can, it's best to buy from a reputable producer, but this is not always possible. That is why it is vital to study the label, paying close attention to the compulsory statements; it is the best guarantee that you are getting the real thing.
Look for Two Magic Words: "Extra Virgin"
Ignore any bottles labeled "100% pure olive oil," "light olive oil," or any similar terminology. Pick up, instead, one which says "extra virgin." These two words tell you that this is the highest grade of olive oil, that it was made without any added chemicals, and that that the producer took extra care during the extraction process to keep the temperature at 27° C or lower. This is sometimes labeled as "first cold pressure." It means that you are getting the maximum nutritional and organoleptic qualities. Heat matters because although excessive heat yields more oil, the quality becomes inferior when processed this way. "Extra virgin" on the label also tells us that only the best olives were used, that the oil was laboratory tested, and that it meets the required chemical and organoleptic standards. It is in the best commercial category approved by the IOC.
Following closely behind the extra virgin category is the description, also a mandatory statement from the IOC:
Virgin olive oils are the oils obtained from the fruit of the olive tree solely by mechanical or other physical means under conditions, particularly thermal conditions, that do not lead to alterations in the oil, and which have not undergone any treatment other than washing, decantation, centrifugation, and filtration.
Check the Dates
The second important thing you should look for is the "best before" date or the "harvesting" date. Shelf life can be variable, and though some producers might give an optimal "use by" date or say "preferentially to two years," a lot depends on the olive variety. The harvesting date is probably most useful for consumers, but bear in mind that you need to think about how the oil was stored in the supermarket, a factor which plays an important part in the lifespan of olive oil. Remember that the younger the oil, the better it is for your health. Don't buy oil that is more than eighteen months past harvest time. Getting one from this year's harvest is the best proof of freshness.
Check the Origin of the Olives
The label is a kind of contract between the producer or the bottling enterprises and the consumer. The source of the fruit and the geographic region for some consumers serve as a guarantee of quality; these days, consumers like to see the clear, direct path right up to the production process. However, if the identification of the supplier is necessary, the origin of the fruit is not always mentioned. In most cases, the supplier is not the person who owns the olive plantation. Take, for example, the case of Italy, one of the world's major importers of olive oil. You might be surprised to learn that much of the fruit comes from orchards in Spain, Greece, and Tunisia. Also, Italians consume some of the oil imported into Italy, but much of it is blended, packed, and re-exported. Legislation laws in Europe now protect consumers with a more detailed traceability chain, with regulations that stipulate that either the origin of the olives or the place of harvesting be mentioned on the label.
Look for the Certification
Although rules and regulations concerning certification vary from country to country, here's what you need to know.
Are you buying olive oil from Europe?
The United States is the second biggest importer of olive oil (after the EU), and, not surprisingly, most of the olive oil sold in the United States comes from Europe. Certification on the label is the best guarantee that your olive oil comes from where it says on the bottle.
A Protected Designation of Origin (PDO) on the label is the European certificate that confirms that the product comes from a particular European country. This means that the olive oil was produced and processed in the same place. If your oil comes from Europe, it may carry one or two of the following:
AOC: The French term for PDO is Appellation d'Origine Contrôlée.
DPO: The Italian version of PDO is Denominazione d'Origine Protetta.
PGI: Protected Geographic Indication. This shows that at least one stage of production, processing, or preparation took place in the geographic location.
The benefits of olive oil can now be highlighted through labeling and marketing. The European Food Safety Authority (EFSA) allows the following health claim on extra-virgin olive oil labels: "Olive oil polyphenols contribute to the protection of blood lipids from oxidation stress." The European Commission stipulates, however, that the claim may be used only for olive oil which contains at least 5 milligrams of hydroxytyrosol and its derivatives.
Are you buying olive oil from the United States?
Although federal standards are very similar to the IOC, these are only voluntary and not mandatory.
If you are buying from the United States, look out for the following:
The COOC Seal: This is the California Olive Oil Council seal, which shows that the oil has passed both the chemical and sensory tests to be sold as extra-virgin olive oil.
The NAOOA Seal: The North American Olive Oil Association seal shows that the oil was imported and meets IOC standards. Many leading brands in the retail industry carry this seal.
Interestingly, you may see the following claim on an olive oil bottle, allowed since 2004 by the United States Food and Drug Administration (FDA):
Limited and not conclusive scientific evidence suggests that eating 2–3 tablespoons (23–50 grams) of olive oil daily may reduce the risk of coronary heart disease due to the mono-unsaturated fat in olive oil. To achieve this possible benefit, olive oil is to replace a similar amount of saturated fat and not increase the total number of calories you eat in a day. One serving of this product contains [x] grams of olive oil.
Storing Extra Virgin
You've carefully selected your precious green nectar, but how do you store it?
Don't save it for special occasions! Contrary to wine, olive oil does not get better with time. Extra-virgin olive oil is freshly squeezed fruit juice—it's seasonal, perishable, and best at its freshest (within the first week of pressing). As with all fresh food products, both the taste and the aroma of olive oil lose quality with time. The overall quality starts deteriorating just a few months after milling, declining rapidly when the oil is bottled and even faster when the bottle is opened.
That is why it is increasingly common to have recommendations or advice on how to store your olive oil on the label.
Here's what you should know:
Auto-oxidation and photo-oxidation are the two culprits responsible for causing the oil to oxidize. Unlike other cooking oils, extra-virgin olive oil contains natural antioxidants, which protect it somewhat from auto-oxidation (oxidation that occurs in the presence of oxygen). It is, however, very sensitive to photo-oxidation (oxidation caused by light). To avoid your extra virgin going bad, store the oil away from light, heat, and air (oxygen). A sealed bottle of extra-virgin olive oil lasts between nine and twenty months, but once opened, it is best to use it up within three months. Although the oil is still okay to use, oxidation might cause an unpleasant odor and taste. More importantly, the sensory and nutritional qualities start diminishing rapidly.
To limit oxidation and to conserve your extra virgin, you should ideally keep your bottle between 15° C and 20° C (59–68° F). The following is advisable:
Store the bottle in a cupboard away from direct light, such as from a window. You need to protect it from ultraviolet rays.
It might look pretty in a clear bottle, but choose a tinted glass container. Also suitable is a nonreactive metal such as stainless steel.
You should not use plastic. Although it is light, plastic is porous and lets in light, air, and humidity as well as ultraviolet rays.
Heat is just as dangerous when it comes to storing olive oil. Keep it away from any heat source such as above the oven or stove.
Try to buy your oil in small amounts, ideally in containers of about 250 or 500 milliliters (1 or 2 cups), or transfer it to smaller containers to use for the week.
Remember: olive oil starts degrading quickly, so cover it immediately after each use. You could invest in an olive oil pourer; they come with long metal spouts, ideal for continuous pouring. If you can get a ceramic or stainless steel one, that's even better; it will keep the light out, thus protecting your oil from oxidation.
DIFFERENT CATEGORIES OF OLIVE OIL
With growing awareness, new research, and concerns for our health, the Mediterranean basin is no longer the only center of olive oil production. The demand for the olive fruit is growing, with about eighty-six liters of olive oil produced every second in the world. Unfortunately, not every liter produced is real, natural juice; only two categories can be classed and sold as pure olive oil.
To ensure quality standards in olive oil production, the International Olive Council has established very specific regulations for all types of olive oil. To make it easier for the consumer, we focus here on the most common categories, those we find in supermarkets. To understand the complexity of this product, we have added at the end of this chapter the ten categories of olive oil defined by the IOC as well as those of the United States. These are somewhat similar but aren't compulsory for producers in the United States.
Before bottling, the producer or the person in charge of the commercialization must submit a sample of his oil for testing. Based on the results of physicochemical tests and sensory test, the oil will be assigned a category. This is where the story gets complicated, because the classification of olive oils is anything but simple.
The first family of olive oils extracted exclusively from olives by mechanical or physical processes is called "virgin olive oil." Any other oils obtained using solvents, high temperatures, or any mixture with oils of other kinds are excluded from this category.
Extra-virgin olive oil is obviously the most superior olive oil, the best quality you can get. As production methods are carefully surveyed, the oil retains all its nutritional and aromatic properties. It is a pure fruit juice, just as fresh and nutritious as freshly squeezed orange juice. The label often includes the phrases "first cold pressed" or "extracted cold," which signifies that this was the first time the fruit was extracted and that the temperatures during the mixing stage did not exceed 27° C (80.6° F).
According to the criteria of the IOC, extra-virgin olive oil must meet three quality settings: it must have a low acidity (below 0.8%), a nice, pleasant aroma that reminds us of healthy fruit (such as the olive or another similar fruit or vegetable), and no sensory defects such as fermentation or oil rancidity.
If the chemical test for acidity of the oil exceeds 0.8%, it will be noted as having a slight defect; the oil then gets put in the "virgin olive oil" category, still classed as first cold-pressed oil but with a slight defect which the consumer does not always notice. The acidity is a chemical parameter that shows the freshness and quality of the oil; it is not something that the consumer or the professional taster can distinguish. A low acidity shows that manufacturing was carried out in optimum conditions and in a very short time between harvest and pressure, with natural and mechanical methods, without chemical additives.
Why does acidity exceed the limit?
The virgin olive oil, from the tree to bottling, may suffer some alterations of fruit or oil. The 2014–2015 harvest in some parts of Europe, for example, was catastrophic due to unfavorable climate conditions: a rainy summer and a mild winter that caused an invasion of olive flies. These flies lay their eggs directly on the fruit, allowing the larvae to feed on the flesh of the olive. From the outside, it is difficult for a grower to see the larvae; only the extracted oil with a high acidity reading, as well as an unpleasant taste in the mouth and nose, will tell. The oil cannot be consumed in this state and must be refined before it can be sold.
Some other factors can damage the fruit and oil. The list is long but includes very long storage of the fruit before extraction, using fruit that fell to the ground, diseases, prolonged contact with oil and organic materials, and temperatures being too high during malaxing.
According to IOC regulations, if a virgin olive oil has acidity above 2% or a marked sensory defect, it becomes lampante oil and cannot be sold for consumption. Growers strive for low acidity levels; they do not want to see their oils downgraded and sent to the refinery.
Let's now consider the second family. These are virgin olive oils with defects or second-pressed oils. Pressing a second time extracts another 3%–8% of oil that could not be obtained from the first press.
These oils cannot be sold and are sent to the refinery to be chemically treated. This process removes all the smell, taste, and color. Although the first press renders a natural product, the second is an industrial product that has nothing to do with fruit juice.
This classification makes it tough for the consumer. Education and promotion are essential for extra-virgin olive oil. It is the only natural juice—the only one which has all the health benefits and endless cooking options.
The second challenge for the International Olive Council is the harmonization of quality standards for consumer countries which are not producers, such as Brazil and China. Issues today concern protection from fraudulent practices (such as the mixing of olive oil with other vegetable oils or the sale of a lower quality at the price of extra-virgin olive oil, for example). Strangely, it's an increasingly regulated sector and often quoted in the press for fraud.
A few years ago, olive oil was mostly a Mediterranean product and a Mediterranean way of life; this is no longer the case. Extra-virgin olive oil has become a global product, present in 150 countries all over the world. Consumption is increasing in nonproducer countries, too. According to the IOC, the United States is the biggest importer of olive oil, followed by these nine countries: Brazil, Italy, Japan, China, Canada, Australia, Russia, Spain, and Switzerland. Nonmember countries account for 82% of the worldwide imports.
Food products must have a label that complies with the legislation of the country where they will be sold and distributed. The information supplied guarantees health protection and also protects consumers against fraud. These rules of labeling and classification of oils, according to chemical and organoleptic parameters, must be applied by all member countries of the International Olive Council, which carry 98% of world production. How else can we be sure that the contents are the same as what it says on the label?
Rules to protect the consumer in nonmember countries vary from country to country.
Mandatory Standards
For nonmember countries, the legislation to protect the consumer varies according to the country and each competent institution. The International Olive Council is working closely with the Codex Alimentarius to homogenize the mandatory standards.
Some nonmember countries, like Brazil and Japan, work closely with IOC. With such collaboration, these countries are gradually adopting the official standard. In the United States, the federal standard is voluntary, not mandatory, but very similar to the IOC's guidelines. In Australia, the Australian Olive Association, which represents Australian growers, sets and maintains quality standards through the Code of Practice (COP).
Designations and Definitions of Olive Oils from the International Olive Oil Council
Virgin olive oils are the oils obtained from the fruit of the olive tree (Olea europaea L.) solely by mechanical or other physical means under conditions, particularly thermal conditions, that do not lead to alterations in the oil, and which have not undergone any treatment other than washing, decantation, centrifugation, and filtration. Virgin olive oils are fit for consumption.
Extra-virgin olive oil: virgin olive oil which has a free acidity, expressed as oleic acid, of not more than 0.8 grams per 100 grams, and the other characteristics of which correspond to those fixed for this category in the IOC standard.
Virgin olive oil: virgin olive oil which has a free acidity, expressed as oleic acid, of not more than 2 grams per 100 grams and the other characteristics of which correspond to those fixed for this category in the IOC standard.
Ordinary virgin olive oil: virgin olive oil which has a free acidity, expressed as oleic acid, of not more than 3.3 grams per 100 grams and the other characteristics of which correspond to those fixed for this category in the IOC standard. This designation may only be sold directly to the consumer if permitted in the country of retail sale. If not permitted, the designation of this product has to comply with the legal provisions of the country concerned.
Virgin olive oil not fit for consumption is classed as lampante virgin olive oil, virgin olive oil which has a free acidity, expressed as oleic acid, of more than 3.3 grams per 100 grams and/or the organoleptic characteristics and other characteristics of which correspond to those fixed for this category in the IOC standard. It is intended for refining or technical use.
Refined olive oil is the olive oil obtained from virgin olive oils by refining methods which do not lead to alterations in the initial glyceridic structure. It has a free acidity, expressed as oleic acid, of no more than 0.3 grams per 100 grams, and its other characteristics correspond to those fixed for this category in the IOC standard. This designation may only be sold directly to the consumer if permitted in the country of retail sale.
Olive oil is the oil consisting of a blend of refined olive oil and virgin olive oils fit for consumption as they are. It has a free acidity, expressed as oleic acid, of no more than 1 gram per 100 grams, and its other characteristics correspond to those fixed for this category in the IOC standard. The country of retail sale may require a more specific designation.
Olive pomace oil is the oil obtained by treating olive pomace with solvents or other physical treatments, to the exclusion of oils obtained by esterification processes and of any mixture with oils of other kinds. It is marketed by the following designations and definitions:
• Crude olive pomace is olive pomace oil whose characteristics correspond to those fixed for this category in the IOC standard. It is intended for human consumption, or it is intended for technical use.
• Refined olive pomace is the oil obtained from crude olive pomace oil by refining methods which do not lead to alterations in the initial glyceridic structure. It has a free acidity, expressed as oleic acid, of no more than 0.3 grams per 100 grams, and its other characteristics correspond to those fixed for this category in the IOC standard. This product may only be sold directly to the consumer if permitted in the country of retail sale.
• Olive pomace is the oil comprising the blend of refined olive pomace oil and virgin olive oils fit for consumption as they are. It has a free acidity of no more than 1 gram per 100 grams, and its other characteristics correspond to those fixed for this category in the IOC standard. The country of retail sale may require a more specific designation.
Olive Oil Grade Descriptions from the United States International Trade Commission
Virgin oils: oils obtained from olives that have not undergone any treatment other than washing, decanting, centrifugation, and filtration. Virgin oil is obtained during the first pressing of whole olives.
Extra-virgin: The highest grade of virgin olive oil, which exhibits some fruitiness and is free of taste defects.
Virgin: Oil that is fit for human consumption, has reasonably good flavor, but may exhibit taste defects.
Lampante virgin: Virgin oil that is not fit for human consumption without further processing and contains flavor and odor defects.
Olive oil: Oil that is made from blending virgin and refined olive oil. This oil is fit for human consumption and makes up the majority of global olive oil sales.
Refined olive oil: Olive oil that has been refined into an edible product yet maintains the initial glyceridic structure of olive oil. The refining process heats the oil to get rid it of flavor flaws such as rancidity. Refined oil does not contain the same beneficial nutrients as virgin oil.
Olive pomace oil: Olive pomace is the solid remains (skins, pulp, seeds, etc.) left over from the first press of olives for oil. Olive pomace oil is extracted from the pomace using chemical solvents and must be refined to be fit for human consumption. Once refined, it is mixed with virgin oils for consumption.
Olive Oil Tasting
Appreciating good olive oil requires an understanding of two essential senses: taste and smell. Tasting olive oil is like wine tasting; it is a sensory experience, an acquired taste. A newbie to wine tasting cannot differentiate between an inexpensive bottle and an excellent vintage one unless he understands how grapes from different regions picked at various harvests can influence quality. He must also be able to recognize positive attributes of the wine. The same goes for olive oil. Both contain aromatic molecules and high antioxidant polyphenols; both are rich, fragrant, and full of flavor.
Unlike other vegetable oils, olive oil must be tested for three positive attributes before it can be graded. Once you learn how to recognize the subtle but unique aromas and how to identify the different tastes, it will be less of a mystery. Laboratory analyses and official techniques are required for official tasting, but with time, you too can evaluate extra-virgin olive oil in three steps: pouring, smelling, and sipping the oil.
Virgin olive oil is the only food product that must have mandatory sensory tests before bottling. Although the chemical analysis is necessary and takes place at the same time, that alone cannot determine the flavor and aroma of the oil. Professionally trained tasters follow strict guidelines set by the International Olive Council when testing the oil for taste.
The first test is to check the acidity level of the oil. For extra-virgin olive oil, this must not be more than 0.8%. The chemical test takes place in a laboratory, not through tasting by professionals. "Acidity level" does not mean that it leaves an acid taste in the mouth. It indicates good quality, showing the oil has a low level of oleic free acid with no sensory flaws. The condition of the fruit at the time of processing has a direct influence on the acid level—a higher level shows that the fruit might be overripe, that it might have been overheated during production, or that there was too much time between picking and processing.
Official testing usually takes place with eight to twelve professional tasters who sip different oils from numbered samples. They judge according to three attributes: fruitiness, pungency, and bitterness. The oils are first warmed to a temperature of 26.5° C (80° F), and then presented in little blue bottles so that the tasters aren't influenced by color. Extra-virgin olive oil can vary in color, from green to golden, but color does not indicate quality. The tasters classify the oil according to flavor and the presence or absence of defects. "Defects" refer to the negative aspects of the oil. For example, the oil could have a rancid taste because the oil is too old or it could be made with bad olives. Pure extra-virgin olive oil has no defects.
Tasters have to assess the properties of the oil and note the score on the organoleptic profile sheet. After analyzing the data, the oils are then classified into the categories of extra-virgin, virgin, and lampante oil. In excellent oils, these characteristics of fruitiness, pungency, and bitterness are harmoniously balanced, together with complex aromas, flavors, and aftertastes picked up through olfactory, gustatory, and tactile sensations.
You might find yourself in a store where they invite you to taste the oil before you buy. Just like the official tasters, you need to think of the three attributes: fruitiness when you smell and taste the oil and bitterness and pungency recognized by tasting.
To conduct your sensory test, first warm the container with both hands, close your eyes, and smell the oil; this is, by far, the best way to identify the aroma of the extra-virgin olive oil. It is easy to understand why. Scientists say the human nose, with four hundred olfactory sensors, is capable of detecting more than ten thousand odors, so the nose is the best machine for detecting both pleasant and unpleasant smells. Though we underrate our sense of smell, it is probably the most sensitive and evocative of the senses, which we can never replace with a machine.
A good olive oil gives good aromas when you smell it. Most importantly, it should have a pleasant smell, like the olive fruit. In other words, as we inhale, it should evoke a fruity sensation. The word "fruity" is often used to describe the aroma of olive oil.
You'll hear tasters talk of two distinct types of fruity oil. They are referring to the variety and the time at which the fruit was harvested. Olives picked early in the season can be classed as fruity green. These have aromas of herbaceous plants, grass, the tomato plant, banana skin, basil, the leaf of an olive tree, artichoke, the green almond, or green olives. On the other hand, when the fruit is picked when ripe, you get more mature oil, a ripe fruitiness with warmer notes such as banana, red fruit, dried fruit, exotic fruit, apple, cocoa, and tomato.
Above all, seek out freshness; a good extra-virgin olive oil cannot smell or taste moldy, rancid, cooked, greasy, meaty, metallic, or like cardboard.
Once you've recognized the positive fruity aspect by smelling, you are ready to taste. Cup the container in your hands for a few minutes then take in a sip, holding the liquid in your mouth for a few moments, not swallowing straightaway. Rather, let the oil linger in your mouth for a while. Pay attention to mouthfeel—it's better for the oil to be crisp and clean rather than flabby, coarse, or greasy.
As you take it into your mouth, the oil reaches your body temperature of 37° C (98.6° F). Because our mouths connect with our nasal cavities at the back of the throat, aromas and flavors come together, allowing us to experience taste. Sensory scientists call this the "retronasal" sensation.
If it is a good quality olive oil, you will feel the bitterness at the back of the tongue caused by the polyphenols, the strong natural antioxidants present in the oil. The earlier the fruit is harvested, the bitterer and better it is.
Pungency, also due to healthy polyphenols, is not a taste but a stinging sensation, a chemical reaction that we detect in the throat as we swallow. Taking a good mouthful of the oil is vital, as the entire mouth cavity becomes involved in olive oil tasting, from the front of the mouth to the sides of the tongue as well as the palate support and the throat.
All extra-virgin oils should have these positive attributes; if it is very pungent and bitter, we can say the oil is robust, but if the tickle has less of a kick, then consider it a smooth oil.
To summarize, by smelling the oil, we discover the fruitiness, and by tasting, we detect the fruitiness (by retro-olfaction), bitterness, and pungency—all good characteristics of extra-virgin olive oil that describe its style and complexity. Importantly, a high concentration of bitterness and pungency prevents the fats from going rancid, thus allowing the oil to stay fresh for a longer time.
As you smell and taste, here are some points you should be looking for:
Is the aroma pleasant? Is it mild, strong, or between the two?
Is the oil bitter? Is the bitterness strong, medium, or mild?
Look out for pungency, that peppery bite when you swallow. Did it make you cough?
You do not have to be an expert to taste a good olive oil; it must, above all, be pleasurable, and cooking with olive oil must be a joyful experience. Let it be a hedonistic experience.
Learn about the different varieties of olives and terroirs, the various types of fruity oils, and varying intensities and pay attention to what sensory notes emerge. This will pave the way for food and olive oil matches.
With time and practice, you'll come to detect nuances in each of these groups and even start using the same vocabulary as the experts for tasting. You'll be able to describe your olive oil sample as medium fruity green with notes of artichoke and grass. Your taste buds will detect flavors you might describe as medium bitterness and a long and persistent pungency in all the palate. One oil may remind you of June when people mow their lawns when another oil tastes strongly of freshly cut herbs. Oddly enough, freshly cut grass is one of the aromatic notes that can be found in the fruity extra-virgin olive oil.
And the more you taste, the better you become.
OLIVE OIL IN THE KITCHEN
Along time ago, the Minoans, a Bronze Age civilization, lived on the island we now know as Crete. They grew barley, grapes, and olives and, according to archeological evidence, traded their olive oil with neighboring countries. Numerous findings on the Minoan civilization show that they used urns for storing and transporting olive oil. When historians analyzed their cooking utensils, they discovered that the Minoans also cooked with olive oil. This shows just how far back the art of cooking with olive oil goes—from the third millennium to 1100 BC.
Over the years, we have discovered thousands of uses for olive oil, from health and cosmetics to household remedies and cooking, where olive oil plays an undisputed vital role today.
At one time, olive oil was limited to the elite; you could find olive oil only in the Mediterranean region or in certain luxury delicatessens. That is no longer the case. Today, we have become more curious. We want to learn more about the different flavors of olive oil, and we want to find new ways of using it in the kitchen. And, because we are more health conscious, we explore the health benefits.
Olive oil has become associated with Mediterranean cuisine, another reason it has become an essential ingredient in kitchens all over the world. Raw, cooked, fried, or as a natural preservative, olive oil has numerous uses in cooking. Cooks and chefs appreciate its wide range of flavors and aromas; they know that cooking with olive oil means fine cuisine with good flavor and aroma. What's good to know, too, is that even if we cook with olive oil, it still retains its nutritional value.
We get the best nutritional and aromatic properties of extra-virgin olive oil from newly pressed oil obtained from fruit picked green or just when the color is about to change. We want to keep those two properties, but if we increase our heating temperatures whilst cooking, we damage the natural antioxidants present in the oil. The highly prized vitamin E, for example, is one of those precious antioxidants which loses its qualities, and can even be eliminated, through natural oxidation, cooking, and repeated heating.
Drizzling and dipping olive oil over food are easy ways to take in the daily recommended olive oil without cooking or heating. Here are some popular methods of incorporating olive oil into meals:
Add flavor and color to a dish of steamed fish or mashed potatoes with a dash of olive oil.
Use it to provide the finishing touch to a hot dish, a favorite of food lovers.
Drizzle a little over toasted bread with perhaps a hint of garlic.
Drizzle some oil on goat cheese to bring out more flavors.
Use it in a marinade. Thinly sliced beef carpaccio is a good example.
Drizzle just a touch more olive oil on a tomato salad.
Add it to a fresh fruit salad for a subtle, contrasting flavor.
To liven up our dishes, we can choose from a selection of olive oils flavored with different herbs, lemons, and so on; infused porcini mushrooms is a favorite with gourmets. Some mills even grind oranges or lemons along with the olives to give it an acidic flavor. And let's not forget the hot olive oil spiced with pili-pili red peppers, another favorite used for seasoning pizzas.
One renowned French chef, Mauro Colagreco, adds a creative touch to traditional cuisine by using yuzu fruit—a cross between a lemon, a mandarin, and a grapefruit—into his own brand of olive oil, "Chef's Olive Oil with Yuzu." The citrus fruit is hard to grow, expensive, and rare, but it thrives in Menton, where Colagreco lives. To make this different, subtle type of oil, they pick the fruit when it is ripe and macerate the whole fruit in extra-virgin olive oil for about a month. Mauro Colagreco uses his yuzu olive oil at all of his restaurants: The Mirazur, his elegant restaurant in Menton, one in Paris, and two in China.
Although herb-infused olive oil is gaining popularity as an easy, practical option, purists argue that these flavored varieties used in sauces and dressings distort the true flavors of olive oil. They say that olive oil already has its own complex aroma and doesn't need any other additions like garlic and basil; besides, this speeds up the unwanted oxidation and rancidity processes. The International Olive Council has a strict policy on flavored oils. They say that extra-virgin olive oils must be 100% pure and that nothing can be added, which is why flavored oils are not included in their classification of extra-virgin. These are commercially labeled as "dressings made with extra-virgin olive oil" and "flavored olive oils."
These infused olive oils, however, are great in marinades. The technique of marinating food has been around for a long time, used especially in the past to prevent food from going bad. Today, we leave meat to marinate in olive oil or wine before cooking essentially to tenderize and to add flavor.
Preserving food by marinating it was especially popular in Mediterranean countries. Marinating prevents oxidation (the reaction when food is exposed to oxygen) and hydrolysis (exposure to water). One fairly well-known Mediterranean dish, escabeche, is a good example of this ancient cooking method still used today. The olive oil and vinegar mixture creates a sort of seal, protecting and preserving the fish and at the same time inhibiting any contamination by microbes.
Cooking with Olive Oil
Cooking with olive oil opens up a whole new world, but one thing is sure: no matter how good the oil is, it still burns when you heat it. The chemical composition of food changes with cooking, making food easy to digest. This is a positive change, and we want it to take place. But not all changes are good. Some can be detrimental to our health, and a lot depends on the stability of the cooking oil we use.
When we cook, stew, shallow-fry, deep-fry, and sauté in extra-virgin olive oil, we lose some of the oil's nutritional and taste properties because the minor components are destroyed, especially if we use high temperatures and cook for a long time. However, all the categories of olive oil are rich in oleic acid and, therefore, high in monounsaturated fats. It is this important property that makes them more stable and makes the oil more resistant to heat. This is not the case for other vegetable oils that are rich in omega-3s.
When we talk about stability in cooking, we refer to the resistance of oil or fat to high temperatures. Olive oil is a stable oil because it has a high smoke point (the temperature at which the fat changes its chemical composition, gives off smoke, and becomes toxic). All fats, when subjected to high temperatures (180° C or 356° F for most olive oils), undergo changes to their molecular structure. At first, they lose their organoleptic and nutritional qualities, but as the temperature rises, they deteriorate, oxidize, and finally end up being bad for our health. Virgin olive oil reaches its smoke point at 190° C (374° F) on average, but the fresher and less acidic it is, the more stable, with the smoke point sometimes reaching 207° C (404° F).
The smoke point is related not only to the fatty acid content but also to the presence of free fatty acids and the acidity level. That is why refined oils are more stable—the refining process that manufacturers employ eliminates these elements. The smoke point of refined sunflower oil is 232° C (450° F), the same as refined soy oil and peanut oils. Admittedly, they have higher smoke points than olive oil, but that is because they have been refined and have undergone a chemical extraction process which has also eliminated all of their health benefits. Extra-virgin olive oil, by contrast, is a pure, natural fruit juice.
Frying is a popular method of cooking in all Mediterranean countries, regardless of religion. It consists of immersing food in oil that is already heated at a temperature of between 130° C and 180° C (226–356° F)—the ideal temperature for cooking food quickly without reaching the smoke point. (If the oil is heated beforehand, the food absorbs much less oil as it cooks.) Extra-virgin olive oil is ideal because of its composition of fatty acids and low acidity, qualities which make it more stable.
When you fry with olive oil, a crust forms around the food, providing a sort of protection envelope around the food; this means that the oil does not seep in. At the same time, it makes the food crunchy and tasty.
Frying in Olive Oil
To fry or not to fry with olive oil? This has been an ongoing subject of debate for chefs, but a new study shows that even with frying, we can still enjoy the benefits of olive oil. This nutritional discovery suggests that frying with extra-virgin olive oil improves the nutritional profile of our vegetables better than boiling them.
In 2015, Jessica del Pilar Ramírez Anaya, under the supervision of Dr. Cristina Samaniego Sánchez of the University of Granada in Spain, set out to study "phenols and the antioxidant capacity of Mediterranean vegetables prepared with extra-virgin olive oil using different domestic cooking techniques." In other words, the aim of her study was to see what happens to the natural antioxidants and phenols in vegetables when you cook them in extra-virgin olive oil.
Ramírez Anaya chose four common vegetables of the Mediterranean diet—the potato, the tomato, the aubergine (eggplant), and the pumpkin. The vegetables were cooked using three basic methods—deep-fried, sautéed, and boiled. 120 grams of each vegetable without the skin or seeds were boiled in water, boiled in a mix of water and olive oil, or fried or sautéed for ten minutes in extra-virgin olive oil.
The study was performed using good quality extra-virgin olive oil with low acidity which had been adequately stored. The frying time was limited to ten minutes and the cooking temperature to 180° C (356° F).
The study results, published in Food Chemistry in December 2015, showed that frying vegetables in extra-virgin olive oil improved the antioxidant content of the food much more than ordinary cooking methods. The researchers say a larger quantity of the polyphenols present in the extra-virgin olive oil was transferred to the food through frying, much more than through the other cooking methods.
According to Ramírez Anaya: "In the household conditions of the study, using extra-virgin olive oil for frying is recommendable if we wish to increase the level of monounsaturated fatty acids in the lipid profile of food and improve the polyphenol content of vegetables, which is naturally low."
We must remember, though, that deep-frying still increases the caloric density of food. Ramírez Anaya, who earned her doctorate though the study, reminds us that "that is why we insist on the recommendation to limit the consumption of fried food in our daily diet, as it is high in calories; it's also very tempting and not recommended for a modern sedentary lifestyle."
So, as frying increases the caloric density of food, a small amount of fried food is beneficial; it should just be consumed in moderation.
Food and Oil Combinations
We have to remember that there is not one good type of oil, but many. Olive oil is a fruit juice that covers a very broad aromatic palette, with different varieties; various types of fruitiness; and different intensities of fruitiness, bitterness, and spiciness. This great wealth of flavors allows for a myriad of uses and recipes and means that oils can be combined in a dish. To get more information on the organoleptic profile of an extra-virgin olive oil, we need to look at the olive variety or varieties that make up the oil. A combination of different varieties is called a "blend," whereas oil made from only one variety is called "single varietal."
There are more than 1,700 listed varieties in the world, as well as those not yet identified. As with wine, each variety is an ensemble of unique qualities that are expressed physically through the shape of the leaves, the fruit, the color, the chemical composition of the olive, and all the particularities that give it its own unique sensory profile. Some varieties produce a soft and pure juice, while others yield intense fruity green oil, even though they are harvested over the same period and from the same place. The variety is widely responsible for this organoleptic profile.
These days, you'll find the olive variety displayed prominently on the label; this information is helpful to the consumer looking to cook or season a dish. Here are some examples:
If you need soft and delicate oil for a dessert, it is best to choose a Spanish Arbequina or a Portuguese Arbosana.
A spicy Coratina is ideal for an oven-baked pizza.
A French Négrette is great on mashed potatoes.
A Koroneiki works well on a Greek salad.
Ideally, a cook should have oils of different varieties on hand to enhance each dish. This aspect of cooking has, up until now, received little attention, but it offers a wealth of opportunities. We should feel free to experiment with contrasting flavors by adding an intense fruity green olive oil to vanilla ice cream, for example, or we could combine complementary aromas such as tomato-flavored oil with a salad of tomato, mozzarella, and basil. We could also try using mild oil on a very delicate dish or on food that doesn't require intense flavors, such as desserts or mayonnaise.
Matching oils to dishes is a new discipline, so no cookbooks have yet been written on the subject. However, here are a few fundamental and accepted rules:
Good-quality oils improve the taste of all food, whether served hot or cold.
The de-sugaring quality of olive oil allows us to prepare desserts that don't feel fatty or sugary.
The bitterness of olive oil normally does not go well with a dish that already has a bitter flavor (like meat) as it further accentuates the taste, but with some foods, it's an amazing pairing (as in the case of chocolate with a strong Picual).
Extra-virgin olive oil enhances flavors and aromas, intensifying our sensory perception.
As we mentioned before, although olive oil is a healthy fat, it is a fat nevertheless and very high in calories. Like all vegetable oils, olive oil is a high-energy food, providing nine kilocalories per gram—more than butter, which provides 7.5 kilocalories per gram. We should, therefore, consume it in moderation if we wish to stay in shape and follow the recommendations of the World Health Organization. They tell us, broadly speaking, to consume less saturated fat, trans-fatty acids, free sugars, and salt and to eat more fruit, vegetables, whole-grain cereals, and nuts.
The golden rules are to eat fresh and seasonal food and to combine oils but always with a preference for omega-9 and raw omega-3 oils—always in small quantities. Equally important: we should go for quality rather than quantity, keep active, and take our time when cooking. In short, follow the fundamental principles of Mediterranean cuisine.
Recipes We Love
These are some of our favorite recipes, ones that we have made and tried; they're not taken from cookbooks but come from our friends and family. We even have one from Dr. Amal Kaddoumi (Wonder 5: Olive Oil Slows Alzheimer's Disease).
BEAUTIFUL HEALTHY MAYONNAISE
From Cécile
Did you know that most shop-bought mayonnaise is full of bad fats? Making your own mayonnaise is so easy and so much better for your health, especially if you use extra-virgin olive oil (EVOO) and a free-range egg. Mayonnaise will keep for about a week if you store it in a sterilized jar in the fridge. And once you've tasted it, you won't want store-bought mayonnaise again. It's okay to be nervous the first time you make your own mayonnaise, but be patient; it's worth it.
1 free-range egg yolk | Salt and pepper
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1 tablespoon vinegar (15 milliliters) | 1 cup EVOO (1/4 liter or 240 milliliters)
1 heaping teaspoon mustard
In a deep bowl, mix the egg yolk, vinegar, and mustard. Add salt and pepper.
Add a little olive oil in a steady stream, mixing with a whisk until the mixture has thickened. Some use a fork, but you could use a whisk or an electric beater, starting with a little oil in the beginning and beating until the mixture thickens then adding more oil gradually. Taste it, and if you like it and are feeling adventurous, you might want to add a clove of garlic, some fresh herbs, or a bit more salt and pepper.
SHISH KEBAB LAMB
From Alice
This is one of my favorite recipes. Lamb marinated this way is wickedly delicious and so easy to prepare. What's good is that you can do most of it the night before. You can also double the proportions for a crowd.
4 tablespoons honey (60 milliliters)
2 pounds lamb (1 kilogram) | 2 tablespoons Worcestershire sauce (30 milliliters)
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Juice of 1 lemon
2 tablespoons oyster sauce (30 milliliters) | 3 tablespoons Lea & Perrins sauce (45 milliliters)
3 tablespoons EVOO (45 milliliters) | 1 tablespoon garlic powder (15 milliliters)
1 tablespoon seasoning salt (15 milliliters) | 1 small pineapple, cut into bite-size pieces
Microwave the honey for a minute or two until it has melted. Allow to cool. Prepare a marinade of oyster sauce, olive oil, seasoning salt, Worcestershire sauce, Lea & Perrins sauce, garlic powder, and the honey.
Cut the lamb into bite-sized pieces, soak in lemon juice for about 5 minutes, then rinse with cold water and pat dry.
Season the lamb with the marinade and refrigerate overnight.
The following day, once the meat has had time to become tender, heat the barbecue. Drain the marinade from the meat and set aside. On skewers, place lamb and pineapple chunks alternately.
Place the skewers on the barbecue. Cook for about 20–25 minutes or until the lamb is cooked, turning and basting with the marinade from time to time.
This lamb goes well with garlic mashed potatoes.
TAPENADE
From Cécile
I asked Anne-Laure Meunier, French nutritionist (see page 15, "Understanding Fats"), what her favorite recipe was from olives and olive oil. Without hesitating and with a broad smile, she said, "La tapenade!" Tapenade is a Provençal favorite, made with green olives, black olives, or both! It's perfect for picnics and can be served with drinks, at tapas parties, and even as an hors d'oeuvre. Serve it with tiny toasted bread slices or crackers. Traditionally, it is made using a mortar and pestle, but you can get equally good results using a food processor.
1 cup black pitted and minced olives (180 grams) | 1 tablespoon capers (14 grams)
1 garlic clove
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1 cup green pitted and minced olives (180 grams) | 1 anchovy fillet
1/4 cup EVOO(60 milliliters)
Mix all the ingredients together to form a smooth paste or make it coarse and rustic. Once made, put it in the fridge a few minutes before serving.
Extras
Once you create your paste, try adding one or two of these ingredients: fresh chopped parsley, basil, oregano, thyme, or even a bit of lemon juice.
Adding homemade tapenade to baked fish will make it moist and delicious. You can use it as a stuffing, spread it on top of the fish before you put it in the oven, or just spread the tapenade on top of fish fillets, roll them up, and secure with toothpicks. Bake the fish as you would normally.
SALMON FILLETS COOKED NADIA'S WAY
From Alice
This recipe comes from my friend and good neighbor Nadia. This is her very own recipe—a warm, elegant, yet simple starter for chilly days. Fish, ginger, and olive oil can only be healthy and full of goodness, but when I tried to make this dish for the first time, I ran into some difficulties when frying the fillet. I have included some advice to make frying easier.
As salmon is such a tender fish; it is important not to let it stick to the pan when pan-frying. Bear in mind that you want to lift the fillets without tearing them, so make sure that the pan is hot enough, that the fillets are not crowded, and that you have enough oil. Remember, too, that salmon has a tendency to curl up as you fry, so use a flexible metal spatula to hold it firmly in place as you cook the bottom and use the spatula also for flipping over. I preheated the pan then reduced the temperature.
Also, make sure your guests are sitting, ready and waiting, for this starter. This dish needs to be served warm.
3 tablespoons EVOO for the vinai-grette (45 milliliters)
Extra olive oil for frying
Juice of 1 lime
Salt and freshly ground pepper
4 salmon fillets, about 120 grams each | A mix of mesclun and rocket salad
A stick of fresh ginger, about 1 centimeter long
4 cooked prawns
Some spring onions (cleaned and chopped finely)
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Make a vinaigrette using 3 tablespoons olive oil, lime juice, salt, and pepper and set aside.
Prepare the pan and cook the salmon in the remaining oil. First, cook skin side down for 5 minutes. Then flip over and cook the other side for about 3 minutes. They should retain their texture and pink color.
Wrap the fillets in some aluminum foil to keep them warm while you prepare the plates.
Prepare the serving dishes, starting with the salad then placing the fillets in the middle. Finely grate the ginger over the salmon. Decorate with the prawns and spring onions, and then cover it all with the vinaigrette before serving.
MANAKISH
From Dr. Amal Kaddoumi
This is a flatbread from the Eastern Mediterranean region. The dough, Amal says, can be basic pizza dough, either homemade or store bought.
1 package pre-made pizza dough
3 tablespoons EVOO | 1 tablespoon za'atar (roasted thyme)
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Preheat oven to 350° F (177° C) or according to package directions. Spread the dough onto a lightly floured work surface and gently knead one or two times before using. Use your hands to flatten the dough and round the edges. Top the dough with olive oil (a generous amount!). Sprinkle with za'atar and bake until just crisp.
MADELEINES OR SHELL-SHAPED COOKIES
From Cécile
This one comes from Lola, my Spanish mother-in-law. She makes these every week and serves them for breakfast or at coffee time. I especially like these because it's wonderful to be able to replace butter with olive oil in a dessert and it's so simple to put together. Makes 30 madeleine cookies.
3 eggs
1/2 teaspoon salt
1 1/4 cups sugar (250 grams)
1 cup EVOO (1/4 liter or 240 milliliters)
8 ounces milk
Zest of 1 lemon | 2 teaspoons baking powder (1.5 grams)
2 1/2 cups flour (300 grams)
30 paper liners
Optional: chocolate or chocolate chips
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Preheat oven to 355° F (180° C).
Separate the eggs. Put the whites and a pinch of salt in a bowl. Beat the whites until they are stiff and then add the yolks. Add the sugar, oil, milk, lemon zest, baking powder, and flour. Put these ingredients in slowly and stir after each addition.
Using a teaspoon measurer, half fill each paper cup. Sprinkle each madeleine with a little sugar and put them in the preheated oven. Don't open the door until the madeleines are golden brown (about 5–10 minutes). Remove them when they are brown and allow them to cool. They are ready! Lola warns that these madeleines bake very quickly, so you do need to be careful if your oven is a fast one like hers.
If you like chocolate, why not fill the cookies with melted chocolate or chocolate chips? Just add a teaspoon of melted chocolate as you're filling the paper cups or add the chocolate chips at the same time as you sprinkle the sugar before baking.
PROVENÇAL-STYLE YOGURT CAKE WITH QUINCE
From Alice
One of the first things children learn to make in France is a yogurt cake. It's easy because they don't have to measure anything. All they need is a pot of yogurt—they empty it, rinse it, dry it, and use it to measure the rest of the ingredients.
This jazzed-up variation incorporates ground almonds and quince, a recipe that my good friend Madeleine from Cooking in Provence taught in her cooking class. The "pot" refers to a pot of yogurt. In France, our pots of yogurt contain 250 grams, but the recipe works with any size. You might just have to watch the baking powder quantity if your pots of yogurt are over 250 grams.
1 quince | Zest of 1 lemon
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1 pot sugar | 2 pots flour
4 eggs | 3/4 teaspoon baking powder
2 pots plain yogurt | 2 pots ground almonds
1 pot EVOO
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Preheat oven to 355° F (180°C).
Core the quince, then place it in a saucepan with a spoonful of sugar and cover with water. Bring to a boil and cook uncovered until tender.
In a large mixing bowl, whisk the eggs and sugar until the mixture is quite pale. Stir in the yogurt, olive oil, and lemon zest until all the ingredients are fully incorporated. Add the flour, baking powder, and almonds.
Grease a baking tin and pour in the batter. Arrange the quince on top.
Bake for about 45 minutes, checking the center of the cake with a fork to make sure that the cake is cooked.
Remove the cake from the oven, but let cool for 10 minutes or so before removing from the cake tin.
BAKED ARTICHOKE
From Cécile
I love this recipe because it's so simple yet nutritious and delicious! If you plan to serve it as a main course, I recommend three small artichokes per person. As a starter, one artichoke per person will be enough. I learned this recipe in Barcelona, where they have a lot of artichoke fields and just as many ways to cook it.
4 artichokes
Salt and pepper, to taste
4–6 tablespoons EVOO, preferably | a robust one (60–90 milliliters)
Extra ingredients: garlic, paprika,
ham or bacon
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Preheat oven to 355° F (180°C).
Cut away any excess stems, then cut the artichokes in half from top to bottom, making sure you remove any hair that you sometimes get in the middle of an artichoke. Rinse well and place an oven-proof dish (leaves facing down). Sprinkle with salt and pepper and drizzle with olive oil.
Bake the artichokes for 30 minutes. You will know they're cooked when the first leaves become slightly charred. When the artichokes are cooked, remove them, put in a serving dish, and drizzle with a little more olive oil (and any optional toppings). Serve hot.
During the summer, you can follow the same recipe but cook them on a barbecue. Yum!
PROVENÇAL TOMATOES
From Alice
Simple, easy-to-make Provençal tomatoes are perfect with grilled steak or as a starter. I particularly love this recipe; it's one of my standby meals and one I can make almost with my eyes closed. It always goes down well with family, friends, and neighbors who drop in for drinks and get invited to an impromptu dinner. Just double the quantities if you expect a crowd.
4 ripe tomatoes
2 cloves crushed garlic
1 tablespoon fresh parsley (15 milliliters) | 1 tablespoon fresh thyme (15 milliliters)
2 tablespoons bread crumbs (30 mililiters)
EVOO for sprinkling over
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Preheat oven to 400° F (200° C).
Cut the tomatoes in half; remove the seeds and flesh with a spoon and sprinkle with salt. Turn tomatoes upside down to drain for 10 minutes. When drained, place tomatoes in a gratin dish.
Mix the garlic, parsley, and thyme in a bowl. Fill each of the tomatoes with the mixture, top with breadcrumbs, sprinkle with olive oil, and bake for about 20 minutes.
OLIVE OIL QUIZ
Are you an olive oil expert? This simple quiz will test your knowledge. Check the answers at the end to see how well you did.
1. WHAT IS THE PROPORTION OF LIPIDS IN EXTRA-VIRGIN OLIVE OIL?
a. Around 75%
b. Around 100%
c. Around 98%–99%
2. HOW MANY KILOGRAMS OF OLIVES DO YOU NEED TO PRODUCE ONE LITER OF EXTRA-VIRGIN OLIVE OIL?
a. Between 4 and 10 kilograms
b. Less than 4 kilograms
c. Between 10 and 15 kilograms
3. WHICH COUNTRY IS THE BIGGEST PRODUCER OF OLIVES?
a. Greece
b. Spain
c. Italy
4. HOW MANY TABLESPOONS OF EXTRA-VIRGIN OLIVE OIL DO THE EXPERTS RECOMMEND YOU SHOULD TAKE EVERY DAY TO GET THE FULL BENEFIT OF OLEOCANTHAL?
a. 1 tablespoon
b. 3 tablespoons
c. 6 tablespoons
5. WHICH OF THESE STATEMENTS ARE TRUE? (HINT: THERE ARE TWO.)
a. Olive oil contains vitamin E
b. Olive oil contains fluoride
c. Olive oil contains squalene
d. Olive oil contains caffeine
6. "BITTERNESS" AND "PUNGENCY" ARE TERMS USED TO DESCRIBE EXTRA-VIRGIN OLIVE OIL, BUT WHAT ARE THEY?
a. They are defects
b. They are the effects of high acid content
c. They are two positive attributes
7. WHY SHOULD YOUR BOTTLE OF EXTRA VIRGIN BE OPAQUE?
a. To see the label and the nutritional value better
b. Because color isn't a characteristic of quality
c. To protect the oil from the light
8. WHICH IS THE BEST PLACE TO STORE OLIVE OIL?
a. Under the sink
b. In a refrigerator under 7° C (45° F)
c. Out of the light and between 15°C and 20° C (59° F and 68° F)
9. WHICH OF THE FOLLOWING STATEMENTS IS TRUE WHEN WE TALK ABOUT TABLE OLIVES?
a. They can be eaten directly from the tree
b. They do not contain oil
c. Some varieties are better for table olives, others for oil, and some for both
10. HYDROXYTYROSOL IS . . .
a. A phenolic component
b. A fertilizer
c. Strong bacteria which kills olive trees
11. WHICH OF THE FOLLOWING DESCRIBES OLIVE OIL?
a. Has a high level of oleic acid
b. Has a high level of palmitic acid
c. Has a high level of free fatty acid
12. WHAT DOES "COLD" MEAN IN "FIRST COLD PRESS"?
a. The cold chain must not be interrupted while processing the oil
b. To comply with regulations, the temperature must be less than 27° C (81° F) during the olive paste kneading
c. Producers wait until the winter to begin harvesting and processing olives
13. WHICH OF THESE OIL CROPS IS THE MOST PRODUCED OIL IN THE WORLD?
a. Soybeans
b. Rapeseed
c. Palm kernels
14. HOW LONG CAN YOU KEEP YOUR BOTTLE OF OLIVE OIL?
a. Olive oil is like wine; the longer you keep it, the better
b. Around two or three years
c. Around one year
15. OLEOCANTHAL IS ONE OF THE MOST IMPORTANT COMPONENTS IN EXTRA-VIRGIN OLIVE OIL. WHICH OF THESE IS THE BEST METHOD OF CHECKING THE LEVEL OF OLEOCANTHAL IN THE BOTTLE?
a. Read the label; it will state the percentage clearly
b. Take a sip and see if you get a sting at the back of your throat
c. Smell it; it will have a strong but sweet smell
16. SOME FATTY ACIDS ARE GOOD FOR OUR HEALTH, BUT WHICH FATTY OIL SHOULD YOU NOT USE FOR FRYING AND COOKING?
a. Oils rich in omega-9 acids
b. Oils rich in omega-3 acids
c. Oils rich in omega-6 acids
17. WHICH OF THESE STATEMENTS IS CORRECT?
a. Olive oil contains a small amount of calcium
b. Olive oil does not contain any calcium
c. Olive oil contains a lot of calcium
18. WHAT IS THE MARKET SHARE OF OLIVE OIL IN THE GLOBAL EDIBLE OIL MARKET?
a. 1%
b. 3%
c. 10%
19. WHAT HAPPENS WHEN A VEGETABLE OIL REACHES ITS SMOKE POINT?
a. It gives off a pleasant aroma
b. The oil is hot enough and ready for frying foods
c. The oil has been destroyed and has become toxic to our health
20. WHY DOES OLIVE OIL HAVE GOOD STABILITY FOR FRYING OR COOKING?
a. It is composed mostly of monounsaturated fats
b. It is composed mainly of saturated fatty acids
c. It is a natural juice that has not been refined
21. HOW DO WE GET THE MOST NUTRITIONAL AND AROMATIC QUALITIES OUT OF EXTRA-VIRGIN OLIVE OIL?
a. By frying
b. Having it raw
c. En papillote
22. WHAT DOES "EXTRA-VIRGIN OLIVE OIL MATCHING" MEAN?
a. It is something that the renowned Spanish chef Ferran Adrià invented
b. It is a new discipline
c. It is a discipline that has disappeared
23. THE SENSORIAL ANALYSIS TEST FOR OLIVE OIL . . .
a. Is compulsory
b. Is one of two choices: chemical analysis or sensorial analysis
c. Is not necessary
24. WHEN CHOOSING OLIVE OIL, WHAT IS THE FIRST THING YOU MUST LOOK FOR?
a. The category it belongs to
b. The deadline for optimal consumption
c. Where the olives come from
25. WHICH CATEGORY BELOW CAN BE PURCHASED COMMERCIALLY?
a. Pure olive oil
b. Olive pomace oil
c. Lampante olive oil
Answers
Here are the answers to the questions. If you got 20 correct answers, then you are an expert on olive oil matters. If you got between 15 and 19 right answers, it shows you have a reasonably good knowledge.
1. Extra-virgin olive oil is characterized by a range of minor components (total: 0.5–1.5%) and a lipids portion of between 98–99%.
2. This can vary. It depends mainly on the olives; different types of olives produce different yields. In California, for example, producers need 4–10 kilograms of olives of the popular Manzanillo and Mission varieties to produce 1 liter of oil. Other contributing factors are the size of the tree, the age, how it is watered, and the climatic conditions.
3. Currently, Spain is the biggest olive oil–producing country in the world, averaging around 1,536,600 tons annually and accounting for 45% of the total world production. Italy comes second, producing around 450,000 tons annually, and in third place is Greece, with 230,000 tons per year.
4. Dr. Breslin, who conducted research on oleocanthal and cancer cells, thinks we should have 50 milliliters of extra-virgin olive oil per day, the equivalent of 3–4 tablespoons. Dr. Kaddoumi, who did studies on Alzheimer's, recommends the same amount. It might seem like a high fat content, but the majority of the fat is monounsaturated fat—healthy dietary fat—as opposed to saturated fat.
5. Two of these statements are correct. Olive oil contains the vital antioxidant vitamin E. Olive oil also contains squalene. Recent studies show that the squalene content in olive oil has a cancer risk–reducing effect.
6. Bitterness, pungency, and fruitiness are the three positive attributes you should look for when tasting olive oil. Bitterness is the taste you feel on the back of the tongue, and pungency is the sensation at the back of the throat. Bitterness and pungency are due to the phenolic compounds present in extra-virgin olive oil. For the consumer, this means that the olive oil is fresh and full of antioxidants.
7. Extra-virgin olive oil is very sensitive to photo-oxidation caused by light, oxygen, and heat. Glass bottles and opaque packaging are better than transparent and plastic bottles. Also, proper storage conditions will help maintain the nutritional and sensorial properties of extra-virgin olive oil much longer.
8. Extra-virgin olive oil is very sensitive to photo-oxidation caused by light, oxygen, and heat. To limit oxidation and to conserve your extra-virgin olive oil, you should keep it at a temperature between 15° C and 20° C (59° F and 68° F). (Most wine cellars meet these conditions.) You should not store extra-virgin olive oil under the sink, because the humidity will affect the flavor.
9. The correct answer is "Some varieties are better for table olives, other for oil, and some for both." It all depends on the variety of olives and the quantity of oil or pulp. In most cases, you can't eat olives directly from the tree, because they are much too bitter; they have to be treated to remove the bitterness before you add them to your salads. When they're picked young, you get green olives; when they're picked at full maturity, you get black olives.
10. Hydroxytyrosol is a polyphenol, a powerful antioxidant contained in the olive leaf and olive oil. Concentration can vary depending on the variety, oil producing techniques, how ripe the fruit was at picking, storage before extracting the oil, and the method used for extraction.
11. Olive oil is characterized by a high level of oleic acid; between 55% and 83%. Cultivars vary, but the oil will contain 8%–20% of palmitic acid and 4%–21% of linoleic acid. Its oxidative stability is linked to the fatty acid composition of the oil, particularly the high level of oleic acid. This high level of oleic acid allows olive oil to be classed as a monounsaturated fatty acid.
12. The first option is the correct answer—the cold chain must not be interrupted during the entire process. This means that, following regulations, the olives are kept at less than 27° C (80° F) during processing so as to retain the nutrients and lessen the degradation of the oil. Interestingly, there is a difference between cold extraction and cold pressing. Regulations in Europe stipulate that when a centrifuge, the modern method of extraction for large quantities, is used, the label must say "cold extracted," while only a physically pressed olive oil may be marked as "cold pressed." In many parts of the world, you'll see "cold pressed" on the label even if the producers used centrifugal extraction.
13. The outlook for the market season 2015/2016 shows global soybean production yield at 318 million tons followed by rape-seed with 64.3 tons and palm kernels at 15.8 tons.
14. The "use by" date usually guarantees the best organoleptic properties of the olive oil, around one year, but it depends on the variety (some can be kept more than two years), the phenolic compounds, and how it's stored. For example, a transparent bottle in a supermarket can suffer photo-oxidation and last less than one year. Once the bottle is opened, it's best to use it up within three months.
15. The sting at the back of the throat tells us how much oleocanthal is in the oil. Gary Beauchamp, who discovered oleocanthal, accidentally stumbled upon the similarities between ibuprofen and olive oil when he tasted olive oil for the first time.
16. Both omega-3 and -9 are polyunsaturated acids. Our bodies do not produce omega-3s: you find them in some vegetable oils (flax, hemp, rapeseed, nut) and in foods like oily fish, hazelnuts, oysters, mussels, almonds, and flaxseed. Although omega-3s are essential for our health, for proper functioning of our brain and heart, there is one downside: oils containing omega-3 acids do not do well with high temperatures and become toxic. It's best to have them raw.
17. Olive oil contains a small amount of calcium—only 1 milligram per 100 grams of oil.
18. The olive oil market accounts for less than 3% of the world edible oil market.
19. When you heat oil too much and it gives off smoke, it has reached its smoke point. Smoke point is the temperature at which the oil starts breaking down, turns black, and forms smoke. It is an indication of the stability of the oil. You need to be aware of the smoke points of various fats because fat is no longer considered healthy for consumption after it has exceeded its smoke point and has begun to break down. From this critical temperature, the oil produces highly toxic and carcinogenic compounds such as benzopyrene or acrolein.
Pan frying can be done at varying temperatures, ranging from about 50° C (122° F) for the mildest cooking up to 150° C (302° F) and sometimes more. We suggest not exceeding 200° C (392° F). The ideal temperature for a good fry is 190° C (374° F).
Not all oils are the same. They have different chemical compositions and different smoke points. Oils rich in polyunsaturated fatty acids with omega-3s are very sensitive to cooking and are best used for raw preparation. Extra-virgin olive oils with low acidity have a relatively high smoke point (190° C) due to their composition rich in monounsaturated fats. Refined oils (with the impurities removed) tend to have higher smoke points.
20. Olive oil is rich in monounsaturated fatty acids, the type of fatty acid that is considered stable for cooking. Its smoke point is somewhat elevated compared to other fats. Virgin olive oils must be of the best quality, as moisture and fatty acids reduce stability. Refined olive oils are more stable in cooking but must not exceed 220° C (428° F).
Here are the relevant smoke points:
Extra-virgin: 190° C (374° F)
Virgin: 216° C (421° F)
Pomace: 238° C (460° F)
Extra-light: 242° C (468° F)
Remember that repeated use of the oil reduces its stability. Oil should be filtered after each use to remove all excess particles. This should be done once the oil has become cold. It is generally advisable to renew the oil completely after ten frying sessions.
21. Although extra-virgin olive oil has good cooking stability, it is best taken raw to get the maximum health benefits and enjoy its wide aromatic range. Olive oil must be of the highest quality extra-virgin—and its deadline for optimal use must be respected (indicated on the label).
22. It means pairing food with olive oil, a new discipline in the culinary world.
Successful food matches in the kitchen depend upon the variety of the oils, the typical characteristics, and the type of fruitiness of the oil that will accompany the dishes. Although challenging, choosing the right olive oil for the right dish is becoming more and more popular in the kitchen, but there's no cookbook to teach it. Using extra-virgin olive oil allows the chef to develop new flavors and offers spectacular aromas for a hot main course dish as well as in desserts. This cooking art has undoubtedly a good future ahead!
23. It depends on the country where the olive oil is made and sold, but for the IOC members countries—98% of the world's production—both the chemical analysis and the sensorial analysis are compulsory. You cannot have one and not the other. Virgin olive oil is the only food product that needs a compulsory sensorial analysis before bottling.
The sensorial analysis is a real measuring tool. The human nose is the most accurate tool for detecting a sensory defect and level of palatability. At the official tasting of virgin olive oils, between eight and twelve trained experts test the samples and classify them according to presence or absence of sensory defects. These trainers have followed the IOC training course in designated laboratories. Extra-virgin olive oil cannot have any sensory defect and must have a certain fruitiness.
24. The commercial category is the first thing that you should look for when choosing or buying olive oil. You should choose an extra-virgin olive oil so as to have the best quality in terms of both taste and nutrition. Optimal consumption date and place of harvest or bottling of the oil are necessary information, but that does not guarantee that the oil comes exclusively from olives. Also, the oil might not have been cold extracted or stored in the best conditions to avoid damage.
25. You can buy olive pomace oil commercially.
When the oil is first pressed, you get leftovers from the extraction. It consists of skin, olive pits, and pulp. This paste is called the pomace of the oil. To this pomace are added chemical solvents like hexane or some very high heat, processes to get the last of the remaining oil. This is refined pomace oil. Olive pomace oil is a combination of this refined oil, virgin oil, and/or extra-virgin oil. Olive pomace oil is also used in the manufacture of soap.
Lampante oil is oil that has a high acidity level of above 3.3%. It needs to be refined before it can be used. Then it is mixed with virgin olive oil or extra virgin and called "olive oil."
Pure olive oil is a marketing name to create confusion for consumers. It is not legal to sell olive oil under this category. Pure olive oil is, in fact, lampante olive oil refined and assembled with some virgin olive oil or extra virgin. It is no longer a natural product but an industrial one.
GLOSSARY
ACIDITY
Acidity tells us about the quality of the oil; it shows how much care was taken from the time the fruit started blossoming right up to when it was ready for bottling. Acidity is a chemical parameter measured in the chemical analysis to determine the oleic acid content per 100 grams of oil. A low acidity level indicates freshness and high-quality oil. It shows that the manufacturing process was carried out under optimum conditions with a very short time between harvesting and extraction. It also indicates that processing took place using natural or mechanical methods, without any chemical additives. A high acidity level, on the other hand, is not desirable. It could mean that the oil is rancid, that the fruit was infested with fly disease or suffered fungal disease (Gloeosporium and Macrophoma are two examples), or that there was considerable time between the harvesting and extraction of the oil.
This parameter not only provides information on the degradation of the fatty acids in the oil but, more importantly for IOC standards, determines the category or grade of the olive oil. According to IOC, the oil can be classified into one of three categories:
Extra-virgin olive oil must not have an acidity level of more than 0.8 grams of oleic acid per 100 grams.
Virgin olive oil must not have an acidity level of more than 2 grams per 100 grams.
Lampante virgin olive oil must not exceed 3.3 grams per 100 grams.
Acidity has no taste and has nothing to do with the acid flavor that you can feel on the tongue as with eating lemons. It is a chemical parameter measured in the laboratory; it cannot be evaluated in sensory analysis.
ANTIOXIDANTS
Although oxygen is essential for life, some molecules present in the air are directly involved in the aging process and cell damage. These molecules are called "free radicals," and they cause what is known as "oxidative stress." Some foods can help us to counteract their negative effects.
Minor components present in virgin olive oil, such as tocopherols (vitamin E) and some polyphenols (phenolic compounds), are nutritionally essential as well as significant for oil stability when cooking and for storage. The more antioxidants oil contains, the more stable it is over time.
The quantity of antioxidants is determined by several factors. The quality of the fruit before oil extraction, the extraction method, the maturity stage of the fruit, and the type of farming (rain, land, or irrigated) as well as the variety of the olive tree are all factors that determine the presence and intensity of these antioxidants.
AROMA
Aroma refers to the volatile compounds perceived by the olfactory sense. Olive oil has a broad range of aromas, each variety with its own characteristic scent. This sensation is known as the "fruitiness" of the olive oil. The aromatic profile of oil and its flavor intensity depends on the quality, olive variety, and ripeness of the fruit when harvested. One feature of extra-virgin olive oil is the presence and complexity of these aromas. By smelling, we pick up various aromas such as those found in vegetables, flowers, fruit, or spices. There are two types of fruity aromas. The first is a fruity green, which may be perceived as green grass, olive leaf, artichoke, or green olives picked at the beginning of the harvest. The second aroma, from ripe fruit, has olfactory notes that remind us of nuts, exotic fruits or berries, bananas, or apples. A harvested olive starts in a green fruit aroma which, over time, changes to ripe notes, while an olive harvested later directly gives a ripe fruit aroma.
The variety of olive is the main factor that determines the aroma of the oil. Olive oil aroma should be pleasant to the nose and have a fresh fruit scent. However, these flavors can also be unpleasant. The main defects of olive oil are musty, wet, winey, rancid, and metallic aromas, which can be due to fruit ripening, a fatty acid oxidation, or prolonged storage of olives before the extraction of the oil.
BASIC TASTES
Our sense of taste starts in the mouth. The tongue and taste buds are the fundamental instruments that allow us to perceive our sense of taste. There are four major flavors or tastes: sweet (e.g., banana), sour (lemon), salty (cheese, charcuterie, seafood), and bitter (endives, raw artichoke). A fifth flavor was discovered by the Japanese: umami (う ま み) or "tasty" in Japanese. This flavor is produced by amino acids present in protein-heavy foods like meats.
The appetite for sweet is innate in humans and also in mammals. For a feeding infant, the sucking action is related to the recognition of a sweet flavor. The taste buds on either side of our tongue allow us to experience saltiness, the presence of sodium ions. Acid generates displeasure, the bitter taste. Bitterness results in real repulsion, the taste similar to a natural form of protection, because poisons are usually bitter.
Our sense of smell allows us to perceive aromas, olfactory notes, and odors (direct and retronasal) and determine the fruitiness in extra-virgin olive oil.
Bitterness is perceived with the tongue and taste buds, whereas spiciness or pungency are tactile sensations instead of flavors.
BITTERNESS
Bitterness is one of the five basic tastes, specifically felt on the back of the tongue when tasting olive oil. Bitterness, together with fruit-iness and pungency, shows quality in olive oil. It shows that the oil was produced from healthy fruit at the optimum moment of maturity and in the best condition so as not to alter the oil.
As the oil matures, it loses its bitterness; it can even disappear. The sooner the olives are picked, the more pronounced the bitterness.
FILTRATION
Filtration refers to the removal of sediment in the oil after production. It is a mixture of olive pits, pulp, and skin. Filtration extends the shelf life of the olive oil.
FIRST COLD PRESS
A method in which the components are pressed at the lowest possible temperature to extract a part of the oil they contain. This pressing preserves the oil nutrients (vitamins, tocopherols, essential fatty acids, pigments, etc.) and the aromas. Olive pomace oil comes from the second pressing of the olive paste.
Pressure
Applying pressure used to be the main means of obtaining olive oil. These devices, which separate the oil and paste, are gradually being replaced by continuous and centrifugal systems, which allow better temperature and hygiene control.
Cold Press
Cold oil extraction processes refer to manufacturing olive oil without using heat. Under EU rules, extraction processes should be kept below 27° C (80° F). (For other vegetable oils, keep below 50° C [122° F]).
FLAVOR
This is the combination of olfactory, gustatory, and tactile sensations perceived during an olive oil tasting. This is how we can evaluate the sensory appreciation of the oil. The most volatile compounds are detected first, and then, by keeping the oil in the mouth a while longer, we pick up the less volatile compounds.
MINOR COMPONENTS
Extra-virgin olive oil consists of 99% fatty acids, and the remaining 1% includes the minor components. This designation can be imprecise and confusing, since this proportion confers crucial organoleptic and compositional characteristics to the oil. Minor components can range between 1%–2% of the oil, and they include natural antioxidants, pigments that give the color to the oil (from a golden yellow to a deep green), vitamins, and volatile components responsible for the aroma and fruitiness of the olive oil.
With an entirely mechanical extraction process, rigorous temperature control, and monitoring of the developmental stages of the fruit from the tree to the table, virgin olive oil keeps many of these substances, antioxidants, and flavorings that enhance its sensory and nutritional value. Only virgin olive oil with the "extra" mention guarantees that the minor components will be intact up to the deadline of optimal consumption and will have a beneficial effect on our health and palate.
OLEIC ACID
Olive oil comprises around 99% fats. The remaining 1% are minor components. The fats of olive oil are composed of triglycerides, comprising different kinds of fatty acids. Fatty acid composition is different and unique for each vegetable oil. Oleic acid is the primary source of monounsaturated fatty acids or omega-9s. Olive oil is mainly composed of oleic acid oil, and that is the reason we say it is monounsaturated oil.
The name comes from the Roman oleum, which means "oil," and composes 55%–80% of the fatty acids of olive oil. We can find oleic acid in many animal or vegetable species, such as walnut, hazelnut, avocado, sesame, peanut, rapeseed oil, duck fat, mutton, and beef.
Monounsaturated fatty acids have a beneficial effect on cholesterol, lowering the "bad" cholesterol without affecting the "good" cholesterol.
OLEOCANTHAL
(See "Pungency")
OLEUROPEIN
Oleuropein is one of the main phenolic compounds, a powerful antioxidant that you find in the olive leaf, the olive tree, and olive oil. You can find it in argan oil as well. This is what gives extra-virgin olive oil that pungent, bitter taste.
Besides the many health benefits of phenolic compounds, oleuropein also prevents the premature oxidation of the oil.
OLIVE OIL
Olive oil is the fat obtained from the fruit of the olive tree (Olea europaea L.), the olive. Only olive oils that include "virgin" are exclusively obtained by mechanical or physical extraction processes. The oil will be assigned to a category once it has passed the chemical and sensory analyses. Only then can it be bottled.
Extra-Virgin Olive Oil
An extra-virgin olive oil has no sensory defects, with a fruitiness greater than or equal to 0 and less than 0.8% acidity.
Virgin Olive Oil
A virgin olive oil has a fruitiness greater than or equal to 0, a sensory evaluation of the defect below 2.5 (score from 0 to 10, 10 being the maximum intensity of defect), and an acidity between 0.8% and 2%.
Lampante Olive Oil
Lampante olive oil has a sensory defect intensity higher than 3.5 and an acidity higher than 3.3%. These virgin olive oils cannot be consumed in this state and must be refined in order to be marketed.
The commercial category "olive oil" (not to be confused with the name of the family) is actually a lampante olive oil that has been refined and later mixed with a certain amount of extra-virgin olive oil to give the desired color, flavor, and smell.
Olive Pomace Oil
Olive pomace oil is a by-product or residue of olive oil. During the first pressing or first extraction, the centrifuge separates solids from liquids, thus obtaining oil and pomace separately. Pomace will be pressed again, and an oil of lower quality will be extracted and sent to the refinery. This oil is sold under the commercial category of olive pomace oil. Olive growers use pomace as fertilizers.
POLYPHENOLS
Polyphenols refer to the range of phytochemicals in olive oil, natural products which help with oxidation. Research shows they have a multitude of positive health properties. They also contribute to the stability of the oil as well as bitterness and pungency. (See "Antioxidants")
PUNGENCY
Also known as pepperiness, pungency is a tactile sensation that is felt in the throat or at the back of the palate; it's a sort of intense tingling, depending on the oil. Pungency is a sign of quality and freshness; it is one of the three positive attributes which oil tasters measure at the tasting session. Generally, olives picked green produce more pungent oil, but some varieties have a higher polyphenol content, which gives more pungency.
Oleocanthal, a phenolic compound with antioxidant and anti-inflammatory properties present in olive oil, is largely responsible for the pungency of extra-virgin olive oil.
PREDIMED
PREDIMED is an acronym that stands for PREvention con DIeta MEDiterránea (Prevention by Mediterranean Diet), a group of Spanish researchers who carry out studies based on the Mediterranean diet.
REFINING
Refining refers to the chemical and physical processes which remove all the unpleasant characteristics of oil; these include colors, tastes, and odors. No refined oil can legally carry the "extra virgin" label.
Lampante virgin olive oil is not suitable for consumption, and it requires a refining process to remove any chemical or sensory flaws such as free fatty acids, odors, and other impurities. Unfortunately, this chemical process also destroys all the high nutritional value items, such as pigments that give color, volatile compounds, and polyphenols. Once refined, olive oil has no more color, smell, or taste.
Olive oils without the words "virgin" or "pressed" on the label are obtained with chemical processes.
RETRO-OLFACTION OR RETRONASAL SENSATION
Retro-olfaction allows us to perceive the aromas of food once in the mouth. It is also referred to as retronasal or indirect olfaction. Retro means "backward" or "behind," and nasal refers to the nose. To understand retronasal sensations, you have to remember that the back of the mouth and nose are all connected via the retronasal passage; in other words, we carry the aroma into the nasal cavity from the mouth.
We can retronasally identify the volatile compounds in extra-virgin olive oil more easily from 28° C (82° F) onward. It's the increase in temperature of the oil in the mouth, together with the addition of saliva and air, that lets you better appreciate the aromas. When tasting olive oil, retro-olfaction allows the appreciation of new flavors and complements the sensory profile of olive oil.
SENSORY ANALYSIS (SENSORY EVALUATION)
Sensory analysis is an assessment of the organoleptic attributes of a product by human senses. The evaluation of olive oil involves taste, smell, and touch, but not sight. The color of olive oil does not tell you about the quality. That is why olive oil tasting glasses are blue. By doing this, the taster cannot see the contents and cannot be influenced by its color.
For virgin olive oil, tasting is done in parallel with the chemical analysis; both are necessary requirements before bottling can take place. Tasting evaluates the presence or absence of fruitiness, pungency, and bitterness. It allows the oil to be classified into three categories: extra virgin, virgin, and lampante. The analysis is performed by an olive oil tasting panel, a group of eight to twelve experts who follow the IOC protocol.
Currently, the human nose is the most efficient way to detect the palatability of a product or to check for sensory defects. Today, a more hedonistic approach to olive oil tasting is becoming more and more popular among consumers. They look more for pleasure, harmony, food and olive oil pairings, production know-how, and varieties of olives. And just as it is with wine, olive oil characteristics are very much appreciated in this new tasting style.
SHELF LIFE
Extra-virgin olive oil is a fresh product, and, consequently, it deteriorates with time. It loses its nutritional and aromatic qualities and oxidizes. Olive oil shelf life depends on how it is stored and the variety of olives used to make the oil. Shelf life can range from six to twenty-four months for varieties rich in polyphenols if they are stored under proper conditions.
It is important to protect extra-virgin olive oil from light, air, and wide temperature fluctuations, especially heat. It is best to consume a bottle within three to six months after opening.
TASTE
(See "Basic Tastes")
VARIETY
There are more than 1,740 varieties of olives in the world and lots more to be discovered. Just as different grapes create different flavors of wine, olive varieties give typical organoleptic, chemical, and agronomic character to the olive tree and its oil. Here are some of the best-known olive varieties: Picual from southern Spain, Picholine from France, Koroneiki from Greece, Coratina from Italy, and Chemlali from Tunisia.
REFERENCES
We have read and studied thousands of published scientific studies for this book and for our own interest. We cannot list them all here, but here are the references that have directly supported this book. They include the most useful medical and scientific data supporting the benefits of incorporating extra-virgin olive oil into our daily routine.
Part I
OLIVE OIL, THE CORNERSTONE OF MEDITERRANEAN HISTORY
Cardenas, Raphael, and Juan Vilar. The International Olive Oil Production Sector: A Descriptive Study of the Different Producing Countries. GEA Westfalia Separator Ibérica Centre for Excellence in Olive Oil: 2012.
"Olive Oil." Wikipedia.com. <https://en.wikipedia.org/wiki/Olive_oil>.
"Know Everything: History." Huiles-et-Olives.fr. <http://huiles-et-olives.fr/en/> know-everything/history/.
"L'Huile d'Olive." Adelirose.free.fr. <http://adelirose.free.fr/dossier/aliments/> huil_oliv.htm.
Barjol, Jean Louis. "L'économie mondiale de l'huile d'olive." EDP Sciences, 2014. <http://www.ocl-journal.org/articles/ocl/pdf/2014/05/ocl140010.pdf>.
M. Abdellatif Ghedira (the new International Olive Oil Council Executive
Director, who was nominated in January 2016) in discussion with the author, February 2016.
UNDERSTANDING FATS
"Actualisation des Apports Nutritionnels Conseillés pour les Acides Gras." ANSES, May 2011. <https://www.anses.fr/fr/system/files/NUT2006sa0359Ra.pdf>.
Anne-Laure Meunier (dietician nutritionist) in discussion with the author, November 2015. Lectures on nutrition and diet to EDNH (School of Dietetics and Human Nutrition) and the Courts of Paris Diderot. Coauthor of the book Power, Nutrition and Plans to Studyrama editions. <http://www.annelauremeunier.com>.
FROM OLIVE TO OIL
Uceda, Marino, Aguilera Mari Paz, and Mazzucchelli Isabel. Manual de Cata y Maridaje del Aceite de Oliva—Proceso de Elaboración: Del Árbol a la Bodega. Almuzara: 2010. Pp. 23-45.
Juan M. Olivares (producer of olive oil) in discussion with the author during the harvesting time in Hellín, Spain. November 2015. Further information can be found at <http://www.pagodepenarrubia.com/>.
"Valorisation des Sous-Produits de l'Olivier." Fellah-trade.com. <http://www.fellahtrade.com/fr/produire/conseil-technique/sous-produits-olivier>.
"Huesos de Aceituna para Producir Alimentos Funcionales contra la Hipertensión o el Colesterol." Mercacei.com. March 2016. <http://www.mercacei.com/noticia/45298/Actualidad/huesos-aceituna-produciralimentos-funcionales-contra-hipertensi%C3%B3n-o-colesterol.html>.
Part II
WONDER 1: OLIVE OIL CONTAINS OLEOCANTHAL, A NATURAL ANTI-INFLAMMATORY
Gimeno, Iván L. "Researcher to Share Details of 'Thrilling Discovery.'" Olive Oil Times, November 2014. <http://www.oliveoiltimes.com/olive-oil-health-news/beauchamp-shares-details-oleocanthal-discoverycordoba/42667>.
"Olive Oil Reduces Arthritis Inflammation." Arthritis Today Magazine, January 2015. <http://blog.arthritis.org/living-with-arthritis/olive-oil-anti-inflammatory-arthritis-diet/>.
Dr. Roberta Holt (University of California) in communication with authors. For further information from Dr. Holt, see <http://www.oleocanthal.es/un-oceano-de-salud/>.
Jose Antonio Amerigo (retired Spanish physician) and Daniel Garcia Peinado (chef in a restaurant in Málaga, Spain) in e-mail messages to authors, November 2015.
The Oleocanthal International Society: <http://www.oleocanthal.es/oleocanthal-international-society/>.
WONDER 2: OLIVE OIL COMBATS AND PREVENTS CANCER
Branson, Ken. "Ingredient in Olive Oil Looks Promising in the Fight Against Cancer." Rutgers Today, February 2015. <http://news.rutgers.edu/research-news/ingredient-olive-oil-looks-promising-fight-against-cancer/20150211#>. V8fN-bPtbUL.
Paul Breslin (professor in the Department of Nutritional Sciences at Rutgers University, a member at the Monell Chemical Senses Center, and adjunct professor in the Department of Anatomy and Cell Biology of the School of Dental Medicine at the University of Pennsylvania) in an e-mail message to authors, July 2015.
Oreste Gualillo (PharmD, PhD, head researcher at the NeuroEndocrine Interactions in Rheumatology and Inflammatory Diseases laboratory, Santiago University Clinical Hospital) in discussion with the author, March 2016.
"The Endocannabinoid System." Fundación Canna. <http://www.fundacion-canna.es/en/endocannabinoid-system>.
Mauro Maccorrone (professor at the University Campus Bio-Medico of Rome, Italy) in discussion with the author, May 2015.
Lee, Martin A. "Olive Oil, Cancer and the CB-1 Receptor." Projectcbd.org. November 2014. <https://www.projectcbd.org/article/olive-oil-cancer-and-cb-1-receptor>.
Filik, L., and O. Ozyilkan. "Olive-Oil Consumption and Cancer Risk." European Journal of Clinical Nutrition 57 (2003): 191. <http://www.nature.com/ejcn/journal/v57/m/full/1601497a.html>
Newmark, HL. "Squalene, Olive Oil, and Cancer Risk: Review and Hypothesis." Annals of the New York Academy of Sciences 889 (1999): 193–203. <http://www.ncbi.nlm.nih.gov/pubmed/10668494>.
Trichopoulou, Antonia, Christina Bamia, Lagiou Pagona, and Dimitrios Trichopoulo. "Conformity to Traditional Mediterranean Diet and Breast Cancer Risk in the Greek EPIC (European Prospective Investigation into Cancer and Nutrition) Cohort." American Society for Nutrition, 2010. <http://ajcn.nutrition.org/content/92/3/620.full>.
Tjan, Lukas T. S. "Squalene: The Miraculous Essential Omega 2 Oil." Scienceforlife.eu. <http://www.scienceforlife.eu/tekst%20what%20is%20squalenel.htm>.
Martínez-González, Miguel A. "Mediterranean Diet Plus Olive Oil Associated with Reduced Breast Cancer Risk." JAMA Internal Medicine, September 2015. <http://media.jamanetwork.com/newsitem/mediterranean-diet-plus-olive-oil-associated-with-reduced-breast-cancer-risk/>.
Cancer Research UK: <http://www.cancerresearchuk.org/health-professional/breast-cancerstatistics#heading-Zero>.
Michelle Harvie (UK dietician) in discussion with the author, January 2016. For further information, see <https://preventbreastcancer.org.uk/about-us/the-nightingale-centre/>.
José Gaforio (Immunology Division, Department of Health Sciences, University of Jaén, Spain) in an e-mail message to the authors, July 2015.
Sánchez-Quesada C., A. López-Biedma, and José J. Gaforio. "Oleanolic Acid, a Compound Present in Grapes and Olives, Protects against Genotoxicity in Human Mammary Epithelial Cells." Molecules 20, no. 8 (2015): 13670–13688. <http://www.mdpi.com/1420-3049/20/8/13670>.
Moral, R., R. Escrich, M. Solanas, et al. "Diets High in Corn Oil or Extra-Virgin Olive Oil Differentially Modify the Gene Expression Profile of the Mammary Gland and Influence Experimental Breast Cancer Susceptibility." European Journal of Nutrition 55 (2015): 1397. <http://link.springer.com/article/10.1007%2Fs00394-015-0958-2>.
"Les Huiles Végétales de A à Z." Leshuilesvegetales.fr. <http://www.leshuilesvegetales.fr/category/huiles-vegetales-a-z/>.
WONDER 3: OLIVE OIL BEAUTIFIES, PROTECTS, AND REJUVENATES BEAUTIFUL SKIN
Alech, Alice. An Olive Oil Tour of France: Revival of Savon de Marseille. CreateSpace Independent Publishing Platform, 2012. P. 33. <https://www.amazon.com/Olive-Oil-Tour-France/dp/1480208442>.
Passion Savon: <http://www.passion-savon.fr/en/>.
"Squalane." Wikipedia.org. <https://en.wikipedia.org/wiki/Squalane>.
"Independent Research." Bloom. <http://www.bloomassociation.org/en/our-actions/our-activities/independant-research/>.
Najmi, M., Z. Vahdat Shariatpanahi, M. Tolouei, and Z. Amiri. "Effect of Oral Olive Oil on Healing of 10–20% Total Body Surface Area Burn Wounds in Hospitalized Patients." Burns 41, no. 3 (2015): 493–496. <http://www.ncbi.nlm.nih.gov/pubmed/25306088>.
Nia Hafsia (olive oil soap maker) in discussion with the author, 2016.
WONDER 4: OLIVE OIL KEEPS BONES HEALTHY
"DEXA (DXA) scan." Nhs.uk. <http://www.nhs.uk/conditions/DEXA-scan/Pages/Introduction.aspx>.
Benetou, V., P. Orfanos, U. Pettersson-Kymmer, et al. "Mediterranean Diet and Incidence of Hip Fractures in a European Cohort." Osteoporosis International 24, no. 5 (2013): 1587–1598. <http://link.springer.com/article/10.1007/s00198-012-2187-3>.
Saleh, Nermine K., and Hanan A. Saleh. "Olive Oil Effectively Mitigates Ovariectomy-Induced Osteoporosis in Rats." BMC Complement Alternative Med, 2011. <http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3045997/>.
Filip R., S. Possemiers, A. Heyerick, I. Pinheiro, G. Raszewski, M. J. Davicco, and V. Coxam. "Twelve-Month Consumption of a Polyphenol Extract from Olive (Olea europaea) in a Double Blind, Randomized Trial Increases Serum Total Osteocalcin Levels and Improves Serum Lipid Profiles in Postmenopausal Women with Osteopenia." The Journal of Nutrition, Health, and Aging 19, no. 1 (2015): 77–86. <http://www.ncbi.nlm.nih.gov/pubmed/25560820>.
Fernández-Real, José Manuel, Mónica Bulló, José Maria Moreno-Navarrete, Wifredo Ricart, Emilio Ros, Ramon Estruch, and Jordi Salas-Salvadó. "A Mediterranean Diet Enriched with Olive Oil Is Associated with Higher Serum Total Osteocalcin Levels in Elderly Men at High Cardiovascular Risk." The Journal of Clinical Endocrinology and Metabolism 97, no. 10 (2012): 3792–3798. <http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3462931/>.
WONDER 5: OLIVE OIL SLOWS ALZHEIMER'S DISEASE
Amal Kaddoumi (PhD, Associate Professor of Pharmaceutics, Department of Basic Pharmaceutical Sciences, School of Pharmacy, The University of Louisiana at Monroe) in discussion and e-mail messages with the author, December 2015.
Abuznait, Alaa H., Hisham Qosa, Belnaser A. Busnena, Khalid A. El Sayed, and Amal Kaddoumi. "Olive-Oil-Derived Oleocanthal Enhances β-Amyloid Clearance as a Potential Neuroprotective Mechanism against Alzheimer's Disease: In Vitro and in Vivo Studies." ACS Chemical Neuroscience 4, no. 6 (2013): 973–982. <http://pubs.acs.org/doi/abs/10.1021/cn400024q>.
"About Alzheimer's Disease: Alzheimer's Basics." National Institute on Aging. <https://www.nia.nih.gov/alzheimers/topics/alzheimers-basics>.
Netburn, Deborah. "A Mediterranean Diet Could Protect Your Brain from Shrinking in Old Age." Los Angeles Times, October 2015. <http://www.latimes.com/science/sciencenow/la-sci-sn-mediterranean-diet-brain-shrinkage-20151020-story.html>.
Valls-Pedret, Cinta, Aleix Sala-Vila, Mercè Serra-Mir, Dolores Corella, Rafael de la Torre, Miguel Ángel Martínez-González, Elena H. Martínez-Lapiscina, Montserrat Fitó, Ana Pérez-Heras, Jordi Salas-Salvadó, Ramon Estruch, and Emilio Ros. "Mediterranean Diet and Age-Related Cognitive Decline." JAMA Intern Med 175, no. 7 (2015): 1094–1103. <http://archinte.jamanetwork.com/article.aspx?articleid=2293082>.
Rapaport, Lisa. "Mediterranean Diet with Olive Oil, Nuts Linked to Healthier Brain." Reuters.com, May 2015. <http://www.reuters.com/article/us-brain-health-diet-idUSKBN0NW1PN20150511>.
"Mediterranean Diet." Patient.info. <http://www.patient.co.uk/health/health-benefits-of-the-mediterranean-diet>.
Alloush, Abed. "Greek Island Crete: Healthy Diet Key to Long Life." Eu.greekreporter.com, December 2013. <http://eu.greekreporter.com/2013/12/18/greek-island-crete-healthy-diet-key-to-long-life/>.
Fallon, Sally, and Mary G. Enig. Nourishing Traditions: The Cookbook That Challenges Politically Correct Nutrition and the Diet Dictocrats, 2nd Revised Edition. NewTrends Publishing Inc.: White Plains, MD, 2001.
WONDER 6: OLIVE OIL HELPS REDUCE THE RISK OF DIABETES
Sergio López and Francisco J. G. Muriana (scientists from the Institute of Fat in Seville, Spain) in an e-mail message to authors, April 2016.
"Diabetes." World Health Organization, June 2016. <http://www.who.int/mediacentre/factsheets/fs312/en/>.
Kenwright, Sarah. "Olive Oil May Offer Diabetes Protection." Chemistry World, April 2014. <https://www.chemistryworld.com/research/olive-oil-may-offer-diabetes-protection/7252.article>.
Guasch-Ferré, M., A. Hruby, J. Salas-Salvadó, M. A. Martínez-González, Q. Sun, W. C. Willett, F. B. Hu. "Olive Oil Consumption and Risk of Type 2 Diabetes in US Women." The American Journal of Clinical Nutrition 102, no. 2 (2015): 479–486. <http://www.ncbi.nlm.nih.gov/pubmed/26156740>.
Trichopoulou, Antonia, Tina Costacou, Christina Bamia, and Dimitrios Trichopoulos. "Adherence to a Mediterranean Diet and Survival in a Greek Population." The New England Journal of Medicine 348 (2003): 2599–2608. <http://www.nejm.org/doi/full/10.1056/NEJMoa025039>.
M. P. de Bock (author of "Olive (Olea europaea L.) Leaf Polyphenols Improve Insulin Sensitivity in Middle-Aged Overweight Men: A Randomized, Placebo-Controlled, Crossover Trial") in e-mail messages with the authors, April 2016. More information on his research can be found at <http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0057622>.
"L'Industrie du Beurre Finance une Étude Scientifique pour en Redorer le Blason sur le Plan Sanitaire . . . C'est Raté." Nouvelobs.com, November 2015. <http://tempsreel.nouvelobs.com/en-direct/a-chaud/6710-santeindustrie-beurre-finance-etude-scientifique-redorer.html>.
Ferdman, Roberto A. "Study about Butter, Funded by Butter Industry, Finds That Butter Is Bad for You." The Sydney Morning Herald, August 2015. <http://www.smh.com.au/national/health/study-about-butterfunded-by-butter-industry-finds-that-butter-is-bad-for-you-20150809-giuuia.html>.
WONDER 7: OLIVE OIL HELPS REDUCE STROKES AND HEART ATTACKS
"Go Red for Women." American Heart Association. <https://www.goredforwomen.org/>.
Jordi Salas-Salvadó (author of "Olive Oil Intake and Risk of Cardiovascular Disease and Mortality in the PREDIMED Study," results published February 2014) in an e-mail message to the authors, 2015. More information on the research is at <http://bmcmedicine.biomedcentral.com/articles/10.1186/1741-7015-12-78>.
Iannuzzi, Arcangelo, Egidio Celentano, Salvatore Panico, Rocco Galasso, Giuseppe Covetti, Lucia Sacchetti, Federica Zarrilli, Mario De Michele, and Paolo Rubba. "Dietary and Circulating Antioxidant Vitamins in Relation to Carotid Plaques in Middle-Aged Women." The American Journal of Clinical Nutrition 76 (2002): 582–587. <http://ajcn.nutrition.org/content/76/3/582.full.pdf>.
Dr. William Mullen (main researcher of the study) in an e-mail message to the authors, February 2016.
"Regular Consumption of Olive Oil Can Improve Heart Health." University of Glasgow, November 2014. <http://www.gla.ac.uk/research/news/2014archive/headline_376522_en.html>.
"DMU Research on 'Healthiest' Cooking Oils Revealed on BBC's Trust Me, I'm a Doctor." De Montfort University Leicester, July 2015. <http://www.dmu.ac.uk/about-dmu/news/2015/july/dmu-research-on-healthiest-cooking-oils-revealed-on-bbcs-trust-me-im-a-doctor.aspx>.
Hoffman, Richard, and Mariette Gerber. "Dietary Surveys and Nutritional Epidemiology: Can Rapeseed Oil Replace Olive Oil as Part of a Mediterranean-Style Diet?" British Journal of Nutrition 112, no. 11 (2014): 1882–1895. <http://www.ncbi.nlm.nih.gov/pubmed/25322908>.
Pedersen, A., M. W. Baumstark, P. Marckmann, H. Gylling, and B. Sandström. "An Olive Oil-Rich Diet Results in Higher Concentrations of LDL Cholesterol and a Higher Number of LDL Subfraction Particles Than Rapeseed Oil and Sunflower Oil Diets." Journal of Lipid Research 41, no. 12 (2000): 1901–1911. <http://www.ncbi.nlm.nih.gov/pubmed/11108723>.
"Health and Olive Oil." The Olive Oil Source. <http://www.oliveoilsource.com/page/health-and-olive-oil>.
Brackett, Robert E. "Letter Responding to Health Claim Petition Dated August 28, 2003: Monounsaturated Fatty Acids from Olive Oil and Coronary Heart Disease (Docket No 2003Q-0559)." US Food and Drug Administration, November 2004. <http://www.fda.gov/Food/IngredientsPackagingLabeling/LabelingNutrition/ucm072963.htm>.
Gallego Edelfelt, Eric. "Stress oxydatif et antioxydants." Olivae 121 (2015): 31–34. <http://www.internationaloliveoil.org/store/view/101-olivae-121-english>.
"Frying with Olive Oil." International Olive Oil Council. <http://www.internationaloliveoil.org/web/aa-ingles/oliveWorld/aceite3.html>.
Part III
UNDERSTANDING THE LABEL AND BUYING EXTRA-VIRGIN
Granitto, Ylenia. "Police Uncover 7,000-Ton Olive Oil Fraud in Italy." Olive Oil Times, December 2015. <http://www.oliveoiltimes.com/olive-oil-business/police-uncover-7000-ton-olive-oil-fraud-initaly/49929>.
"Règlement d'Exécution (UE) No 1335/2013 de la Commission du 13 Décembre 2013 Modifiant le Règlement d'Exécution (UE) No 29/2012 Relatif aux Normes de Commercialisation de l'Huile d'Olive." Journal Officiel de l'Union Européenne, December 2013. <http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2013:335:0014:0016:FR:PDF>.
"Les Dénominations de Vente Autorisées pour l'Huile d'Olive au Stade du Commerce de Détail." Afidol.org. <http://afidol.org/commercant/letiquette-et-les-mentions/>.
"Huiles d'Olive: Des Améliorations sur l'Étiquetage." Economie. gouv.fr, June 2014. <http://www.economie.gouv.fr/dgccrf/huiles-dolive-des-ameliorations-sur-letiquetage>.
Pouyet, Brigitte, and Véronique Ollivier. "Réglementations sur l'Étiquetage et la Présentation des Huiles d'Olive." Oilseeds & Fats, Crops, and Lipids 21, no. 5 (2014). <http://www.ocl-journal.org/articles/ocl/pdf/2014/05/ocl140005.pdf>.
"Olive Oil Cleared for Heart-Healthy Claim." WebMD Health News, 2004. <http://www.webmd.com/heart/news/20041101/olive-oil-cleared-for-heart-healthy-claim>.
"Olive Oil: Conditions of Competition between U.S. and Major Foreign Supplier Industries." U.S. International Trade Commission. Investigation No. 332-537, USITC Publication 4419, August 2013. <https://www.usitc.gov/publications/332/pub4419.pdf>.
Mueller, Tom. "How to Buy Great Olive Oil." Truthinoliveoil.com. <http://www.truthinoliveoil.com/great-oil/how-to-buy-great-olive-oil/>.
"Recommendation Issued to Members by the IOC Council." International Olive Oil Council, 2014. <http://www.internationaloliveoil.org/news/view/666-year-2014-news/516-recommendation-issued-to-members-by-the-ioc-council>.
DIFFERENT CATEGORIES OF OLIVE OIL
"Designations and Definitions of Olive Oils." International Olive Oil Council. <http://www.internationaloliveoil.org/estaticos/view/83-designations-and-definitions-of-olive-oils?lang=en_US>.
"Olive Oil: Conditions of Competition between U.S. and Major Foreign Supplier Industries." U.S. International Trade Commission. Investigation No. 332-537, USITC Publication 4419, August 2013. <https://www.usitc.gov/publications/332/pub4419.pdf>.
Bushdid, C., L. Vosshall, et al. "Humans Can Discriminate More Than 1 Trillion Olfactory Stimuli." Science 343 (2014): 1370–1372. <http://vosshall.rockefeller.edu/assets/file/BushdidScience2014.pdf>.
"Testing methods." International Olive Oil Council, 2007–2015. <http://www.internationaloliveoil.org/estaticos/view/224-testing-methods>.
OLIVE OIL IN THE KITCHEN
Dra. Jessica del Pilar Ramírez Anaya and Dra. Cristina Samaniego (Pharmacy Department, University of Granada, Spain) in discussion with the author, February 2016.
Del Pilar Ramírez Anaya, Jessica. "Influencia de las Técnicas Culinarias sobre el Contenido de Polifenoles y Capacidad Antioxidante en Hortalizas de la Dieta Meiterránea (Phenols and the Antioxidant Capacity of Mediterranean Vegetables Prepared with Extra Virgin Olive Oil Using Different Domestic Cooking Techniques)." Doctoral thesis, Universidad de Granada, 2014. <http://hera.ugr.es/tesisugr/22622457.pdf>.
Uceda, Marino, Aguilera Mari Paz, and Mazzucchelli Isabel. Manual de Cata y Maridaje del Aceite de Oliva—Proceso de Elaboración: Del Árbol a la Bodega. Almuzara: 2010.
"Variedades de Olivo." Variedadesdeolivo.com. <http://www.variedadesdeolivo.com/>.
"Stratégie Mondiale pour l'Alimentation, l'Exercice Physique et la Santé." Organisation Mondiale de la Santé. <http://www.who.int/dietphysicalactivity/diet/fr/>.
"Fritures, Huiles et Santé." Cerin.org, 2015. <http://www.cerin.org/actualite-scientifique/fritures-huiles-et-sante.html>.
CHARTS AND ILLUSTRATIONS
Barjol, Jean Louis. "L'économie mondiale de l'huile d'olive." EDP Sciences. 2014. <http://www.ocl-journal.org/articles/ocl/pdf/2014/05/ocl140010.pdf>
J.-L. Barjol : OCL 2014, 21(5) D502 Tableau 2. Classement prévu en 2013/2014 des 10 premiers pays consommateurs d'huile d'olive et volumes consommés (en milliers
Illustration "Consommation par personne/an" by Théo Gorjean. AFIDOL. <http://wesavoirfaire.com>
WeSavoirFaire : Visites d'entreprises et Tourisme industriel. wesavoirfaire.com
WesavoirFaire vous souhaite la bienvenue en entreprise : Visites d'entreprises et Tourisme industriel en France. Faites du tourisme de savoir-faire.
Sundeep, Mishra. "Cooking Oils for Heart Health." Department of Cardiology, Sir Ganga Ram Hospital, New Delhi, India. February 2012. http://www.journalofpreventivecardiology.com/pdf/Issue3/Cooking%20oils%20for%20 heart%20health.pdf
"Mediterranean Diet." Dietary Guidelines for Adults in Greece. Ministry of Health and Welfare Supreme Scientific Health Council. 1999. <http://www.nut.uoa.gr/dietaryENG.html>
ACKNOWLEDGMENTS
Almost all of the data in this book comes from talking with the experts—researchers, growers, producers, nutritionists, and decision makers, all experts in their field. We are grateful for all the time they spent with us, for sharing knowledge and valuable insight. We could not rely only on publicized material for 7 Wonders.
At the end of this book, we included a few simple, healthy recipes. We wanted to keep them short and invited only a few people to contribute. We thank all those who shared these gems.
We are grateful to Brooke Jorden for all her encouragement and support right from the beginning of our decision to write this book. Our editor, Lindsay Sandberg, guided us patiently and generously with her time; we thank her for her sharp editing of The y Wonders of Olive Oil. Special thanks also to our publishers and to all the team at Familius for their advice, help, and support. Without them, this book would not have been possible.
FROM ALICE:
I am grateful to those who explained their medical conditions to me. They are all real people, though we've changed their names— from Joanna, who suffered from breast cancer and whom I met as a patient, to Susan, who spent time explaining her fears of the disease and what it is like to be an osteoporosis victim. We did not disclose their real names, for obvious reasons, but through them, we better understood the different diseases.
Thank you to the National Osteoporosis Society based in Manchester in the United Kingdom and to Genesis Breast Cancer Prevention, also in Manchester, for useful information, facts, and figures.
The researchers I contacted came from the United States, Italy, New Zealand, the UK, and Spain. Dr. Amal Kaddoumi from the United States and Dr. Mauro Maccorrone from Italy were particularly patient every time I bombarded them with more questions. I thank them for describing their studies in a way I could follow.
I am grateful to Janet Amos, the specialist nurse practitioner who filled me in on the seriousness of diabetes, explaining the challenges she faces working in a busy practice where almost all of her patients suffer from the disease.
Thank you to Nia Hafsia, a gifted and talented soap maker, for spending precious time explaining her work and the intricacies of healthy soap making.
I would like to thank all my friends and work colleagues for their inspiration and support during the writing of 7 Wonders.
FROM CÉCILE:
Thank you to the International Olive Council for allowing me to follow the training course in virgin olive oils tasting in 2014, and especially to Juliette Cayol, press officer, who was always available and responsive to our various queries and research information.
I am grateful to Mr. Abdellatif Ghedira, the new Executive Director of the International Olive Council, for taking time from his busy schedule to answer my questions.
Thank you to Sylvie Borrat Coirault, a pharmacy technician and herbalist, who very patiently helped me to understand the collective expert report "Updating the recommended dietary allowances for fatty acids" published in 2011 by ANSES, the French national agency for food safety and environment.
I am grateful to Anne-Laure Meunier, dietitian nutritionist, who works in Paris, for all her good advice on the Mediterranean diet and fatty acids.
Thank you to Juan Olivares, his family, and the Dominguez family for hosting me in October 2015 during the olive harvest in the olive grove of Pago de Peñarrubia in Hellín, Spain. I was able to follow for a week the harvesting time and observe the preparation of the mill before the reception of the olives for pressing. I recently had the chance to taste the first production of olive oil under the brand Pago de Peñarrubia. This was a very successful first production and excellent extra-virgin olive oil!
My special thanks also to Dr. Jessica del Pilar Ramírez Anaya for her thesis, "Phenols and the Antioxidant Capacity of Mediterranean Vegetables Prepared with Extra-Virgin Olive Oil Using Different Domestic Cooking Techniques," and to her thesis supervisor at the University of Granada in Spain, Dr. Cristina Sánchez Samaniego. This fascinating study gives us a new approach to the advantages of using extra-virgin olive oil in cooking, especially for frying.
ABOUT THE AUTHORS
ALICE ALECH is a writer and a qualified X-ray technologist (radiographer), specializing in mammography. She has worked and lived in France, the UK, Australia, and the West Indies. She discovered the wonderful world of olive oil when she moved to the South of France, to Provence—olive oil country—where the taste, flavor, and quality of olive oil are crucial. Drawn into olive oil culture, she has been covering olive oil news in France for many years, meeting olive growers, olive oil producers, and chefs—men and women who work hard and believe in what they are doing. It started a real thirst for more knowledge. With her health background, Alice was particularly interested to talk to the researchers studying the health benefits of extra-virgin olive oil. Alice feels strongly that a healthy, well-balanced diet can lower our risk of chronic disease.
Alice is the self-published author of An Olive Oil Tour of France, an appreciation of Provence's and Corsica's contributions to the world of olive oil.
CÉCILE LE GALLIARD comes from the north of France, where butter is traditionally used for cooking. After gaining a diploma in communication, she moved to Spain to continue her studies. This was where her olive oil education started, where she changed her cooking habits and became passionate about olive oil. Cécile Le Galliard is now a French expert in olive oil tasting. She is a recent graduate of the University of Jaén in Spain in virgin olive oil tasting. She was selected by The International Olive Council as the French candidate for the training and is now recognized as a skilled professional in her field. Cécile is a web journalist on www.jusdolive.fr and works as a consultant in olive oil, specializing in the creation of oil cellars, training, and olive oil tasting for professionals. Her olive oil blog, Jus d'olive, written in French, currently attracts 50,000 visitors per year. Today, Cécile is involved in olive oil training and promoting olive oil. She passed the Savantes Tasting Skills test in March 2016, an individual assessment, and is now a Savantes associate.
Cécile feels that the olive oil business is booming, and she is happy to continue learning more about the goodness and flavor of extra-virgin olive oil.
ABOUT FAMILIUS
VISIT OUR WEBSITE: www.familius.com
Join Our Family: There are lots of ways to connect with us! Subscribe to our newsletters at www.familius.com to receive uplifting daily inspiration, essays from our Pater Familius, a free ebook every month, and the first word on special discounts and Familius news.
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The most important work you ever do will be within the walls of your own home.
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{"url":"http:\/\/openstudy.com\/updates\/50c285b3e4b066f22e104f54","text":"## xKingx 2 years ago How do you write this in simplest form?\n\n1. xKingx\n\n2. geoffb\n\n$\\LARGE (b^{x})^{y} = b^{x \\times y}$\n\n3. geoffb\n\nRemember the outside exponent also applies to the 8.\n\n4. mathgirl73\n\neverything must be raised to -2\/3 power . when raise a product by a power you multiply the exponents\n\n5. xKingx\n\n@geoffb Can you explain how I'd use that formula with this problem?\n\n6. mathgirl73\n\n|dw:1354925885023:dw|\n\n7. mathgirl73\n\n|dw:1354925885023:dw|\n\n8. xKingx\n\nCan you explain what you did in each step so I can make sure I understand it?\n\n9. mathgirl73\n\nfrom the image the exponents looks negative.\n\n10. xKingx\n\nThey are\n\n11. mathgirl73\n\neverything has to be raised by the power that is outside the parentheses. so -2\/3\n\n12. mathgirl73\n\neverything has to be raised by the power that is outside the parentheses. so -2\/3\n\n13. mathgirl73\n\n8 raised to the -2\/3 and a^-3 raised to the -2\/3\n\n14. mathgirl73\n\nso you have 8^-2\/3\n\n15. mathgirl73\n\nyou have to multiply a^-3 by -2\/3\n\n16. mathgirl73\n\nso you get in the numerator -3 times -2 = 6 and the denominator 3 times 1=3\n\n17. mathgirl73\n\nso you get in the numerator -3 times -2 = 6 and the denominator 3 times 1=3\n\n18. mathgirl73\n\n6\/3=2 so you have a^2\n\n19. mathgirl73\n\nnow you will have 8^-2\/3 a^2\n\n20. mathgirl73\n\nyou can't have negative exponents so you drop 8-2\/3 down in the denominator and it becomes 8^2\/3\n\n21. mathgirl73\n\nnow you have a^2\/8^2\/3\n\n22. mathgirl73\n\nif you havent learned how to convert fractional exponents to radicals then you can stop there\n\n23. mathgirl73\n\nbut if not you convert 8 2\/3 to a radical\n\n24. mathgirl73\n\n|dw:1354927080454:dw|\n\n25. xKingx\n\nI think I get it lol\n\n26. mathgirl73\n\nok\n\n27. geoffb\n\n\"you can't have negative exponents\" Sure you can. The rest of what you said looks good though.\n\n28. mathgirl73\n\nyou can but 99.9% of the time n Algebra, teachers want the answer using positive exponents.","date":"2015-10-07 02:51:04","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\": 0.8273475170135498, \"perplexity\": 3290.1920717808266}, \"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-2015-40\/segments\/1443736680773.55\/warc\/CC-MAIN-20151001215800-00063-ip-10-137-6-227.ec2.internal.warc.gz\"}"}
| null | null |
Myrmecaelurus medius är en insektsart som beskrevs av Navás 1913. Myrmecaelurus medius ingår i släktet Myrmecaelurus och familjen myrlejonsländor. Inga underarter finns listade i Catalogue of Life.
Källor
Myrlejonsländor
medius
|
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| 3,325
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I finally got to see Grandpa's apartment, it seems to suit him really well. He showed me his WWII photos (glad Brett told me about them!), and Effie Bradley's memoranda.
We played snooker. Badly, but it was fun! Screwed up the rules, apparently you don't get awarded negative points for a foul… I kind of liked that house rule though, takes it in to QI territory. Sam showed us a video of how it's actually supposed to be done.
We met up with Tara Vancil and Paul Frazee in Austin, which was excellent! What lovely people. We scheduled that early, anticipating bad jet lag, but got an unexpected second wind on our way back. Likely related to a slightly odd cab driver. So instead of turning in, we went to The White Horse – very Twin Peaks! need to practice my two step – and Las Trancas.
Made Victorias for the first time with my mom. That was fascinating and fun, with really helpful tips from Grandma's notes and from my uncle.
Also scanned the Haywood family history docs. I didn't know we had so much written down! Need to sit down and read through it, ideally also type up that cursive doc. I wish I had taken the time to go through this when Nan-nan and Pop-pop were still around.
Spent NYE in Boston with good friends, lots of food, drinks, and games. Can't ask for much more, it was great to be able to see so many people in one place.
This book is more about the journey than than a climactic set-piece, as his books usually seem to be. From a Guardian review (don't read the review, it gives too much away and kills the vibe): "Significantly, Murakami's painter likes to leave some of his portraits unfinished. One has the sense that the author does, too." Count me in as well. I really enjoyed it. Still thinking about the main character's musing that maybe a real piece of art is something that is necessary. Necessary is in the eye of the beholder, but that's part of the point. What about when applied to other work, careers, creative endeavours?
I get it, I get why this book is considered foundational. Took me too long to get round to it. Glad to have read it and look forward to reading more of her work.
This was always going to be a winner for me. Gaiman has been one of my favourites for a long time, I love mythology, and I love short story collections. Also SB is playing God of War right now so this book is perfectly timed. I think Gaiman struck the perfect tone, a relatively "straight" telling with little inklings of his own voice here and there.
I had never heard of Jemisin before this holiday and had two very close friends (that don't know each other) recommended her to me in the span of two days. Extremely impressive world building, unlike anything else I've read before. I'm unable to anticipate most of her next storytelling steps (can be unusual in popular fantasy). Very excited about her body of work. I may have started too many series now though.
I got back in to a good reading habit in part because I renewed my Boerne library card, and also because I discovered that Libby can now send books to your Kindle (US libraries only). Pretty cool.
|
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| 9,452
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/* EXPERIMENTAL (really) */
/* Copyright (c) 2009 Google Inc.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package com.google.orkut.client.api;
import org.json.me.JSONObject;
/**
* Represents an incoming friend request entry.
*
* @author Sachin Shenoy
*/
public class FriendRequestEntry {
private String userId;
private String message;
private OrkutPerson userProfile;
public FriendRequestEntry(JSONObject json) {
userId = Util.JSONUtil.getRequiredStringField(Fields.FROM_USER_ID, json);
message = json.optString(Fields.BODY);
JSONObject userProfileJson =
Util.JSONUtil.getRequiredJSONObjectField(Fields.FROM_USER_PROFILE, json);
userProfile = new OrkutPerson(userProfileJson);
}
/**
* Returns Id of the user who sent the friend request.
*/
public String getUserId() {
return userId;
}
/**
* Returns the message received along with the friend invite.
*/
public String getMessage() {
return message;
}
/**
* Returns profile of the user who sent the friend invite.
*/
public OrkutPerson getUserProfile() {
return userProfile;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 4,127
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Colban, 7. Earl of Fife (auch Colban Macduff, Earl of Fife) († um 1270) war ein schottischer Magnat.
Colban entstammte der schottischen Clan Macduff. Er war ein Sohn von Malcolm, 6. Earl of Fife. Beim Tod seines Vaters 1266 war er noch minderjährig, als er den Titel Earl of Fife erbte. Seine Mutter heiratete in zweiter Ehe einen Sohn des Earl of Mar. Dieser gewann dadurch erheblichen Einfluss in Fife, bis Colban Ende der 1260er Jahre volljährig wurde. Deshalb spielte er bis zu seinem frühen Tod in der schottischen Politik kaum eine Rolle und diente nur gelegentlich als Zeuge von königlichen Urkunden.
Colban war offenbar bereits als Jugendlicher verheiratet worden, denn bei seinem Tod war sein Sohn bereits acht Jahre alt. Seine Frau Anna war vielleicht eine Tochter von Alan Durward. Mit ihr hatte er mindestens zwei Kinder:
Duncan, 8. Earl of Fife (1262–1289)
Isabel (um 1270–nach 1313) ⚭ John Comyn, 7. Earl of Buchan
Sein Erbe wurde sein Sohn Duncan. Die Verwaltung von Fife vergab König Alexander III. während der Minderjährigkeit des Erben an seinen eigenen Sohn Alexander.
Weblinks
Einzelnachweise
Familienmitglied des Clan MacDuff
Earl of Fife
Schotte
Geboren im 13. Jahrhundert
Gestorben im 13. Jahrhundert
Mann
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| 2,882
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Q: How to make a file run on startup using batch or VBScript? I need to schedule a task using a batch file that will make my program run on startup. However, I can't find out how to do that using any scripts that are pre-installed on Windows.
A: Use: Control panel -> Administrative tools -> Task scheduler
A: Simply put a shortcut to your batch file within the startup group
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 6,562
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Q: Radio Button Showing as a Hyperlink? I have four radio buttons, and although they display correctly on desktop web browsers when I view them on the iOS simulator and iPhone they "size3" radio label displays as a hyperlink? Why is this or how can I correct the issue?
http://jsfiddle.net/jasonniebauer/eLMqk/
HTML
<div id="surv_radio3">
<p>
Square Footage
</p>
<div>
<input type="radio" id="size1" name="sqft"/>
<label for="size1">
0-250
</label>
<input type="radio" id="size2" name="sqft"/>
<label for="size2">
251-500
</label>
</div>
<div>
<input type="radio" id="size3" name="sqft"/>
<label for="size3">
501-2000
</label>
<input type="radio" id="size4" name="sqft"/>
<label for="size4">
2000+
</label>
</div>
</div>
CSS
#surv_radio3 > div:nth-of-type(1) input[type="radio"] + label,
#surv_radio3 > div:nth-of-type(2) input[type="radio"] + label {
box-sizing:border-box;
padding: 1rem 3rem 1rem 3rem;
width: 100%;
display: inline-block;
color: #BDC3C7;
background-color: #F2F2F2;
text-align: center;
vertical-align: middle;
cursor: pointer;
border-radius: 3px;
font-size: 1.125em;
margin-bottom: .5rem;
}
#surv_radio3 input[type="radio"] {
display: none;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 35
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\section{Introduction }
\label{introduction}
Given two binary addends $A = (a_n \dots a_1)$ and $B = (b_n \dots b_1)$,
where index $n$ denotes the most significant bit, their sum $S = A+B$
has $n+1$ bits.
We are looking for a logic circuit, also called an \emph{adder}, that
computes $S$. Here, a \emph{logic circuit} is a non-empty connected
acyclic directed graph consisting of nodes that are either
\emph{gates} with incoming and outgoing edges, \emph{inputs} with at
least one outgoing edge and no incoming edges, or \emph{outputs} with
exactly one incoming edge and no outgoing edges. Gates represent one
or two bit Boolean functions, specifically \textsc{And}, \textsc{Or},
\textsc{Xor}, \textsc{Not} or their negations. A small example is
shown on the right side of Figure~\ref{fig:pfx-to-gate3}. The
\emph{fan-in} is the maximum number of incoming edges at a vertex, and
it is bounded by two for all gates.
The main characteristics in adder design are the {\it depth}, the {\it size},
and the {\it fan-out} of a circuit.
The depth is defined as the maximum length of a directed path in the logic
circuit and is used as a measure for its speed. The lower the
depth, the faster is the adder. The size is the total
number of gates in the circuit, and is used as a measure for the space and power
consumption of the adder, both of which we aim to minimize. The
fan-out is the maximum number of outgoing edges at a vertex. High
fan-outs increase the delay and require additional repeater gates (implementing the identity function) in physical design.
Thus, when comparing the depth of adder circuits, their fan-out should be considered as well; we will focus on the usual fan-out bound of two. Circuits with higher fan-outs can be transformed into fan-out two circuits by replacing each interconnect with high fan-out by a balanced binary {\it repeater tree}, \.i.e. the underlying graph is a tree and all gates are repeater gates. However, this increases the size linearly and the depth logarithmically in the fan-out.
Hoover, Klawe, and Pippenger [1984] gave a smarter way to bound the fan-out of a given circuit,
but it would also triple the size and depth in our case of gates with two inputs.
Using logic circuit depth as a measure for speed
is a common practice in logic synthesis that simplifies many aspects of physical hardware.
In CMOS technology, \textsc{Nand}/\textsc{Nor} gates are faster than
\textsc{And}/\textsc{Or} gates and efficient implementations exist for
integrated multi-input {\sc And-Or}-Inversion gates and {\sc Or-And}-Inversion
gates.
We assume that a {\it technology mapping} step
\cite{Chatterjee+techmap2006,keutzer88} translates the adder circuit
after logic synthesis using logic gates that are best for the given
technology.
Despite its simplicity, the depth-based model is at the core of
programs such as {\sc BonnLogic} \cite{bonnlogic} for refining carry bit circuits,
which is an integral part of the current IBM microprocessor design flow.
Recently, we reduced the running time for computing such carry bit circuits
significantly from $\mathcal{O}(n^3)$ to $\mathcal{O}(n \log n)$
\cite{held+spirkl:2014}.
Exemplary, for all newly proposed adder circuits in this paper we will demonstrate
how to efficiently transform them into equivalent circuits
using only \textsc{Nand}/\textsc{Nor} and \textsc{Not} gates.
Like most existing adders, we use the notion of generate and propagate signals, see \cite{sklansky,brent,knowles}.
For each position $1 \leq i \leq n$, we compute a
\emph{generate signal} $y_i$ and a \emph{propagate signal}
$x_i$, which are defined as follows:
\begin{equation}
\begin{array}{rl}
x_i &= a_i \oplus b_i,\\
y_i &= a_i \wedge b_i,
\end{array}
\label{eqn:generate-and-propagat}
\end{equation}
where $\wedge$ and $\oplus$ denote the binary \textsc{And} and
\textsc{Xor} functions. The \emph{carry bit} at position $i+1$ can be
computed recursively as $c_{i+1} = y_i \vee (x_i \wedge c_i)$, since
there is carry bit at position $i+1$ if the $i$-th bit of both inputs
is $1$ or, assuming this is not the case, if at least one (hence
exactly one) of these bits is $1$ and there was a carry bit at
position $i$.
The first carry bit $c_1$ can be used to represent the
\emph{carry-in}, but we usually assume $c_1 = 0$. The last carry bit
$c_{n+1}$ is also called the \emph{carry-out}. From the carry bits, we
can compute the output $S$ via
\begin{equation}
s_i = c_i \oplus x_i \text{ for } 1 \leq i \leq n \text{ and } s_{n+1} = c_{n+1}.
\label{eqn:sum-computation}
\end{equation}
With this preparation of constant depth, linear size, and fan-out two at the inputs $a_i, b_i$ and fan-out one at the carry bits $c_{i+1}$ ($i=1,\dots,n$), the binary addition is reduced to the problem of
computing all carry bits $c_{i+1}$ from $x_i,y_i$
($i=1,\dots,n$).
{\flushleft{\bf Convention}: }
{\it From now on, we will omit the preparatory steps (\ref{eqn:generate-and-propagat}) and (\ref{eqn:sum-computation}) and
consider a circuit an adder circuit if it computes all $c_{i+1}$ from $x_i,y_i$ ($i=1,\dots,n$).}
\vspace*{0.5cm}
Expanding the recursive formula for $c_{i+1}$ as in equation (\ref{eqn:and-or-form}) results in a logic circuit that is a path of alternating \textsc{And} and \textsc{Or} gates. It corresponds to the long addition method and has linear depth $2(n-1)$.
\begin{eqnarray}
c_{i+1} = \ &y_i \vee \left(x_i \wedge (y_{i-1} \vee (x_{i-1} \wedge \dots \wedge (y_2 \vee (x_2 \wedge y_1)).
\dots ))\right) \label{eqn:and-or-form}
\end{eqnarray}
\subsection{Prefix Graph Adders}
\label{sec:prefix-gate-adders}
For two pairs $z_i = (x_i,
y_i)$ and $z_j = (x_j, y_j)$, we define the associative {\it prefix operator} $\circ$
as
\begin{equation}
{x_i \choose y_i} \circ {x_j \choose y_j} = {x_i \wedge x_j \choose y_i \vee (x_i \wedge y_j)}.
\label{eqn:prefix-operator}
\end{equation}
A circuit computing (\ref{eqn:prefix-operator}) can be implemented as a logic circuit consisting of three gates and with depth two as shown in Figure~\ref{fig:pfx-to-gate3}.
For $i=1,\dots,n$, the result of the prefix computation $z_i \circ \dots \circ z_1$ of the expression $z_n \circ \dots \circ z_1$ contains the carry bit $c_{i+1}$:
\begin{equation}
{x_i \wedge x_{i-1} \wedge \dots \wedge x_1 \choose c_{i+1}} = {x_i
\choose y_i} \circ {x_{i-1} \choose y_{i-1}} \circ \dots \circ {x_1
\choose y_1}.\label{eqn:prefix-gate-carry}
\end{equation}
A circuit of $\circ$-gates computing all prefixes $z_i \circ \dots \circ z_1$ ($i=1,\dots,n$) for an associative operater $\circ$ is called a \emph{prefix graph}. A prefix graph yields an adder by expanding each $\circ$-gate as in Figure~\ref{fig:pfx-to-gate3}, and extracting the carry bits as in~\eqref{eqn:prefix-gate-carry}.
\begin{figure}
\begin{subfigure}[b]{0.49\linewidth}
\tikzc{Prefix gate and underlying logic circuit}{0.85}{pfx-to-gate3}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\linewidth}
\tikzc{Kogge-Stone prefix graph}{1}{kogge-pfx}
\end{subfigure}
\caption{Prefix gate and graph}
\end{figure}
Most previous constructions for adders are based on prefix graphs of small depth, size and/or fan-out.
Sklansky [1960] developed a prefix graph of minimum depth
$\ld n$, size $\frac{1}{2}n \ld n$, but high fan-out $\frac{1}{2}n+1$.
The first prefix graph with logarithmic depth ($2\log n -1$) and linear size ($3n - \log n - 2$) was developed by Ofman [1962], exhibiting a non-constant fan-out of $\frac{1}{2}\log n$.
Kogge and Stone [1973] introduced the {\it recursive doubling algorithm}
which leads to a prefix graph with depth $\log_2 n$ and fan-out two (see Figure~\ref{fig:kogge-pfx}).
Since we will use variants of it in our construction, we describe it in detail.
For $1\le s\le t\le n$, let $Z_{s,t} := z_t\circ\dots\circ z_s,$
and for $x \in \Rset$, let $(x)^+:= \max\{x,0\}$.
The graph has ${\ld n}$ levels, and on level $i$ it computes for every input $j$ ($1\le j \le n$)
the prefix $Z_{1+\nneg{j - 2^i},j} = z_j \circ\dots\circ z_{1+\nneg{j - 2^i}}$
according to the recursive formula
\begin{equation}Z_{1+(j - 2^i)^{+},j} = Z_{1+(j - 2^{i-1})^{+} ,j} \circ Z_{1+(j - 2^i)^{+} ,(j - 2^{i-1})^{+}}, \label{eq:ks}
\end{equation}
from the prefixes of sequences of $2^{i-1}$ consecutive inputs computed in the previous level.
The fan-out is bounded by two, since every intermediate result is used exactly twice: once as the ``upper half'' and once as
the ``lower half'' of an expression of the form $z_j \circ \dots \circ
z_{1+\nneg{j - 2^i}}$.
Note that for level $i$ ($1 \leq i \leq \ld n$), a repeater gate (which computes the identity function) is used instead of a $\circ$-gate if $j \leq 2^i$, i.\ e.\ in the case that the right input in \eqref{eq:ks} is empty. Repeaters are shown as blue squares in Figure~\ref{fig:kogge-pfx}.
The Kogge-Stone prefix
graph minimizes both depth and fan-out. On the other hand, since there
is a linear number of gates at each level, the total size in terms of prefix gates is $n{\ld n}
- \frac{n}{2}$.
Ladner and Fischer [1980] constructed a prefix graph of depth $\log_2 n$ but
high fan-out. Brent and Kung found a linear-size prefix graph with fan-out two, but twice the depth of the other constructions.
Finally, Han and Carlson [1987] described a hybrid between a Kogge-Stone adder and a Brent-Kung adder
which achieves a trade-off between depth and size.
Lower bounds for the trade-off between the depth and size of a prefix graph can be found in \cite{fich,sergeev}.
The above prefix graphs can be used for prefix computations with respect to any associative
operator $\circ$. In fact, we will later use a prefix graph
in which the operator $\circ$ represents an {\sc And} gate.
When turning one of the above prefix
graph adders into a logic circuit for addition such that each prefix gate is implemented as in Figure~\ref{fig:pfx-to-gate3}, the depth of the logic circuit is
twice the depth of the prefix graph and the number of logic gates is
three times the number of prefix gates.
The fan-out of the underlying logic circuit can increase
by one compared to the prefix graph, because the left propagate signal $x_i$ is used twice within a prefix gate.
In Section~\ref{sec:brent-kung-step}, we will see that in the case of the Brent-Kung adder
a fan-out of two can be achieved by using reduced prefix gates.
Any adder constructed from a prefix graph has a logic gate depth of at
least $\loq n -1 > 1.44 \ld n - 1$, where
$\varphi = \frac{1+ \sqrt{5}}{2}$ is the golden section
\cite{held+spirkl:2014}, see also \cite{code-trees}. In
\cite{held+spirkl:2014} an adder of size $\mathcal{O}(n\ld\ld n)$
asymptotically attaining this depth bound is described, however with a
high fan-out of $\sqrt{n}+1$.
\subsection{Non-Prefix Graph Adders}
Since none of the $2n$ inputs $x_i,y_i$ ($1 \le i\le n$) except for $x_1$ are redundant for $c_{n+1}$, the depth of any adder circuit using 2-input gates
is at least $\ld n+1$, which would be attained by a balanced binary
tree with inputs/leaves $x_i,y_i$ ($1 \le i\le n$).
With adders that are not based on prefix graphs, this bound is asymptotically tight.
Krapchenko showed that any formula (a circuit with tree topology) for computing $c_{n+1}$ has depth at least $\ld n +0.15\ld\ld\ld n + \mathcal{O}(1)$ \cite{krapchenkoLB}.
Brent [1970] gives an approximation scheme for a single carry bit circuit attaining an
asymptotic depth of $(1+\vare)\ld n +o(\ld n)$ for any given
$\vare>0$.
The best known depth for a single carry bit circuit is $\ld n +\ld \ld n + \mathcal{O}(1)$, due to Grinchuk [2008].
However, \cite{Grinchuk-ShallowCarryBit2009} and \cite{brent} did not address how to overlay circuits for the different
carry bits to bound the size and fan-out of an adder based on their circuits.
One problem in sharing intermediate results is that this can create high fan-outs.
Krapchenko [1967] (see
\cite[pp. 42-46]{wegener}) presented an adder with
asymptotically optimum depth $\ld n +o(\ld n)$ and linear
size.
It was refined for small $n$ by \cite{Gashkov+ImprovingKrapchenko2007}.
However, the fan-out is almost linear.
\subsection{Our Contribution}
\label{sec:our_contribution}
In this paper, we present the first family of adders of asymptotically optimum
depth, linear size, and fan-out bound two:
\begin{restatable}[Main Theorem]{theorem}{maintheorem}
\label{thm:central-theorem}
Given two $n$-bit numbers $A$,$B$, there is a logic circuit computing
the sum $A+B$, using gates with
fan-in and fan-out two and that has depth $\ld n + o(\log n)$
and size $\mathcal{O}(n)$.
\end{restatable}
The rest of the paper is organized as follows. In
Section~\ref{sec:min-depth-fan-out-two}, we develop a family of adders
of asymptotically minimum depth, fan-out two, but super-linear
size $\mathcal{O}\left(n\ceil{\sqrt{\ld n}}^2 2^{\sqrt{\ld n}}\right)$. In Section~\ref{sec:linearizing-size}, using reductions
similar to \cite{krapchenko}, this adder is transformed into an adder
of linear size with the asymptotically same depth, proving Theorem~\ref{thm:central-theorem}.
While all of the above adders use only \textsc{And}/\textsc{Or} gates and repeaters,
we show in Section~\ref{sec:technology-mapping}
that Theorem~\ref{thm:central-theorem}
holds also if only \textsc{Nand}/\textsc{Nor} and \textsc{Not} gates are available.
\section{Asymptotically Optimum Depth and Fan-Out Two}
\label{sec:min-depth-fan-out-two}
For $1\le s\le t \le n$, let $X_{s,t}$ and $Y_{s,t}$ denote the propagate and generate signal
for the sequence of indices between $s$ and $t$, i.e.\
\begin{equation}
\begin{array}{rl}
X_{s,t} &= \bigwedge_{i=s}^t x_i\\
Y_{s,t} &= y_t \vee \left(x_t \wedge (y_{t-1} \vee (x_{t-1} \wedge \dots \wedge (y_{s+1} \vee (x_{s+1} \wedge y_s))\dots ))\right)\\
\end{array}
\end{equation}
The adders based on prefix graphs as in Section~\ref{sec:prefix-gate-adders} impose
a common topological structure on the computation of intermediate results $X_{s,t}$ and
$Y_{s,t}$.
In the adder described by Brent [1970], on the other hand, intermediate results $X_{s,t}$ and $Y_{s,t}$ are computed separately within larger blocks.
Let $n=2^{rk}$ for $r \in \mathbb{N}$ and $k \in \mathbb{N}$ to be chosen later.
A central idea of generating a faster adder is to use multi-fan-in (also called high-radix) subcircuits within a Kogge-Stone prefix graph.
While all the prefix gates in Figure~\ref{fig:kogge-pfx} have fan-in two, we want to use prefix gates with fan-in $2^r$,
so that the number of levels reduces from $\ld n$ to $\log_{2^r} n$ = $\frac{1}{r}\ld n$.
Each prefix gate with fan-in $2^r$ represents a logic circuit with fan-in and fan-out bounded by two. Since the output of each prefix gate will be used in $2^r$ prefix gates at the next level, our approach also requires to duplicate the intermediate result at the output of a prefix gate $2^{r-1}$ times.
To accomplish this, we consider the computation of generate and
propagate sequences separately.
Our adder consists of two global Kogge-Stone type prefix graphs. The
first such graph uses 2-input \textsc{And}-gates and computes
propagate signals used in the other prefix graph. This graph uses
$2^r$-input subcircuits that are arranged in the same way as the
Kogge-Stone graph, and it computes the generate (carry) signals.
Both
graphs are modified to duplicate intermediate generate signals $2^{r-1}$ times
and intermediate propagate signals $2^r$
times so that the overall constructions obeys the fan-out bound of two.
\subsection{Multi-Input Generate Gates}
\label{sec:multi-input-generate-gate}
We now introduce {\it multi-input generate gates}, which are the main
building block for computing the generate signals.
Given $2^r$ propagate and generate pairs $(\tilde{x}_{2^r},\tilde{y}_{2^r}),\dots, (\tilde{x}_{1},\tilde{y}_{1})$,
a multi-input generate gate computes the generate signal
$$\tilde{Y}_{1,2^r} = \tilde{y}_{2^r} \vee \left(\tilde{x}_{2^r} \wedge (\tilde{y}_{2^r-1} \vee (\tilde{x}_{2^r-1} \wedge \dots \wedge (\tilde{y}_{2} \vee (\tilde{x}_{2} \wedge \tilde{y}_1))\dots ))\right).$$
The input pairs $(\tilde{x}_i,\tilde{y}_i)$ $(i\in \{1,\dots, 2^r\})$ are not necessarily the input pairs of the adder; they can be intermediate results.
Each multi-input generate gate has $2^{r-1}$ outputs, each of which provides the result $\tilde{Y}_{1,2^r}$, because later we want to reuse this signal $2^r$ times and bound the fan-out of each output by two. In contrast to two-input prefix gates computing (\ref{eqn:prefix-operator}), multi-input generate
gates do not compute the propagate signals $\tilde{X}_{1,2^r}=\bigwedge_{i=1}^{2^r} \tilde{x}_i$ for the given input pairs.
All required propagate signals will be computed by
the separate {\sc And}-prefix graph, described in
Section~\ref{sec:and-pfx-graph}.
\begin{figure}
\tikzc{A $2^r$-input $2^{r-1}$-output generate gate for $r=3$}{0.5}{fo-pfx-small}
\end{figure}
Figure~\ref{fig:fo-pfx-small} shows an example of a multi-input generate gate with $8$ inputs. A $2^r$-input prefix gate computes $\tilde{Y}_{1,2^r}$ as in the disjunctive normal form
$$ \tilde{Y}_{1,2^r} = \displaystyle \bigvee_{j=1}^{2^r} \left( \tilde{y}_j \wedge\left(\bigwedge_{i=j+1}^{2^r} \tilde{x}_i\right)\right),$$
first computing all the minterms $m_j:=\tilde{y}_j \wedge\left(\bigwedge_{i=j+1}^{2^r} \tilde{x}_i\right)$ $(j=1,\dots,2^r)$, and then
the disjunction $\bigvee_{j=1}^{2^r} m_j$.
The terms $\bigwedge_{i=j+1}^{2^r} \tilde{x}_i$ are computed as a Kogge-Stone {\sc And}-suffix graph, which arises
from a Kogge-Stone prefix graph by reversing the ordering of the inputs.
A single stage of (red) {\sc And}-gates and one repeater concludes the computation of the minterms. Each input $\tilde{y}_i$ is used exactly once within this circuit. The repeater is dispensable but simplifies the size formula and will become useful in Section~\ref{sec:technology-mapping}.
Finally, instead of computing the disjunction $\bigvee_{j=1}^{2^r}
m_j$ by a balanced binary {\sc Or} tree and duplicating the results
$2^{r-1}$ times through a balanced repeater tree, the duplication is accomplished by $r$
rows of $2^{r-1}$ {\sc Or}-gates as shown in Figure~\ref{fig:fo-pfx-small}. Formally, let $M_{i,j} = \bigvee_{i'=i}^{j} m_{i'}$ be the conjunction of minterms $i, i+1, \dots, j$. Then, on level $l \in \{1,\dots, r\}$, we compute each signal of the form $M_{i2^{l} + 1, (i+1)2^l}$, $i=0, \dots, 2^{r-l}-1$, from the previous level, and we compute $2^{l-1}$ copies of it. By using $M_{i2^{l} + 1, (i+1)2^l} = M_{{2i 2^{l-1} + 1, (2i+1)2^{l-1}}} \vee M_{{(2i+1)2^{l-1} + 1, (2i+2)2^{l-1}}}$, and since each preceding signal is available $2^{l-2}$ times ($l\ge 2$), we can ensure that each of them has fan-out two. On the last level, we will have computed $2^{r-1}$ copies of $M_{1,2^r} = \tilde{Y}_{1,2^r}$. Each level uses $2^{r-1}$ \textsc{Or}-gates.
Note that a similar construction for reducing fan-out has been used by Lupanov
when extending his well-known bounded-size representation of general Boolean functions to circuits with bounded fan-out \cite{LupanovBoundedBranching62}.
\begin{lemma}
The multi-input generate gate has $2^r$ generate/propagate pairs as input and
$2^{r-1}$ outputs. Each propagate input has fan-out two and each generate input has fan-out one.
The gate consists of $r2^r + (r+1)2^{r-1}$ logic gates
which have fan-out at most two.
The depth for the propagate inputs $\tilde{x}_i$ is $2r+1$
and the depth for the generate inputs $\tilde{y}_i$ is $r+1$
($i\in \{1,\dots,2^r\}$).
\end{lemma}
\begin{proof}
All the terms $\bigwedge_{i=j+1}^{2^r} \tilde{x}_i$ are computed as
a Kogge-Stone {\sc And}-suffix graph (blue and yellow gates in
Figure~\ref{fig:fo-pfx-small}) of size
$$2^r\lceil \ld 2^r\rceil -\frac{2^r}{2} = (r-1)2^r + 2^{r-1}.$$
Then, there is a level of $2^r$ (red) {\sc And} gates and one
repeater, concluding the computation of the minterms. Finally,
there are $r2^{r-1}$ (green) {\sc Or}-gates to compute the
disjunction $\bigvee_{j=1}^{2^r} m_j$ $2^r$ times, for a total of
$$r2^r + (r+1)2^{r-1} $$ gates.
By construction, no gate and propagate input has fan-out larger than two, and all generate inputs have fan-out one.
The depth is $r$ for the {\sc And}-suffix graph,
one for the red gates, and $r$ for the disjunctions, yielding the desired depths of $2r+1$ for the propagate inputs
and $r+1$ for the generate inputs.
\end{proof}
\subsection{Augmented Kogge-Stone \textsc{And}-Prefix Graph}
\label{sec:and-pfx-graph}
The second important component of our construction is the {\it augmented
Kogge-Stone \textsc{And}-prefix graph}. It is used to compute
$X_{s,t}= \bigwedge_{i=s}^t x_i$ for all $1 \leq t \leq n$ and $s
=1+\nneg{t-2^{rl}}$ with $0 \leq l < k$, providing each output
$2^r$ times through $2^r$ individual gates.
It is constructed as follows. First, we take a Kogge-Stone
[1973] prefix graph, where the prefix operator is an {\sc And}-gate, i.e.\ $\circ = \wedge$. It consists of
$\ld n$ levels, and on level $i$ it computes for every input $j$
($1\le j \le n$) the prefix $X_{1+\nneg{j - 2^i},j}$
from the prefixes of sequences of $2^{i-1}$ consecutive inputs computed in the previous level.
Each of the results $X_{s,t}$ from level $rl$ will later be used in $2^r$ multi-input generate gates
for all $0\le l < k$, $s =1+ \nneg{t-2^{rl}}$ and $ 1\le t \le n$.
In order to achieve a fan-out bound of two, starting at the inputs, we insert one row of $n$ repeaters after every $r$ levels of {\sc And}-gates. This allows to use the repeaters as the inputs for the next level, and to extract the signals $X_{s,t}$ once at the \textsc{And}-gates before the repeaters. The construction is shown in Figure~\ref{fig:kogge-aug} with the extracted outputs $X_{s,t}$ shown as red arrows.
\begin{figure}
\tikzc{Augmented Kogge-Stone \textsc{And}-prefix graph for $r=k=2$.}{1.0}{kogge-aug}
\end{figure}
The last block of $r$ rows of gates (hatched gates in Figure~\ref{fig:kogge-aug}) of the Kogge-Stone prefix graph can be omitted in our construction to reduce the size.
Each output signal $X_{s,t}$ will be input to a multi-input generate gate,
where it is immediately duplicated. Thus, each output $X_{s,t}$
of the augmented Kogge-Stone \textsc{And}-prefix graph has to be provided through an individual gate.
To this end, at each of the $nk$ outputs, we add $2^{r+1}-1$ repeater gates as the vertices of a
balanced binary tree to create $2^r$ copies of the signal with a single repeater serving each leaf. For simplicity
these repeaters are hidden in Figure~\ref{fig:kogge-aug}.
\begin{lemma}\label{lem:size-augmented-kogge-stone-and-prefix-graph}
The total size of the augmented Kogge-Stone \textsc{And}-prefix graph is $ nr(k-1) + nk2^{r+1}$.
\end{lemma}
\begin{proof}
Each binary repeater tree at one of the $nk$ outputs consists of $2^{r+1}-1$ repeaters, summing up to $nk(2^{r+1}-1)$
repeaters in these repeater trees.
The remaining circuit consists of $r(k-1)+k$ rows ($r(k-1)$ rows of {\sc And}-gates and $k$ rows of repeaters) of $n$ gates each,
summing up to $n(r(k-1)+k)$ gates.
Altogether, the circuit contains $ nr(k-1) + nk2^{r+1}$ gates.
\end{proof}
\begin{lemma}
The signal $X_{s,t}$ for $1 \leq t \leq n$ and $s = 1+ \nneg{t-2^{rl}}$ for $0 \leq l < k$
is available $2^r$ times with internal fan-out one at a depth of $(l+1)(r+1)$.
\end{lemma}
\begin{proof}
The functional correctness is clear by construction. For the depth
bound, let $1 \leq t \leq n$ and $0 \leq l < k$. Then, for
$s =1+\nneg{t-2^{rl}}$, the signal $X_{s,t}$ is available at the
bottom of the $l$-th block at a depth of $l(r+1)$. Subsequently, we
create $2^r$ copies of the signal in a repeater tree of depth $r+1$.
Together, this gives the desired depth $(l+1)(r+1)$.
\end{proof}
\subsection{Multi-Input Generate Adder}
We now describe the multi-input generate adder for $n = 2^{rk}$.
It consists of an augmented Kogge-Stone {\sc And}-prefix graph from the previous
section and a circuit consisting of multi-input generate gates similar
to a radix-$2^r$ Kogge-Stone adder.
The construction uses $k$ rows with $n$ multi-input generate gates or repeater trees (see Figure~\ref{fig:fixed-size2}).
The $t$-th multi-input generate gate in level $l\in\{1,\dots,k\}$ computes $Y_{1+\nneg{t-2^{rl}},t}$ according to the formula $Y_{1+\nneg{t-2^{rl}},t} = $
\begin{equation}
\bigvee_{j=1}^{2^r} \left( Y_{1+\nneg{t-j2^{r(l-1)}},\nneg{t-(j-1)2^{r(l-1)}}} \wedge \left(\bigwedge_{k=j+1}^{2^r}X_{1+\nneg{t-k2^{r(l-1)}},\nneg{t-(k-1)2^{r(l-1)}}} \right)\right).
\label{eqn:multi-input-generate-adder-recursion}
\end{equation}
If $\nneg{t-2^{rl}} < \nneg{t-2^{r(l-1)}}$ (yellow circuits in Figure~\ref{fig:fixed-size2}),
this computation is carried out using a multi-input generate gate from Section~\ref{sec:multi-input-generate-gate}.
As its inputs, it uses generate signals from the previous level,
$l-1$, and propagate signals obtained from the augmented Kogge-Stone
\textsc{And}-prefix graph.
Except for the last level, each intermediate generate signal will be used $2^r$ times as in (\ref{eqn:multi-input-generate-adder-recursion})
in the next level. As the fan-out of each generate input inside a multi-input generate gate is one,
we need to provide $2^{r-1}$ copies through individual gates to serve $2^r$ multi-input generate gates with fan-out two.
If $\nneg{t-2^{rl}} = \nneg{t-2^{r(l-1)}}$ (blue squares in
Figure~\ref{fig:fixed-size2}), $Y_{1+\nneg{t-2^{rl}},t}$ is already
computed in the previous level, and in this level it is sufficient to
duplicate the signal $2^{r-1}$ times using a balanced binary repeater tree.
The augmented Kogge-Stone {\sc And}-prefix graph provides each signal $2^r$ times with individual repeaters.
Thus, it can be distributed to $2^r$ multi-input generate gates, where the fan-out of each propagate input is two.
For the first level of multi-input generate gates, we duplicate each generate signal $y_i$ at an input $i\in
\{1,\dots,n\}$ using a balanced binary repeater tree of depth $r-1$
and size $2 + 2^2 + \dots + 2^{r-1} = 2^{r}-2$. Again, we can distribute each copy to two multi-input generate gates,
maintaining fan-out two.
In the last level of multi-input generate gates, we do not need to duplicate the signals
anymore. Instead of the $r$ rows of $2^{r-1}$ {\sc Or}-gates each, we can compute the single outputs
using a balanced binary tree of $2^r-1$ {\sc Or}-gates and depth $r$.
\begin{figure}
\tikzc{Multi-input multi-output generate gate adder for $r=k=2$}{1}{fixed-size2}
\end{figure}
\begin{lemma}
The multi-input generate adder for $n=2^{rk}$ bits obeys a fan-out bound of two, contains less than
$$3 nk(r+2)2^{r-1} + n2^{r} + nrk$$
gates, and has depth
$$ kr + 2r + k + 1.$$
\label{lem:depth+size-lemma}
\end{lemma}
\begin{proof}
Inside each multi-input generate gate, the fan-out of propagate inputs is two and the fan-out of generate inputs is one.
Thus, it suffices to observe that in each non-output level there are $2^{r}$ copies of each propagate signal and
$2^{r-1}$ copies of each generate signal, and that the
fan-out of two holds within the augmented Kogge-Stone graph and within each multi-input generate gate.
By Lemma~\ref{lem:size-augmented-kogge-stone-and-prefix-graph}, the
size of the augmented Kogge-Stone {\sc And}-prefix graph is $nr(k-1) + nk2^{r+1}$.
The size of the $n$ balanced binary trees duplicating the input generate signals is $n (2^{r}-2)$.
The remainder of the graph consists of $k$ rows of $n$ $2^r$-input multi-input generate gates or repeater trees.
The size of a repeater tree (blue boxes in Figure~\ref{fig:fixed-size2}) is at most $2^{r-1}-1 \le r2^r + (r+1)2^{r-1}$ ($r\ge 1$), which is
the size of a multi-input generate gate.
Thus, the size of all these multi-input generate gates is at most $nk\left(r2^r + (r+1)2^{r-1}\right)$.
Summing up, the total size is at most
\begin{equation*}
\begin{array}{rl}
& nr(k-1) + nk 2^{r+1} + n (2^{r}-2) + nk \left(r2^r + (r+1)2^{r-1}\right) \\
= & nk2^{r+1} + nkr2^r + n2^r + nk(r+1)2^{r-1} + nkr - n(r + 2) \\
= & nk\left(4+ 2 r + (r+1)\right)2^{r-1} + n2^{r} + nkr - n(r + 2) \\
< & 3nk\left(r+2\right)2^{r-1} + n2^{r} + nkr.\\
\end{array}
\end{equation*}
For a simpler depth analysis, we assume that the input generate signals
$y_i$ arrive delayed at a depth of $r+2$.
The generate input signals traverse a binary tree of depth $r-1$ and
the propagate input signals traverse a binary tree of depth $r+1$
before reaching the first multi-input generate gate, i.\ e.\ generate signals $y_i$
become available at depth $2r+1$ and propagate signals at depth $r+1$.
Thus, the first row of multi-input generate gates has depth
$$3r+2 = \max \{2r+1 + 1+ r,r+1+r+1+r\},$$
where the first term in the maximum is caused by the delayed
generate signals $y_i$ and the second term by the propagate signals
$x_i$ ($1\le i \le 1$).
For the next level, the propagate signals are available at time $2r+2$, and the generate signals at time $3r+2$, and the propagate signals again arrive $r$ time units before the corresponding generate signals, so at the next level, both signals arrive $r+1$ time units later than they did before. Inductively, we know that for each level $2 \le l \le k$, the generate and propagate
signals arrive at a depth of $(l-1)(r+1)$ more than than they did for at the first
level. Consequently, the total depth of the adder is $(k-1)(r+1) + 3r + 2 = kr
+ 2r + k +1$.
\end{proof}
If ${\sqrt{\log n}}\in \Nset$, we can choose $r=k=\sqrt{\log n}$ and
receive the following result.
\begin{cor}
If $\sqrt{\log n}\in \Nset$, there is a multi-input generate adder
for $n$ bits with fan-out two, size at most
$$3 n({\log n}+2\sqrt{\log n})2^{\sqrt{\log n}-1} + n2^{\sqrt{\log n}} + n\log n,$$ and depth
$$ \log n + 3{\sqrt{\log n}} + 1.$$
\label{cor:depth+size-integral-sqrt-log-n}
\end{cor}
In general, ${\sqrt{\log n}}\not \in \Nset$, and we get the following result.
\begin{theorem}
Let $n\in\Nset$. For input pairs $(x_i,y_i)$ ($i\in\{1,\dots,n\}$),
there is a circuit, computing all carry bits
with maximum fan-out $2$, depth at most $$\ld n + 5\ceil{\sqrt{\ld n}} + 2.$$ The size is at most
\begin{equation}
\label{eqn:mulit-input-generate-adder-large-n}
4 n \ceil{\sqrt{\ld{n}}}^22^{\ceil{\sqrt{\ld {n}}}}
\end{equation}
if $n\ge 16$, and at most
\begin{equation}
\label{eqn:mulit-input-generate-adder-small-n}
8 n \ceil{\sqrt{\ld{n}}}^2 2^{\ceil{\sqrt{\ld {n}}}}
\end{equation}
if $n\le 15$.
\label{thm:generalized-adder}
\end{theorem}
\begin{proof}
We choose $r = k = \ceil{\sqrt{\ld n}}$ and apply Lemma~\ref{lem:depth+size-lemma}, obtaining
\begin{equation}
\begin{array}{rl}
3nk(r+2)2^{r-1} + n2^{r} + nrk
& = n\left(3(r^2+2r)2^{r-1} + 2^{r} + r^2\right).\\
\label{eqn:multi-input-adder-size-common-bound}
\end{array}
\end{equation}
Now, if $n\ge 16$, we have $r=k\ge 2$. Thus, we can
use $2r \le r^2$ and $2^{r}+r^2 \le r^22^{r}$
to bound the right hand side by
\begin{equation*}
\begin{array}{rl}
n\left(3\left(r^2 + r^2\right) 2^{r-1} + r^22^{r}\right)
& \displaystyle = 4nr^22^{r},\\
\end{array}
\end{equation*}
implying (\ref{eqn:mulit-input-generate-adder-large-n}).
Otherwise, $n\le 16$, $r=k\le 2$, $r^2\le 2r$, $r^2\le 2^r$, and the right hand side of (\ref{eqn:multi-input-adder-size-common-bound})
is bounded by
\begin{equation*}
\begin{array}{rl}
\displaystyle n\left(3\left(2r + 2r\right)\right) 2^{r-1} + 2^{r} + 2^{r} & \displaystyle = 8nr2^{r} \le 8nr^22^r,\\
\end{array}
\end{equation*}
The resulting depth is
\begin{equation*}
\begin{array}{rl}
kr + 2r + k +1 & = \ceil{\sqrt{\ld n}}^2 + 3 \ceil{\sqrt{\ld n}} +1 \\
& \leq (\floor{\sqrt{\ld n}}+1)^2 + 3 \ceil{\sqrt{\ld n}} + 1\\
& \le \ld n + 5 \ceil{\sqrt{\ld n}} +2.
\end{array}
\end{equation*}
\end{proof}
If $\sqrt{\ld n} \not \in \Nset$, the adder in Theorem~\ref{thm:generalized-adder}
is larger than necessary since it has $n' = 2^{ \ceil{\sqrt{\ld n}}^2}> n$ inputs.
If for example $n=32$, we choose $r=k=3$ and $n'= 512$.
Thus, if $\ceil{\sqrt{\ld n}}^2 \geq n + \ceil{\sqrt{\ld n}}$, choosing $r = \ceil{\sqrt{\ld n}} - 1$ instead still yields an adder with at least $n$ inputs and outputs and reduces the size and depth significantly. For $n = 32$, we would still obtain a $64$-input adder using this method.
The analysis can be refined further by noticing that the columns $n'$ down to $n+1$ in the augmented Kogge-Stone {\sc And}-prefix graph and the multi-input gate graph can be omitted, since they are not used for the computations of the first $n$ output bits. This reduces the size of the construction. If $n' > n$, we can omit the left half of the construction and notice that the right half of lowest row of multi-input generate gates only has $2^{r-1}$ inputs, so we can actually use $2^{r-1}$-input generate gates and reduce the depth by 1. This process can be iterated until $n' = n$, which decreases the rounding error incurred in Theorem~\ref{thm:generalized-adder}; the depth is decreased by $\ceil{\sqrt{\ld n}}^2 - \log_2 n$.
In this section, we have achieved a depth bound of $\log_2 n + \mathcal{O}(\sqrt{\log n}) = \ld n + o(\ld n)$, which is asymptotically optimal, since the lower bound is $\ld n$.
\section{Linearizing the Size of the Adder}
\label{sec:linearizing-size}
To achieve a linear size while keeping the adder asymptotically
fastest possible, we adopt a technique similar to the construction by Brent and Kung
[1982], which was first used as a size-reduction tool by Krapchenko [1967]
(see \cite[pp. 42-46]{wegener}).
\subsection{Brent-Kung Step}
\label{sec:brent-kung-step}
Brent and Kung [1982] construct a prefix graph recursively as shown in Figure~\ref{fig:krap-reduc}.
If $n$ is a at least two, it
computes the $n/2$ intermediate results $z_{n} \circ z_{n-1}; \dots;
z_{2} \circ z_{1}$ (see Section~\ref{sec:prefix-gate-adders} for the
definition of $z_i$).
A prefix graph
for these $n/2$ inputs is used to compute the prefixes $Z_{1,2i}$
for all even indices $i\in\{1,\dots, n/2\}$. For odd indices, the
prefix needs to be corrected by one more prefix gate as $Z_{1,2i+1} = z_{2i+1} \circ Z_{1,2i}$
($i\in\{1,\dots, n/2-1\}$).
We call this method of input halving and output correction a {\it Brent-Kung step}.
Note that the propagate signals are not needed after the correction step.
Thus, we can use reduced prefix gates (Figure~\ref{fig:pfx-to-bk-output-gate}) in the output correction step.
In these prefix gates, the left propagate signal $x_i$ is used only once. Thus, the underlying logic circuit
inherits the fan-out of two from the prefix graph.
\begin{figure}
\begin{subfigure}[b]{0.49\linewidth}
\tikzc{Brent-Kung (reduction) step}{1}{krap-reduc}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.48\linewidth}
\tikzc{Brent-Kung prefix graph}{1}{brent-pfx}
\end{subfigure}
\caption{Brent-Kung Step and Prefix Graph}
\end{figure}
The Brent-Kung step reduces the instance size by a factor of two, but it increases the depth of the construction by four and the size by $5/2n$ in terms of logic gates.
Applying these Brent-Kung steps recursively, Brent and Kung obtain a
prefix graph that has prefix gate depth $2 \ld n -1$
and logic gate depth $4 \ld n -2$.
The prefix gate depth is not optimal anymore,
but the adder has a comparatively small size of $\frac{1}{2}(5n - \ld
n - 8)$ gates, and its fan-out is bounded by two at
all inputs and gates. It is shown in Figure~\ref{fig:brent-pfx}.
Brent-Kung steps were actually known before the paper by Brent and Kung [1982]
e.g.\ they were already used in \cite{krapchenko}.
But the Brent-Kung adder is based solely on these steps.
\subsection{Krapchenko's Adder} \label{sec:krap}
Krapchenko's adder is a non-prefix adder
computing all carry bits with asymptotically optimal depth and linear
size. Its fan-out, on the other hand, is almost linear as well, which
makes it less useful in practice. Krapchenko's techniques can be used
to derive the following reduction, based on Brent-Kung steps.
\begin{lemma}[{\cite{krapchenko}, see \cite[pp. 42-46]{wegener}}]
\label{lem:krap} Let $\tau \leq \ld n - 1$, then given a family of
adders computing $k$ carry bits with depth $d(k)$, maximum fan-out
$f(k)$ and size $s(k)$, there is a family of adders computing $n$
carry bits with depth $d({n/2^{\tau}}) + 4\tau$,
maximum fan-out $\max\left\{2,
f({n/2^\tau})\right\}$ and size
$s({n/2^\tau}) + 5n$.
With size $s({n/2^\tau}) + {5.5n}$, we can achieve the
same depth and a maximum fan-out of at most $\max\sset{2,
f({n/2^\tau})}$.
\end{lemma}
\begin{proof}
We apply $\tau$ Brent-Kung steps
and construct the remaining adder for ${n/2^{\tau}}$ from the given adder family.
Figure~\ref{fig:krap-reduc} shows the situation for $\tau = 1$. The
simple application of $\tau$ Brent-Kung steps would achieve the
claimed depth and fan-out result, except with at most $2n$
additional $2$-input prefix gates (because we will never add more
prefix gates than are present in the Brent-Kung prefix graph) and
thus with $6n$ additional logic gates.
\begin{figure}
\tikzc{Reduced output correction prefix gate of a refined Brent-Kung step}{0.33}{pfx-to-bk-output-gate}
\end{figure}
To see that $5n$ logic gates are enough, we show that we can omit
the propagate signal computation for the parity-correcting part of the
Brent-Kung step.
Such a reduced output prefix gate is shown in Figure~\ref{fig:pfx-to-bk-output-gate}.
With this construction, note that for $i$ even, we
have computed $(x, y) = z_i \circ \dots \circ z_1$. For $z_{i+1} =
(y_{i+1}, x_{i+1})$, the carry bit arising from position $i+1$ is
$c_{i+2} = x_{i+1} \vee (y_{i+1} \wedge y)$, which uses two
gates. It follows that a Brent-Kung step uses only the propagate signals
at the inputs. For the next Brent-Kung step, the inputs are the
$n/2$ pairs $z_n \circ z_{n-1}; \dots; z_2 \circ z_1$, therefore we
need three logic gates per prefix gate for the reduction step.
Note that in Figure~\ref{fig:brent-pfx}, the propagate signal at a gate is used
if and only if there is a vertical line from this gate to another
prefix gate (and not to an output or repeater). These lines
exist only in the ``upper half'' of the adder, i.\ e.\ the parts
with depth $\leq {\ld n}$. Since parity correction occurs
exclusively in the lower half with depth $> {\ld n}$, the
propagate signals from parity correction steps are never used.
As in the Brent-Kung prefix graph, $\frac{n}{2}$ repeaters can be
used to distribute the fan-out and reduce the maximum fan-out of the
parity-correcting gates to two (see also Figure~\ref{fig:brent-pfx}).
\end{proof}
The fact that the refined Brent-Kung step does not require the inner adder to
provide the propagate signals, which a prefix graph adder would
provide, allows us to use the multi-input generate adder with the size
and depth bounds stated in Theorem~\ref{thm:generalized-adder}, and which
omits the last $r$ rows of {\sc And} gates (hatched gates in
Figure~\ref{fig:kogge-aug}) in the augmented Kogge-Stone {\sc
And}-prefix graph.
Lemma~\ref{lem:krap} can be used to achieve different trade-offs. In particular,
constructions for all carry bits of size up to $n^{1+o(1)}$ can be
turned into linear-size circuits with the same asymptotic depth or
depth guarantee, since we could choose $\tau = o(1) \ld n$. This works for prefix graphs and logic circuits; for
example with $\tau = \ld \ld n$, the Kogge-Stone prefix graph will
have size $3n$, depth $\ld n + 2\ld \ld n$ and fan-out bounded by two
in terms of prefix gates \cite{han+carlson}.
While the technique in Lemma~\ref{lem:krap} is essentially a $2$-input
prefix gate construction, the main result of \cite{krapchenko}
cannot be constructed using only prefix gates.
\subsection{Adders with Asymptotically Minimum Depth, Linear Size, and Fan-Out Two}
By combining Theorem~\ref{thm:generalized-adder} and
Lemma~\ref{lem:krap}, we get an adder of asymptotically minimum depth,
linear size and with fan-out at most two.
\begin{theorem}
There is an adder for $n$ inputs of size bounded by ${13.5 n}$ with
depth $$\ld n + 8 \ceil{\sqrt{\ld n}} + 6 \ceil{\ld\ceil{\sqrt{\ld n}}} + 2$$
and maximum fan-out two.
If $n\ge 4096$, the size can be bounded by ${9.5}n$.
\label{all carry bits}
\end{theorem}
\begin{proof}
We apply Lemma~\ref{lem:krap} with $\tau = \ceil{\sqrt{\ld n}+ 2 \ld{\ceil{\sqrt{\ld n}}}}$
and use an adder for $n/2^{\tau}$ inputs according to Theorem~\ref{thm:generalized-adder} as an inner adder.
From the proof of Lemma~\ref{lem:krap}, we have seen that the output correction of the Brent-Kung step does not
require propagate signals from the inner adder. So the fan-out is indeed two.
Using (\ref{eqn:mulit-input-generate-adder-small-n}), this results in an adder of size
\begin{equation*}
\begin{array}{rl}
& \ceil{8\frac{n}{2^\tau} 2^{\ceil{\sqrt{\ld{\frac{n}{2^\tau}}}}} \ceil{\sqrt{\ld{\frac{n}{2^\tau}}}}^2} + {5.5n}\\
\le & \ceil{8\frac{n}{2^{\ceil{\sqrt{\ld n}}+ 2 \ld{\ceil{\sqrt{\ld n}}}}} 2^{\ceil{\sqrt{\ld{n}}}} \ceil{\sqrt{\ld{n}
}}^2} + {5.5n} \\
\le &8n+{5.5n} = {13.5 n}.
\end{array}
\end{equation*}
If $n\ge 4096$ we have $n/2^{\tau} \ge 16$ that allows us to
apply the alternative bound (\ref{eqn:mulit-input-generate-adder-large-n}) to achieve a size bound of $9.5n$.
The depth is
\begin{equation*}
\begin{array}{rl}
\ld{\frac{n}{2^\tau}} + 5\ceil{\sqrt{\ld \frac{n}{2^\tau}}} + 2 + 4\tau
& =\ld{n} + 5\ceil{\sqrt{\ld \frac{n}{2^\tau}}} + 2 + 3\tau \\
&\le \ld n + 8 \ceil{\sqrt{\ld n}} + 6 \ceil{\ld\ceil{\sqrt{\ld n}}} + 2,
\end{array}
\end{equation*}
where we are using
$\tau \le \ceil{\sqrt{\ld n}}+ 2 \ceil{\ld{\ceil{\sqrt{\ld n}}}}$
for the inequality.
\end{proof}
From Theorem~\ref{all carry bits}, we can easily conclude our main result
stated in Section~\ref{sec:our_contribution}:
\maintheorem*
\section{Technology Mapping}
\label{sec:technology-mapping}
In this section we show that our construction from Theorem~\ref{all
carry bits} can be transformed into an adder using only
\textsc{Nand}/\textsc{Nor}, and \textsc{Not} gates, which are faster
in current CMOS technologies. This increases the depth by one and the size by a small constant factor.
\begin{theorem}
There is an adder for $n$ inputs using only \textsc{Nand}, \textsc{Nor}, and \textsc{Not} gates.
Its size is bounded by ${(18+\frac{1}{3})n}$,
the depth is at most
$$\ld n + 8 \ceil{\sqrt{\ld n}} + 6 \ceil{\ld\ceil{\sqrt{\ld n}}} + 3,$$
and the maximum fan-out is two.
\label{all carry bits inverted}
If $n \ge 4096$, the size is bounded by ${(15+\frac{5}{6})n}$,
\end{theorem}
In the next two sections, we show how to transform the two main components of our construction, the Brent-Kung steps and the multi-input multi-output generate gate adder, into circuits using only the desired gates.
\subsection{Mapping Brent Kung Steps}
\label{sec:mapping-brent-kung}
Brent-Kung steps can be implemented using \textsc{Nand}/\textsc{Not} prefix gates as shown in Figure~\ref{fig:pfx-nandnot}
in the reduction step.
Similarly, the reduced output reduction gate in Figure~\ref{fig:pfx-to-bk-output-gate}
can be realized by two \textsc{Nand} gates and one \textsc{Not} gate, i.e.\ by
eliminating the two rightmost gates in Figure~\ref{fig:pfx-nandnot}.
The modified prefix gates do not increase the depth of the Brent-Kung step, and increase the size
by a constant factor less than $ \frac{5}{3}$.
\begin{figure}[htb]
\tikzc{A \textsc{Nand}/\textsc{Not} prefix gate used in the reduction step}{0.33}{pfx-nandnot}
\end{figure}
\subsection{Mapping Multi-Input Multi-Output Generate Adders}
We want to transform a multi-input multi-output generate gate adder using DeMorgan's laws.
For easier understanding, we first insert repeaters so that the gates can be arranged in rows, such
all input signals for gates in odd rows are computed in even rows and vice versa.
This bipartite structure is already given in the augmented Kogge-Stone {\sc And}-prefix graph (see Figure~\ref{fig:kogge-aug}).
Let us now consider a multi-input generate gate shown in Figure~\ref{fig:fo-pfx-small}.
By inserting $2^r/2$ repeaters gates in the last row of the \textsc{And}-suffix graph, we
achieve a uniform depth of this first stage.
The red row of \textsc{And} gates and the final $2^{r-1}$-output \textsc{Or} already have a uniform depth.
The additional repeaters increase the size by less than a factor of $\frac{5}{3}$.
Except for the first row of generate gates, the depth of the generate signals equals the depth
of the propagate signals when they are merged in the red row of \textsc{And} gates.
In the first row of
generate gates, the propagate signals arrive there at depth $2r+1$, while the generate signals arrive at
depth $r-1$ (see the proof of Lemma~\ref{lem:depth+size-lemma}). Thus, if $r$ is odd, we add one additional repeater at every
generate input signal so that it arrives at an odd depth at the red level of \textsc{And} gates.
Note that we can do this without increasing the overall depth, as we already assumed that the generate signals are delayed
by $r+1$ in the proof of Lemma~\ref{lem:depth+size-lemma}.
At most $n$ repeaters are inserted this way.
Some generate gates of the multi-input generate gate adder are just buffer trees, i.e.\ blue boxes in Figure~\ref{fig:fixed-size2}.
They have depth $r-1$, which is odd if and only if the depth $r+1$ of the corresponding paths of generate signals through multi-input generate gates is odd. Thus, they preserve the bipartite structure.
Now we can use the bipartite structure to transform the multi-input multi-output generate adder into a circuit consisting of \textsc{Nand},
\text{Nor}, and \textsc{Not} gates.
In our construction we will maintain the following characteristics.
Inputs to an odd row, i.e.\ outputs of an even row, will be the original function values, while
inputs to an even row, i.e.\ outputs of an odd row, will be the negated original function values.
We achieve this by transforming gates as follows:
Repeaters are always transformed into \textsc{Not} gates.
In odd rows, we translate
\textsc{And} gates into \textsc{Nand} gates and \textsc{Or} gates into \textsc{Nor} gates.
In even rows, we translate
\textsc{And} gates into \textsc{Nor} gates and \textsc{Or} gates into \textsc{Nand} gates.
If the number of rows is odd, we add one row of \textsc{Not} gates to correct the otherwise negated outputs of the adder.
Together with the $n$ repeaters that we insert behind each generate input signal if $r$ is odd,
this makes $2n$ gates that can by accounted for by the size of the
augmented Kogge-Stone {\sc And}-prefix graph (see Figure~\ref{fig:kogge-aug}),
which is at least $3n$ if $r\ge 1$.
Thus, the overall size of the generate adder increases by at most a factor of $\frac{5}{3}$.
Together with the mapping of the Brent-Kung step in Section~\ref{sec:mapping-brent-kung}, this proves Theorem~\ref{all carry bits inverted}.
\section*{Conclusion}
We introduced the first full adder with an asymptotically optimum
depth, linear size and a maximum fan-out of two. Asymptotically, this
is twice as fast and significantly smaller than the Kogge-Stone adder, which
is often considered the fastest adder circuit, as well as most other prefix graph adders.
For small $n$, Theorem~\ref{all carry bits} will not immediately improve upon existing adders.
When focusing on speed for small $n$, one would rather omit the size reduction from Section~\ref{sec:linearizing-size}.
Without the size reduction, our results in Lemma~\ref{lem:depth+size-lemma} match the depth of the Kogge-Stone adder for 512 inputs and improve on it for 2048 inputs, where $r = 3, k = 4$ yields an adder with depth 21 for our construction, but the adder of Kogge-Stone will have depth 22.
Today's microprocessors contain adders for a few hundred bits.
However, adders for 2048 bit numbers exist already today on cryptographic chips.
Thus we expect that adders based on our ideas will find their way into hardware.
\bibliographystyle{ACM-Reference-Format-Journals}
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Q: "neither of whom" vs "none of whom" Here is the sentence:
Several prisoners escaped last night, none of whom/neither of whom
has been caught so far.
In my opinion, I'd use 'none of whom', since we are talking about several people who escaped.
Which one would you rather use in this particular context?
A: Yes, you're correct.
Neither of whom would apply to two individuals.
None of whom would apply to three or more.
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Q: Getting values from a multi column listbox VBA I am in need of help, I am trying to get all three columns of my multi column list box in a for loop, it should be pretty simple but I cannot figure it out,
I have googled for hours and to be honest haven't gotten anywhere as everyone wants it to be able to select and then loop through where as i just want to loop through all of them and take them to separate strings,
Hope this makes sense,
Thank you,
A: You can cycle through entries in the list by using the script below and fill a new array which can be pasted to an assigned range on the sheet of the same size as the array
dim arrayctr as integer, i as integer, myarray() as variant
With mylistbox
'below sizes array to size of list in listbox.
'use arrctr to populate selected items to array without blank entries in between.
redim myarray(1 to .listcount, 1 to 1) as variant
'.listcount gives total entries, but list starts from 0 so use i-1 to get values
For i = 1 to .listcount
if .selected(i-1) = true then myarray(arrayctr) = .list(i-1): _
arrctr = arrctr + 1
Next i
End with
you can use For next loop through entries and use Exit For when the value of the array = "" so it handles only entries with data in and avoids pointless calculations.
I have had similar troubles with finding online answers to some questions such as this, I hope it helps!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,777
|
Opel will reorganize Corsa, Adam, Mokka output in Europe
David Jolley
The Corsa, shown, is Opel's best-selling model in Europe.
FRANKFURT -- Opel is reorganizing output of key small cars at its plants in Germany and Spain to improve production efficiency.
The General Motors' unit will move production of its Adam minicar to its factory in Zaragoza, Spain, from Eisenach, Germany.
The Corsa subcompact hatchback, the brand's best-selling model, will in future be built exclusively in Spain. It is currently manufactured in Zaragoza and Eisenach.
Production of the next-generation Mokka X subcompact crossover will move to Eisenach from Zaragoza.
"The revised allocations will make production considerably more efficient," Philip Kienle, Opel's manufacturing chief, said in a statement. "The focused allocation of individual cars to a production site enables synergies in assembly and improved logistics coordination," he said.
Through October, European sales of the Corsa fell 0.4 percent to 227,337, according to JATO Dynamics market researchers. In the same period, Mokka sales were flat at 138,037, while Adam sales slipped 6 percent to 44,945.
The Corsa is the No. 5 best-selling model in Europe in the first 10 months, according to JATO, after the VW Golf and Polo, the Ford Fiesta and the Renault Clio.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,330
|
package org.apereo.cas.support.saml.metadata.resolver;
import org.apereo.cas.support.saml.BaseMongoDbSamlMetadataTests;
import org.apereo.cas.support.saml.services.SamlRegisteredService;
import org.apereo.cas.support.saml.services.idp.metadata.SamlMetadataDocument;
import org.apereo.cas.util.junit.EnabledIfContinuousIntegration;
import org.apereo.cas.util.junit.EnabledIfPortOpen;
import lombok.val;
import org.apache.commons.io.IOUtils;
import org.junit.jupiter.api.Test;
import org.springframework.core.io.ClassPathResource;
import org.springframework.test.context.TestPropertySource;
import java.io.IOException;
import java.nio.charset.StandardCharsets;
import static org.junit.jupiter.api.Assertions.*;
/**
* This is {@link MongoDbSamlRegisteredServiceMetadataResolverTests}.
*
* @author Misagh Moayyed
* @since 5.2.0
*/
@TestPropertySource(properties = {
"cas.authn.samlIdp.metadata.mongo.databaseName=saml-idp-resolver",
"cas.authn.samlIdp.metadata.mongo.dropCollection=true",
"cas.authn.samlIdp.metadata.mongo.collection=samlResolver",
"cas.authn.samlIdp.metadata.mongo.host=localhost",
"cas.authn.samlIdp.metadata.mongo.port=27017",
"cas.authn.samlIdp.metadata.mongo.userId=root",
"cas.authn.samlIdp.metadata.mongo.password=secret",
"cas.authn.samlIdp.metadata.mongo.authenticationDatabaseName=admin",
"cas.authn.samlIdp.metadata.mongo.idpMetadataCollection=saml-idp-metadata-resolver",
"cas.authn.samlIdp.metadata.location=file:/tmp"
})
@EnabledIfPortOpen(port = 27017)
@EnabledIfContinuousIntegration
public class MongoDbSamlRegisteredServiceMetadataResolverTests extends BaseMongoDbSamlMetadataTests {
@Test
public void verifyResolver() throws IOException {
val res = new ClassPathResource("sp-metadata.xml");
val md = new SamlMetadataDocument();
md.setName("SP");
md.setValue(IOUtils.toString(res.getInputStream(), StandardCharsets.UTF_8));
resolver.saveOrUpdate(md);
val service = new SamlRegisteredService();
service.setName("SAML Service");
service.setServiceId("https://carmenwiki.osu.edu/shibboleth");
service.setDescription("Testing");
service.setMetadataLocation("mongodb://");
assertTrue(resolver.supports(service));
val resolvers = resolver.resolve(service);
assertEquals(1, resolvers.size());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,927
|
<?php
namespace Symfony\Component\Form\Extension\Core\Type;
use Symfony\Component\Form\AbstractType;
use Symfony\Component\OptionsResolver\Options;
use Symfony\Component\OptionsResolver\OptionsResolver;
class RadioType extends AbstractType
{
/**
* {@inheritdoc}
*/
public function configureOptions(OptionsResolver $resolver)
{
$resolver->setDefaults([
'invalid_message' => function (Options $options, $previousValue) {
return ($options['legacy_error_messages'] ?? true)
? $previousValue
: 'Please select a valid option.';
},
]);
}
/**
* {@inheritdoc}
*/
public function getParent()
{
return CheckboxType::class;
}
/**
* {@inheritdoc}
*/
public function getBlockPrefix()
{
return 'radio';
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,172
|
{"url":"https:\/\/nbviewer.ipython.org\/github\/barbagroup\/AeroPython\/blob\/master\/lessons\/05_Lesson05_InfiniteRowOfVortices.ipynb","text":"# Infinite row of vortices\u00b6\n\nThe objective of this assignment is to visualize the streamlines around an infinite row of vortices. First, you will consider the case of a finite number of vortices, obtained by simple superposition. By adding more and more vortices, you should be able to see how the flow pattern approaches that of an infinite row of vortices. But there will always be some differences (pay attention to what these may be).\n\nIt's possible to derive an analytical expression for the infinite case, and the derivation is provided below. With this analytical expression, you can visualize the streamlines for the infinite case. Observe and think: how are the streamlines different from one case to the other?\n\nIn this notebook, there is no Python code. Your job is to study the theory (and follow the mathematics on your own handwritten notes), to think how you could implement it in an efficient manner and finally to code it and visualize the results.\n\n## Vortex flow (from previous lesson)\u00b6\n\nYou might not suspect it, but the vortex has a very important role in classical aerodynamic theory. You'll discover some of its uses in this assignment.\n\nFirst, a little review of the basics. As seen in a previous lesson, a vortex of strength $\\Gamma$ has a stream-function:\n\n$$\\psi\\left(r,\\theta\\right) = \\frac{\\Gamma}{2\\pi}\\ln r$$\n\nand a velocity potential\n\n$$\\phi\\left(r,\\theta\\right) = -\\frac{\\Gamma}{2\\pi}\\theta$$\n\nWe can now derive the velocity components in a polar coordinate system, as follows:\n\n$$u_r\\left(r,\\theta\\right) = 0$$$$u_\\theta\\left(r,\\theta\\right) = -\\frac{\\Gamma}{2\\pi r}$$\n\nIn a Cartesian coordinate system, the velocity components at $\\left(x,y\\right)$ around a vortex of strength $\\Gamma$ located at $\\left(x_\\text{vortex},y_\\text{vortex}\\right)$, are given by\n\n$$u\\left(x,y\\right) = +\\frac{\\Gamma}{2\\pi}\\frac{y-y_\\text{vortex}}{(x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2}$$$$v\\left(x,y\\right) = -\\frac{\\Gamma}{2\\pi}\\frac{x-x_\\text{vortex}}{(x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2}$$\n\nand the stream-function is written as\n\n$$\\psi\\left(x,y\\right) = \\frac{\\Gamma}{4\\pi}\\ln\\left((x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2\\right)$$\n\n## Superposition of many vortices\u00b6\n\nIn this assignment, we consider a useful example to illustrate the concept of a vortex sheet: an infinite row of vortices of equal strength $\\Gamma$ (same sign and magnitude) evenly spaced by a distance $a$. But let's start with a finite row of vortices first.\n\nThe stream-function $\\psi_i$ of the $i^{th}$ vortex at a distance $r_i$ is given by:\n\n$$\\psi_i = \\frac{\\Gamma}{2\\pi}\\ln r_i$$\n\nApplying the principle of superposition, the stream-function of $N$ vortices is, then\n\n$$\\psi = \\frac{\\Gamma}{2\\pi} \\sum_{i=1}^N \\ln r_i$$\n\nAnd the velocity field (in Cartesian coordinates) of the row of vortices is\n\n$$u\\left(x,y\\right) = + \\frac{\\Gamma}{2\\pi} \\sum_{i=1}^N \\frac{y-y_i}{(x-x_i)^2+(y-y_i)^2}$$$$v\\left(x,y\\right) = - \\frac{\\Gamma}{2\\pi} \\sum_{i=1}^N \\frac{x-x_i}{(x-x_i)^2+(y-y_i)^2}$$\n\nwhere $\\left(x_i,y_i\\right)$ are the Cartesian coordinates of the $i^{\\text{th}}$ vortex.\n\nHere is a diagram of the situation:\n\n(In the next section, we'll show you how to derive a closed-form expression for the infinite sum of vortices. But hang on.)\n\n#### Compute the velocity field and plot the streamlines of a row of vortices.\u00b6\n\nYou will place $N$ vortices aligned on the horizontal axis and visualize the flow pattern. Do the following:\n\n\u2022 Using the equations derived above, compute the velocity components of each vortex on a mesh grid.\n\n\u2022 Remember that a finite number of vortices can be represented by a list or a NumPy array. Think and decide which one to use.\n\n\u2022 Define functions to avoid code repetition (and why not classes, if you prefer and are familiar with the concept).\n\n\u2022 Once you have all the velocities, apply the principle of superposition and plot the resulting flow pattern.\n\n\u2022 Play around with the size of your vortex row and the plotting range. Make your plot of publishable quality!\n\nIn the end you should get something similar to this:\n\n## Infinite row of vortices\u00b6\n\nThis will be a fun mathematical exercise! Follow along, and take your own notes.\n\nTo derive the closed-form functions, we use the complex representation:\n\n$$z = x + jy$$\n\nwhere $j^2=-1$. (We don't use $i$ to not confuse it with our indices!)\n\nThe complex potential is defined as $w = \\phi + j\\psi$, where $\\phi$ is the potential velocity and $\\psi$ is the stream-function. If we differentiate the complex potential $w$ with respect to the complex variable $z$, we get the complex velocity\n\n$$\\frac{dw}{dz} = u-jv$$\n\nwhere $u$ and $v$ are the Cartesian velocity components.\n\nThe complex potential representing a vortex of strength $\\Gamma$ located at the origin is\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\ln z$$\n\nWhy?\n\nBecause $z=re^{j\\theta}$ and $w$ becomes\n\n$$w = -\\frac{\\Gamma}{2\\pi}\\theta + j\\frac{\\Gamma}{2\\pi}\\ln r = \\phi + j\\psi$$\n\nConsider a second vortex located at $\\left(a,0\\right)$ with the same strength $\\Gamma$. Its complex potential is given by\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\ln \\left(z-a\\right)$$\n\nA additional vortex located $a$ further will have a complex potential of the form\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\ln \\left(z-2a\\right)$$\n\nand so on...\n\nTherefore, the complex potential representing an infinite row of vortices (on the line $y=0$) is given by\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\sum_{m=-\\infty}^{+\\infty} \\ln \\left( z-ma \\right)$$\n\nWhen we integrate the velocity components, in the previous notebooks, to find the stream-function and the potential, we dropped the constant of integration. Here, we decide to add a constant (not a random one!). Why not, since the differentiation will give the same results.\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\sum_{m=-\\infty}^{+\\infty} \\ln \\left( z-ma \\right) + \\text{constant}$$\n\nwhere\n\n$$\\text{constant} = -\\frac{j\\Gamma}{2\\pi} \\sum_{m=-\\infty, m\\neq 0}^{+\\infty} \\ln \\left( -ma \\right)$$\n\nso that, the complex potential can be cast in the following form\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\sum_{m=-\\infty,m\\neq 0}^{+\\infty} \\ln \\left( \\frac{z-ma}{-ma} \\right) + \\frac{j\\Gamma}{2\\pi}\\ln z$$\n\nNow, it is time do some mathematics...\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\sum_{m=-\\infty,m\\neq 0}^{+\\infty} \\ln \\left( 1-\\frac{z}{ma} \\right) + \\frac{j\\Gamma}{2\\pi}\\ln z$$$$w = \\frac{j\\Gamma}{2\\pi}\\sum_{m=1}^{+\\infty} \\left\\lbrace \\ln\\left(1-\\frac{z}{ma}\\right) + \\ln\\left(1+\\frac{z}{ma}\\right) \\right\\rbrace + \\frac{j\\Gamma}{2\\pi}\\ln z$$$$w = \\frac{j\\Gamma}{2\\pi}\\sum_{m=1}^{+\\infty} \\ln\\left(1-\\frac{z^2}{m^2a^2}\\right) + \\frac{j\\Gamma}{2\\pi}\\ln z$$$$w = \\frac{j\\Gamma}{2\\pi} \\ln\\left(\\prod_{m=1}^{+\\infty}\\left(1-\\frac{z^2}{m^2a^2}\\right)\\right) + \\frac{j\\Gamma}{2\\pi}\\ln z$$$$w = \\frac{j\\Gamma}{2\\pi} \\ln\\left(z\\prod_{m=1}^{+\\infty}\\left(1-\\frac{z^2}{m^2a^2}\\right)\\right)$$$$w = \\frac{j\\Gamma}{2\\pi} \\ln\\left(z\\prod_{m=1}^{+\\infty}\\left(1-\\frac{\\left(\\frac{z\\pi}{a}\\right)^2}{m^2\\pi^2}\\right)\\right)$$\n\nThe product is actually the formula for the sine and the complex potential becomes\n\n$$w = \\frac{j\\Gamma}{2\\pi}\\ln\\left(\\sin\\left(\\frac{z\\pi}{a}\\right)\\right)$$\n\nWe derive the complex potential with respect to the complex variable to get the complex velocity\n\n$$\\frac{dw}{dz} = u-iv = \\frac{j\\Gamma}{2a}\\cot\\left(\\frac{z\\pi}{a}\\right)$$$$u-jv = \\frac{j\\Gamma}{2a}\\frac{\\cos\\left(\\frac{\\pi x}{a}+j\\frac{\\pi y}{a}\\right)}{\\sin\\left(\\frac{\\pi x}{a}+j\\frac{\\pi y}{a}\\right)}$$\n\nApplying trigonometric identities, we find the following expression\n\n$$u-jv = \\frac{j\\Gamma}{2a}\\frac{\\cos\\left(\\frac{\\pi x}{a}\\right)\\cosh\\left(\\frac{\\pi y}{a}\\right) - j\\sin\\left(\\frac{\\pi x}{a}\\right)\\sinh\\left(\\frac{\\pi y}{a}\\right)}{\\sin\\left(\\frac{\\pi x}{a}\\right)\\cosh\\left(\\frac{\\pi y}{a}\\right) + j\\cos\\left(\\frac{\\pi x}{a}\\right)\\sinh\\left(\\frac{\\pi y}{a}\\right)}$$\n\nwhich can be cast in the form (try it!)\n\n$$u-jv = \\frac{\\Gamma}{2a}\\frac{\\sinh\\left(\\frac{2\\pi y}{a}\\right)}{\\cosh\\left(\\frac{2\\pi y}{a}\\right) - \\cos\\left(\\frac{2\\pi x}{a}\\right)} + j \\frac{\\Gamma}{2a}\\frac{\\sin\\left(\\frac{2\\pi x}{a}\\right)}{\\cosh\\left(\\frac{2\\pi y}{a}\\right) - \\cos\\left(\\frac{2\\pi x}{a}\\right)}$$\n\nTherefore, the Cartesian velocity components of an infinite row of vortices are given by\n\n$$u\\left(x,y\\right) = +\\frac{\\Gamma}{2a}\\frac{\\sinh\\left(\\frac{2\\pi y}{a}\\right)}{\\cosh\\left(\\frac{2\\pi y}{a}\\right)-\\cos\\left(\\frac{2\\pi x}{a}\\right)}$$$$v\\left(x,y\\right) = -\\frac{\\Gamma}{2a}\\frac{\\sin\\left(\\frac{2\\pi x}{a}\\right)}{\\cosh\\left(\\frac{2\\pi y}{a}\\right)-\\cos\\left(\\frac{2\\pi x}{a}\\right)}$$\n\n#### Compute the velocity field and plot the streamlines of an infinite row of vortices\u00b6\n\nNow that we have derived the functions for the velocity components, implement them in a code cell and plot the streamlines.\n\nCan you notice the differences with the previous case where the number of vortices was finite?\n\nPlay around with your plotting settings for the finite row of vortices, until you can get a plot that looks more like the infinite case. When can you say the finite case is a good approximation of the infinite case?\n\n##### Think\u00b6\n\nNotice that the streamline pattern is parallel to the vortex sheet: there is no net normal flow. Therefore, the sheet can model a solid surface in potential flow. We end up with a slip velocity at such an interface: how is this consistent with potential flow?\n\nPlease ignore the cell below. It just loads our style for the notebook.\nIn\u00a0[1]:\nfrom IPython.core.display import HTML\ndef css_styling(filepath):","date":"2021-09-17 14:09:23","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\": 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\": 0.7771624326705933, \"perplexity\": 442.48055640842165}, \"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-2021-39\/segments\/1631780055645.75\/warc\/CC-MAIN-20210917120628-20210917150628-00241.warc.gz\"}"}
| null | null |
Q: model binding nested objects I'm using a ViewModel to get the data from the HTML form to the MVC controller.
In the ActionResult parameter, there is the ViewModel that is filled, it's working fine.
Now, I want to get one field of the viewmodel separated in addition to the viewmodel. Is it possible? In another word, I want a copy of MyViewModel.MyModel.Id into another actionresult parameter.
Here is the working Controller Method:
public ActionResult Edit(AuditViewModelCritereViewModel model){}
Now I want to have this kind of method
public ActionResult Edit(AuditViewModelCritereViewModel model, int auditId){}
Here are the Models:
public class AuditViewModelCritereViewModel
{
public AuditViewModel audit { get; set; }
...
}
public class AuditViewModel
{
public int auditId { get; set; }
...
}
I could achieve this by adding HiddenField but clean is the best, I would like to make this prettier than copiing HTML id value.
In the HTML form, here is the field:
<input data-val="true" id="audit_auditId" name="audit.auditId" type="hidden" value="10">
This is HTTP request content that interest us:
This is what I've tried so far as int paramater:
*
*int auditId
*int audit_auditId
Thank you for your help.
A: Yes it is possible
<input data-val="true" id="audit_auditId" name="auditId" type="hidden" value="10">
Take same name of input field where you want to get the data in action parameter.
Action Parameter name and Field name should be same. It will works fine
OR
You can get this things with using Bind Attribute in Controller
public ActionResult Edit(AuditViewModelCritereViewModel model, [Bind(Prefix = "audit.auditId")]int auditId) { }
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,090
|
\section{Introduction}
Microorganisms have adapted their locomotion to the harsh environment of low Reynolds number regime by invoking different swimming strategy \cite{Eric_lauga_2009}. For example, the E. Coli moves moves by rotating its helical flagellum \cite{Ecoli_nature_1973,berg2005coli}, Chlamydomonas flagella have a breast stroke kind of motion \cite{Chlamydomona_2000}, African Trypanosome has a helical flagellum attached to the cell body with a planar wave passing through it \cite{Oberholzer_2010,Babu_trypanosome} etc. Swimming of these kind of natural swimmers have been investigated for the last half-century \cite{Eric_lauga_2009,Ecoli_nature_1973,bray2001cell,berg2005coli,Chlamydomona_2000,Oberholzer_2010,Babu_trypanosome,gray1955propulsion,pacey1994acquisition}. As a result of these studies, artificial swimmers have also been proposed, like Taylor sheet \cite{Taylor_1951}, Purcell's two-hinge swimmer \cite{Purcell_1977,Avron_2008}, three-linked spheres swimmer \cite{Ali_2008}, Elastic two-sphere swimmer \cite{PhysRevFluids.2.043101} and Three-sphere with a passive elastic arm \cite{Montino2015} etc, which have further enhanced our understanding about low Reynolds number swimmers. One of the challenges in proposing an artificial swimmer lies in the fact that the proposed movement stroke should not be reciprocal otherwise it cannot propel itself due to the Scallop theorem. In Scallop theorem, Purcell had argued that a swimmer with one-hinge or one degree of freedom is bound to perform reciprocal motion and thus will not be able to swim in the Stokes regime \cite{Eric_lauga_2009, Purcell_1977}.
\subparagraph{}
Purcell proposed two possible ways to elude from Scallop theorem, one is "corkscrew" \cite{Ecoli_nature_1973,purcell1997efficiency} motion and second is "flexible oar" \cite{Goldstien_1998,lagomarsino2003simulation} motion. Using the concept of flexible oar, Dreyfus et al. \cite{magnetic_swimmeri_nature_2005} reported a micro swimmer that exploit elastic property of a slender filament made up of paramagnetic beads. To break the time inversion symmetry, a passive head was attached to the flexible arm. The passive head reduces the velocity of the flexible swimmer, bigger the head, higher is the drag force experienced by the swimmer. The head is essential for swimming because without it the tail performs a reciprocal motion and the velocity of the swimmer reduces to zero \cite{lauga2007floppy}.
\subparagraph{}
In the present study, we design and simulate a two-dimensional swimmer having two symmetric arms joined by a single hinge that looks like a two-dimensional scallop. The arms of the swimmer are semi-flexible and thus behave as an elastic scallop. To design an elastic scallop swimmer we use bead spring model \cite{babu2011dynamics} for the two arms joined by a common bead which acts as a hinge and to introduce flexibility to the two symmetric arms of the swimmer, we use bending potential \cite{allen-tildesley-87}. The actuation happens only at the hinge and the rest of the arm relaxes depending on the strength of the actuation similar to the one-armed flexible swimmer \cite{Goldstien_1998}. In case of the elastic scallop the time inversion symmetry is broken because through the viscous drag term time enters into the equation of the filament shape, similar to the one-armed swimmer. Hydrodynamic interactions between the arms of the swimmer are implemented using the particles based simulation method for the fluid called as multi-particle collision dynamics (MPC) \cite{Gompper2009}. The advantage of MPC is that it solves the Navier-Stokes equation and also has inherent thermal fluctuations \cite{malevanets1999mesoscopic,malevanets2000solute} resembling real fluids. By this method a variety of hydrodynamic problems have been solved, for example, swimming of sperm cells \cite{Sperm_mpc_2008}, African Trypanosome \cite{Babu_trypanosome}, E. coli \cite{hu2015modelling}, Taylor line \cite{munch2016taylor}, Spheroidal microswimmer \cite{theers2016modeling}, Squirmer \cite{downton2009simulation,zottl2012nonlinear} etc. This is the method of our choice as it is very easy to implement and is also shown to be one of the most efficient method in the Stokes limit \cite{Gompper2009}.
\subparagraph{}
We know that the dynamics of the one-armed swimmer is described by the non dimensional hyper diffusion equation given by Wiggins et al. \cite{wiggins1998trapping,Goldstien_1998}. They have already shown that a flexible slender filament is propelled by periodic actuation, which is characterized by a dimensionless number called the Sperm number $S_p$ \cite{Goldstien_1998}. Sperm number is the ratio of the length of the swimmer to its hydrodynamic penetration length. When $S_p\ll 1$, the penetration length is larger than the length of the swimmer, which means that the arms of the swimmer are stiff and perform near reciprocal motion, because of that we get very small velocity. When $S_p\gg 1$, we have the length of the swimmer to be very large compared to the penetration length, and hence the drag forces acting on the swimmer increase, thus the velocity of the swimmer again reduces. The interplay between these two effects leads to a maximum in the velocity at a point when the hydrodynamics penetration length is approximately equal to the length of the swimmer as shown by Wiggins et al. \cite{wiggins1998trapping,Goldstien_1998}.
\subparagraph{}
The paper is arranged as follows. In section \ref{sec:2} we describe the model of our two-dimensional swimmer using bead spring and bending potentials. Then we briefly describe the particle based simulation technique MPC and also explain how we couple the swimmer with the solvent particles to properly resolve hydrodynamic interactions. In section \ref{sec:3} we discuss how our one-hinge swimmer breaks time inversion symmetry with flexible arms. We also study the velocity of the swimmer as a function of bending potential, frequency of actuation and amplitude of the actuating wave. We are able to define the dimensionless Sperm number for this swimmer and also discuss how our dimensionless velocity vary with respect to $S_p$. In section \ref{sec:4} we discuss how this artificial swimmer can be realized experimentally, followed by the conclusions in section \ref{sec:5}.
\section{COMPUTATIONAL METHODS}
\label{sec:2}
\subsection{Multiparticle collision dynamics (MPC)}
To simulate our artificial swimmer in two-dimensional Newtonian fluid, we make use of the coarse-grained simulation technique called multi-particle collision dynamics (MPC). Although there are different variation of MPC, we use the technique MPC with Anderson thermostat, where both linear and angular momentum are conserved \cite{Gompper2009}. We know that MPC solves the Navier-Stokes equation and has been shown to be computationally efficient method for low Reynolds number hydrodynamics \cite{Gompper2009}. MPC is a particle based method where we use fictitious point particles of mass $m_0$ to mimic the fluid. MPC consists of two steps, one is called the streaming step and another one is called the collision step. In streaming step, we update the position $\textbf{r}_i(t)$ of all the solvent particles at time $t$ according to
\begin{equation}
\textbf{r}_i(t+\delta t)= \textbf{r}_i(t) + \delta t \textbf{v}_i(t).
\label{e1}
\end{equation}
Where $\textbf{v}_i(t)$ is the velocity of $i_{th}$ particle at time $t$ and $\delta t$ is the MPC time step.
\subparagraph{}
The collision step is performed by dividing the simulation box into the square cells of length $a_0$. In the present study, we keep the number of fluid particles per cell $\rho=10$ with periodic boundary condition. The initial velocities of the particles are assigned from a Gaussian distribution with variance $k_BT/m_0=1$. In collision step, the velocity of each particle $\textbf{v}_i(t)$ is updated according to
\begin{equation}
\begin{split}
\textbf{v}_i(t&+\delta t) =\textbf{u}(t) + \textbf{v}_i^{ran} - \sum_{i\in cell} \textbf{v}_i^{ran}/N_c + \\
&\quad \left\{m_0\textbf{I}^{-1} \sum_{j\in cell} \textbf{[r}_{j,c} \times (\textbf{v}_j - \textbf{v}_j^{ran})\textbf{]}\times \textbf{r}_{i,c}\right\}.
\label{e2}
\end{split}
\end{equation}
Where $\textbf{u}(t)$ is the center of mass velocity of all particles in a cell, $\textbf{v}_i^{ran}$ is a random velocity taken from a Gaussian distribution with variance $k_BT/m_0=1$, $N_c$ is the number of particles in a cell, $\textbf{I}$ is the moment of inertia of all the particles in a cell, $\textbf{r}_{i,c} = \textbf{r}_i - \textbf{R}_c$ is the relative position of $i_{th}$ particle in a cell with respect to the center of mass position $\textbf{R}_c$ of all the particles in a cell.
\subparagraph{}
The swimmer keeps on actuating thereby pumping energy into the fluid which increases the kinetic energy of the fluid particles, and therefore the temperature of the fluid also increases known as viscous heating. The Anderson's thermostat which is inherent in the simulation technique, keeps the temperature of our system constant. In collision rule MPC conserves both linear as well as angular momentum. In MPC simulation, we know that when $\delta t \ll 1$, fluid particles become correlated thereby the system no longer remains Galilean invariant. In order to restore Galilean invariance we perform a random shift of the simulation box in the interval $[-a_0/2,a_0/2]$ as proposed by Ihle et al. \cite{ihle2001stochastic,ihle2003stochastic}.
\subparagraph{}
In present work, we have measured quantities in MPC units, where length is normalised with $a_0$, mass with $m_0$, energy with $k_BT$ and time with $\tau_0=a_0\sqrt{m_0/k_BT}$ \cite{padding2006hydrodynamic}. We have used $a_0=1$, $m_0=1$ and $k_BT=1$, which makes time unit equals to unity. One of the advantage of MPC is that, we have analytical expressions for the calculation of transport coefficients \cite{Gompper2009}. The total kinematic viscosity $\mu$ is the sum of the kinetic viscosity $\mu^{kin}$ and collision viscosity $\mu^{col}$, where $\mu^{kin} = 0.61 \delta t$ and $\mu^{col} = 0.036 / \delta t$. We have used $\delta t =0.01$, which keeps the kinematic viscosity $\mu\approx3.6$.
\subsection{Modelling two-dimensional scallop}
\graphicspath {}
\begin{figure}
\centering
\includegraphics[scale=0.18]{flexible_swimmer}
\caption{The snapshot of a model one-hinge artificial swimmer with flexible arms. Three mass points are given blue color to show that the bending waves pass through only these three points. We have magnified the bond vectors $t_i$ and $t_{i+1}$, which connect neighboring mass points and make an angle $\theta$. $\varphi$ is the angle between two arms near the hinge.}
\label{fig1}
\end{figure}
In order to model the two-dimensional swimmer with one-hinge, we consider a string of $N$ odd numbered mass points of mass $m_0=1$ connected with spring \cite{babu2011dynamics}. The middle bead acts as a hinge for the two-armed swimmer, so that the length of each arm is the same as shown is figure \ref{fig1}. The spring potential between the mass points is given by
\begin{equation}
U_s = \frac{1}{2} K_s (l-l_0)^2.
\label{eq.3}
\end{equation}
Where $K_s$ is the spring constant, $l$ is the distance between two mass points at any given time and $l_0$ is the equilibrium distance between the two mass points. In our simulation we keep the equilibrium spring length $l_0 = 0.5 a_0$. To make sure that the springs remain relatively stiff during the simulations, we use a high spring constant $K_s = 10^8$ in the present study. The arms of the swimmer are made semi-flexible in nature by introducing a three body bending potential
\begin{equation}
U_b = K_b (1-\cos \theta).
\label{eq.4}
\end{equation}
Where $K_b$ is the stiffness constant, $\theta$ is an angle between two bond vectors $\textbf{t}_i$ and $\textbf{t}_{i+1}$, as illustrated in figure \ref{fig1}(magnified bond vectors) \cite{babu2011dynamics}. We apply the bending potential along both the arms of the swimmer except for the hinge points. When $K_b \rightarrow \infty$, we have rigid arms for the swimmer and for $K_b \rightarrow 0$, we have completely flexible arms. We vary the value of $K_b$ from $10^4$ to $10^7$, thereby vary the rigidity along the arms of the swimmer.
\subparagraph{}
In order to simulate a two-dimensional scallop we define an angle $\varphi$ (see figure \ref{fig1}) between the two arms of the swimmer, which oscillates between a minimum and maximum angle. In order to achieve this, we introduce a three body bending wave potential along the three blue colored beads given by
\begin{equation}
U_w = \frac{1}{2} K_w [t_{i+1}-\textbf{R}(l_0 \alpha)t_i]^2.
\label{eq.5}
\end{equation}
Where $K_w$ is a bending stiffness constant that decides the strength of the potential, $\textbf{R}$ is the rotation matrix, $\phi = l_0 \alpha $, $\alpha = A\sin^2 {(2\pi\nu t)}$ is the spontaneous curvature between the mass points where the bending wave potential is applied, $A$ is the amplitude of the wave and $\nu$ is the frequency with which the potential makes the beads to beat \cite{Babu_trypanosome}. The matrix $\textbf{R}$ rotates one bond vector $\textbf{t}_i$, against the neighboring bond vector $\textbf{t}_{i+1}$, about a unit vector which is perpendicular to $\textbf{t}_i$ and $\textbf{t}_{i+1}$, by an angle $\phi$. As the curvature $\alpha$ is function of square of the sine wave, we will have only positive values for curvature, thus the angle $\varphi$ between the arms can vary only between $0 - \pi$. When two arms are completely opened (i.e. zero curvature position), the angle $\varphi$ will be the maximum $\varphi_{max}= \pi$ at that time, the value of $K_w=4 \times 10^5$ is used in the present work. The total force on the $i_{th}$ bead of the swimmer, due to spring and bending potentials is given by $\textbf{F}_i = - {\nabla}_i(U_s+U_b+U_w)$. The movement of the swimmer is implemented by molecular dynamics (MD) technique. We update the velocities and positions of the beads using a leap frog velocity Verlet algorithm \cite{allen-tildesley-87}, where the integration time step is always kept at $\delta t_{MD}=10^{-4}$.
\subsection{Incorporation of swimmer with fluid particles}
We perform $n$ number of MD steps, where $n=\delta t /\delta t_{MD}$ before a MPC step. In streaming step of MPC, we only consider the fluid particles, and during this step we allow the fluid particle to pass through the swimmer as well. We in-cooperate the mass points of the swimmer with the fluid particles in the collision step, which give the correct hydrodynamics for the swimmer \cite{Yeomans,noguchi2005dynamics,babu2011dynamics}.
\section{RESULTS}
\label{sec:3}
\subsection{Breakdown of the time inversion symmetry}
From the Scallop theorem, we know that the movement undergoing time reversal symmetry will not be able to propel in a low Reynolds number fluid. In the present work we simulate a two-dimensional one-hinge swimmer similar to a two-dimensional scallop. Figure \ref{fig2}(a) shows the shape conformation when we have flexible arms and figure \ref{fig2}(b) is when we have the rigid arms, for one complete cycle. During one complete cycle the curvature of the hinge undergoes a change as $\sin^2 {(2\pi\nu t)}$, which means we have only positive cycles. In figure \ref{fig2}(a) when we look from right to left, first half of the figure, (till the solid black line), $\sin^2 {(2\pi\nu t)}$ changes from $0-1$, shows the closing of the arms and the second half shows the opening of the arms with $\sin^2 {(2\pi\nu t)}$ changing from $1-0$. The curvature of the hinge changes as a function of the time ($\alpha = A\sin^2 {(2\pi\nu t)}$) as given in the equation \ref{eq.5}. When $t=0$ we have zero curvature that means arms of the swimmer is opened to the maximum possible extent or we start from a straight line. As time progresses, $\alpha$ increases and the arms of the swimmer start to close from the center of the swimmer, similar to a two-dimensional scallop. When the arms start to close, the center of mass of the swimmer moves in the backward direction. The rest of the arms follow the actuation of the hinge points. As the actuation only happens on the hinge, there will be a delay for the actuation to reach the end of the arms. This will cause the points close to the hinge of the swimmer to close faster compare to the points further away. So as the arms close, the ends bend in an outward direction and resembles ${\bf V}$ shaped conformation. When it reaches $\sin^2 {(2\pi\nu t)}=1$, we get a maximum value of curvature equal to the amplitude $A$. After that the curvature at the hinge starts decreasing and the arms of the swimmer start to open up. Again the point away from the hinge will be moving in the closing direction, due to the delay in the propagation of the actuation. By that time the points close to the hinge would have already started to move in the opening direction. Due to this the ends of the swimmer are now bend inward as can be observed in the latter half of the figure \ref{fig2}(a). The center of mass of the swimmer also starts moving in the forward direction and the ends of the arms start to retract now in a ${\bf U}$ shaped conformation. At a later time, when the arms reach $\sin^2 {(2\pi\nu t)}=0$ or $\alpha=0$, the arms of the swimmer are opened at maximum possible extent. As the swimmer has a $V$ shape conformation during closing of the arms while in case of opening it has a $U$ shaped conformation, thus breaking the time inversion symmetry and the one-hinge swimmer move ballistically from right to left as shown by the black arrow in figure \ref{fig2}, which was also predicted by E. Lauga \cite{lauga2007floppy}. The propulsion of the one-hinged swimmer is illustrated in the supplementary materials movie1.
\begin{figure}
\centering
\includegraphics[scale=0.32]{skeleton}
\caption{Sequence of conformation of the arms (here different colors are used just for the better visualization) of the one-hinged swimmer over one complete beat cycle, (a) represents the case for a flexible arms and (b) for the case of rigid arms. From right to left, we observe the sequence of closing, from the center bold conformation shows the opening of the arms of the swimmer. The black arrow at the top shows the direction of swimming for the flexible swimmer.}
\label{fig2}
\end{figure}
When we keep the bending stiffness $K_b=10^7$, in our simulation we obtain rigid arms for one-hinge swimmer. In the case of rigid arms, the oscillations at the hinge are followed by every point of the arms and thus the swimmer performs reciprocal motion. There is no difference in the opening and closing conformation, as can be observed in figure \ref{fig2}(b). As a consequence the center of the mass of the swimmer does not undergo ballistic motion, as illustrated in the supplementary material movie2.
\subsection{Mean squared displacement}
\subsubsection{Finite size effect}
In problems involving hydrodynamics effect the simulation box has to be choosen appropriately to avoid finite size effect \cite{PhysRevE.85.021901,Sperm_mpc_2008}. To verify the effect of finite box size we have calculated the mean square displacement of the center of mass (MSD) of the rigid arm swimmer
\begin{equation}
\langle {\bf R}^2 \rangle= \langle {\left({{\bf r}_{cm}(t)-{\bf r}_{cm}(t_0)}\right)^2} \rangle.
\end{equation}
Where ${\bf r}_{cm}(t_0)$ and ${\bf r}_{cm}(t)$ are the position of the center of mass of the swimmer at beginning of the simulation $t=0$ and at time $t$ respectively. In figure \ref{fig01} we have plotted the MSD as a function of time $t$ for a swimmer with length of one arm $L=15$ for $4$ different box sizes. The rigid arms scallop will only diffuse and we expect slope of $1$ of MSD, due to the inherent thermal fluctuation of the MPC fluid. In figure \ref{fig01}, we can see that in all the cases, till $t\approx 10$ swimmer does hardly move, because bending waves take some time to propagate from hinge to the end of the arms. As we have discussed in the previous section, initially a swimmer has open arms and with time it starts to close the arms due to that center of the mass displaces, we get an upswing in MSD. Once the curvature reaches the maximum, arms starts to open again and center of the mass tries to come back to the previous position and due to that we get a down swing in MSD. Because of the oscillatory stroke that the swimmer performs, MSD of the swimmer always oscillates. We observe that when the box size is $60$ or $4$ times the length of the arm $L$, after $1-2$ cycles of oscillation MSD attains slope of $1$. When the MSD reaches a time of $t>300$, we observe a deviation from the slope of $1$. When the size of the box was increased to $90$ or $6$ times of $L$, we observe that the deviation of MSD from slope of $1$ happens at a later time $t>10^3$, indicating that the finite size effect happens at a much later time. When we further increased the size to $8$ and $10$ times the length of the arm of the swimmer we did not observe a deviation from a slope of $1$ upto the time of our observation. So in the present work we have always kept the size of the simulation box $10$ times the length of the arm of the swimmer to avoid finite size effect.
\graphicspath { {Result/1/} }
\begin{figure}
\centering
\includegraphics[scale=0.23]{Box_size_effect}
\caption{MSD of a rigid scallop of arm length $L=15$ in four different boxes (box sizes are mentioned in figure). Here solid black line has slope one.}
\label{fig01}
\end{figure}
\subsubsection{Ballistic motion}
In order to obsereve the difference between a flexible and a rigid scallop we have calculated the MSD as a function of time for both the case as shown in figure \ref{fig3}. Here we observe that after $3-4$ cycles of oscillation the flexible swimmer starts to undergo ballistic motion characterized by the slope of $2$, with the swimmer moving toward the left as indicated by the arrow in the figure \ref{fig2}(a), as predicted by E. Lauga \cite{lauga2007floppy}. As mentioned before for the rigid arms the center of mass of the swimmer undergoes diffusive motion characterized by $<\textbf R^2> \enspace \propto \textbf t$.
\graphicspath {}
\begin{figure}
\centering
\includegraphics[scale=0.175]{MSD}
\caption{Mean squared displacement of the center of mass position of the swimmer is plotted as a function of time $t$ for the flexible arms (blue) and rigid arms (red). The solid line has slope of $1$ indicating diffusive motion and dotted line has slope of $2$ representing directed motion of the swimmer.}
\label{fig3}
\end{figure}
\subsection{Effect of Flexibility}
From the previous section we know in the present work flexibility plays a major role in the swimming of the two arms swimmer. To study the effect of flexibility we measure the reduced velocity after the swimmer has attained ballistic motion which is defined as $V_{cm}/L \nu$, where $V_{cm}=\sqrt{\langle R^2 \rangle}/t$. The Reynolds number is defined $Re = V_{max}L/\mu$ \cite{Purcell_1977,Sperm_mpc_2008,padding2006hydrodynamic}, we use $ \mu=3.6$ and the maximum velocity we have studied for the scallop is $V_{max} \approx 0.03$, the length of the arm of the scallop is $L=20$, so the Reynolds number of our system $Re<0.16$, which is consistent with the previous studies of low Reynolds number regime using MPC simulations \cite{Sperm_mpc_2008,AndreasPhysRevLett,munch2016taylor}. The Peclet number is defined as $Pe = V_{cm}L/D$, where $D$ is the diffusion coefficient of the rigid swimmer \cite{padding2006hydrodynamic} and in our case the range of $Pe = 16–80$, which means the thermal fluctuations do not play a significant role in our simulations \cite{Elgeti}. In MPC simulation the Mach number of the fluid should be very small in order for the fluid to be incompressible. The Mach number is defined as $Ma = V_{max}/V_{sound}$, here $V_{sound}= \sqrt{2}$ is the speed of sound in $2$ dimension for MPC fluid. In our case for the maximum frequency we have used the $V_{max} \approx 0.03$ and so $Ma \approx 0.021$ \cite{PhysRevE.85.021901}, which is within the incompressible limit of MPC. In order to verify that MPC fluid is in the incompressible limit at all length scales in our simulation we calculated the mach number for the tip of the swimmer which moves with the maximum velocity $\approx 0.24$. The mach number for the tip of the arm turns out to be $Ma \approx 0.17$, which is again within the accepted MPC limit for having an incompressible fluid \cite{padding2006hydrodynamic,re0231fId0,refId0}. In figure \ref{fig4}, we plot the reduced velocity as a function of bending rigidity ${K_b}$ of the arms of the swimmer. When $K_b=0$ and $K_w=0$ our swimmer becomes passive and behaves as a Gaussian polymer chain. When we increase ${K_b}$ from $10^{5}$ onward, keeping $K_w=4 \times 10^5$, we observe a nominal increment in speed. The reason being that here the arms of the swimmer are very flexible and the wave which passes from the hinge is damped before reaching the end point. Here, the configurations of the swimmer do not change appreciably during the entire beat cycle. The arms of the swimmer become progressively rigid by increasing $K_b$, the speed increases as shown in figure \ref{fig4}. Here we have a competition between the elastic forces of the swimmer and viscous drag because of the fluid particles. The speed starts to increase till it reaches a maximum when the elastic forces cancel the frictional forces, for intermediate bending stiffness $K_b= 3.25 \times 10^5$. On further increasing the rigidity of the arms, the velocity goes down. Near $K_b=10^7$ it starts behaving as a conventional scallop and eventually the speed of the scallop goes towards zero as we keep on increasing the rigidity of the arms.
\graphicspath {}
\begin{figure}
\centering
\includegraphics[scale=0.25]{Kb_vs_v}
\caption{The reduced velocity of the swimmer is plotted as a function of bending rigidity of the arms of the swimmer.}
\label{fig4}
\end{figure}
\subsection{Beating frequency}
\graphicspath { {Result/1/} }
\begin{figure}
\centering
\includegraphics[scale=0.178]{freq_vs_V}
\caption{The reduced velocity is plotted as a function of frequency of the actuation. The solid line has slope of one as predicted for the small amplitude approximation.}
\label{fig5}
\end{figure}
In figure \ref{fig5}, we have plotted the reduced velocity of the swimmer with respect to the beating frequency $\nu$ imparted on the hinge. In the present study for the one hinge swimmer, we are considering only large amplitudes, the reduced velocity scales linearly with $\nu$, as shown by the straight dashed line in the figure \ref{fig5} which has slope of $1$. For a single-armed swimmer Wiggins et al. had derived the equation for velocity using small amplitude approximation, where it was shown that the reduced velocity scales linearly with $\nu$. In our case, the amplitude we have considered $A=4.25$ is large, still we observe that the scaling is in agreement with that of Wiggins et al. \cite{Goldstien_1998}.
\subsection{Amplitude of the actuation}
For closing and opening of the arms of the swimmer, the parameter that we are changing is the amplitude $A$ of the bending wave given in the equation \ref{eq.5}. The amplitude is chosen in such a way whereby we make sure that the mass points of the swimmer close to the hinge will never overlap. As we change the amplitude of the beating, the angle created between the arms $\varphi$ of the swimmer changes as well, see figure \ref{fig1}. The maximum angle between the arms in the simulation nearly stays around $\varphi_{max}= \pi$, while the minimum angle varies between $\varphi_{min}=\pi/3 - 2\pi/3$. In the present study the amplitude is related to the difference in the minimum and maximum angle made by the arms of the swimmer. In figure \ref{fig6}, we plot the amplitude $A$ as a function of $\Delta \varphi= \varphi_{max}-\varphi_{min}$, which is the difference between the minimum and maximum angle during the opening and closing of the arms of the swimmer.
When $\Delta \varphi = \pi$, then the maximum angle will be $\pi$ and the minimum angle will be $0$ between the arms of the swimmer, which means the arms of the swimmer is going to overlap at the minimum angle. For measuring $\varphi$, we calculate a unit vector between the hinge and the fourth bead from the hinge for the left arm $\hat{p}_l$ and the right arms $\hat{p}_r$ respectively. Then the angle $\varphi=\cos^{-1} ({\hat{p}_l \cdot \hat{p}_r})$, also note that for the beating of the arms we are changing the spontaneous curvature of the beads close to the hinge by $\sin^2{(2\pi \nu t)}$, which means the angle between the arms will not go beyond $\pi$. The dashed line in figure \ref{fig6} is drawn from straight line equation which shows that $A \propto \Delta \varphi$. In the present work we have kept $K_w\approx 10^5$, the we observe that arms can open and close in such a way $\Delta \varphi> \pi/3$.
\subparagraph{}
In figure \ref{fig7}, we plot the reduced velocity as a function of the $\Delta \varphi$. Here we observe that as the difference in the angle between the arms increases the speed of the swimmer also increases. We observe that the reduced velocity scales as the square of $\Delta \varphi$, the dashed line in the inset of figure \ref{fig7} has slope of two. The equation for the velocity has been deduced from the small amplitude approximation for the single-armed swimmer \cite{Goldstien_1998} as well as for the one-hinge swimmer \cite{lauga2007floppy}. In these works they had shown that the reduced velocity scales with the square of the amplitude of the actuation. Even though our amplitudes are larger than that were considered in these work, we observe that the reduced velocities have similar kind of scaling as that of small amplitude approximation. We were not able to further increase the amplitude of the actuation as that will lead to the overlap of the arms of the swimmer, which is not a physical scenario for an artificial swimmer.
\graphicspath {}
\begin{figure}
\centering
\includegraphics[scale=0.175]{Phi_vs_amp}
\caption{$\Delta\varphi$ (in radian) is plotted as a function of the amplitude of actuation. The dashed line shows the linear fit given by $(0.43 A/a_0 + 0.1)$.}
\label{fig6}
\end{figure}
\graphicspath {}
\begin{figure}
\centering
\includegraphics[scale=0.1825]{Phi_vs_V}
\caption{Reduced velocity of the swimmer is plotted as a function of $\Delta\varphi$ (in radian), the dashed line (inset) has slope of two as predicted for small amplitude approximation.}
\label{fig7}
\end{figure}
\subsection{Sperm number}
\graphicspath {}
\begin{figure}
\centering
\includegraphics[scale=0.185]{Sp_vs_V}
\caption{The reduced velocity of the swimmer is plotted as a function of Sperm number. The value of $S_p$ was varied, changing $\nu$ (square), $L$ (circle) and $k$ (triangle) keeping all other parameters the same.}
\label{fig8}
\end{figure}
Wiggins et al. had shown that if a flexible rod is perturbed at one end by $y_0 \cos \omega t$, for small amplitudes, the dynamics of the one arm follows the dimensionless hyper diffusion equation $\frac{dy}{dt}=S_p^4 \frac{d^4y}{dx^4}$, where $S_p$ is a dimensionless number called Sperm number. The Sperm number is given by
\begin{equation}
S_p = L \left( \frac{\xi_\perp 2 \pi \nu}{k} \right)^{\frac{1}{4}}.
\label{e.7}
\end{equation}
Where $L$ is the length of the arms of the swimmer, $k=K_b l_0$ is stiffness of the arms of the swimmer, $\nu$ is the frequency and $\xi_\perp$ is the frictional coefficient per unit length in perpendicular direction of motion defined to be consistent with the one-armed swimmer. In the present study, we calculate the perpendicular friction coefficient using $\xi_\perp=4\pi\eta / [\ln{(L/r)}+1/2]$ \cite{pak2015theoretical}, where $\eta$ is the viscosity of the fluid and $r$ is the radius of the arm. In the present work, the spring that connects two beads behave as a rigid rod of length $l_0=0.5 $, as a result we have considered the radius of the rod to be $r=0.25$. In figure \ref{fig8} we show the reduced velocity of the swimmer as a function of the Sperm number. For the small value of the $S_p$ we observe that the elastic forces dominate and we get a small velocity, while in the high Sperm number region the drag forces acting on the arms of the swimmer dominate and again the velocity goes down. As there is a competition between the viscous and elastic forces in the arms of the swimmer, there should be one particular $S_p$ value where we should observe a maximum for the velocity, which is around $ \sim 1.8$ in our case. For the one-armed swimmer with small amplitude approximation, it was shown that the maximum velocity is when $S_p\approx 4$, while Lagomarsino et al. have demonstrated using simulations that for the same one-armed swimmer with large amplitude, the maximum velocity happens when $S_p\approx 2$ \cite{lagomarsino2003simulation,lowe2003dynamics}. This is similar to our present work, where we consider the amplitude larger than the small amplitude approximation. In figure \ref{fig8}, $S_p$ is varied by changing the length of the arms, the frequency of actuation and the stiffness $K_b$ of the arms of the swimmer keeping the rest of the parameter constant of the equation \ref{e.7}. Here we can see that the peak positions of the reduced velocities remain almost same. For the small values of the Sperm number till $S_p< 1.6$, we find continuous increment in scaled velocity with $S_p$ as expected. After the maximum, we observe that the scaled velocity continuously decreases as we increases the $S_p$ and after $S_p>2$ we observe a slow drop in scaled velocity. For small amplitude approximation Wiggins et al. had shown that the scaled velocity attains a plateau for higher $S_p$ value. Our results are similar to the results obtained by Lagomarsino et al. \cite{lagomarsino2003simulation}, where they showed that for large values of $S_p$, when large amplitudes are considered, the velocity does not stay constant, it slowly decreases with $S_p$.
\section{DISCUSSIONS}
\label{sec:4}
It was shown that the one-armed flexible swimmer \cite{magnetic_swimmeri_nature_2005,Gauger_2006} will undergo ballistic motion only if a passive head is attached to the arm. The two-armed swimmer with a head was initially considered by Lauga \cite{lauga2007floppy}, for simplicity of the calculation he ignored the hydrodynamic interactions between the arms of the swimmer and also the velocity of the swimmer was derived under the infinitely small amplitude approximation. Lauga has also shown analytically that if two arms are actuated similar to a one-armed swimmer a passive head is not required to break the time inversion symmetry. In the present work we are able to show that if two filaments are joined by a hinge and actuated only at the hinge the artificial swimmer starts to behave as a self propelled object. We obtain the direction of the swimmer as predicted by Lauga \cite{lauga2007floppy} but the magnitude of velocity is not in agreement with our work. This may be because in the present work there are hydrodynamic interactions between the arms of the swimmer as well as the amplitude of the actuation in the present work is higher than that considered for the small amplitude approximation. Also recently Tian Quie et al. \cite{qiu2014swimming} demonstrated that a micro scallop can swim in a non Newtonian fluid. In this work they have used a rigid arm scallop in a non Newtonian fluid while in the present work our scallop or one-hinge swimmer have flexible arms thereby performing the ballistic motion even in Newtonian fluid. It is known that swimmer African Trypanosome velocity is increased approximately $8$ times when it swims in a fluid (non Newtonian) having obstacles of size and spacing of RBC in blood \cite{heddergott2012trypanosome} and it will be interesting to study the swimming of the one hinge swimmer in these environments.
Recently Haug et al. have shown that they can create soft robotic materials using light actuated materials \cite{huang2015miniaturized}. The light-driven liquid-crystal (LDLC) material \cite{yu2003photomechanics}, which are sensitive to ultra-violet light, can convert light into the mechanical energy with quick response and large deformation. When we shine UV light the LDLC undergoes a phase transition by which it gets converted into a shorter molecule, while shining visible light it recovers its original conformation. We believe using these materials experimental realization of the one-hinge swimmer is possible. They have already demonstrated using these materials that they are able to recover the shape conformation for a one-armed swimmer and to extend it to a one-hinge swimmer should not be too challenging.
\section{CONCLUSIONS}
\label{sec:5}
In this paper, we have modeled a two-dimensional scallop or a one-hinge swimmer. We have shown that the hydrodynamic interactions between the mass points of the swimmer and the fluid particles can be simulated using multi-particle collision dynamics (MPC). We have also demonstrated that if the arms of swimmer are very rigid, it follows the Scallop theorem and the swimmer is not able to propel itself through the viscous fluid. While when the arms are made flexible the time inversion symmetry is broken and swimmer performs ballistic motion. We have also shown that the velocity of the swimmer has a maximum, for intermediate bending rigidity along the arms of the swimmer. For small bending rigidity the arms are very flexible and the actuation is not able to produce the desired shape conformations, while for stiff arms we are closer to the Scallop theorem and the velocity goes towards zero.
The reduced velocity is also studied as a function of frequency as well as the amplitude of actuation and we have shown a similar scaling relation as predicted by elastohydrodynamic within the small amplitude approximation even though we have considered large amplitude in the present work. We were also able to define Sperm number for the swimmer and also showed that the reduced velocity had a maximum at $S_p\sim 1.8$ consistent with what is expected for large amplitude actuation.
\subsection*{Acknowledgements}
We would like to thank DST for funding and also would like to acknowledge IIT Delhi HPC facility for computational resources.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Before stating the key goal of our investigations and stating our results, we will briefly review some relevant background material on matroids and simplicial complexes. For further information, we refer the reader to the books of Oxley \cite{Oxley-book}, White \cite{White-book}, and Stanley \cite{Stanley-book}.
Recall that a matroid $M = (E(M),{\mathcal{I}}(M))$ consists of a \textit{ground set} $E(M)$ and a family of subsets $\mathcal{I}(M) \subseteq 2^{E(M)}$ called \textit{independent sets} such that
\begin{enumerate}
\item[(I1):] $\emptyset \in {\mathcal{I}}(M)$;
\item[(I2):] If $A \in {\mathcal{I}}(M)$ and $A' \subset A$, then $A' \in {\mathcal{I}}(M)$; and
\item[(I3):] If $A,A' \in {\mathcal{I}}(M)$ with $|A| < |A'|$, then there is some $e \in A'\setminus A$ such that $A \cup e \in {\mathcal{I}}(M)$.
\end{enumerate}
Equivalently, the independent sets of a matroid $M$ on the ground set
$E(M)$ form a simplicial complex, called the \textit{independence complex of $M$}, with the property that the
restriction $\mathcal{I}(M)|_{E'}$ is pure for any subset $E'
\subseteq E(M)$. A \textit{basis} of $M$ is a maximal independent set
under inclusion. The \textit{rank} of a subset $E' \subseteq E(M)$ is the size of the largest independent set $A \subseteq E'$; in particular, the \textit{rank} of $M$ is the cardinality of a basis. A \textit{loop} is a singleton $\{e\} \notin \mathcal{I}(M)$. Since the loops of a matroid are not seen by the independence complex, no generality will lost in only considering loopless matroids.
If $M$ is a loopless matroid, elements $e,e' \in E(M)$ are \textit{parallel} if $\{e,e'\} \notin \mathcal{I}(M)$. The \textit{parallelism classes} of $M$ are maximal subsets $E_1,\ldots,E_t \subseteq E(M)$ with the property that all elements in each set $E_i$ are parallel. It can be easily checked that if $\{e_{i_1},\ldots,e_{i_k}\} \in \mathcal{I}(M)$ with $e_{i_j} \in E_{i_j}$, then $\{e'_{i_1},\ldots,e'_{i_k}\} \in \mathcal{I}(M)$ for any choice of $e'_{i_j} \in E_{i_j}$. Alternatively, the parallelism classes of $M$ are maximal rank-one subsets of $E(M)$.
Given a matroid $M$ on the ground set $E(M)$ with bases ${\mathcal{B}}(M)$, we define its \textit{dual matroid}, $M^*$, to be the matroid on $E(M)$ whose bases are ${\mathcal{B}}(M^*) = \{E\setminus B: B \in {\mathcal{B}}(M)\}$. We say that $\{e\}$ is a \textit{coloop} in $M$ if $\{e\}$ is a loop in $M^*$ or, equivalently, if each basis of $M$ contains $e$.
If $M$ is a matroid of rank $d$, the \textit{$f$-vector} of $M$ is $f(M):=(f_{-1}(M),f_0(M),\ldots,f_{d-1}(M))$, whose entries are $f_{i-1}(M):=|\{A \in {\mathcal{I}}(M): |A| = i\}|$. Oftentimes, it is more convenient to study the \textit{$h$-vector} $h(M):=(h_0(M),\ldots,h_d(M))$ whose entries are defined by the relation $$\sum_{j=0}^dh_j(M)\lambda^{j} = \sum_{i=0}^df_{i-1}(M)\lambda^i(1-\lambda)^{d-i}.$$ See \cite{Stanley-book} for more on $h$-vectors and the combinatorics of simplicial complexes.
It should not be expected that the $h$-numbers of a general simplicial complex are nonnegative; however, the $h$-numbers of a matroid $M$ may be interpreted combinatorially in terms of certain invariants of $M$. Fix a total ordering $\{v_1<v_2<\ldots<v_n\}$ on $E(M)$. Given a basis $B \in {\mathcal{I}}(M)$, an element $v_j \in B$ is \textit{internally passive in $B$} if there is some $v_i \in E(M) \setminus B$ such that $v_i<v_j$ and $(B\setminus v_j)\cup v_i$ is a basis of $M$. Dually, $v_j \in E(M)\setminus B$ is \textit{externally passive in $B$} if there is an element $v_i \in B$ such that $v_i<v_j$ and $(B \setminus v_i) \cup v_j$ is a basis. (Alternatively, $v_j$ is externally passive in $B$ if it is internally passive in $E(M) \setminus B$ in $M^*$.) It is well known (\cite[Equation (7.12)]{White-book}) that
\begin{equation} \label{h-nums-matroid}
\sum_{j=0}^dh_j(M)\lambda^j = \sum_{B \in {\mathcal{B}}(M)}\lambda^{ip(B)},
\end{equation}
where $ip(B)$ counts the number of internally passive elements in $B$. This proves that the $h$-numbers of a matroid complex are nonnegative. Alternatively,
\begin{equation} \label{h-nums-dual}
\sum_{j=0}^dh_j(M)\lambda^j = \sum_{B \in {\mathcal{B}}(M^*)}\lambda^{ep(B)},
\end{equation}
where $ep(B)$ counts the number of externally passive elements in $B$. Since the $f$-numbers (and hence the $h$-numbers) of a matroid depend only on its independent sets, equations \eqref{h-nums-matroid} and \eqref{h-nums-dual} hold for \textit{any} ordering of the ground set of $M$. It is worth
remarking that the $h$-polynomial above is actually a specialization of the well-known \emph{Tutte polynomial} of the corresponding matroid
(see \cite{White-book}).
An \textit{order ideal} $\mathcal{O}$ is a family of monomials (say of degree at most $r$) with the property that if $\mu \in \mathcal{O}$ and $\nu|\mu$, then $\nu \in \mathcal{O}$. Let $\mathcal{O}_i$ denote the collection of monomials in $\mathcal{O}$ of degree $i$. Let $F_i(\mathcal{O}):=|\mathcal{O}_i|$ and $F(\mathcal{O}) = (F_0(\mathcal{O}),F_1(\mathcal{O},\ldots,F_r(\mathcal{O}))$. We say that $\mathcal{O}$ is \textit{pure} if all of its maximal monomials (under divisibility) have the same degree. A vector $\mathbf{h} = (h_0,\ldots,h_d)$ is a \textit{pure $O$-sequence} if there is a pure order ideal $\mathcal{O}$ such that $\mathbf{h} = F(\mathcal{O})$.
A longstanding conjecture of Stanley \cite{Stanley-CM-complexes} suggests that matroid $h$-vectors are highly structured.
\begin{conjecture} \label{Stanley-conjecture}
For any matroid $M$, $h(M)$ is a pure $O$-sequence.
\end{conjecture}
Conjecture \ref{Stanley-conjecture} is known to hold for several
families of matroid complexes, such as paving matroids
\cite{Merino-paving}, cographic matroids \cite{Merino-cographic},
cotransversal matroids \cite{Oh-contransversal}, lattice path matroids
\cite{Schweig-LP}, and matroids of rank at most three
\cite{Ha-Stokes-Zanello, Stokes-thesis}. The purpose of this paper is
to present a proof of Stanley's conjecture for all matroids on at
most nine elements, all matroids of corank two and all matroids of
rank at most three. While Stanley's conjecture was known to hold for
matroids of rank two \cite{Stokes-thesis} and rank three
\cite{Ha-Stokes-Zanello}, we use the geometry of the independence
complexes of matroids of small rank to provide much simpler shorter proofs in
these cases. Our results show that any counterexample to Stanley's
conjecture must have at least ten elements, rank at least four, and corank at least three.
This article will use several ideas from the theory of multicomplexes
and monomial ideals. Although a general classification of matroid
$h$-vectors or pure $O$-sequences seems to be an incredibly difficult
problem, some properties are known and will be used in the proofs
below:
\begin{theorem}\cite{Brown-Colbourn, Chari, Hibi}
Let $\mathbf{h} = (h_0,h_1,\ldots,h_d)$ be a matroid $h$-vector or a pure $O$-sequence with $h_d \neq 0$. Then
\begin{enumerate}
\item $h_0 \leq h_1 \leq \cdots \leq h_{\lfloor \frac{d}{2}\rfloor}$,
\item $h_i \leq h_{d-i}$ for all $0 \leq i \leq \lfloor \frac{d}{2} \rfloor$, and
\item for all $0 \leq s \leq d$ and $\alpha \geq 1$, we have
\begin{equation}\label{BC-ineq}
\sum_{i=0}^s(-\alpha)^{s-i}h_i \geq 0.
\end{equation}
\end{enumerate}
\end{theorem}
Inequality \eqref{BC-ineq} is known as the Brown-Colbourn inequality \cite[Theorem 3.1]{Brown-Colbourn}.
\section{Rank-$2$ Matroids}
Let $M$ be a loopless matroid of rank $2$. The independence complex of $M$ is a complete multipartite graph whose partite sets $E_1,\ldots,E_t$ are the parallelism classes of $M$. Let $s_i:=|E_i|$. Choose one representative $e_i \in E_i$ from each parallelism class of $M$ so that the simplification of $M$ is a complete graph on $\{e_1,\ldots,e_t\}$, and let $\widetilde{E}_i = E_i \setminus e_i$. Clearly
\begin{eqnarray*}
f_0(M) &=& \sum_{i=1}^t(s_i-1) + t \\
\text{and } f_1(M) &=& \sum_{1 \leq i<j \leq t}(s_i-1)(s_j-1) + (t-1)\sum_{i=1}^t(s_i-1) + {t \choose 2},
\end{eqnarray*}
and hence,
\begin{eqnarray*}
h_1(M) &=& \sum_{i=1}^t(s_i-1) + (t-2) \\
\text{and } h_2(M) &=& \sum_{1 \leq i<j \leq t}(s_i-1)(s_j-1) + (t-2)\sum_{i=1}^t(s_i-1) + {t-1 \choose 2}.
\end{eqnarray*}
Consider the pure $O$-sequence $\mathcal{O}$ with
\begin{eqnarray*}
\mathcal{O}_1 &=& \{x_1,\ldots,x_{t-2}\} \cup \{x_e: e \in \widetilde{E}_i, 1 \leq i \leq t\} \\
\mathcal{O}_2 &=& \{x_ex_{e'}: e \in \widetilde{E}_i, e' \in \widetilde{E}_j, 1 \leq i<j\leq t\} \\
&& \cup \{x_ix_e: e \in \widetilde{E}_j, 1 \leq i \leq t-2, 1 \leq j \leq t\} \\
&& \cup \{\text{degree $2$ monomials in } x_1,\ldots,x_{t-2}\}.
\end{eqnarray*}
We see that $h(M) = F(\mathcal{O})$, which proves the following theorem.
\begin{theorem}
Let $M$ be a matroid of rank $2$. Then $h(M)$ is a pure $O$-sequence.
\end{theorem}
\section{Corank-$2$ Matroids}
In this section, we aim to prove Conjecture \ref{Stanley-conjecture} for corank-$2$ matroids.
\begin{theorem}\label{corank2}
Let $M$ be a matroid of rank 2. Then $h(M^*)$ is a pure $O$-sequence.
\end{theorem}
\begin{proof}
As before, let $E_1,\ldots,E_t$ denote the parallelism classes of $M$. Impose a total order on the ground set $E(M)$ so that $v_i < v_j$ for all $v_i \in E_k$ and $v_j \in E_{\ell}$ with $1 \leq k < \ell \leq t$.
For each basis $B = \{v_i,v_j\}$ of $M$ with $v_i \in E_k$, $v_j \in E_{\ell}$, and $k<\ell$, let
\begin{eqnarray*}
a_1(B)&:=&\#\{i'>i: v_{i'} \in E_k \cup \cdots \cup E_{\ell-1}\} \\
\text{and } a_2(B) &:=& \#\{j'>j: v_{j'} \in E_{\ell} \cup \cdots \cup E_t\},
\end{eqnarray*}
and set $\mu_B:= x_1^{a_1(B)}x_2^{a_2(B)}$. We claim that $\mathcal{O}:=\{\mu_B: B \in {\mathcal{B}}(M)\}$ is a pure order ideal and that $F(\mathcal{O}) = h(M^*)$.
\begin{figure}[ht]
\begin{center}
\epsfig{file=corank2.ps, height = 7cm, bbllx=110, bblly=355, bburx=525, bbury=670}
\end{center}
\caption{The bases $B = \{v_i,v_j\}$ (left) and $\widetilde{B} = \{u_1,u_{\ell}\}$ (right) with their externally passive elements shaded.}
\label{corank2-picture}
\end{figure}
We see that $a_1(B)$ counts the number of elements $v \in E(M) \setminus B$ that are externally passive in $B$ for which $v_i < v < v_j$ (shown in Figure \ref{corank2-picture} (left) shaded with lines of slope $1$); and $a_2(B)$ counts the number of elements $v \in E(M) \setminus B$ that are externally passive in $B$ for which $v_j < v \leq v_n$ (shown in Figure \ref{corank2-picture} (left) shaded with lines of slope $-1$). Since $a_1(B)+a_2(B)$ counts the number of externally passive elements in $B$, Equation \eqref{h-nums-dual} shows that $h(M^*) = F(\mathcal{O})$.
To see that $\mathcal{O}$ is an order ideal, we need only show that if $\nu|\mu_B$ and $\deg(\nu) = \deg(\mu_B)-1$, then $\nu \in \mathcal{O}$. Let $B = \{v_i,v_j\}$ as before. If $a_1(B) > 0$, consider $B' = \{v_{i+1},v_j\} \in {\mathcal{I}}(M)$. Clearly $a_1(B') = a_1(B)-1$ and $a_2(B') = a_2(B)$ so that $\mu_{B'} \in \mathcal{O}$ and $\deg(\mu_{B'}) = \deg(\mu_B)-1$. If $a_2(B)>0$, we must consider two possible cases. If $v_{j+1} \in E_{\ell}$, then consider $B'' = \{v_i,v_{j+1}\} \in {\mathcal{I}}(M)$. Again $a_1(B'') = a_1(B)$ and $a_2(B'') = a_2(B)-1$ so that $\mu_{B''} = x_1^{a_1(B)}x_2^{a_2(B)-1}$. On the other hand, if $v_{j+1} \in E_{\ell+1}$, then $v_{j-a_1(B)} \in E_{k'}$ for some $k' \leq \ell$, and so $B''' = \{v_{j-a_1(B)},v_{j+1}\} \in {\mathcal{I}}(M)$. Again we see that $\mu_{B'''} = x_1^{a_1(B)}x_2^{a_2(B)-1}.$ This establishes that $\mathcal{O}$ is an order ideal.
Finally, we must show that $\mathcal{O}$ is pure. For each $1 \leq i \leq t$, let $u_i$ denote the smallest element of $E_i$. For any basis $B = \{v_i,v_j\}$ as above, let $\widetilde{B} = \{u_1,u_{\ell}\}$. As Figure \ref{corank2-picture} (right) indicates, $a_1(B) \leq a_1(\widetilde{B})$ and $a_2(B) \leq a_2(\widetilde{B})$, and hence $\mu_B|\mu_{\widetilde{B}}$. Moreover, $\deg(\mu_{\widetilde{B}}) = |E_1| + \cdots + |E_t|-2$, and hence each such monomial $\mu_{\widetilde{B}}$ has the same degree.
\end{proof}
The techniques used to prove Theorem \ref{corank2} can be easily extended to prove that $h(M^*)$ is a pure $\mathcal{O}$-sequence for any matroid $M$ whose simplification is a uniform matroid. The reader may easily check, however, that these techniques may not be used to prove Stanley's conjecture when $M$ is the Fano matroid, thus these techniques may not be extended to corank 3.
\section{Rank-3 Matroids}
Our goal for this section is to give a simple, short, geometric-combinatorial
proof of the following theorem, first proved in
\cite{Ha-Stokes-Zanello} for the case that $d=3$ using the language of
commutative algebra.
\begin{theorem} \label{rank3-o-seq}
Let $M$ be a loopless matroid of rank $d \geq 3$. The vector $(1,h_1(M),h_2(M),h_3(M))$ is a pure $O$-sequence.
\end{theorem}
\begin{lemma} \label{h-vec-upper}
For any positive integers $s_1,\ldots,s_t$, the vector $\mathbf{h} = (1,h_1,h_2,h_3)$ with
\begin{eqnarray*}
h_1 &=& \sum_{i=1}^t (s_i-1)+(t-d), \\
h_2 &=& \sum_{1 \leq i < j \leq t} (s_i-1)(s_j-1) + (t-d)\sum_{i=1}^t(s_i-1) + {t-d+1 \choose 2},\\
h_3 &=& \sum_{1 \leq i<j<k \leq t}(s_i-1)(s_j-1)(s_k-1) + (t-d)\sum_{1 \leq i<j \leq t}(s_i-1)(s_j-1) \\
&&+ {t-d+1 \choose 2}\sum_{i=1}^t(s_i-1)+{t-d+2 \choose 3},
\end{eqnarray*}
is a pure $O$-sequence.
\end{lemma}
\begin{proof}
Consider disjoint sets $\widetilde{E}_1,\ldots,\widetilde{E}_t$ with $|\widetilde{E}_i| = s_i-1$ for all $i$. We will construct a pure order ideal $\mathcal{O}$ with $F(\mathcal{O}) = \mathbf{h}$ whose degree-one terms are $$\mathcal{O}_1 = \{x_1,\ldots,x_{t-d}\} \cup \{x_e: e \in \widetilde{E}_i\}_{i=1}^t.$$ We explicitly construct such an order ideal by setting
\begin{eqnarray*}
\mathcal{O}_2 &=& \{x_ex_{e'}: e \in \widetilde{E}_i, e' \in \widetilde{E}_j, 1 \leq i <j \leq t\} \\
&& \cup \{x_jx_e: e \in \widetilde{E}_i, 1 \leq i \leq t, 1 \leq j \leq t-d\} \\
&& \cup \{\text{all degree 2 monomials in } x_1,\ldots,x_{t-d}\}
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal{O}_3 &=& \{x_ex_{e'}x_{e''}: e \in \widetilde{E}_i, e' \in \widetilde{E}_j, e'' \in \widetilde{E}_k, 1 \leq i<j<k \leq t\} \\
&& \cup \{x_kx_ex_{e'}: e \in \widetilde{E}_i, e' \in \widetilde{E}_j, 1 \leq k \leq t-d, 1 \leq i<j \leq t\} \\
&& \cup \{x_jx_kx_e: e \in \widetilde{E}_i, 1 \leq j<k \leq t-d, 1 \leq i \leq t\} \\
&& \cup \{x_j^2x_e: e \in \widetilde{E}_i, 1 \leq i \leq t, 1 \leq j \leq t-d\} \\
&& \cup \{\text{all degree 3 monomials in } x_1,\ldots,x_{t-d}\}.
\end{eqnarray*}
\end{proof}
\begin{lemma} \label{h-vec-lower}
For any positive integers $s_1,\ldots,s_t$, the vector $\mathbf{h}' = (1,h_1,h_2,h_3)$ with
\begin{eqnarray*}
h_1 &=& \sum_{i=1}^t (s_i-1)+(t-d), \\
h_2 &=& \sum_{1 \leq i < j \leq t} (s_i-1)(s_j-1) + (t-d)\sum_{i=1}^t(s_i-1) + {t-d+1 \choose 2},\\
h_3 &=& \sum_{1 \leq i<j \leq t} (s_i-1)(s_j-1) + (t-d-1)\sum_{i=1}^t(s_i-1) + {t-d \choose 2}+1,
\end{eqnarray*}
is a pure $O$-sequence.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{h-vec-upper}, let $\widetilde{E}_1,\ldots,\widetilde{E}_t$ be disjoint sets with $|\widetilde{E}_i| = s_i-1$. Recall the order ideal $\mathcal{O}$ constructed in the proof of Lemma \ref{h-vec-upper}. We will construct a pure order ideal $\widetilde{\mathcal{O}}$ with $F(\widetilde{\mathcal{O}}) = \mathbf{h}'$ such that $\widetilde{\mathcal{O}}_1 = \mathcal{O}_1$, $\widetilde{\mathcal{O}}_2 = \mathcal{O}_2$, and $\widetilde{\mathcal{O}}_3 \subseteq \mathcal{O}_3$. We set
\begin{eqnarray*}
\widetilde{\mathcal{O}}_3 &=& \{x_1x_ex_e': e \in \widetilde{E}_i, e' \in \widetilde{E}_j, 1 \leq i<j\leq t\} \\
&& \cup \{x_j^2x_e: e \in \widetilde{E}_i, 1 \leq i \leq t, 2 \leq j \leq t-d\} \\
&& \cup \{x_i^2x_j: 1 \leq i<j \leq t-d\} \cup \{\mu_0\},
\end{eqnarray*}
where $\mu_0$ is a monomial defined as follows: if $\widetilde{E}_1 \cup \cdots \cup \widetilde{E}_t$ is nonempty, choose some $e_0 \in \widetilde{E}_1\cup \cdots \cup \widetilde{E}_t$ and set $\mu_0 = x_1^2x_{e_0}$. Otherwise, set $\mu_0 = x_1^3$. This distinction in the monomial $\mu_0$ is necessary for handling the cases in which $|\widetilde{E}_1 \cup \cdots \cup \widetilde{E}_t| \leq 1$.
\end{proof}
\textit{Proof: (Theorem \ref{rank3-o-seq})}
Let $E_1,\ldots,E_t \subseteq E(M)$ denote the parallelism classes of $M$, and set $s_i:=|E_i|$. Choose one representative $e_i$ from each class $E_i$, and let $W = \{e_1,\ldots,e_t\}$. Observe that $\Delta:= M|_W$ is a simple matroid of rank $d$. Let $\widetilde{E}_i = E_i \setminus \{e_i\}$, and notice that for any choice of $\widetilde{e}_{i_j} \in E_{i_j}, \{\widetilde{e}_{i_1},\ldots,\widetilde{e}_{i_k}\} \in \mathcal{I}(M)$ if and only if $\{e_{i_1}, \ldots, e_{i_k}\} \in \Delta$. Thus
\begin{eqnarray*}
f_0(M) &=& \sum_{i=1}^ts_i \text{ and hence}\\
h_1(M) &=& \sum_{i=1}^t(s_i-1) + (t-d); \\
f_1(M) &=& \sum_{1 \leq i<j \leq t}s_is_j \\
&=& \sum_{1 \leq i<j \leq t}(s_i-1)(s_j-1) + (t-1)\sum_{i=1}^t(s_i-1) + {t \choose 2} \text{ and hence} \\
h_2(M) &=& \sum_{1 \leq i<j \leq t}(s_i-1)(s_j-1) + (t-d)\sum_{i=1}^t(s_i-1) + {t-d+1 \choose 2}; \\
f_2(M) &\leq& \sum_{1 \leq i < j < k \leq t} s_is_js_k \text{ and hence} \\
h_3(M) &\leq& \sum_{1 \leq i<j<k \leq t}(s_i-1)(s_j-1)(s_k-1) + (t-d) \sum_{1 \leq i < j \leq t}(s_i-1)(s_j-1) \\
&& + {t-d+1 \choose 2}\sum_{i=1}^t(s_i-1) + {t-d+2 \choose 3}.
\end{eqnarray*}
On the other hand, by the Brown-Colbourn inequality \eqref{BC-ineq},
\begin{eqnarray*}
h_3(M) &\geq& h_2(M) - h_1(M) + h_0(M) \\
&=& \sum_{1 \leq i<j \leq t} (s_i-1)(s_j-1) + (t-d-1)\sum_{i=1}^t(s_i-1) + {t-d \choose 2}+1.
\end{eqnarray*}
We construct a pure order ideal $\mathcal{O}'$ with $F(\mathcal{O}') = h(M)$ as follows. Following the notation used in Lemmas \ref{h-vec-upper} and \ref{h-vec-lower}, we set $\mathcal{O}'_1 = \mathcal{O}_1$; $\mathcal{O}'_2 = \mathcal{O}_2$, and choose $\widetilde{\mathcal{O}}_3 \subseteq \mathcal{O}'_3 \subseteq \mathcal{O}_3$ with $|\mathcal{O}'_3| = h_3(M)$.
\hfill{$\Box$}
\section{Matroids on at most 9 elements}
This part of our paper is experimental and is crucially based on the data provided to us by Dillon Mayhew and Gordon Royle. They constructed a computer database of all 385,369 matroids on at most nine elements \cite{Mayhew-Royle}. We used this data to generate a list of all possible $h$-vectors of matroid complexes on at most nine elements. Given a loopless, coloopless matroid $M$ of rank $d$ on $n$ elements, we searched for a pure $O$-sequence $\mathcal{O}$ with $h(M) = F(\mathcal{O})$ in the following way: we know that $h_d(M)$ counts the number of top-degree monomials in $\mathcal{O}$, and $h_1(M) = n-d$ counts the number of variables (degree-one terms) in $\mathcal{O}$. By sampling the space of monomials of degree $h_d(M)$ on $h_1(M)$ variables, we can generate thousands of pure $O$-sequences that are candidates to be $h$-vectors of matroid complexes. Of course, because of the tremendous restrictions that the basis exchange axioms place on matroids, and hence also on their $h$-vectors, we often generated pure $O$-sequences that were not matroid $h$-vectors. For example $(1,5,15,27,22)$ and $(1,5,15,27,35)$ are both valid pure $O$-sequences which were generated, but the only $h$-vectors of matroid complexes of rank 4 with initial value $(1,5,15,27,*)$ are
\begin{verbatim}
(1 5 15 27 0) (1 5 15 27 19) (1 5 15 27 20) (1 5 15 27 21) (1 5 15 27 24)
(1 5 15 27 25) (1 5 15 27 26) (1 5 15 27 27) (1 5 15 27 30) (1 5 15 27 36).
\end{verbatim}
To generate the $O$-sequences, we used a combination of {\tt Perl} and {\tt Maple} code available at \url{www.math.ucdavis.edu/~ykemper/matroids.html}.
The key idea is that $m=h_d(M)$ provides us with the size of a monomial set to be sampled in a given number of variables $k=h_1(M)$.
Specifically, we started with an initial set of $m$ monomials within the simplex $\{ (x_1,x_2,\dots,x_k) : \sum_i x_i=d, \ x_i \geq 0 \}$, then calculated the corresponding pure $O$-sequence by counting the number of monomials of each degree less than or equal to $d$ which divide one or more of the initial monomials. One approach we used to generate large numbers of $O$-sequences was to sample randomly within the lattice points of this simplex. Another was to perform ``mutation'' operations based on the idea that within the simplex, all
lattice points are connected by the vectors $e_i-e_j$ of the root system $A_n$. We could therefore move ``locally'' from one pure order ideal to the next. In addition, we partially adapted a simulated annealing
type method to search for particular $h$-vectors (program labeled {\tt Boxy}) not found in our random sampling. {\tt Boxy} is also quite useful for computing the $O$-sequence of a family of
monomials given the top-degree monomials of that family. For example, by entering $[[0,0,0,5], [0,0,2,3], [1,3,0,1]]$, one can obtain the
corresponding $O$-sequence $(1,4,7,7,6,3)$.
The data we present in the web site is grouped by rank and corank. The largest groups are concentrated around rank four and corank five.
We have decided not to include the cases of rank one, two, and three, and corank one and two
because they are consequences of theorems presented earlier. Note that we have not listed monomials for matroids with coloops: a matroid having $j$ coloops has an $h$-vector with $j$ zeros at the end, and the non-zero entries correspond to the $h$-vector of the same matroid with all coloops contracted. Since this new matroid also has a ground set of at most nine elements, a family of monomials has been provided for it elsewhere in the table. The total number distinct matroid $h$-vectors (including $h$-vectors corresponding to matroids with coloops) and the total number of matroids per rank and corank are listed below. When the rank plus corank is greater than nine, we have no information on the quantities of matroids or distinct $h$-vectors, and have indicated this with `--.' \\
\begin{table}[hbt]
\begin{tabular}{|c||p{1cm}|p{1.05cm}|p{1.1cm}|p{1.2cm}|p{1.2cm}|p{1.1cm}|p{1.1cm}|p{1.05cm}|p{1cm}|p{1cm}|}
\hline
\textbf{Rank}/\textbf{Corank} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} \\
\hline
\hline
\textbf{0} & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\hline
\textbf{1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & --\\
\hline
\textbf{2} & 1 & 2 & 4 & 6 & 8 & 12 & 17 & 20 & -- & --\\
\hline
\textbf{3} & 1 & 3 & 9 & 22 & 49 & 101 & 196 & -- & -- & --\\
\hline
\textbf{4} & 1 & 4 & 18 & 67 & 244 & 816 & -- & -- & -- & --\\
\hline
\textbf{5} & 1 & 5 & 31 & 186 & 1132 & -- & -- & -- & -- & --\\
\hline
\textbf{6} & 1 & 6 & 51 & 489 & -- & -- & -- & -- & -- & -- \\
\hline
\textbf{7} & 1 & 7 & 79 & -- & -- & -- & -- & -- & -- & --\\
\hline
\textbf{8} & 1 & 8 & -- & -- & -- & -- & -- & -- & -- & --\\
\hline
\textbf{9} & 1 & -- & -- & -- & -- & -- & -- & -- & -- & --\\
\hline
\end{tabular}
\begin{center}
\caption{Number of distinct matroid $h$-vectors for particular rank and corank}
\end{center}
\end{table}
\begin{table}[h]
\begin{tabular}{|c||p{1cm}|p{1.05cm}|p{1.1cm}|p{1.2cm}|p{1.2cm}|p{1.1cm}|p{1.1cm}|p{1.05cm}|p{1cm}|p{1cm}|}
\hline
\textbf{Rank}/\textbf{Corank} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} \\
\hline
\hline
\textbf{0} & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\hline
\textbf{1} & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & --\\
\hline
\textbf{2} & 8 & 14 & 24 & 30 & 40 & 42 & 42 & 29 & -- & --\\
\hline
\textbf{3} & 7 & 18 & 45 & 100 & 210 & 434 & 950 & -- & -- & --\\
\hline
\textbf{4} & 6 & 20 & 72 & 255 & 1664 & 189274 & -- & -- & -- & --\\
\hline
\textbf{5} & 5 & 20 & 93 & 576 & 189889 & -- & -- & -- & -- & --\\
\hline
\textbf{6} & 4 & 18 & 102 & 1217 & -- & -- & -- & -- & -- & -- \\
\hline
\textbf{7} & 3 & 14 & 79 & -- & -- & -- & -- & -- & -- & --\\
\hline
\textbf{8} & 2 & 8 & -- & -- & -- & -- & -- & -- & -- & --\\
\hline
\textbf{9} & 1 & -- & -- & -- & -- & -- & -- & -- & -- & --\\
\hline
\end{tabular}
\begin{center}
\caption{Total number of matroids, for particular rank and corank}
\end{center}
\end{table}
\section*{Acknowledgements}
The first author was partially supported by NSF grant DMS-0914107, the second and third authors were supported by NSF VIGRE grant DMS-0636297.
We are truly grateful to Dillon Mayhew and Gordon Royle for the opportunity to use their data in our investigations. We are also grateful to
David Haws for his help on getting this project started. We are incredibly grateful to Jonathan Browder and Ed Swartz for a number of thoughtful
and insightful conversations. Criel Merino was kind enough to give us several useful comments and references.
|
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Юрий Афанасьев:
Афанасьев, Юрий Николаевич (1934—2015) — советский и российский учёный и политический деятель, основатель Российского государственного гуманитарного университета.
Афанасьев, Юрий Николаевич (1926—2015) — разработчик СВЧ-аппаратуры зенитных ракетных систем ПВО, лауреат Ленинской премии.
Афанасьев, Юрий Павлович (род. 1938) — советский игрок в хоккей с мячом.
|
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International Slavery Museum
Albert Dock, Liverpool, United Kingdom
Visit our homepage
Museums nearby
Beatles Story
Her Majesty Customs and Excise National Museum
Suggest update
The International Slavery Museum in Liverpool is part of the National Museums Liverpool group. It is concerned specifically with the Atlantic Slave Trade, rather than the history of slavery in other regions and eras.
The museum is located on the third floor of the building that houses the Merseyside Maritime Museum, with the UK Border Agency National Museum, 'Seized! The Border and Customs uncovered', located in a gallery in the basement.
Text: http://en.wikipedia.org/wiki/International_Slavery_Museum
Photo: http://commons.wikimedia.org/wiki/File:Merseyside_Maritime_Museum.JPG
127 Dale Street, Liverpool, United Kingdom
Lady Lever Art Gallery
Port Sunlight Village, Liverpool, United Kingdom
William Brown Street, Liverpool, United Kingdom
World Museum
Margaret Mitchell House
Atomic Testing Museum
Posjet Nuklearnoj elektrani Krško
Učna ura Pisana Loka, ki se odvija se v starem...
20th Century Media Collection
Nationaal Monument Kamp Vught
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,091
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android:layout_width="fill_parent"
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\section{Introduction}
\label{sec:introduction}
Competition between self-focusing (SF) and self-defocusing (SDF)\
nonlinearities occurs in various physical media, playing an
important role in the creation of self-trapped modes (in particular,
solitons). A well-known example is the competition between
quadratic (second-harmonic-generating, alias $\chi ^{(2)}$) and SDF cubic
nonlinear interactions in optics, in the case when the proper choice of the
mismatch constant makes the $\chi ^{(2)}$ interaction effectively
self-focusing \cite{chi2_1,chi2_2,chi2_3,chi2_4,chi2_5,chi2_6,chi2_7,chi2-review,chi2-review2}.
A large number of publications have addressed systems featuring the competition
between SF cubic and SDF quintic terms. Such combinations of nonlinear terms
frequently occur in optics, including liquid waveguides \cite%
{liquids,liquids2,liquids3}, special kinds of glasses \cite%
{liquids,quintic-experiment,quintic-experiment2,quintic-experiment3} and
ferroelectric films \cite{ferroelectric}. Especially flexible are colloidal
media formed by metallic nanoparticles, in which the cubic-quintic (CQ)
optical nonlinearity can be adjusted within broad limits by selecting the
size of the particles and the colloidal filling factor \cite%
{colloid,colloid2,colloid3,colloid4}.
The SF-SDF CQ nonlinearity has a great potential for the creation of stable
multidimensional solitons, including two- \cite%
{Manolo,Manolo2,Manolo3,Pramana} and three- \cite{nine-authors,nine-authors2}
dimensional (2D and 3D) solitary vortices, as reviewed in Ref. \cite{review}
and recently demonstrated experimentally in the 2D setting, in the
colloidal waveguide, in Ref. \cite{liquids3}. Indeed, the cubic-only SF
nonlinearity cannot create stable multidimensional solitons, as the
corresponding 2D self-trapped modes (alias \textit{Townes' solitons}) and 3D
solitons are subject to instabilities related to the critical collapse in
2D [recently, it was demonstrated that stable 2D composite
(half-fundamental - half-vortical) solitons can be
created in a two-component system combining the cubic self-attraction
and linear mixing of the components through first-order
spatial derivatives, which represent the spin-orbit coupling\cite{HS1}], and supercritical collapse in 3D \cite{Berge,Berge2}(spatial inhomogeneity, in the form of a finite jump of the SF Kerr coefficient between an inner circle and the surrounding area, may stabilize fundamental solitons \cite{HS2}). The additional SDF quintic term arrests the collapse, imposing soliton stability \cite{Manolo,Manolo2,Manolo3,Pramana,review,nine-authors,nine-authors2}.
The study of the competing CQ nonlinearities is also relevant in 1D
settings. A remarkable fact is that, although the 1D nonlinear Schr\"{o}%
dinger (NLS) equation with the combined CQ nonlinearity is not integrable,
it admits well-known exact solutions for the full soliton family \cite%
{Pushkarov,Pushkarov2}. Furthermore, these solitons are stable not only in
the case of the competition between the SF cubic and SDF quintic terms, but
also when both terms have the SF sign \cite{Pelin}. The latter fact is
surprising, because the 1D solitons created by the quintic-only SF term are,
as a matter of fact, a 1D version of the Townes' solitons \cite%
{Salerno,Salerno2,Salerno3}, and, accordingly, are unstable against the 1D
variety of the critical collapse. When both cubic and quintic SF terms are
present, the 1D solitons may still be pushed into the collapse by strong
perturbations (sudden compression), but, somewhat counter-intuitively, they
are stable against small perturbations.
In the case of the combination of the SDF cubic and SF
quintic terms, the entire family of 1D solitons is completely unstable. We stress that this
combination of the competing nonlinearities may be physically relevant.
In addition to the optical examples mentioned above, it appears naturally as a result of the reduction of the 3D
Gross-Pitaevskii (GP) equation to the 1D form for cigar-shaped traps filled
with atomic Bose-Einstein condensates (BECs) \cite%
{1D-BEC,1D-BEC2,1D-BEC3,1D-BEC4,1D-BEC5,1D-BEC6}. In the latter case, the
sign of the cubic terms is determined by the sign of the scattering length
in the BEC, and typically corresponds to the SDF, while the quintic term is
generated by the tight confinement of the condensate in the transverse
plane, and always corresponds to the SF.
Thus, search for physically realistic settings that would allow the
stabilization of 1D solitons in the NLS equation with the SDF-SF combination
of the cubic and quintic terms is an interesting problem, which is the subject
of the present work. It is well known that 2D Townes solitons and solitary
vortices, created by the SF cubic nonlinearity, can be readily stabilized by
an harmonic-oscillator (OH) trapping potential \cite%
{2D-HO-stabilization,2D-HO-stabilization2,2D-HO-stabilization3,2D-HO-stabilization4,2D-HO-stabilization5}%
, or by periodic potentials provided by optical lattices \cite%
{2D-OL-stabilization,2D-OL-stabilization2}. In this work, we consider the
stabilization of 1D solitons provided by the HO potential, and also by the
delta-functional attractive potential. In the optical waveguide, effective
traps may be induced by a transverse profile of the refractive index
\cite{Kivshar-Agrawal}, while in the BEC setup traps are created by means of
properly tailored magnetic and/or optical fields \cite{Pitaevskii,Randy}.
In these contexts, both broad HO traps and narrow ones, which may be
approximated by the Dirac's delta-function, are in principle relevant settings.
The NLS equation with the SDF-SF CQ nonlinearity and a trapping potential
offers a possibility to explore peculiarities of the stabilization of 1D
solitons under the action of the competing nonlinear terms, which is
especially interesting as concerns the central issue of this topic, i.e.,
identification of the stability of the so produced solitons. Indeed, it is
commonly known that the necessary stability criterion for families of
solitons supported by the SF nonlinearity is provided by the Vakhitov-
Kolokolov (VK) criterion \cite{VK,Berge,Berge2}, which is simply formulated
in terms of the dependence between the soliton's norm (total power), $N$,
and its propagation constant, $k$: $dN/dk>0$. In some cases, this stability
criterion may actually be a sufficient one, too. On the other hand, for
solitons supported by the interplay of SDF nonlinearity and trapping
potentials, a necessary stability condition is provided by the \textit{%
anti-VK criterion}, $dN/dk<0$ \cite{HS}. Therefore, a natural question,
which we address in the present work, is what criterion determines the
stability of solitons in the case of the competition between the SDF cubic
and SF quintic terms. In the opposite case of the combination of the SF
cubic and SDF quintic nonlinearities, the former one dominates the formation
of 1D solitons, their entire family being stable in accordance with the VK
criterion.
The rest of the paper is organized as follows. The model is formulated in
Section II. Findings for the soliton trapped in the HO potential are
collected in Section III, which includes both approximate analytical
results, obtained by means of the variational and Thomas-Fermi approximations
(VA and TFA correspondingly), and systematically generated numerical results for the
existence and stability of the solitons. A significant conclusion is that the
usual VK criterion works, as the necessary and sufficient one, for \textit{%
forward-going} branches of $N(k)$ curves of soliton families, in either case
of the positive and negative local slope ($dN/dk\gtrless 0$), while the
anti-VK criterion works equally well, but for \textit{backward-going}
branches. Both these types of branches are produced by the analytical
and numerical results under the action of the HO trap. Findings for the
delta-functional potential are reported in Section IV, including both
exponentially localized solitons and weakly localized but normalizable
trapped modes. In fact, they are obtained in a fully analytical form, which
is verified by dint of the numerical analysis, and it is again demonstrated
that the usual VK criterion is correct for all the solitons in the case of
forward-going branches, while the anti-VK is proper for backward-going ones.
The paper is concluded by Section V.
\section{The model}
\label{sec:model} In the free space, the 1D NLS equation with the SDF cubic
and SF quintic terms is
\begin{equation}
i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial ^{2}u}{\partial
x^{2}}-g|u|^{2}u+|u|^{4}u=0, \label{NLSE}
\end{equation}%
where $g\geq 0$ is the strength of the cubic nonlinearity. Here, Eq. (\ref%
{NLSE}) is written in the form of the paraxial equation for the spatial
evolution of the optical beam in a nonlinear planar waveguide, $z$ and $x$
being the longitudinal and transverse coordinates, respectively \cite%
{Kivshar-Agrawal}. In physical units, the solitons considered below may have
the transverse width $\lesssim 50$ $\mathrm{\mu }$m, while the
experimentally relevant transmission distance may be a few cm. If Eq. (\ref%
{NLSE}) is considered as the reduced GP equation for the BEC, the solitons
may be composed of several thousands of atoms, with the characteristic size $%
\lesssim 100$ $\mathrm{\mu }$m \cite{Randy}.
Equation (\ref{NLSE}) has a family of exact soliton solutions with
propagation constant $k\geq 0$, which can be easily obtained from the
well-known solitons for the opposite case of the SF-SDF CQ nonlinearity \cite%
{Pushkarov} by means of the analytical continuation:
\begin{equation}
u\left( x,z\right) =e^{ikz}\sqrt{\frac{2\sqrt{3}k}{\sqrt{4k+3g^{2}/4}\cosh
\left( {2\sqrt{2k}x}\right) -(g/2)\sqrt{3}}} \label{solution}
\end{equation}%
(in the case of the GP equation, $-k$ is the chemical potential of the BEC).
The norm of the soliton (\ref{solution}), which represents the total power of
the spatial optical solitons, or the total number of atoms in matter-wave
solitons in BEC, is
\begin{widetext}
\begin{equation}
N(k)\equiv \int_{-\infty }^{+\infty }|u(x)|^{2}dx=\sqrt{6}\arctan \left( {%
\frac{2\sqrt{k}}{\sqrt{4k+3g^{2}/4}-(g/2)\sqrt{3}}}\right) . \label{N}
\end{equation}%
\end{widetext}
This soliton family is completely unstable. Notice that $N(k)$, as given by Eq. (%
\ref{N}) yields $dN/dk<0$, hence it does not satisfy the VK criterion \cite%
{VK}.
As said above, solitons can be stabilized by means of an external
potential, $W(x)$, which added to Eq. (\ref{NLSE}) gives:%
\begin{equation}
i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial ^{2}u}{\partial
x^{2}}-g|u|^{2}u+|u|^{4}u=W(x)u. \label{U}
\end{equation}%
The stationary version of Eq. (\ref{U}), corresponding to $u\left(
x,z\right) =\exp \left( ikz\right) U(x)$, is
\begin{equation}
-kU+\frac{1}{2}U^{\prime \prime }-gU^{3}+U^{5}=W(x)U.
\label{stationary equation}
\end{equation}%
In the optical waveguide, the potential represents an inhomogeneous profile
of the refractive index, while in the BEC trap it may be induced by an external laser beam. In either type of the physical situation imposed potential may be either broad
(width measured by dozens of $\mathrm{\mu }$m) or narrow o(squeezed to a few $\mathrm{\mu }$m). Here we consider two
basic types of the trapping potential, harmonic trap,
\begin{equation}
W(x)=({1/2})\Omega ^{2}x^{2}, \label{HO}
\end{equation}%
and the Dirac's delta-function,
\begin{equation}
W(x)=-\epsilon \delta (x) \label{delta}
\end{equation}%
with $\epsilon >0$. By the means of rescaling, we fix $\Omega \equiv 0.5$ in
Eq. (\ref{HO}), and $\epsilon =0.5$ in Eq. (\ref{delta}).
The delta-functional attractive potential (\ref{delta}) implies that the $x$%
-derivative of the wave field is a subject to a jump condition at $x=0$:%
\begin{equation}
u_{x}\left( x=+0\right) -u_{x}\left( x=-0\right) =-2\epsilon u(x=0),
\label{jump}
\end{equation}%
while the field itself is continuous at this point. In fact, a full family of
solitons pinned to the ideal delta-functional potential embedded into the
medium with the CQ nonlinearity can be found in an exact analytical form, as
shown below in Section IV.
\section{The harmonic-oscillator potential}
\label{sec:harmonic}
In this section we present results obtained for Eq. (\ref{stationary equation}) with the HO
potential (\ref{HO}). We first use analytical approximations to gain a first
insight as to where one may expect solitons, and then we concentrate on
numerical analysis in these regions.
\subsection{The variational approximation (VA)}
\label{subsec:VA}
The stationary NLSE equation is now given by
\begin{equation}
-kU+\frac{1}{2}U^{\prime \prime }-gU^{3}+U^{5}=\frac{1}{2}\Omega ^{2}x^{2}U,
\label{stationary NLSE}
\end{equation}%
where $k$ is the propagation constant, the prime stands for $d/dx$, and the
HO potential is introduced as per Eq. (\ref{HO}). The starting point of the
VA is that Eq. (\ref{stationary NLSE}) can be derived from the Lagrangian
density,
\begin{equation}
2\mathcal{L}=\frac{1}{2}\left( U^{\prime }\right) ^{2}+kU^{2}+\frac{1}{2}%
gU^{4}-\frac{1}{3}U^{6}+\frac{1}{2}\Omega ^{2}x^{2}U^{2}.
\label{Lagrangian density}
\end{equation}
We approximate the fundamental trapped mode by the usual Gaussian \textit{%
ansatz} \cite{ansatz,ansatz2},
\begin{equation}
u(x)=A\exp \left( -x^{2}/w^{2}\right) , \label{VA solution}
\end{equation}%
with amplitude $A$, width $w$, and norm%
\begin{equation}
N_{\mathrm{VA}}=\sqrt{\pi /2}A^{2}w. \label{NVA}
\end{equation}%
The substitution of ansatz (\ref{VA solution}) into Eq. (\ref{Lagrangian
density}) and integration over $x$ yields the corresponding effective
Lagrangian, $L=\int_{-\infty }^{+\infty }\mathcal{L}(x)dx$,
\begin{widetext}
\begin{equation}
\frac{2L}{\sqrt{\pi }}=\frac{1}{2}\left( \frac{2}{w^{4}}+\frac{1}{2}\Omega
^{2}\right) A^{2}\left( \frac{w}{\sqrt{2}}\right) ^{3}+\frac{w}{\sqrt{2}}%
kA^{2}+\frac{1}{4}gA^{4}w-\frac{1}{3}A^{6}\frac{w}{\sqrt{6}},
\label{Lagrangian}
\end{equation}%
\end{widetext}
which gives rise to the respective Euler-Lagrange equations, $\partial
L/\partial A^{2}=\partial L/\partial w=0$, i.e.,
\begin{equation}
\frac{1}{2}\left( \frac{2}{w^{4}}+\frac{1}{2}\Omega ^{2}\right) \left( \frac{%
w}{\sqrt{2}}\right) ^{3}+\frac{kw}{\sqrt{2}}+\frac{1}{2}gA^{2}w-A^{4}\frac{w%
}{\sqrt{6}}=0, \label{dL/dA2}
\end{equation}%
\begin{equation}
-\frac{1}{w^{2}\sqrt{2}}+\frac{3}{4}\Omega ^{2}\frac{w^{2}}{(\sqrt{2})^{3}}+%
\frac{k}{\sqrt{2}}+\frac{1}{4}gA^{2}-\frac{1}{3}\frac{A^{4}}{\sqrt{6}}=0.
\label{dL/dw}
\end{equation}%
Equation (\ref{dL/dw}) produces two solutions for the squared amplitude,
\begin{equation}
A_{\pm }^{2}=\sqrt{\frac{3}{8}}g\pm \sqrt{\frac{3}{8}g^{2}+\frac{\sqrt{3}}{4}%
\left( \frac{2}{w^{2}}+\frac{1}{2}\Omega ^{2}w^{2}+4k\right) }. \label{A^2}
\end{equation}%
Solution $A_{+}^{2}$ is physical (positive) under condition $k\geq -\frac{%
\sqrt{3}}{8}g^{2}-\frac{1}{2}w^{-2}-\frac{1}{8}\Omega ^{2}w^{2}$, while $%
A_{-}^{2}$ is relevant only for $-\frac{\sqrt{3}}{8}g^{2}-\frac{1}{2}w^{-2}-%
\frac{1}{8}\Omega ^{2}w^{2}\leq k<-\frac{1}{2}w^{-2}-\frac{1}{8}\Omega
^{2}w^{2}$. In Fig. (\ref{ITM+VA2}) a typical fundamental-mode profile
produced by the VA is compared to its numerical counterpart.
\begin{widetext}
\begin{figure}[th]
\centering
\includegraphics[scale=0.22]{shooting-VA2.eps}
\caption{The wave function of the fundamental mode trapped in the HO
potential (\protect\ref{HO}) with $\Omega =0.5$, as obtained in the
numerical form, by means of the shooting method (the black, taller profile),
and as predicted analytically by the VA (the blue, lower profile) for $g=2$,
$k=1.309$, $N=2.69$. The norm is calculated for the numerical solution.}
\label{ITM+VA2}
\end{figure}
\end{widetext}
\subsection{The Thomas-Fermi approximation (TFA)}
For negative values of $k$ in Eq. (\ref{stationary NLSE}), another
analytical approach may be applied, in the form of the TFA, which neglects
the diffraction term, $d^{2}U/dx^{2}$, in the equation. This approximation,
which is relevant for confined modes when the SDF nonlinear term is the
dominant one \cite{Borovkova,Borovkova2}, yields the stationary solution in the form of
\begin{equation}
U_{\mathrm{TFA}}^{2}(x)=\left\{
\begin{array}{c}
\frac{g}{2}-\sqrt{\frac{g^{2}}{4}-\left( |k|-\frac{1}{2}\Omega
^{2}x^{2}\right) },~\mathrm{at}~~x^{2}<2|k|/\Omega ^{2}, \\
0,~\mathrm{at}~~x^{2}>2|k|/\Omega ^{2}.%
\end{array}%
\right. \label{TF}
\end{equation}%
As seen from Eq. (\ref{TF}), the TFA solution is a physically relevant
provided that the wavenumber satisfies condition $0<-k<g^{2}/4$. There exists also
a solution with the positive sign in front of the square root in Eq. (%
\ref{TF}), but it is irrelevant, as it does not vanish at $|x|\rightarrow
\infty $. The comparison of typical solution profiles predicted by the TFA
with their numerical counterparts is presented in Fig. \ref{ITM+TF for g=2,4}%
. It can be concluded that the TFA predicts the solutions in a qualitatively
correct form only when the cubic SDF term dominates in Eq. (\ref{stationary
NLSE}), which occurs at $g\gtrsim 2$.
\begin{widetext}
\begin{figure}[th]
\centering\subfigure[]{\includegraphics[scale=0.2]{shooting-TF1.eps}}%
\subfigure[]{\includegraphics[scale=0.2]{shooting-TF2.eps}}
\caption{Examples of the fundamental mode trapped in the HO potential, as
obtained in the numerical form (the black, lower profiles), and by means of
the TFA (the blue profiles with truncated tails), for $g=2,k=-0.72,N=1.17$
(a) and $g=4,k=-3.041,N=6.15$ (b).}
\label{ITM+TF for g=2,4}
\end{figure}
\end{widetext}
\subsection{Numerical results}
\label{sec:numerical}
\subsubsection{Fundamental solutions}
\label{subsubsec:symmetric solutions}
Numerical stationary solutions for HO potential were
found by means of the imaginary-time-integration \cite{IT} and shooting \cite%
{shooting} methods. The results are collected in Fig. \ref{N(k)-symmetric},
where the corresponding dependences $N(k)$ are displayed for different
values of the cubic-SDF coefficient, $g$, and compared to the results
produced by the VA and TFA. All the branches stem, at $N=0$, from the point which corresponds to the fundamental mode of the HO potential in linear quantum mechanics.
\begin{widetext}
\begin{figure}[th]
\centering
\includegraphics[scale=0.33]{Nk-VA.eps}
\caption{The norm of the fundamental modes, trapped in the HO potential (%
\protect\ref{HO}) versus $k$ for different values of $g$. Here and in Fig.
\protect\ref{N-antisymmetric} below, full and empty marks designate,
respectively, stable and unstable solutions found numerically: black (squares) for $g=0
$, red (circles) for $g=1$, and blue (triangles) for $g=2$. Solid and dashed lines depict
VK-stable and unstable states predicted by the VA: black (the lowest line in the graph) for $g=0$, orange (the middle one)
for $g=1$ and cyan (the highest position in the graph) for $g=2$. Results obtained for $g=2$ by the means of TFA are depicted by navy-blue stars. The black dotted horizontal line here and in Fig. \protect\ref{N-antisymmetric} corresponds to the limit value of $N(k=\infty)=(\sqrt{6}/4)\pi\approx 1.924$, see Eq. (\ref{N}), which does not depend on $g$. Higher order modes are not included here.}
\label{N(k)-symmetric}
\end{figure}
\end{widetext}
The maximum value of dependence $N(k)$ increases with the growth of $g$. The turning point, where the two branches of the VA solution meet, can be
estimated from Fig. \ref{N(k)-symmetric}. For instance, it is $k\approx -1$
at $g=2$. Only a small part of the top (blue) solution branch in Fig. \ref%
{N(k)-symmetric} satisfies the Vakhitov-Kolokolov (VK) criterion, $dN/dk>0$,
which, as said above, is a necessary condition for the stability of
localized modes supported by the SF nonlinearity \cite{VK}. On the other
hand, the necessary stability condition for localized modes dominated by the
SDF nonlinearity may switch from the VK form to the \textit{anti-VK} one, $%
dN/dk<0$ \cite{HS}. Because the present model features the competition of
the quintic SF and cubic SDF terms, it is not obvious which one plays the
dominant role at different values of $k$ and $N$, hence a numerical stability analysis is necessary.
To explore the stability of the solutions, eigenfrequencies of small
perturbations around the stationary solutions have been computed, using the
standard linearization procedure \cite{Yang}. For stable solutions,
imaginary parts of these frequencies are zero (or negligibly small, in terms
of the numerical computation). All instabilities detected by this method
feature pure-imaginary eigenvalues (i.e., the respective instability mode is
expected to grow exponentially without oscillations). The latter finding
suggests that the VA and anti-VK criteria may be sufficient for the
stability analysis, because what they cannot detect, are complex unstable
eigenvalues \cite{VK,Berge}, which do not exists in the present setting
anyway. Eventually, the full results for the (in)stability eigenvalues
demonstrate that the VA criterion \emph{correctly} predicts the
(in)stability for the \textit{forward-going} segments of the $N(k)$
branches, with either sign of the slope, $dN/dk\gtrless 0$, see Fig. \ref%
{N(k)-symmetric}, while the anti-VK criterion is also \emph{correct}, but for
the \textit{backward-going} branches. In fact, the latter ones always have $%
dN/dk<0$, being completely stable, accordingly. Our findings imply a
conclusion that may be relevant for other models too, \textit{viz}., that
the SF and SDF nonlinearities determine the stability, severally, for the
forward- and backward-going segments of families of trapped states.
The stability analysis presented above has been corroborated by direct
simulations of the perturbed evolution of the corresponding trapped states.
The simulations were run by dint of the finite-difference algorithm. In
particular, the solutions predicted to be unstable indeed blow up in the
course of the evolution (i.e., exhibit the collapse driven by the SF quintic
term \cite{Berge}; not shown here in detail).
\subsubsection{Antisymmetric trapped states}
\label{subsubsec:antisymmetric solutions}
A family of the lowest excited states with an antisymmetric profile,
has been found and
investigated for the HO trap. Results are presented in Fig. \ref{antisymmetric profile}. Numerical solutions were constructed by
means of the shooting method, and then their stability was tested
through the computation of perturbation eigenvalues. In Fig.
\ref{N-antisymmetric} for the sake of comparison, branches
of the antisymmetric states are shown along with their ground-state
(symmetric) counterparts. Similar to the situation shown above, for the
ground-state branches in Fig. \ref{N(k)-symmetric}, all the curves
representing the antisymmetric states stem, at $N=0$, from the point
corresponding to the first excited state of the HO potential in
linear quantum mechanics.
\begin{widetext}
\begin{figure}[th]
\centering
\includegraphics[scale=0.2]{antisymmetric.eps}
\caption{The profile of the antisymmetric excited state, trapped in the HO
potential for $g=2$, $k=-1$, and $N=0.673$.}
\label{antisymmetric profile}
\end{figure}
\end{widetext}
\begin{widetext}
\begin{figure}[th]
\centering
\includegraphics[scale=0.35]{antisym-Nk.eps}
\caption{Numerically obtained $N(k)$ dependences for symmetric modes trapped in the HO potential,
at $g=0$ (black, squares) and $g=2$ (red, circles), and for antisymmetric ones at $g=0$ (blue, triangles)
and $g=2$ (green, turned squares). The TFA solutions for $g=2$ are designated by pink stars. The third and higher order modes are not included here.}
\label{N-antisymmetric}
\end{figure}
\end{widetext}
The largest value of dependence $N(k)$ increases with the mode's number,
i.e., the maximum of the green branch (antisymmetric mode) exceeds the
maximum of the red one (symmetric mode). Furthermore, the plots presented in Fig. \ref{N-antisymmetric} demonstrate additional
instability of the antisymmetric modes, in comparison with their fundamental
symmetric counterparts (which is not surprising, as excited states in
nonlinear systems are usually more prone to instability \cite{Borovkova,Borovkova2}).
This instability concerns the forward-going branches, that satisfy the VK
criterion. The respective perturbation eigenvalues are complex, on the
contrary to the pure imaginary ones that could account for the instability
of the symmetric states, as mentioned above, hence they definitely cannot be
detected by the VK criterion, which is, generally, less relevant for excited
states, in comparison with the ground state. The predicted instability was
verified by direct simulations. It was found that the instability destroys
the antisymmetry of the stationary mode, and one of the two resulting peaks
develops the collapse, see Fig. \ref{antisymmetric-evolution-HO}.
\begin{widetext}
\begin{figure}[th]
\centering
\subfigure[]{\includegraphics[scale=0.19]{antisymmetric_stable.eps}} %
\subfigure[]{\includegraphics[scale=0.19]{antisymmetric_unstable.eps}}
\caption{Stable (a) and unstable (b) evolution of an antisymmetric state
trapped in the HO potential, for $g=2$, $k=-1.53$, $N=3.067$, and $g=2$, $%
k=-1.53$, $N=5.657$, respectively (both solutions have the same $k$ but
belong, severally, to different branches: \textit{backward-going} and
\textit{forward-going}). The initial slightly perturbed stationary state and
the final one are shown by the black and blue curves, respectively.}
\label{antisymmetric-evolution-HO}
\end{figure}
\end{widetext}
Results concerning stability of the fundamental and
antisymmetric states, obtained numerically, are collected in the diagram displayed in Fig. \ref%
{stability for harmonic}, in the plane of the norm and SDF cubic
coefficient, $g$. Note that the stability area expands with
the increase of $g$. Indeed, in the limit of the dominant SDF nonlinearity,
there is no apparent reason for destabilization of the fundamental and
excited states, which are obviously stable in the linear system.
Furthermore, the strengths of the destabilizing SF and stabilizing SDF terms
in Eq. (\ref{NLSE}), for modes with width $a_{\perp }\sim \Omega ^{-1/4}$
and large squared amplitude $A^{2}\sim N/a_{\perp }$, balance each other at $%
N\sim g$, which explains the asymptotically linear shape of the stability
boundaries at large $N$ in Fig. \ref{stability for harmonic}. On the other
hand we observe that the stability intervals for both the ground
state and the antisymmetric mode do not vanish in the case the SF
quintic-only interaction, $g=0$.
\begin{figure}[th]
\centering
\includegraphics[scale=0.36]{stability.eps}
\caption{Stability boundaries, $N_{\mathrm{cr}}(g)$, for the fundamental
(black) and first antisymmetric (red) modes trapped in the HO potential.}
\label{stability for harmonic}
\end{figure}
\section{Analytical solutions for the delta-functional trapping potential}
\label{sec:delta}
If the HO trapping potential in Eq. (\ref{U}) is replaced by the
delta-functional potential (\ref{delta}), then the respective stationary
equation (\ref{stationary NLSE}) is replaced by%
\begin{equation}
-kU+\frac{1}{2}U^{\prime \prime }-gU^{3}+U^{5}=-\epsilon \delta (x)U
\label{stat-eps}
\end{equation}%
(recall we have fixed $\epsilon =0.5$ by means of rescaling), which implies
that the jump condition (\ref{jump}) for symmetric solutions, with $%
U(-x)=U(x)$, reduces to%
\begin{equation}
U^{\prime }(x=+0)=-\epsilon U(x=0). \label{jump-eps}
\end{equation}%
Using exact solution (\ref{solution}) of Eq. (\ref{stat-eps}) in the free
space (at $x\neq 0$), one can easily construct the following exact solution
for the mode pinned to the delta-functional potential:
\begin{widetext}
\begin{equation}
U(x,k)=\sqrt{\frac{2\sqrt{3}k}{\sqrt{4k+3g^{2}/4}\cosh {[2\sqrt{2k}(|x|+\xi(k)
)]}-\left( \sqrt{3}/2\right) g}}, \label{delta solution}
\end{equation}%
\end{widetext}
where $\xi (k)>0$ is determined by the boundary
condition (\ref{jump-eps}):%
\begin{widetext}
\begin{gather}
\xi (k)=\frac{1}{2\sqrt{2k}}\arcsin \left[ -\frac{\sqrt{3\left[ 8k^{2}+({3/2}%
)g^{2}k\right] }g\epsilon }{\left[ 8k+({3/2})g^{2}\right] (2k-\epsilon ^{2})}%
\right. \notag \\
\left. \pm \frac{\sqrt{3\left[ 8k^{2}+({3/2})g^{2}k\right] g^{2}\epsilon
^{2}+16k\epsilon ^{2}\left[ 8k^{2}+({3/2})g^{2}k-4k\epsilon ^{2}-({3/4}%
)g^{2}\epsilon ^{2}\right] }}{\left[ 8k+({3/2})g^{2}\right] (2k-\epsilon
^{2})}\right] . \label{xi}
\end{gather}
\end{widetext}
While at $k>0$ solution (\ref{delta solution}) is exponentially localized,
in the limit of $k=0$ remains relevant, and it produces a pair of weakly
localized pinned modes:%
\begin{eqnarray}
U(x,k &=&0)=\frac{1}{\sqrt{g\left( |x|+\xi \right) ^{2}+\frac{2}{3g}}},
\notag \\
\xi &=&\frac{1}{2\epsilon }\pm \sqrt{\frac{1}{4\epsilon ^{2}}-\frac{2}{%
3g^{2}}}. \label{k=0,general}
\end{eqnarray}%
Obviously, this pair exists if $g$ is large enough:%
\begin{equation}
g\geq g_{\mathrm{cr}}=2\sqrt{2/3}\epsilon \approx 0.816, \label{cr}
\end{equation}%
where the value $\epsilon \equiv 0.5$ adopted above is substituted. The
weakly localized modes are meaningful ones as their norm converges, see
below. Note that the weakly localized state (\ref{k=0,general}) with $\xi =0$
is a valid solution to Eq. (\ref{stat-eps}) in the free space, i.e., with $%
\epsilon =0$, but in the latter case this solution is unstable \cite{Kivshar}%
.
Examples of generic solutions (\ref{delta solution}), and the weakly
localized ones (\ref{k=0,general}), featuring a cusp at $x=0$, are displayed
in Figs. \ref{Dirac delta solution}(a) and \ref{Dirac delta solution}(b),
respectively.
\begin{widetext}
\begin{figure}[th]
\centering
\subfigure[] {\includegraphics[scale=0.18]{DD_solutions.eps}} \subfigure[] {%
\includegraphics[scale=0.18]{DD_solutions_k0.eps}}
\caption{Exact stable solutions for the delta-functional potential (\protect
\ref{delta}), with $\protect\epsilon =0.5$, (a) given by Eqs. (\protect\ref%
{delta solution}) and (\protect\ref{xi}) with negative (black) and positive
(blue) signs for $g=1,k=0.1$, as well as (b) given by Eq. (\protect\ref%
{k=0,general}) for $g=1,k=0$ with both signs (black and blue profiles
correspond to the solutions with \textquotedblleft -" and \textquotedblleft
+", respectively).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ .}
\label{Dirac delta solution}
\end{figure}
\end{widetext}
At $k>0$, the analytical solution given by Eqs. (\ref{delta solution}) and (%
\ref{xi}) seems quite complex. It takes substantially simpler form for the
particular case of the quintic-only nonlinearity, $g=0$:%
\begin{eqnarray}
U_{g=0}(x,k) &=&\left( 3k\right) ^{1/4}\sqrt{\mathrm{sech}\left( 2\sqrt{2k}%
\left( |x|+\xi \right) \right) }, \notag \\
\xi &=&\frac{1}{2\sqrt{2k}}\mathrm{Artanh}\left( {\frac{\epsilon }{\sqrt{2k}%
}}\right) . \label{g=0}
\end{eqnarray}%
Obviously, this solution exists for sufficiently large propagation
constants,
\begin{equation}
k>k_{\min }\equiv \epsilon ^{2}/2. \label{kmin}
\end{equation}
In the opposite limit of the dominating SDF\ cubic term with large $g$, when
the quintic term may be omitted in Eq. (\ref{stat-eps}), the general
solution given by Eqs. (\ref{delta solution}) and (\ref{xi}) also simplifies:%
\begin{eqnarray}
U_{g\rightarrow \infty }(x) &=&\sqrt{\frac{2k}{g}}\frac{1}{\sinh \left(
\sqrt{2k}\left( |x|+\xi \right) \right) }, \notag \\
\xi &=&\frac{1}{\sqrt{2k}}\mathrm{Artanh}\left( {\frac{\sqrt{2k}}{\epsilon }%
}\right). \label{large-g}
\end{eqnarray}%
Its existence region is complementary to that given by Eq. (\ref{kmin}):
\begin{equation}
k\leq k_{\max }\equiv \epsilon ^{2}/2. \label{epsilon}
\end{equation}
Solution (\ref{large-g}) with zero propagation constant, $k=0$, corresponds
to a weakly localized mode,%
\begin{equation}
U_{g\rightarrow \infty }(x,k=0)=\frac{1}{\sqrt{g}\left( |x|+\epsilon
^{-1}\right) }, \label{k=0}
\end{equation}%
which is the limit case (for large $g$) of the above solution (%
\ref{k=0,general}).
The norm of the general solution, given by Eqs. (\ref{delta solution}) and (%
\ref{xi}), is equal to
\begin{widetext}
\begin{equation}
N(k)=\frac{2\sqrt{2k}B}{g}\left\{ \frac{1}{\sqrt{1-B^{2}}}\left[ \arcsin
B+\arcsin \left( \frac{1-B\cosh (2\sqrt{2k}\xi )}{\cosh (2\sqrt{2k}\xi )-B}%
\right) \right] \right\}, \label{general norm}
\end{equation}%
\end{widetext}
where $B\equiv g\sqrt{3/\left( 16k+3g^{2}\right) }$. In the limit of $k=0$
[provided that $g$ exceeds the critical value (\ref{cr})], the norm of the
weakly localized solution (\ref{k=0,general}) can be found in a more
explicit form,%
\begin{equation}
N(k=0)=\sqrt{6}\left[ \frac{\pi }{2}-\arctan \left( \sqrt{\frac{3}{2}}g\xi
\right) \right] . \label{N(k=0)}
\end{equation}
Furthermore, in the case of the quintic-only nonlinearity, $g=0$, when the
solution amounts to Eq. (\ref{g=0}), expression (\ref{general norm}) reduces
to
\begin{equation}
N_{g=0}(k)=\sqrt{6}\left\{ \frac{\pi }{2}-\arctan \left[ \exp \left( \mathrm{%
Artanh}\left( {\frac{\epsilon }{\sqrt{2k}}}\right) \right) \right] \right\} .
\label{special norm}
\end{equation}%
In the other above-mentioned limit, of large $g$, when the solution
takes the form of Eq. (\ref{large-g}), the respective simplification of
expression (\ref{general norm}) for the norm reads%
\begin{equation}
N_{g\rightarrow \infty }(k)=\frac{2}{g}\left( \epsilon -\sqrt{2k}\right) .
\label{N-large-g}
\end{equation}%
Note that Eq. (\ref{N-large-g}) at $k=0$ matches Eqs. (\ref{N(k=0)}) and (%
\ref{k=0,general}) in the limit of large $g$.
To check the stability of these solutions, the VK criterion can be applied
first. For the case of the purely SF (quintic-only) nonlinearity, $g=0$,
when this criterion is definitely relevant, Eq. (\ref{special norm}) yields
\begin{equation}
\frac{dN}{dk}=\frac{\sqrt{3}\epsilon }{4\sqrt{1-(\epsilon ^{2}/2k)}}k^{-3/2},
\label{norm derivative}
\end{equation}%
which is always positive in its existence range (\ref{kmin}), hence the
corresponding family of solution is entirely stable. In the general case, with $%
g>0$, the $N(k)$ dependence is plotted in Fig. \ref{N(k) comparison}(a)
and \ref{N(k) comparison}(b) (in a smaller range).
\begin{widetext}
\begin{figure}[th]
\centering
\subfigure[] {\includegraphics[scale=0.32]{DD_Nk.eps}} \nolinebreak
\subfigure[] {\includegraphics[scale=0.32]{DD_Nk1.eps}}
\subfigure[] {\includegraphics[scale=0.32]{DD_solexisting.eps}} \nolinebreak
\subfigure[] {\includegraphics[scale=0.32]{DD_solstability.eps}}
\caption{(a) The norm of the exact solution trapped in the delta-functional
potential (\protect\ref{delta}) vs. the propagation constant for different
values of strength $g$ of the SDF cubic term. Unstable branches are shown by
dashed lines. (b) The same as (a) in a smaller range of $k$. (c) The
existence area for the solutions in the $\left( k,g\right) $ plane. (d) The
stability boundary, shown as the critical norm vs. the SDF strength, $g$.}
\label{N(k) comparison}
\end{figure}
\end{widetext}
We notice in Figs. \ref{N(k) comparison}(a)
and \ref{N(k) comparison}(b) that the branches produced by $``+"$ sign in Eqs. (\ref{delta solution}) and (\ref{xi}%
),which are located below the blue line corresponding to the critical value $%
g=g_{\mathrm{cr}}$ given by Eq. (\ref{cr}), entirely satisfy the VK
criterion, while at $g>$ $g_{\mathrm{cr}}$ they are entirely VK-unstable,
featuring $dN/dk<0$. In all \textit{forward-going} branches
predicted by Eqs. (\ref{delta solution}) and (\ref{xi}) with the $``+"$
sign, point $k=k_{\mathrm{th}}\equiv \epsilon ^{2}/2=0.125$ is a singular
one. On the other hand, \textit{backward-going} branches
corresponding to $``-"$ in Eq. (\ref{xi}) do not satisfy the VK criterion,
hence they are either completely unstable or satisfy the anti-VK\textit{\ }%
criterion, similar to the HO case. Their stability has been confirmed by the
numerical verification, hence for them the anti-VK criterion indeed
guarantees the stability. All the existing solutions (stable and unstable)
are plotted in Fig. \ref{N(k) comparison}(c), including the special ones
predicted by Eqs. (\ref{k=0,general}) for $k\rightarrow 0$. Below $k=k_{%
\mathrm{th}}$, the existence area is limited by
\begin{equation}
k=\frac{\epsilon ^{2}}{2}-\frac{3g^{2}}{16} \label{asymptote}
\end{equation}%
which can be derived from Eq. (\ref{xi}) as the boundary of physical
solutions. Above $k_{\mathrm{th}}$, only the solution given by Eq. (\ref{xi}%
) with the positive sign exists, its negative counterpart being unphysical.
In the limit of $g\rightarrow \infty $, the solutions with $``+"$ sign in
Eq. (\ref{xi}) become irrelevant, as they require condition $\xi
_{(+)g\rightarrow \infty }\rightarrow 0$, which can be satisfied only in the
system without the delta-functional potential. The only physical solution in
this regime is the one expressed by Eqs. (\ref{delta solution}) and (\ref{xi}%
) in the case of the $``-"$ sign. In other words, in this limit there is
only one physical solution which can be obtained from Eq. (\ref{large-g}),
but, as said above, the solution with the negative sign can exist only for $%
k<k_{\mathrm{th}}$, in agreement with condition (\ref{epsilon}).
\begin{widetext}
\begin{figure}[t]
\centering\centering\subfigure[]{%
\includegraphics[height=2.5in]{DD_stable.eps}} \nolinebreak \centering
\subfigure[]{\includegraphics[height=2.5in]{DD_unstable.eps}}
\caption{Examples of the perturbed evolution of stable (a) and unstable (b)
modes pinned to the delta-functional potential (\protect\ref{delta}), for $%
g=1,k=0.1,N=0.109$ and $g=1,k=0.1,N=2.193$, respectively (the same $k$, but
the solution belongs to the different branches, with the $"-"$ and $"+"$
signs). The initial and final configurations are shown by the black and blue
curves, respectively.}
\label{antisymmetric-evolution}
\end{figure}
\end{widetext}
The predictions of the VK criterion for the present setting were confirmed
in the case of the branches corresponding to the positive sign by numerical
evaluation of eigenvalues for small perturbations, approximating, in the
framework of the numerical scheme, the ideal delta-function by the
Kronecker's delta, subject to the same normalization as the delta-function.
This approximation, used for the numerical evaluation of perturbation
eigenvalues, as well as for direct simulations, gives very reliable results.
For the stable branches, e.g., at $g=0$, all the eigenfrequencies are real,
while in the case of an instability, such as the \textit{forward-going }%
branch with the positive sign at $g\geq g_{\mathrm{cr}}$, e.g., at $g=1$,
there are pure-imaginary ones. The imaginary part of the perturbations
eigenvalues vanishes for \textit{backward-going} branches, corresponding to
the negative sign, both for $g\geq g_{\mathrm{cr}}$ and $g<g_{\mathrm{cr}}$.
That conclusion confirms the anti-VK criterion for the latter branches, as
they obey condition $dN/dk<0$. Furthermore, all \textit{forward-going}
branches below $g_{\mathrm{cr}}$ asymptotically tend to the same limit which
can be easily calculated by taking the $U_{g=0}$ solution from Eq. (\ref{g=0}%
) in the limit of $k\rightarrow \infty $ and its norm given by Eq. (\ref{special norm}), $N_{g=0,k\rightarrow \infty}=\sqrt{6}\left(
\pi /4\right) \approx 1.924$.
This limit value is a stability boarder for all the branches with the
positive and negative signs at $g<g_{\mathrm{cr}}\approx 0.816$. At $g\geq g_{\mathrm{cr}}$ the stability boundary can be
found from Eq. (\ref{N(k=0)}) with $\xi _{(-)}$ taken as per Eq. (\ref%
{k=0,general}). For given $g$ this is a stable solution with the largest
possible value of norm (the largest-norm stable solution belongs to the \textit{%
backward-going} branch). Obviously, there exist other solutions with still
larger norms for the same $g$, but they belong to the \textit{forward-going}
branch, which is entirely unstable. The stability analysis presenting total
norm $N$ versus the SDF strength, $g$, is summarized in Fig. \ref{N(k)
comparison}(d).
The results were also verified by direct simulations\ of the perturbed
evolution of the modes under consideration. Typical examples of the stable
and unstable evolution are presented in Fig. \ref{antisymmetric-evolution}.
The unstable solution starts collapsing in the course of the evolution,
similar to the system with the HO trapping potential considered above. The failure of the delta-functional pinning potential to stabilize the
trapped modes at $g\rightarrow +\infty $, see Eq. (\ref{cr}), is a drastic
difference from the case of the HO potential, where the stability of the
trapped modes monotonously enhances with the increase of the strength of the
cubic SDF term, $g$, see Fig. \ref{stability for harmonic}.
Lastly, stable solutions for solitons pinned to the attractive
delta-functional potential in the model with the opposite combination of the
nonlinearities, SF cubic and SDF quintic, can also be found in an analytical
form. In particular, this setting gives rise to a bistability, with two
different pinned states corresponding to a common value of the propagation
constant \cite{GMW}.
\section{Conclusions}
The aim of this work is to study, in the analytical and numerical form, the
stabilization of 1D solitons by means of the external potential, under the
influence of the competing SF (self-focusing) quintic and SDF (self-defocusing)
cubic terms. Two standard trapping potentials were considered, the
harmonic-oscillator one, and the Dirac's delta-function. In the former
case, both fundamental symmetric and the lowest antisymmetric modes were
studied. In the latter, the results were obtained in the completely
analytical, although somewhat cumbersome, form. The most essential result
concerns the way the competing necessary stability criteria, namely, the VK
(Vakhitov-Kolokolov) and anti-VK ones, divide the regions of their validity
in the system with the competing nonlinear terms. In particular, in both
models with the HO and delta-functional trap, the forward- and
backward-going soliton branches, in terms of their dependence $N(k)$, between
their norm and propagation constant, precisely obey the VK and anti-VK
criteria, respectively.
It is relevant to extend the analysis for more generic forms of the trapping
potential, taking into regard that the cases of the broad HO and narrow
delta-functional traps produce very different results. In the 2D geometry,
it may be interesting to consider the stabilization provided by trapping
potentials in models combining the SF cubic nonlinearity (recall it leads to
the collapse in 2D) and effectively SDF quadratic interactions. This system
may be a relevant model of an optical bulk waveguide.
\nocite{*}
\begin{acknowledgments}
B.A.M. and M.A.K. acknowledge partial support from the National Science Center of Poland in the frame of HARMONIA program no. 2012/06/M/ST2/00479. K.B.Z. acknowledges support from the National Science Center of Poland in project ETIUDA no. 2013/08/T/ST2/00627. M.T. acknowledges the support of the National Science Center grant N202 167840.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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June 17, 2019 dissertion
It seems probable, though I have not found the fact brought out explicitly, that much of the amusement derived by these simple folk from their nightly talks, which are made gay with laughter, consists in teasing attacks on the bodily defects or peculiarities of certain members; though, from the evidence forthcoming, one would infer that a choral laughter over the stranger is the more usual feature in these social entertainments. To avoid, therefore, an infinite progress, he supposed that the matter which any body pushed before it, rolled immediately backwards, to supply the place of that matter which flowed in behind it; and as we may observe in the swimming of a fish, that the water which it pushes before it, immediately rolls backward, to supply the place of what flows in behind it, and thus forms a small circle or vortex round the body of the fish. But this is a ganglion whose sensitiveness is in inverse proportion to its size; in one case the exaction of a cent means the confiscation of the possessor's entire fortune; in another the delinquent could part with a hundred dollars without depriving himself of a necessity or a pleasure. If we enter into conversation upon equal terms with the lowest of the people, unrestrained by circumstance, unawed by interest, we shall find in ourselves but little superiority over them. For Shakespeare it is less than madness and more than feigned. Had a short-hand writer taken down his ravings, it would have proved that this picture is far from being an overcharged one. He was executed, and the countess, seeking an audience of the emperor, disproved the calumny by carrying unharmed the red-hot iron, when Otho, convinced of his rashness by this triumphant vindication, immediately repaired his injustice by consigning his empress to the stake.[937] When Edward the Confessor, who entertained a not unreasonable dislike for his mother Emma, listened eagerly to the accusation of her criminal intimacy with Alwyn, Bishop of Winchester, she was condemned to undergo the ordeal of the burning shares, and, walking over them barefooted and unharmed, she established beyond peradventure the falsehood of the charge.[938] So when in 943 Arnoul of Flanders had procured the assassination of William Longsword, Duke of Normandy, at Pecquigny, he offered to Louis d'Outremer to clear himself of complicity in the murder by the ordeal of fire.[939] Robert Curthose, son of William the Conqueror, while in exile during his youthful rebellion against his father, formed an intimacy with a pretty girl. The natural alliance of laughter with the play-mood has already been touched on.[83] We may now go a step farther and say that these spurts of joyous consciousness which, in simple natures untrammelled by thought of appearances, express themselves in laughter are of the essence of Play. On the other hand, simplicity of manner reduces the person who cannot so far forego his native disposition as by any effort to shake it off, to perfect insignificance in the eyes of the vulgar, who, if you do not seem to doubt your own pretensions, will never question them; and on the same principle, if you do not try to palm yourself on them for what you are not, will never be persuaded you can be any thing. Where pride and vanity, angry passions, and love of power, are active, we cannot, with impunity, force them to work against their inclination; at the same time, it is our duty to lay the axe to the root of the evil, and restrain, and if possible subdue, these inordinate passions; but what I assert, is, that these are very difficult and dangerous passions to encounter, and they are not, with this class, to be restrained and subdued by the mere authority of a tax-master. And if it does so in this one case, I would ask, why not in every other? Footnote 83: The sum of the matter is this. The fact that librarians do not yield, in this case, to the suggestion of a change that would benefit them and all their assistants, is, of course, due to the obviousness of the other fact that it would be bad for the public. These things apply of course to the library worker as to all others, especially to librarians in small towns where tools and materials are apt to be not of the best. This was a remark of Rousseau's, and it is a very true one. We have tried to find out what he is driving at and to help a little–to stock the kind of information that he wants and to help him get at it. Each of these had a regular and constant, but a peculiar movement of its own, which it communicated to what was properly the Sphere of the Planet, and thus occasioned that diversity of motions observable in those bodies. The bodies which excite them may be moved to a greater or to a smaller distance. One may describe this change by saying that the standard of ideas tends gradually to gain ground, hemming in if not narrowing that of custom. It varied considerably in England alone; but when one day it was discovered lifeless, subsequent forms which had enjoyed a transitory life were dead too. The prisoner who refused to plead, whether there was any evidence against him or not, could be tortured until his obstinacy gave way.[1719] Even witnesses were not spared, whether in civil suits or criminal prosecutions.[1720] It was discretionary with the judge to inflict moderate torture on them when the truth could not otherwise be ascertained. To prevent these consequences, I shall state all that I think ought to be done, in another number of this work; which I conceive is the most interesting part in the treatment of insanity. wide are the waves, you see; Shall I come, if I fly, my dear Love, to thee? They will indulge it so far as to allow him to be more anxious custom phd essay writing for hire for college about, and to pursue with more earnest assiduity, his own happiness than that of any other person. It would seem then to be a reasonable view, that if laughter in ordinary cases involves superiority, and is so regarded by its object, the enjoyment of it by its subject will be very apt to bring with it a taste of superiority. That the terrors of religion should thus enforce the natural sense of duty, was of too much importance to the happiness of mankind, for nature to leave it dependent upon the slowness and uncertainty of philosophical researches. Those two situations are the chief which interest us upon the theatre; because, in spite of all that reason and experience can tell us to the contrary, the prejudices of the imagination attach to these two states a happiness superior to any other. The case is not much better with those semi-civilized American nations, the Mayas and Nahuas, who possessed a partially phonetic alphabet, or with the Quichuas, who preserved their records by the ingenious device of the quipu. It looks as if only the more solid material interests now moved the mind, as if sport had to have its substantial bait in the shape of stakes, while comedy must angle for popularity with scenic splendours which are seen to cost money. This, I say, would probably, or rather certainly happen; but it would happen without any intention or foresight in those who first set the example, and who never meant to establish any general rule. But when the observations of Cassini had established the authority of those laws, which Kepler had first discovered in the system, the philosophy of Des Cartes, which could afford no reason why such particular laws should be observed, might continue to amuse the learned in other sciences, but could no longer satisfy those that were skilled in Astronomy. There are other peculiarities in the Egyptian script, which are to be explained by the same historic reason. Association is an habitual relation between continuations of the same ideas which act upon one another in a certain manner simply because the original impressions were excited together. As he is conscious how much he is observed, and how much mankind are disposed to favour all his inclinations, he acts, upon the most indifferent occasions, with that freedom and elevation which the thought of this naturally inspires. To be glad with the gaiety of laughter, to throw off the stiff and wearing attitude of seriousness and to abandon oneself to mirth and jollity is, in truth, to begin to play. How far from the curb may vehicles be parked in St. The cubit, _chumay_, was measured from the point of the elbow to the extremities of the fingers. They think little indeed of Racine. It was not a question of depth or learning, but an instinctive feeling, prompted by a certain generous warmth of blood in every one worthy the name of Briton. What a number of parties and schools have we in medicine,—all noisy and dogmatical, and agreeing in nothing but contempt and reprobation of each other! This they were unable to accomplish, so all unpleasant doubts were settled by promptly having him burnt.[1308] Even after the Lateran Council of 1215, some miracles related by C?sarius of Heisterbach show that the conviction of heretics by the hot iron was regarded as a matter of course,[1309] and a penitential of a somewhat later period complains that suspected heretics on trial had no other means of proving their orthodoxy or their conversion to the true faith. Buoyed up by their garments seven floated, and these were only saved from stoning by the archimandrite Eustache, who administered to them a solemn oath of abstinence from witchcraft. On the one hand, librarians and libraries seem to be shy of religion. But the two gold-heads together would not if taken off at all answer the purpose of a cane, and the two canes together would be more than I should want. This organized violence assumed for itself the sanction of a religion of love and peace, and human intelligence seemed too much blunted to recognize the contradiction. The man of the most perfect virtue, the man whom we naturally love and revere the most, is he who joins, to the most perfect command of his own original and selfish feelings, the most exquisite sensibility both to the original and sympathetic feelings of others. In short, to attempt accounting at all for the nature of consciousness from the proximity of custom phd essay writing for hire for college different impressions, or of their fluxional parts to each other in the brain seems no less absurd than it would be to imagine that by placing a number of persons together in a line we should produce in them an immediate consciousness and perfect knowledge of what was passing in each other's minds. The desire of the esteem and admiration of other people, when for qualities and talents which are the natural and proper objects of esteem and admiration, is the real love of true glory; a passion which, if not the very best passion of human nature, is certainly one of the best. From custom phd essay writing for hire for college that invaluable document, therefore, the "Book of the Dead," we learn what this ancient people expected to happen to the soul when it left the body. I believe this to be the reason why a love for books is so little considered among the modern qualifications of librarianship; it appears in acts, not in words; it cannot be ascertained by asking questions. In many cases the entertainment in observing character comes, not so much through a perception of the juxtaposition of something worthy and something slightly unworthy, as through a detection of some discrepancy between the character and the _role_ assumed at the moment, as when a self-assertive sense of justice, in "a child of larger growth," reveals itself in the quaint exaggeration of doing more than justice to oneself. Yet the standardization of "privileged" denominational consciences is strongly disavowed! If a man had a face like one of Raphael's or Titian's heads, he might be proud of it, but not else; and, even then, he would be stared at as a _non-descript_ by 'the universal English nation.' Few persons who have seen the Antinous or the Theseus will be much charmed with their own beauty or symmetry; nor will those who understand the _costume_ of the antique, or Vandyke's dresses, spend much time in decking themselves out in all the deformity of the prevailing fashion. He had now, therefore, it would seem, become completely master of the language of Vision, and he had become so in the course of a year; a much shorter period than that in which any person, arrived at the age of manhood, could completely acquire any foreign language. The wenches are in like manner neat and clean in their own persons, but insipid. A favorite theme with the writers of the "Books of Chilan Balam" was the cure of diseases. Hence, probably, the fact noted by historians of medi?val manners that the coarseness of the jocosity appeared to increase with the magnitude of the feast. Excellent authorities, however, such as Woodham and Lower, have shown that these devices were frequent in the remotest ages of heraldry.[211] For instance, in the earliest English Roll of Arms extant, recorded in the reign of the third Henry, about the year 1240, nine such charges occur, and still more in the Rolls of the time of Edward the Second. This is illustrated by a further regulation, according to which, under the Gwentian code, in an accusation of theft, with positive evidence, the thief was directed to clear himself with twenty-four raith=men of his own _cantrev_ or district, in equal number from each _cymwd_ or sub-district.[89] Under a different social organization, it is evidently impossible that a kindred sufficiently large could have been assembled in the most numerous families, and even when the requirements were more reasonable, the same difficulty must frequently have occurred. Most critics have some creative interest—it may be, instead of an interest in any art, an interest (like Mr. I am sorry the creator of that epigram put his teacher on a log. Pierre de Fontaines, for instance, directs that in cases of appeal from a roturier to a gentleman the combat shall take place on foot between champions;[447] and I find a case recorded in 1280, in which a _femme de corps_ of Aimeri de Rochechouart accused the Sire de Montricher of burning her houses, and as the duel was adjudged she placed in the lists an armed and mounted knight as her champion, to whom no objection seems to have been made.[448] Throughout both Northern and Southern Germany, where the minute distinctions of birth were guarded with the most jealous care from a very early period, the codes of the thirteenth century, including even the burgher laws, provided that a difference of rank permitted the superior to decline the challenge of an inferior, while the latter was obliged to accept the appeal of the former. It is well when such {322} self-scrutiny can be carried on without any risk of encountering forms of ugliness and of ill omen, which would make speedy end of the amusing exercise. He is too well aware of the truth of what has been said, that 'the wisest amongst us is a fool in some things, as the lowest amongst men has some just notions, and therein is as wise as Socrates; so that every man resembles a statue made to stand against a wall, or in a niche; on one side it is a Plato, an Apollo, a Demosthenes; on the other, it is a rough, unformed piece of stone.'[41] Some persons of my acquaintance, who think themselves _teres et rotundus_, and armed at all points with perfections, would not be much inclined to give in to this sentiment, the modesty of which is only equalled by its sense and ingenuity. ESSAY XXX ON DEPTH AND SUPERFICIALITY I wish to make this Essay a sort of study of the meaning of several words, which have at different times a good deal puzzled me. There is no doubt, however, that in this and most other libraries the demand in this class is too small and needs stimulation. —– CHAP. Instead of patriots and friends of freedom, I see nothing but the tyrant and the slave, the people linked with kings to rivet on the chains of despotism and superstition. The selection of books is well thought-out and adapted to the community in which it is. In turning to the word for love in the Maya vocabulary, we are at once struck with the presence of a connected series of words expressing this emotion, while at the same time they, or others closely akin to them and from the same root, mean pain, injury, difficulty, suffering, wounds and misery. Thus the black race was to show the predominance of the vegetative system; the yellow race, the osso-muscular system; the white race, the nervous system.[36] As Bichat had not discovered any more physiological systems, so there could be no more human races on the earth: and thus the custom phd essay writing for hire for college sacred triplets of the Comtian philosophy could be vindicated. His warning has met with scant heed because experience has not brought it home to most of us. Is it not a wonder that anyone succeeds in composing original music? The tittering in a church at a small _contretemps_ has been our illustration. What is the matter with the books in the average small library? In all European tongues, the mere letters of the alphabet, by themselves, have no meaning and convey no idea; furthermore, their value in a word is fixed; and, thirdly, arranged in a word, they are sufficient to convey its sound and sense to one acquainted with their values. A whole street bowing regularly to a man every time he rides out, may teach him how to pull off his hat in return, without supposing a particular genius for bowing (more than for governing, or any thing else) born in the family. The failure of the Landa alphabet left many scholars total skeptics as to the phonetic values of any of the Maya characters. There is a portrait of a young gentleman striving to get into the boat, while the crew are pushing him off with their oars; but at last he prevailed with them by his perseverance and entreaties to take him in. This incorporative character is still more marked in the objective conjugations, or "transitions." The object, indeed, follows the verb, but is not only incorporated with it, but in the compound tense is included within the double tense signs. In this way a modified admiration attaches itself to a new kind of object, _e.g._, works of art, virtuous actions, when these come to be perceived and reflected on in such a way as to disclose their admirable side. Spurzheim contends elsewhere that one organ can perform only one function, and brings as a proof of the plurality of the organs the alternate action and rest of the body and mind. He will be pleased that the children in his library have learned to wash their hands, but chiefly because he hopes that what they have learned may react upon the physical cleanliness–and perhaps on the moral cleanliness, too–of the community. So far as the Church was concerned its condemnation was irrevocable. Robinson tells us that about 10 per cent. The painting _A Diana and Nymphs_ is like plunging into a cold bath of desire: to make a statue of a _Venus_ transforms the sculptor himself to stone. We may suppose therefore that the habitable area of the New World was notably less at that period, and that the existing tribes were confined to a much narrower space. Temperance, in short, was, according to the Epicureans, nothing but prudence with regard to pleasure. The sentence so frequently quoted in this essay will serve for an example of this process as well as any, and may be profitably contrasted with the opening phrases of the _Posterior Analytics_. It is thought a piece of rudeness to differ from them: it is not quite fair to ask them a reason for what they say. 'Sir,' said he, 'I deny that Mr. Guidance is your business and your high privilege; and if some of you have in the past guided as the jailer guides his prisoners–for a walk around the prison yard with ball and chain–let us be thankful that this oppressive view is giving place to the freer idea of a guide as a counselor and friend. By the middle of the thirteenth century the church had succeeded in virtually establishing the claim, for which it had long striven, that ecclesiastics were not subject to secular law in either civil or criminal matters. Because the quantity of matter, and consequently the attracting power of each of them, would be as the cubes of their diameters. A temper sullen or active, shy or bold, grave or lively, selfish or romantic, (to say nothing of quickness or dulness of apprehension) is manifest very early; and imperceptibly, but irresistibly moulds our inclinations, habits, and pursuits through life. But at least Marlowe has, in a few words, concentrated him into a statement. It requires, however, a preliminary selection and generally the obtaining of books on approval, which is easier in a large place than a small one. Mill, "is a thing which may be exacted from a person as one exacts a debt. {17} The tides at Tonquin are the most remarkable in the world. The same things that tell, perhaps, best, to a private circle round the fireside, are not always intelligible to the public, nor does he take pains to make them so. The common Italian Heroic Poetry being composed of double rhymes, it can admit both of single and of triple rhymes; which seem to recede from the common movement on opposite sides to nearly equal distances. These considerations will commonly make no great impression. Thus the actual feeling of warmth must have a tendency to call up any old ideas of the same kind: _e.g._ to-day being a very warm day put me in mind of a walk I took in a hot day last summer. It is known that the brain acts upon the external world by means of voluntary motion, of the voice, and of the five external senses. 8 Chalk The entire series of these cliffs bears evidence of great and successive changes; the strata, in many places, are folded and bent, and superimposed upon others, which have undergone no dislocation whatever. essay writing for custom hire for phd college.
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High speed, 2 needle chainstitch Features For sewing light to medium weight fabrics, knits and denim Suitable for such operations as lap seam felling, hemming, binding, joining and plain stitching Designed for sewing stretchable seams For stitching such products as blouses, shirts, suits, knitted goods, ladies and children%u2019s dresses and skirts, coats, pants, uniforms, pajamas, work clothes, etc. Automatic lubrication Cam-type take-up lever minimizes tension on the lower threads to allow stitches to stretch Moving presser foot eliminates slippage and puckering Specifications Speed, Max.: 5600 Clearance Under Foot: 5/16%u201D Needle Bar Stroke: 1 3/16%u201D Stitch Length, Max.: 5 1/2 SPI Needle: 149×17 Bed Size: 7%u201D x 18 7/8%u201D Work Space: 10%u201D Stitch Type: 401 US Warranty: 90 Days labor on defects in materials and workmanship. Non-US Warranty: 30 days parts and labor Freight Charges for this machine are $200.00.
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This is cinematic perfection, it's truly one of the finest pieces of animation I've seen. I've praised Allison Schulnik's animations in the past (here and here) but I think this film is her strongest work to date. It is absolutely gorgeous, and also terrifying, and I guess that's sort of how I feel about all of her work.
Watch the Allison Schulnik's animation "Eager" below!
Turn up the volume, turn off the lights and make full screen (if your computer can handle it, it's worth it).
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Windows.Forms;
using System.Drawing;
using System.Reflection;
using AdamsLair.WinForms.Drawing;
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string val = property.GetValue(item, null) as string;
this.Text = val;
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bool ITiledViewItemEditor.ApplyValueToItem(object item)
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if (string.IsNullOrEmpty(this.editedPropertyName)) return true;
if (item == null) return true;
Type itemType = item.GetType();
PropertyInfo property = itemType.GetProperty(this.editedPropertyName, BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic);
if (property == null) return true;
if (property.PropertyType != typeof(string)) return true;
property.SetValue(item, this.Text, null);
return true;
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private void Accept()
{
this.accepted = true;
if (this.StopEditing != null) this.StopEditing(this, EventArgs.Empty);
}
private void Reject()
{
this.accepted = false;
if (this.StopEditing != null) this.StopEditing(this, EventArgs.Empty);
}
protected override void OnKeyDown(KeyEventArgs e)
{
base.OnKeyDown(e);
if (e.KeyCode == Keys.Escape)
this.Reject();
else if (e.KeyCode == Keys.Return)
this.Accept();
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protected override void OnLostFocus(EventArgs e)
{
base.OnLostFocus(e);
this.Accept();
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In this reply to James H. Fetzer's "Minds and Machines: Limits to Simulations of Thought and Action", the author argues that computationalism should not be the view that (human) cognition is computation, but that it should be the view that cognition (simpliciter) is computable. It follows that computationalism can be true even if (human) cognition is not the result of computations in the brain. The author also argues that, if semiotic systems are systems that interpret signs, then both humans and computers are semiotic systems. Finally, the author suggests that minds can be considered as virtual machines implemented in certain semiotic systems, primarily the brain, but also AI computers. In doing so, the author takes issue with Fetzer's arguments to the contrary.
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{"url":"https:\/\/stats.stackexchange.com\/questions\/424252\/how-to-decide-the-boundary-of-a-2d-kd-tree","text":"# How to decide the boundary of a 2d kd-tree?\n\nPeople often cites this figure when talking about k-d tree\n\nin the context of plotting, this is a $$10 \\times 10$$ square, considering this particular dataset, the maximum of x axis is 9, the minimum is 2, [1,7] along the y axis.\n\nis there a rule or algorithm to induce the $$10 \\times 10$$ square based on the dataset?","date":"2020-04-03 02:54:49","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\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.563867449760437, \"perplexity\": 700.281939010436}, \"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-2020-16\/segments\/1585370509103.51\/warc\/CC-MAIN-20200402235814-20200403025814-00150.warc.gz\"}"}
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A list of films produced in Italy in 1976 (see 1976 in film):
References
Footnotes
Sources
External links
Italian films of 1976 at the Internet Movie Database
1976
Films
Lists of 1976 films by country or language
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{"url":"https:\/\/www.semanticscholar.org\/paper\/Decay-of-solutions-to-the-Maxwell-equation-on-the-Andersson-Backdahl\/3a2a9e658fe084476b40677dc892beb62c1fc858","text":"# Decay of solutions to the Maxwell equation on the Schwarzschild background\n\n@article{Andersson2015DecayOS,\ntitle={Decay of solutions to the Maxwell equation on the Schwarzschild background},\nauthor={L. Andersson and T. Backdahl and P. Blue},\njournal={arXiv: Analysis of PDEs},\nyear={2015}\n}\n\u2022 Published 2015\n\u2022 Physics, Mathematics\n\u2022 arXiv: Analysis of PDEs\nA new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor $H_{ab}$ defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current $H_{ab} (\\partial_t)^b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore\u2026\u00a0Expand\n14 Citations","date":"2021-04-21 05:14:12","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\": 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\": 0.9612563252449036, \"perplexity\": 961.0686512074794}, \"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-2021-17\/segments\/1618039508673.81\/warc\/CC-MAIN-20210421035139-20210421065139-00183.warc.gz\"}"}
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